Holder Continuous Solutions of Active Scalar Equations
aa r X i v : . [ m a t h . A P ] M a y H ¨older Continuous Solutions of Active Scalar Equations
Philip Isett ∗ and Vlad Vicol † Abstract
We consider active scalar equations ∂ t θ + ∇ · ( u θ ) = 0 , where u = T [ θ ] is a divergence-free velocityfield, and T is a Fourier multiplier operator with symbol m . We prove that when m is not an odd function offrequency, there are nontrivial, compactly supported solutions weak solutions, with H¨older regularity C / − t,x .In fact, every integral conserving scalar field can be approximated in D ′ by such solutions, and these weaksolutions may be obtained from arbitrary initial data. We also show that when the multiplier m is odd, weaklimits of solutions are solutions, so that the h -principle for odd active scalars may not be expected. Contents ∗ Department of Mathematics, MIT, Cambridge, MA 02139, USA. ([email protected]). † Department of Mathematics, Princeton University, Princeton, NJ 08544, USA. ([email protected]). Estimates for the New Stress 38 C bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.2 Spatial derivative bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3 Advective Derivative bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 k ≥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.3 Verifying Claim 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10 Proof of Corollary 1.1 4911 Proof of Theorem 1.2 5112 Proof of Weak Rigidity for Odd Active Scalars 5113 Conservation of the Hamiltonian for Odd Active Scalars 5414 Constraints on Weak Limits of Degenerate Active Scalars in Higher Dimensions 5615 Concluding Discussion 58
The present paper is concerned with existence, nonuniqueness and results of h -principle type for H¨older contin-uous, weak solutions to inviscid active scalar equations with a divergence free drift velocity. These equationshave the form ∂ t θ + ∂ l ( θu l ) = 0 u l = T l [ θ ] ∂ l u l = 0 . (1.1)The operator T l [ · ] defining the drift velocity u l in (1.1) is represented in frequency space by a multiplier ˆ u l ( ξ ) = b T l [ θ ]( ξ ) = m l ( ξ )ˆ θ ( ξ ) . (1.2)We assume that m l ( ξ ) is defined on the whole frequency space as a tempered distribution and is homogeneousof degree so that T l is an operator of order . The multiplier must satisfy m l ( − ξ ) = m l ( ξ ) so that the drift-velocity u l is real-valued whenever the scalar θ is real-valued, and we assume that m l ( ξ ) is smooth away fromthe origin. The requirement that u l is divergence free corresponds to the requirement that m l ( ξ ) takes valuesperpendicular to the frequency vector ξ , i.e. ξ · m ( ξ ) = 0 for ξ = 0 .Active scalar equations arise from the full Navier-Stokes, Euler, or magneto-hydrodynamic equations in anumber of physical regimes, such as stratification, rapid rotation, hydrostatic, and geostrophic balance. Physi-cally motivated examples include:1. The surface quasi-geostrophic (SQG) equation [17, 30]. Here m ( ξ ) = i h− ξ , ξ i| ξ | − is an odd symbol, bounded and smooth on the unit sphere. The SQG equation belongs to a general classof active scalar equations (with odd constitutive law T ) satisfied by the vorticity of a generalized two-dimensional Euler equation on a Lie algebra (´a la Arnold [1]) with a specific inner product [43] (see also[47] for a more recent account). 2. The incompressible porous media (IPM) equation with velocity given by Darcy’s law [5, 21]. Here m ( ξ ) = h ξ ξ , − ξ i| ξ | − is an even symbol, bounded and smooth on the unit sphere. Note that the IPM equation has a three-dimensional analogue, with symbol m ( ξ ) = h ξ ξ , ξ ξ , − ξ − ξ i| ξ | − , which is again even. Our proofapplies to this three-dimensional case as well, cf. Remark 1 below.3. The magneto-geostrophic (MG) equation [39, 38, 28]. This is a three-dimensional active scalar equation,with symbol given by m ( ξ ) = D ξ ξ | ξ | + ξ ξ ξ , − ξ ξ | ξ | + ξ ξ , − ξ ( ξ + ξ ) E ( ξ | ξ | + ξ ) − for all ξ ∈ Z ∗ with ξ = 0 , and by m ( ξ , ξ ,
0) = 0 . The symbol of the MG equation is even andzero-order homogenous, but as opposed to the previous examples, it is not bounded. This unboundednessmay be seen by evaluating the symbol on a parabola m ( ζ , ζ, , and passing | ζ | → ∞ . Nonetheless, theproof in our paper still applies to the MG equations as we only require smoothness in a neighborhood offinitely many points, cf. Remark 2 below.Remarkably, from the mathematical point of view these scalar equations retain some of the same essentialdifficulties of the full fluid equations. In particular, the global well-posedness for the 2D SQG and IPM equationsremains open, in analogy to the 3D Euler equations. More relevant for this paper, the regularity class in whichthe conservation of the energy k θ k L may be established for weak solutions of (1.1), is H¨older continuity withexponent / , as for 3D Euler. However, due to their more rigid geometry (e.g. no known analogue for Beltramiflows), their non-local nature, and the presence of infinitely many conservation laws (the L p norms of θ , for any p ≥ ), the construction of weak solutions that fail to conserve energy appears to be more restrictive than for3D Euler.The pair ( θ, u l ) is called a weak solution of (1.1) if the equations (1.1) are satisfied on R × T in the senseof distributions. When ( θ, u l ) are continuous, it is equivalent to require the balance laws ddt Z Ω θ ( t, x ) dx = Z ∂ Ω θ u ( t, x ) · n dσ, Z ∂ Ω u ( t, x ) · n dσ = 0 to be satisfied as continuous functions of time for all subdomains Ω with smooth boundary and inward unitnormal n . The definition of weak solution implies immediately that the integral R T θ ( t, x ) dx is a conservedquantity, but this definition does not immediately imply the other conservation laws that hold for classicalsolutions (see also [4, 2] for comparisons with other notions of non-classical solutions for the Euler equations).The study of weak solutions in fluid dynamics, including those which fail to conserve energy, is naturalin the context of turbulent flows. The power spectrum predicted by Kolmogorov [36] implies that solutionswhich arise in the inviscid limit of the 3D Navier-Stokes equations have H¨older / regularity on average, andin particular are not classical. Such flows are expected to exhibit anomalous dissipation of energy, rather thanconserving energy. The exponent / is the same regularity threshold conjectured by Onsager [42] to be criticalfor energy conservation in the 3D Euler equations (see [3, 26, 44] for recent reviews). For power spectra inactive scalar turbulence, we refer to Kraichnan [37] and Constantin [13, 14].Our first main result, Theorem 1.1, shows that if the symbol of the multiplier m l ( ξ ) is not an odd functionof ξ for ξ = 0 , there exist nontrivial, space-periodic solutions in two dimensions with compact support in time,having any H¨older regularity θ ∈ C αt,x with α < / . In contrast, the energy R | θ | ( t, x ) dx is a conservedquantity for solutions with H¨older regularity above α > / and for classical solutions the quantity θ is alsoadvected by the drift velocity u l = T l [ θ ] , whereas both these properties clearly fail for our solutions. This resultgives the first proof of nonuniqueness of continuous weak solutions for any active scalar equation of this type.3 heorem 1.1 (Weak Solutions to Active Scalar equations) . Consider the active scalar equation (1.1) withdivergence free drift velocity, and assume that the multiplier m l ( ξ ) defining the operator T l is not an oddfunction of ξ for ξ = 0 . Let α < / and let I be an open interval. Then there exist nontrivial solutions to (1.1) with H¨older regularity θ, u l ∈ C αt,x ( R × T ) which are identically outside of I × T .Moreover, if f : R × T → R is a smooth scalar function with compact support on I × T which satisfiesthe conservation law ddt R T f ( t, x ) dx = 0 , then there exists a sequence of weak solutions θ n : R × T → R to (1.1) in the above regularity class such that θ n converges to f in the L ∞ weak-* topology, and each θ n hascompact support in I × T . The above result builds upon the recent works by C´ordoba, Faraco, Gancedo [19], Shvydkoy [45], andSz´ekelyhidi [46] which establish the non-uniqueness of L ∞ t,x weak solutions to the IPM equations and D active scalar equations with even symbols m . These previous works are based on a variant of the method ofconvex integration introduced for the Euler equations in [23] that provides an effective and elegant approach toproducing bounded solutions, but which faces a major obstruction to producing continuous solutions. For theEuler equations, this obstruction was overcome in [22, 24, 11] to produce continuous and C α solutions on T and T . A crucial idea to overcome this obstruction is a key cancellation coming from the use of special familiesof stationary, plane wave solutions which allows for the control of interference terms between different waves inthe construction. For D Euler, these solutions are Beltrami flows (eigenfunctions of the curl operator), whilefor D Euler they are rotated gradients of Laplace eigenfunctions.There is an obstruction to generalizing these ideas to obtain continuous solutions to active scalar equations,which is that analogous families of stationary, plane wave solutions do not exist in general for active scalarequations. Furthermore, as we explain more precisely in Section 2.1, there is a sense in which no analogouscancellation is ever available under the assumptions of Theorem 1.1. The same difficulty has also prohibited thisapproach from generalizing to the Euler equations in higher dimensions, even though similar results in principlecould be expected to hold in any dimension. (The conservation of energy for regularity above / holds in anydimension, and the approach of [23] for constructing L ∞ t,x solutions applies in any dimension.)The main idea that forms the starting point of our work is a new, more general, mechanism for obtainingthe cancellation of interference terms in the construction, which arises without any special Ansatz in the con-struction. Our observation is that the interference terms which arise when an individual wave interacts withitself must always cancel thanks to the divergence free structure of the equation, even though we lack a generalmethod for controlling the interference between waves which oscillate in different directions. This observationopens the door to a serial iteration scheme based on one-dimensional oscillations, as in the original scheme ofNash [40]. The same observation applies to both the Euler equations and to general active scalar equations re-gardless of the dimension (c.f. Remark 1). Our proof therefore gives a new approach to constructing continuousand C α weak solutions to these equations that is independent of the use of Beltrami flows or the analogue.Although the regularity obtained in Theorem 1.1 is strictly worse than the results which have been obtainedfor the Euler equations, the exponent / is the best result we can hope to obtain from our method. For the Eulerequations, solutions in the class C / − t,x were constructed in [31], with another proof given by Buckmaster, DeLellis and Sz´ekelyhidi [7]. The construction in [7] has recently been refined in [8] to give continuous solutions inthe class L t C / − x , improving significantly a result of Buckmaster [6]. A main obstruction to higher regularityfaced by all of these works and also the present paper is the presence of anomalously sharp time cutoffs. Thesecutoffs lead to bounds on advective derivatives which are inferior to the bounds that hold for solutions withhigher regularity, cf. [32, Sec. 9] and [34, Sec. 1.1.3]. In our case, we face an additional loss of regularity whichcomes from our inability to eliminate more than one component of the error in a given stage of the iteration.The same obstruction to regularity arises for the isometric embedding equation [18]. For active scalars, we mustdeal with both obstructions at the same time, and improving on either one seems to be a difficult problem.Our approach to proving Theorem 1.1 also yields the following result, which shows that our constructioncan realize arbitrary smooth initial data. Theorem 1.2.
Let I = ( − T, T ) be a finite open interval containing the origin, let α < / and let ( θ (0) , u l (0) ) e a smooth solution to (1.1) on I × T . Then there exists a global, weak solution ( θ, u l ) to (1.1) in the class ( θ, u l ) ∈ C αt,x ( R × T ) which coincides with ( θ (0) , u l (0) ) on the time interval θ ( t, x ) = θ (0) ( t, x ) ( t, x ) ∈ ( − T / , T / × T and which coincides with a constant θ ( t, x ) = ¯ θ for ( t, x ) / ∈ ( − T / , T / × T . To the best of our knowledge, Theorem 1.2 gives the first proof of global existence of weak solutions for(1.1) with multipliers m which are not odd, from arbitrary smooth initial data [19]. The global existence ofweak solutions appears to be only known for odd symbols [43, 9], or for patch-type initial datum in the IPMequations [20]. Thus, in view of the known existence result for odd multipliers, we show that all active scalarequations with smooth constitutive law have global in time weak solutions (see also Remark 2).Our method of construction demonstrates not only the existence of weak solutions, but also the abundanceand flexibility of solutions in the class C / − ǫt,x . This point is emphasized by the following result of “ h -principle”type, which follows from Theorem 1.1, and completely characterizes the weak-* closure of these solutions in L ∞ . The result illustrates that, within this regularity class, the conservation of the integral is the only source ofrigidity for solutions to the equations that is stable in the weak-* topology . We refer to [25, 12] for more on h -principles for fluid equations. Corollary 1.1 ( h -principle for Active Scalar Equations) . Consider the D active scalar equation (1.1) as inthe hypotheses of Theorem 1.1, with multiplier m that is not odd. Then for any α < / and for any openinterval I , the closure in the weak-* topology on L ∞ ( I × T ) of the set of C αt,x solutions to (1.1) with compactsupport in I × T is equal to the space of real-valued f ∈ L ∞ ( I × T ) which satisfy the conservation law R T f ( t, x ) dx = 0 as a distribution in time. While Theorems 1.1-1.2 and Corollary 1.1 illustrate an utter lack of rigidity for multipliers which are notodd, we find a much more rigid situation for weak solutions in the case of odd multipliers. The following resultimplies that, when the multiplier is odd, every weak limit of solutions in L ∞ t,x must also be a solution to thesame active scalar equation, in stark contrast to Theorem 1.1 and Corollary 1.1. This theorem generalizes thestatement at the end of [25] concerning weak rigidity for SQG, and makes precise the assumptions necessaryfor this rigidity. Theorem 1.3 (Weak Rigidity for Active Scalars with Odd Multipliers) . Consider the active scalar equa-tion (1.1) in any dimension, with divergence free drift velocity, and assume that the multiplier m l ( ξ ) definingthe operator T l is an odd function of ξ for ξ = 0 . Suppose that f = lim n θ n is a weak limit of solutions to (1.1) in L p ( I ; L ( T n )) , for some p > . Then f ( t, x ) must be a weak solution to (1.1) . We note that the L p time integrability condition on θ n is by no means restrictive. Indeed, due to the in-compressible transport nature of (1.1), weak solutions constructed via smooth approximations (e.g. vanishingviscosity) are in fact bounded, or even weakly continuous in time.The proof of Theorem 1.3 is based on the approach of [43], where global L ∞ t L x weak solutions of thesurface quasi-geostrophic equations are constructed. The main idea is that odd multipliers m induce a certaincommutator structure in the nonlinear term, which yields the necessary compactness. In fact, the oddness of m implies that the equations are well-posed, even if the operator T l is not of degree (see [9]), and in such casesthe oddness appears to be necessary [27, 29].In addition to the weak rigidity of Theorem 1.3, in the following theorem we show that every active scalarequation in D with odd symbol has a Hamiltonian that is conserved for solutions in the class L t,x . One must be cautious that Corollary 1.1 below does not assert that integral-conserving L ∞ functions can be approximated by a weak-*convergent sequence of solutions as in the statement of Theorem 1.1. Such a statement would be false, since the functions f obtained asweak-* limits of sequences will also inherit time regularity of the type ∂ t f ∈ L ∞ t W − ,px from the equation (1.1). heorem 1.4 (Conservation of the Hamiltonian for Active Scalars with Odd Multipliers) . Consider the activescalar equation (1.1) in two dimensions with divergence free drift velocity and odd multiplier as in Theorem 1.3.Define the operator L = ( − ∆) − ( ∇ · T ⊥ ) = ( − ∆) − / ( R T − R T ) (1.3) where R i is the i th Riesz transform. The fact that m is odd, implies that L is self-adjoint. Define the Hamiltonian H ( t ) = Z T θ ( t, x ) Lθ ( t, x ) dx. (1.4) Then, if θ is a solution to (1.1) in the class θ ∈ L t,x , the function H ( t ) is constant in time. We note that due to the transport structure of (1.1), solutions which are obtained by smooth approximations,such as viscosity approximations, Galerkin truncations, etc, will automatically lie in L ∞ t,x , and thus also in L t,x .Theorem 1.3 precludes any results such as Theorems 1.1-1.2 from holding in the case of the SQG equation,in which case L = ( − ∆) − / and we obtain the conservation of the H − / norm for solutions in L t,x . Notehowever that in general the operator L need not be coercive, as is the case when m vanishes somewhere on theunit sphere. We refer to [43, 47] for an exposition of how the quantity H ( t ) serves as a Hamiltonian for theequation.We conclude our introduction by remarking on how our method extends to higher dimensions, and to thecase of multipliers which are not smooth. Remark . Our proof generalizes to active scalar equations in arbitrary dimensions (c.f.Section 3.2 for the relevant modifications). In this case, however, there are two further restrictions. First ofall, the regularity we obtain becomes worse as the dimension increases. The same type of loss (for essentiallythe same reason, see Section 2.2.1 below) is also seen in the case of the isometric embedding equations [18].Second, we cannot obtain our result for all smooth multipliers whose symbols are not odd, and we require anondegeneracy condition on the even part of the multiplier.The precise result we obtain is the following:
Theorem 1.5 (Multi-dimensional Case) . Consider the active scalar equation (1.1) with divergence free driftvelocity on T d . Assume also that the image of the even part of the multiplier contains d vectors A ( i ) = m ( ξ ( i ) ) + m ( − ξ ( i ) ) , i = 1 , , . . . , d (1.5) such that the vectors A (1) , . . . , A ( d ) span R d . Then Theorems 1.1-1.2 and Corollary 1.1 hold as stated, but withthe condition α < on the H¨older exponent being replaced by α <
11 + 4 d .
Theorem 1.5 applies in particular to the D IPM equation, and in that case yields weak solutions with H¨olderregularity α < / . Note also that Theorem 1.5 generalizes the two dimensional case of Theorem 1.1. Namely,if the even part m ( ξ (1) ) + m ( − ξ (1) ) = 0 is nonzero at a single point, it follows already from incompressibility(i.e. the condition m ( ξ ) · ξ = 0 ) that the span of the image of the even part of m has dimension at least .The assumption (1.5) in Theorem 1.5 arises quickly from the proof and turns out to be necessary for theconclusion of Theorem 1.1. That is to say, when the assumption (1.5) fails, there are in general additionalconstraints on weak limits of solutions besides the conservation of the mean value. In the case where themultiplier is even, such constraints arise from the conservation of the integrals ddt Z T n θ ( t, x )Ψ( x ) dx = 0 for functions Ψ whose gradients take values perpendicular to the image of the multiplier. More generally, wehave the following theorem which can be applied to every multiplier that fails to satisfy (1.5):6 heorem 1.6 (Constraints on Weak Limits of Degenerate Multipliers) . Consider the active scalar equation (1.1) on a torus T n of any dimension and suppose that the image of the even part of the multiplier lies in a hyperplaneperpendicular to some nonzero vector ξ (0) ∈ c T n in the dual lattice. Then there exists a smooth function ofcompact support f ∈ C ∞ ( R × T n ) which is real-valued and satisfies the conservation law R T n f ( t, x ) dx = 0 such that f cannot be realized as a weak-* limit in L ∞ of any sequence of bounded weak solutions to (1.1) . The proof of Theorem 1.6 draws on the proof of weak compactness in Theorem 1.3. One can comparecondition (1.5) to criteria for having a large Λ -convex hull in the theory of differential inclusions (e.g. [35, 23,46]). Remark . In view of the example of the MG equation, it is important to remark thatour proof applies also to multipliers which are not smooth. In fact, the only regularity condition we requirein our proof is that the multiplier should be smooth in a neighborhood of the points ξ (1) , ξ (2) , . . . , ξ ( d ) and − ξ (1) , − ξ (2) , . . . , − ξ ( d ) appearing in (1.5). Thus Theorem 1.5 applies to the MG equation, if we take forexample the points ξ (1) = h , , i , ξ (2) = h , , i , ξ (3) = h , , i . The proof of Theorem 1.5 contains a number of new ideas in the method of convex integration, which wesummarize before we begin the proof.As stated earlier in the Introduction, our main idea is a new mechanism for obtaining cancellations ininterference terms between overlapping waves. This allows us to get around the lack of Beltrami flows, or theiranalogues, as the type of cancellation given by such flows is entirely unavailable in our setting (cf. Section 2.2).This idea gives a new and general approach to constructing continuous weak solutions which generalizes alsoto Euler. The idea is based on the observation that self-interference terms vanish automatically thanks to theincompressible nature of the equation.The above idea opens the door to a multi-stage iteration scheme based on one-dimensional oscillations, as inthe original scheme of Nash for isometric embeddings applied in [40, 18]. This type of scheme had previouslyappeared unavailable in the setting of the Euler equations (see [22, Section 1.3, Comment 2]). On the other hand,while implementing a scheme exactly of this type now appears to be possible, it also appears to be relativelycomplicated, requiring the addition of several iterations of waves (each with their own time, length scale andfrequency parameters) before the error improves in the C norm. We manage to avoid these complications bydefining a space of approximate solutions by a compound scalar stress equation . This concept allows us toobtain a C improvement after only one iteration, which simplifies the iteration and gives estimates which aremuch closer to the bounds familiar from the case of Euler.The main new technical difficulty in obtaining continuous solutions to active scalar equations lies in howto deal with the integral operator in the equation which determines the drift velocity u l = T l [ θ ] . The wholeconstruction is based on high frequency, plane-wave type corrections of the form e iλξ I ( t,x ) θ I ( t, x ) , and it isnecessary to understand very precisely how adding such waves will affect the drift velocity. Furthermore, theconvex integration schemes for producing H¨older continuous Euler flows all use heavily C type estimates onall error terms. From this point of view, the failure of C boundedness of T l suggests some serious trouble.Our main technical device for addressing this difficulty is a “Microlocal Lemma” (Lemma 4.1). This lemmamakes precise how a convolution operator behaves to leading order like a multiplication operator when given ahigh-frequency plane wave input, allowing for the use of nonlinear phase functions. In the case of the operator T l , represented on the Fourier side by the multiplier m l ( ξ ) , our lemma gives a statement of the form u l = T l [ e iλξ ( x ) θ ( x )] = e iλξ ( x ) ( θm l ( ∇ ξ ( x )) + δu l ) and gives an explicit formula for the error term δu l (which also allows us to estimate its spatial and advectivederivatives). We expect that this technique should be of independent interest for other applications. We note, however, this idea alone obtains a lesser H¨older regularity compared to the Beltrami flow approach to Euler.
7o address the lack of C boundedness of T l , our proof makes additional use of the frequency localizationin the construction, which allows for the effective application of the Microlocal Lemma. A number of othersimplifications in the argument arise from the use of frequency localized waves. For instance, many error termscan be estimated in a simpler way than in previous works, and we remove the need for nonstationary phasearguments in solving the relevant elliptic equations.In connection with our space of approximate solutions, we introduce a family of estimates we call compoundfrequency energy levels . These estimates generalize to active scalars the frequency energy levels introduced in[31]. These bounds have the key feature that they carry C type estimates for derivatives of the drift velocityalong the iteration. Otherwise, the lack of C boundedness of T l would prohibit us from deducing theseestimates from the bounds on the scalar field. The overall strategy for the construction is outlined in Section 2. The bulk of the paper then consists of provingthe “Main Lemma”, Lemma 3, which is stated in Section 3. After the statement of the Main Lemma, Section 4 isdevoted to the proof of a “Microlocal Lemma”, which is one of the main technical tools in the paper. Sections 5-8 are then devoted to proving Lemma 3.In Section 9, we explain how the Main Lemma implies the results stated in Theorem 1.1 and Corollary 1.1.Section 11 provides an outline of how Theorem 1.2 also follows from the same Lemma. The modifications usedto prove Theorem 1.5 regarding higher dimensions are explained in Section 3.2.Sections 12 and 13 are devoted to the rigidity properties of weak solutions in the case of odd multipliers. InSection 12, we give a proof of Theorem 1.3 on the rigidity of solutions under weak limits when the multiplier isodd. Section 13 is then devoted to the proof of Theorem 1.4 on the conservation of the Hamiltonian for activescalars with odd multipliers in dimension .The last Section 14 is devoted to proving Theorem 1.6, which shows that the nondegeneracy condition inTheorem 1.5 is necessary in general for the weak limit statement of Theorem 1.1 to apply in higher dimensions.In Section 15 we give a conclusion to the paper and state some open questions. We use the Einstein summation convention of summing over indices which are repeated. We take the conventionthat vectors are written with upper indices, whereas covectors are written with lower indices; thus, for a vectorfield u l and function ξ , we write u · ∇ ξ = u l ∂ l ξ and div u = ∂ l u l .We use the notation X E Y to indicate an inequalities X ≤ Y which have not been proven, but will beproven later on in the course of the argument. We sometimes refer to such inequalities as “goals”. In this Section, we give a technical outline of the main ideas of the construction which includes a list of theimportant error terms and provides a comparison to the cases of the Euler and isometric embedding equations.This section provides the basic ideas to motivate the statement of the Main Lemma of Section 3.We will perform the construction in a space of approximate solutions to the active scalar equation which wenow define.We say that ( θ, u l , R l ) satisfy the scalar-stress equation if ( ∂ t θ + ∂ l ( θu l ) = ∂ l R l u l = T l ( θ ) (2.1)8his system is the analogue for active scalar equations of the Euler-Reynolds system introduced in [22] forthe Euler equations. Here R l is a vector field on T that we call the “stress field” (by analogy with the stresstensor R jl in the Euler-Reynolds equations) which measures the error by which θ fails to solve the active scalarequation.Recall that the operator T l [ θ ] = Z R K l ( h ) θ ( x − h ) dh is a convolution operator with a real-valued kernel K l which is homogenous of degree − as a distribution. Thecorresponding Fourier multiplier m l ( ξ ) = ˆ K l ( ξ ) (2.2)is homogeneous of degree , satisfies m l ( − ξ ) = m l ( ξ ) , and we assume that m l ( ξ ) is smooth on | ξ | = 1 (and therefore smooth away from the origin). To ensure that u l = T l [ θ ] satisfies the divergence free condition ∂ l u l = 0 , we require that m ( ξ ) · ξ = 0 (2.3)At a high level, the basic idea of the convex integration construction is to start with a given solution ( θ, u l , R l ) to (2.1), and proceed to add a (high-frequency) correction Θ to the scalar field θ , so that the corrected scalarfield and drift velocity θ = θ + Θ , u l = u l + U l , U l = T l [Θ] (2.4)satisfy the scalar stress equation (2.1) with a new stress field R l that is significantly smaller than the originalstress field R l . These corrections are added in an iteration to obtain a sequence of solutions to (2.1) ( θ ( k ) , u l ( k ) , R l ( k ) ) such that R l ( k ) → as the number of iterations k tends to infinity. From dimensional analysis and experiencewith the isometric embedding and Euler equations, we expect an estimate k Θ ( k ) k C ≤ C k R ( k ) k / C for thesize of the corrections, so that we will obtain continuous solutions in the limit provided k R ( k ) k C tends to at a reasonable rate . On the other hand, the C norms of the corrections k∇ Θ ( k ) k C will diverge as thefrequencies in the iteration grow to infinity, and we prove convergence of the iteration in H¨older spaces byinterpolating between the bounds for k Θ ( k ) k C and k∇ Θ ( k ) k C after the construction has been optimized toreduce the stress field k R ( k ) k C at the most efficient rate possible. Although this description explains how thescheme works at a high level, we must study the equation and the scheme in much more detail before it is clearthat there is any hope of reducing the stress field R l in this manner.As in [31], we will consider corrections built from rapidly oscillating “plane waves” where we allow forphase functions ξ I and amplitudes θ I which depend on space and time Θ = X I Θ I (2.5) Θ I = e iλξ I ( θ I + δθ I ) (2.6)The amplitude θ I and the phase functions ξ I are scalar functions of our choice, which vary slowly comparedto the frequency parameter λ . The term δθ I is a small correction term which will be made precise later. Eachwave Θ I has a conjugate wave Θ ¯ I = Θ I with opposite phase function ξ ¯ I = − ξ I and amplitude θ ¯ I = ¯ θ I so thatthe overall correction is real valued. In our case, the error will converge to zero exponentially fast: k R ( k ) k C ≤ C e − C k for some constants C , C > .
9e now proceed to calculate the equation satisfied by the corrected scalar field θ = θ + Θ . This requires usto calculate the new drift velocity u l = T l [ θ ] = u l + U l , where U l = T l [Θ] . Our main tool for this calculationis a Microlocal Lemma, which in this case guarantees that each wave Θ I gives rise to a velocity field U lI = T l [Θ I ] = e iλξ I ( u lI + δu lI ) (2.7) u lI = m l ( ∇ ξ ) θ I (2.8)with amplitude determined by the Fourier multiplier m l ( ξ ) in the definition of T l .The amplitude u lI thus has the size comparable to θ I , while the term δu I is a small correction of the sameorder as δθ I . Thus, given a highly oscillatory input such as Θ I = e iλξ I θ I , the operator T l behaves to leadingorder like a multiplication operator on the amplitude. (For our purposes, the simplest way to achieve equation(2.7) will be to use phase functions defined on the whole torus T , but this will not be a serious restriction.)From the Ansatz (2.5) and equation (2.1), we see that the corrected scalar field θ = θ + Θ satisfies theequation ∂ t θ + ∂ l ( u l θ ) = ∂ t Θ + ∂ l ( u l Θ) + ∂ l ( U l θ ) + ∂ l ( U l Θ + R l ) (2.9)We now expand Θ and U into individual waves using (2.5) to derive ∂ t θ + ∂ l ( u l θ ) = ∂ t Θ + ∂ l ( u l Θ) + ∂ l ( U l θ ) (2.10) + X J =¯ I ∂ l ( U lJ Θ I ) + ∂ l ( X I U lI Θ ¯ I + R l ) (2.11)Our goal is to design the correction Θ so that the forcing terms on the right hand side of (2.10)-(2.11) can berepresented in divergence form ∂ l R l for a vector field R l which is significantly smaller in C than the previouserror R l . Our first goal is to cancel out the term R l appearing in the rightmost term of (2.11), which is the only term inequations (2.10)-(2.11) that has low frequency. We expand this term using (2.7)-(2.8) as X I U lI Θ ¯ I + R l = 12 X I ( U lI Θ ¯ I + U l ¯ I Θ I ) + R l ≈ X I ( u lI θ ¯ I + u l ¯ I θ I ) + R l = 12 X I | θ I | ( m l ( ∇ ξ I ) + m l ( −∇ ξ I )) + R l (2.12) = 12 X I | θ I | ( m l ( ∇ ξ I ) + m l ( ∇ ξ I )) + R l (2.13)where the error terms are lower order, involving δθ I and δu lI . Here we can see already why we are restrictedto multipliers m l ( · ) which are not odd. Namely, for an odd multiplier m l ( − ξ ) = − m l ( ξ ) , the high frequencyinteractions fail to leave a nontrivial low frequency part. In other words, the obstruction is that we lack ahigh-low frequency cascade.We therefore assume now that the multiplier m l is not odd. Together with the divergence free property ξ l m l ( ξ ) = 0 and the degree zero homogeneity of the symbol m l ( · ) , this condition implies that there are linearlyindependent vectors in the image of the even part of the multiplier A l = m l ( ξ (1) ) + m l ( − ξ (1) ) , B l = m l ( ξ (2) ) + m l ( − ξ (2) ) (2.14)10here ξ (1) , ξ (2) ∈ Z = b T are nonzero frequencies with integer entries.At this point, since we now have two vectors A l and B l in the image of the even part of m l that are linearlyindependent, there is some hope to get the terms in (2.12) to cancel out. Namely, one should first make sure thatthe phase gradients ∇ ξ I are perturbations of the directions ξ (1) , ξ (2) so that each wave yields a velocity fieldtaking values in the direction ( m l ( ∇ ξ I ) + m l ( −∇ ξ I ) ≈ A l or in the direction ( m l ( ∇ ξ I ) + m l ( −∇ ξ I ) ≈ B l .One would then like to choose coefficients θ I so that terms | θ I | ( m l ( ∇ ξ I ) + m l ( −∇ ξ I )) in (2.12) form theappropriate linear combinations of A l and B l needed to cancel out R l .However, there is an immediate difficulty in implementing the above approach. Namely, although we knowthat A l and B l are linearly independent, it may not be case that R l can be written as a linear combination of A l and B l with non-negative coefficients | θ I | . To get around this difficulty, we take advantage of a degree offreedom which already played an important role in the arguments of [19] and [45]. Namely, observe that we donot need to solve the equation (2.12) exactly, but need only ensure that (2.12) is divergence free. This freedomallows us to subtract from (2.11) any vector field e ( t ) δ l which is constant in space and depends only on time.Therefore the equation we actually solve is more similar to X I | θ I | ( m l ( ∇ ξ I ) + m l ( ∇ ξ I )) = e ( t ) δ l − R lǫ (2.15)where R lǫ is a regularized version of R l and δ l is a constant vector field. If we choose δ l = A l + B l and makesure that e ( t ) is bounded below by, say, e ( t ) ≥ k R ǫ k C on the support of R ǫ , then the coefficients | θ I | solving (2.15) can be guaranteed to be non-negative. Observe also that the equation (2.15) leads to the bounds k θ I k C ≤ C k R ǫ k / C for the amplitudes.The role played by the function e ( t ) δ l is the same as the role played by the low frequency part of the pressurecorrection in the scheme for Euler [31, Section 7.3]. This device in some way appears to limit our proof to theperiodic setting. Controlling the interference terms between high frequency waves is a fundamental difficulty in convex integra-tion. In our case, the interference terms require solving the elliptic equation ∂ l R lH = X J =¯ I ∂ l ( U lJ Θ I ) = X J =¯ I U lJ ∂ l Θ I (2.16) = 12 X J =¯ I ( U lJ ∂ l Θ I + U lI ∂ l Θ J ) (2.17)To leading order, these terms have the form ∂ l R lH = 12 ( iλ ) X J =¯ I e iλ ( ξ I + ξ J ) ( u lJ ∂ l ξ I θ I + u lI ∂ l ξ J θ J ) + . . . (2.18) = 12 ( iλ ) X J =¯ I e iλ ( ξ I + ξ J ) θ I θ J ( m l ( ∇ ξ J ) ∂ l ξ I + m l ( ∇ ξ I ) ∂ l ξ J ) + . . . (2.19)We expect to a gain a factor of λ − while inverting the divergence in (2.18); however, solving (2.18) leads inprinciple to a solution R lH of size k R H k C ≤ k P I | θ I | k C ≤ k R k C , which is not even an improvement onthe size of the previous error R l . These terms therefore seem to already prohibit the construction of continuoussolutions by convex integration. The same difficulty also arises for the Euler equations.For the Euler equations, the key idea introduced in [22] which made it possible to handle high frequencyinterference terms similar to (2.18) was to construct the high frequency building blocks using a family of sta-tionary solutions to the Euler equations known as Beltrami flows. Specifically, the basic building blocks in the11onstruction [22] are constructed using vector fields of the form B l e ik · x where B l is a constant vector ampli-tude, k · x is a linear phase function, and we have ( ik ) × B l = | k | B l so that the experession B l e ik · x is aneigenfunction of curl and hence a stationary solution to Euler. The idea of using Beltrami flows was adaptedin [31] to building blocks V I = e iλξ I v I with nonlinear phase functions ξ I by imposing a “microlocal Bel-trami flow” condition that ( i ∇ ξ I ) × v I = |∇ ξ I | v I pointwise. Viewed from this latter approach, the role of theBeltrami flow condition is to ensure that the leading term in (2.19) cancels out.For the active scalar equations we consider here, such a family of stationary solutions is not available, andmoreover we do not have any method to control interference terms between waves which oscillate in distinctdirections. For instance, suppose that the multiplier m l ( ξ ) is even, and suppose that ξ , ξ ∈ ˆ R are linearlyindependent frequencies for which the terms in (2.19) cancel m ( ξ ) · ξ ± m ( ξ ) · ξ = 0 It then follows from the conditions m ( ξ ) · ξ = 0 , m ( ξ ) · ξ = 0 that both m ( ξ ) and m ( ξ ) must be equal to .More generally, one can show that the even part of the multiplier must vanish when applied to both frequencies m l ( ξ ) + m l ( − ξ ) = m l ( ξ ) + m l ( − ξ ) = 0 if we assume that all of the interference terms in (2.19) cancel. This vanishing of the even part would prohibitany nontrivial contribution to (2.12). In contrast, in the case of the surface quasigeostraphic equation where thedrift velocity is given by u = ∇ ⊥ ( − ∆) − / θ , the set of Laplace eigenfunctions provides a large family of highfrequency, stationary solutions. However, in this case the multiplier m ( ξ ) = i h− ξ , ξ i| ξ | − is odd and we havealready seen that such multipliers are out of reach of our method.Our main observation which allows us to handle these terms is the fact that the interference terms whicharise when an individual wave interacts with itself always vanish to leading order from the structure of theequations. Namely, if we look at a single index J = I , then from the divergence free condition for the symbol m ( ξ ) · ξ = 0 we see that the leading term in (2.19) gives no contribution ( m l ( ∇ ξ I ) ∂ l ξ I + m l ( ∇ ξ I ) ∂ l ξ I ) = 0 Therefore, while we lack a method to control interference terms between waves which oscillate in differentdirections, we can still pursue an approach where in each step of the iteration we use corrections Θ containingwaves waves which oscillate in only a single direction and thus do not interfere with each other. In this Section, we remark on how our observation also gives a new approach to building weak solutions tothe Euler equations which is independent of Beltrami flows, and explain why we expect a loss of regularity bycomparing to analogous considerations in the case of the isometric embedding equations.Our observation of vanishing self-interference terms applies in the case of the Euler equations as well. Forthe Euler equations, an individual wave is a velocity field which takes the form V I = e iλξ I ( v I + δv I ) , and werequire that the amplitude takes values in v I ∈ h∇ ξ I i ⊥ in order to ensure the divergence free condition for V I .In this case, the high frequency interference terms between an individual wave and itself have the form V jI ∂ j V lI = ( iλ ) e iλξ I v jI ∂ j ξ I v lI + lower order terms (2.20)Observe that the requirement v I · ∇ ξ I = 0 forces the the main contribution to cancel. Thus, the method weapply here in principle generalizes to give a new approach to producing H¨older continuous weak solutions tothe Euler equations which entirely avoids the use of Beltrami flows and applies in arbitrary dimensions. Ourobservation appears to be quite natural in that the key cancellation we exploit comes immediately from thestructure of the equations themselves without imposing any particular Ansatz in the construction. On the other12and, in contrast to the use of Beltrami flows for Euler, we are restricted here to removing one component ofthe error at a time during the iteration, which ultimately results in a loss of regularity in the solutions obtainedfrom the construction.The reason we expect to lose regularity from the restriction of removing one component of the error eachstage comes from experience with the isometric embedding equations from the work of Conti, De Lellis andSz´ekelyhidi [18]. For these equations, there is currently no method available for controlling the relevant inter-ference terms between high frequency waves for embeddings of codimension , and this obstruction leads to aloss of regularity for the solutions obtained through convex integration. Namely, without a method to controlinterference terms between distinct waves, it is only possible to eliminate a single, rank one component of themetric error in each step of the iteration from the addition of a single wave. Consequently, it is necessary toincrease the frequencies of the waves multiple times before any C improvement in the metric error can be real-ized, which leads to a loss of regularity. In contrast, the use of Beltrami flows for the Euler equations allows forthe addition of waves which oscillate at the same frequency level in several different directions, and the stresserror can be made smaller in C after only one step of the iteration. Since our scheme suffers from the samedeficiency as in the case of isometric embeddings (that is, we cannot use waves at equal frequency levels whichoscillate in multiple directions), it turns out that our scheme is limited to a H¨older exponent which is inferior tothe exponent / achieved for the Euler equations.The restriction to eliminating a single component of the error in each step of the iteration also threatens tomake our proof considerably more complicated than the scheme used for Euler. While we are unable to avoidthe loss of regularity, we are at least able to keep the overall complexity of the argument to be essentially nomore complicated than the scheme used for Euler. This simplification is accomplished by introducing a newtechnique, which we explain in the following Section. From the discussion in Section 2.2.1, we can now consider a serial convex integration scheme wherein wecannot reduce the size of the error term R l until we have added a series of two corrections θ = θ + Θ (1) + Θ (2) (2.21)Following the original scheme of Nash [40] in the isometric embedding problem, we should first decompose R l into components as R l = c A A l + c B B l where A l and B l are linearly independent vectors in the image of m l ( ξ ) + m l ( − ξ ) defined in (2.14). The firstcorrection Θ (1) to θ should oscillate in the ξ (1) direction in order to eliminate the A l component of the error R l by the method described in Section 2.1. Then, the second correction Θ (2) should have an even larger frequencythan Θ (1) , but the same amplitude | Θ (1) | ∼ | Θ (2) | ∼ | R | / , since its purpose is to eliminate the B l componentof the error R l . Thus, one stage of the convex integration is completed after two steps, where each step involveseliminating one component of the error, and the error R l is smaller in C only at the end of the stage.It appears that such a serial convex integration scheme should be possible for active scalar equations andshould lead to the same H¨older exponent / that we achieve here. On the other hand, such a serial proof seemsto be somewhat complicated compared to the “one-step” scheme used for Euler or to the case of the isometricembedding equations. In our case, a serial proof would involve treating a larger number of error terms havingunfamiliar estimates, and optimizing a larger number of time, frequency and length scale parameters. We avoidthese additional complexities by making a simple observation that allows us to reduce the C norm of the errorin a single step of the iteration rather than several. It turns out that this idea also causes most of the terms in theconstruction to obey estimates which are familiar from experience with the Euler equations, amounting to anoverall more transparent proof. 13ur observation which allows us to reduce the error in every stage of the iteration and thereby simplify ourproof is the following. First, note that the addition of the first correction Θ (1) results in a remaining error R l (1) of the form R l (1) = c B B l + R lE (2.22)where R lE is much smaller than the original error R l , whereas the term | c B B l | ∼ | R l | has the same size. Ratherthan using the second correction Θ (2) to eliminate the term c B B l as discussed previously, we observe that wecan simultaneously get rid of the B l component of the small term R lE , thus leaving an error of the form R l = c A A l + R lJ (2.23)where c A A l is the remaining A l component of R lE , and the term R lJ is an even smaller error term. For our nextcorrection, we can repeat the same idea and eliminate the A l component of (2.23), leaving an error of the form(2.22). Continuing in this way, we see that each correction now causes an improvement in the size of the errorin the C topology, just as in the situation for Euler.The above discussion has been based on the hope that we can really eliminate the A l and B l componentsof the error, which is not entirely justified at this point. In fact, there are some further difficulties which standin our way before this task can be accomplished which will become more clear as we specify the construction.One such difficulty is the appearance of low frequency interference terms. It turns out that the most straightforward approach to the construction based on the ideas Section 2.2 givesrise to certain interference terms of low to intermediate frequency which apparently prohibit the success of ourscheme. Thus, while the idea introduced in Section 2.2 allows us to control the high frequency interferenceterms in a sactifactory manner, we must incorporate one additional idea into the construction before our schemecan handle every type of error term which arises.The ideas in Section 2.2 suggest that a natural approach to the construction is to use waves of the form Θ I = e iλξ I ( θ I + δθ I ) where the phase functions ξ I oscillate in the direction ± ξ (1) (or ± ξ (2) ) in the sense thatthe gradients remain close to their common initial values ∇ ξ I ≈ ± ξ (1) (2.24)For an index I , let us write f ( I ) ∈ {±} to denote the sign appearing in (2.24).According to Section 2.2, we have a method to ensure that high frequency nonlinear interference terms obeygood bounds. Thus, every interaction term of the form ∂ l (Θ I U lJ + Θ J U lI ) (2.25)which arises between waves of the same sign f ( I ) = f ( J ) ∈ {±} can be handled by our method, as theseterms are all of high frequency.A new difficulty arises when we consider interference terms between waves of opposite signs f ( I ) = − f ( J ) , which we call “Low-Frequency Interference Terms”. In this case, the terms of the form Θ I U lJ + Θ J U lI as in (2.25) can be expressed to leading order as Θ I U lJ + Θ J U lI ≈ e iλ ( ξ I + ξ J ) ( θ I u lJ + θ J u lI ) (2.26)When we consider indices with opposite signs f ( I ) = − f ( J ) , the term (2.26) cannot be viewed as a highfrequency error term. In the worst case it may even be true that ∇ ( ξ I + ξ J ) = 0 thanks to the initial conditionssatisfying (2.24). 14t turns out that having low frequency interference terms of the form (2.26) prevents us from solving thequadratic equation to determine the amplitudes θ I . To see this difficulty, note that the left hand side of theequation analogous to (2.15), which includes all low frequency interactions, would have to include terms of theform X I,Jf ( I )=+ , f ( J )= − Θ I U lJ + Θ J U lI = X I,Jf ( I )= f ( J )=+ e iλ ( ξ I − ξ J ) ( θ I ¯ θ J + θ J ¯ θ I ) A l + . . . (2.27)Remarkably, the right hand side of (2.27) appears to obey all the estimates we would require for obtainingsolutions with H¨older regularity / − , despite the appearance of the parameter λ . The problem is that the righthand side of (2.27) must remain bounded from in order to solve the quadratic equation for the amplitudes.On the other hand, there is no way to preclude the possibility that the series (2.27) cancels completely at points ( t, x ) on which the amplitudes θ I ( t, x ) and θ J ( t, x ) have essentially the same size, due to the presence of theoscillating factors e iλ ( ξ I − ξ J ) in the cross terms arising from distinct indices J = I .At first sight, this difficulty would seem to completely prevent us even from achieving continuous solutions,as we are left with no way to obtain a C improvement in the size of the error on the regions where distinctindices interact. We overcome this obstruction by making one more adjustment to the construction. Roughlyspeaking, our idea is to allow the condition (2.24) to be satisfied by “half” the waves in our construction, whereasthe other “half” of the waves in the construction involve phase functions with initial data satisfying ∇ ξ I ≈ ± ξ (1) (2.28)Furthermore, we ensure that every nonlinear interaction which takes place between nonconjugate waves involvesone wave satisfying (2.24), and a second wave satisfying (2.28). In this way, every interference term of the form(2.26) is actually a high frequency error term. Moreover, every wave oscillates in a direction essentially parallelto ξ (1) , so that the idea of Section 2.2 still applies to treat these high frequency interference terms.With these ideas in hand, we are now ready to proceed with the formal construction in detail, beginning withthe statement of the Main Lemma. In order to state the main lemma, let us recall that we have fixed once and for all a choice of linearly independentvectors A l = m l ( ξ (1) ) + m l ( − ξ (1) ) , B l = m l ( ξ (2) ) + m l ( − ξ (2) ) (3.1)where ξ (1) , ξ (2) ∈ Z = b T are nonzero (integral) frequencies. The existence of these vectors is guaranteed bythe condition that m l ( ξ ) is not odd, and the orthogonality condition ξ l m l ( ξ ) = 0 . Definition 3.1.
For a constant vector A l , we say that ( θ, u l , c A , R lJ ) satisfy the Compound Scalar-Stress equation (with vector A l ) if ( ∂ t θ + ∂ l ( θu l ) = ∂ l ( c A A l + R lJ ) u l = T l ( θ ) (3.2)In this case, we will refer to the tuple ( θ, u l , c A , R lJ ) as a compound scalar-stress field .For a solution to the compound scalar-stress equation (2.1), we define compound frequency-energy levels tobe the following 15 efinition 3.2. Let L ≥ be a fixed integer. Let Ξ ≥ , and let e v , e R and e J be positive numbers with e J ≤ e R ≤ e v . We say that ( θ, u l , c A , R l ) have frequency and energy levels below (Ξ , e v , e R , e J ) to order L in C if ( θ, u l , c A , R l ) solve the system (3.2) and satisfy the bounds ||∇ k u || C + k∇ k θ k C ≤ Ξ k e / v k = 1 , . . . , L (3.3) k∇ k ( ∂ t + u · ∇ ) u k C ≤ Ξ k +1 e v k = 0 , . . . , L − (3.4) ||∇ k c A || C ≤ Ξ k e R k = 0 , . . . , L (3.5) ||∇ k ( ∂ t + u · ∇ ) c A || C ≤ Ξ k +1 e / v e R k = 0 , . . . , L − (3.6) ||∇ k R J || C ≤ Ξ k e J k = 0 , . . . , L (3.7) ||∇ k ( ∂ t + u · ∇ ) R J || C ≤ Ξ k +1 e / v e J k = 0 , . . . , L − (3.8)Here ∇ refers only to derivatives in the spatial variables.Note that we assume bounds (3.3)-(3.4) on the drift velocity u l which do not in general follow from thecorresponding bounds on ( θ, c A , R l ) and the transport equation (3.2). We assume these bounds on u l in orderto to avoid logarithmic losses in our estimates which would arise otherwise from the lack of C boundedness ofthe operator u l = T l θ defining the velocity.We now state the Main Lemma of the paper, which summarizes the result of one step of the convex integra-tion procedure. The statement of this lemma involves two constants: K ≥ (specified in Line (5.30) of theconstruction) and K ≥ (determined in Line (5.25) of the construction, see also Section 8.1). These constants K and K depend only on the operator T l in the statement of the Main Theorem. Lemma 3.1 (The Main Lemma) . Suppose that L ≥ and let K, M ≥ be non-negative numbers such that K ≥ K . There is a constant C depending only on L , K , M and the operator T l such that the followingholds:Let ( θ, u l , c A , R lJ ) be any solution of the compound scalar-stress system whose compound frequency andenergy levels are below (Ξ , e v , e R , e J ) to order L in C , and let I ⊆ R be a nonempty closed interval such that supp R J ∪ supp c A ⊆ I × T (3.9) Define the time-scale ˆ τ = Ξ − e − / v , and let e ( t ) : R → R ≥ be any non-negative function for which the lower bound e ( t ) ≥ Ke R for all t ∈ I ± ˆ τ (3.10) is satisfied in a ˆ τ -neighborhood of the interval I , and whose square root satisfies the estimates || d r dt r e / || C ≤ M (Ξ e / v ) r e / R , ≤ r ≤ (3.11) Now let N be any positive number obeying the bound N ≥ (cid:18) e v e R (cid:19) / (3.12) and define the dimensionless parameter b = (cid:18) e / v e / R N (cid:19) / . hen there exists a solution ( θ , u l , c B , R l ) of the form θ = θ + Θ , u = u + U to the CompoundScalar-Stress Equation (3.2) with vector B l whose frequency and energy levels are below (Ξ ′ , e ′ v , e ′ R , e ′ J ) = CN Ξ , e R , K e J , e / v e / R N / ! = CN Ξ , e R , K e J , e / v e / R N ! / e R = CN Ξ , e R , K e J , b − e / v e / R N ! (3.13) to order L in C , and whose stress fields R and c B are supported in supp c B ∪ supp R ⊆ supp e × T (3.14) The correction
Θ = θ − θ is of the form Θ = ∇ · W . This correction and the correction to the velocity field U l = T l [Θ] can be guaranteed to obey the bounds || Θ || C + k U l k C + ≤ Ce / R (3.15) k∇ Θ k C + k∇ U k C ≤ CN Ξ e / R (3.16) k ( ∂ t + u j ∂ j )Θ k C + k ( ∂ t + u j ∂ j ) U k C ≤ C b − Ξ e / v e / R (3.17) || W || C ≤ C Ξ − N − e / R (3.18) k∇ W k C ≤ Ce / R (3.19) || ( ∂ t + u j ∂ j ) W || C ≤ C b − N − e / v e / R (3.20) The energy increment from the correction is prescribed up to errors bounded by (cid:12)(cid:12)(cid:12)(cid:12)Z T | Θ | t, x ) dx − Z T e ( t ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T e ( t ) dx + e R N (3.21) and the incremental energy variation satisfies an estimate (cid:12)(cid:12)(cid:12)(cid:12) ddt Z T | Θ | ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C b − Ξ e / v e R (3.22) uniformly in time. Finally, the space-time support of the correction Θ is contained in supp e × T . The overall structure of Lemma 3.1 is based on the Main Lemma of [31, Lemma 10.1]. The most importantdifference in our Lemma lies in the difference in the definition of the compound frequency energy levels.The bounds implicit in (3.13), which state the rate at which we are able to reduce the stress error, are themost essential point the main lemma and dictate the regularity of the solutions we obtain. Another noticeabledifference between Lemma 3.1 compared to the Lemmas [31, Lemma 10.1] and [34, Lemma 4.1] is that theestimate (3.21) gives us worse control over the increment of energy. In those Lemmas, the term R T e ( t ) dx isnot present, and the error in prescribing the energy increment is of size O ( N − ) .This weaker estimate on the energy increment is still sufficient for the applications considered in thosepapers. In [31] and [34], the same estimate is applied to prove the nontriviality of solutions, by proving that17he energy strictly increases during the iteration at each fixed time slice on which the corrections are nontrivial.The same statement can be obtained here, although in our case the nontriviality of solutions follows alreadyfrom the weak-* approximation statement in Theorem 1.1. In [34], it was shown that a localized version of theestimate (3.21) can be combined with the bounds (3.18)-(3.19) to prove that that the construction necessarilyresults in solutions which fail to have any kind of improved C / ǫx ( B ) local regularity (or even local W / ǫ, regularity) on every open ball B and every time slice contained in the support of the iteration (see [34, Theorem1.2] for a precise statement). This lack of higher regularity is an automatic consequence of the construction,as the same proof shows the failure of local regularity above C / − ǫx regularity for the earlier constructions of C / − ǫt,x solutions in [31, 24]. The same result applies in our setting by the same proof, using the estimates (3.18)-(3.19) and the localized version of (3.21). Namely, our solutions in dimension fail to belong to C / ǫx ( B ) onevery open ball B and every time slice contained in the support of the iteration, and in dimension d fail to haveany local regularity C / (1+4 d )+ ǫx ( B ) in a similar way. In this subsection we make some remarks about how to modify our proof to apply in higher dimensions.In order to prove Theorem 1.5 regarding the case of higher dimensions, the relevant Main Lemma has aslightly different formulation, as one must modify the definitions of the compound scalar stress equation andthe compound frequency energy levels. In the case of dimension d , we assume given a linearly independent setof vectors A (1) , . . . , A ( d ) in the image of the even part of the multiplier. A typical solution to the CompoundScalar Stress equation will then be a solution to the equation ∂ t θ + ∂ l ( θu l ) = ∂ l ( c A, (1) A l (1) + . . . + c A, ( d − A l ( d − + R lJ ) u l = T l ( θ ) (3.23)A single step of the iteration will remove the A (1) component of the error, giving a solution θ and a new errorof the form ∂ t θ + ∂ l ( θ u l ) = ∂ l ( c A, (2) A l (2) + . . . + c A, ( d − A l ( d − + c A, ( d ) A l ( d ) + R lJ, ) (3.24) u l = T l ( θ ) (3.25)At the step above (or even earlier when writing (3.23)) we can absorb the A (2) , . . . , A ( d − components of R J, into the other terms. (To say it in a slightly different way, one can assume from the start in writing (3.23) that R J is a multiple of A ( d ) by absorbing the other components of R J into the other terms.)The Definition 3.2 of compound frequency energy levels now should include d + 1 different energy levels e v ≥ e R, [1] ≥ . . . ≥ e R, [ d − ≥ e J . The Main Lemma then takes as an input a compound scalar stress field withgiven frequency energy levels and outputs another scalar stress field with compound frequency energy levels (Ξ , e v , e R, [1] , . . . , e R, [ d − , e J ) ( CN Ξ , e R, [1] , K e R, [2] , . . . , K e R, [ d − , K e J , e ′ J ) (3.26) e ′ J = e / v e / R, [1] N / e R, [1] (3.27)as in (3.13). All the bounds of the Main Lemma then hold with e R replaced by e R, [1] , since we are eliminatingthe first and largest component of the error, and leaving the other terms for the next stages.The proof of the Main Lemma is then performed similarly as below, but naturally involves more terms andnotation. The Main Lemma is applied to prove Theorem 1.5 in a similar way as is done in Section 9 below, whereone maintains a constant ratio of the consecutive energy levels with size bounded by e v e R, [1] , e R, [ i ] e R, [ i +1] ≤ K Z . Thedifference in the iteration then is the choice of N ( k ) ∼ Z (4 d +1) / instead of (9.14) at later stages k . Comparing18he growth of frequencies Ξ ( k ) ∼ Z (4 d +1) k/ to the decay in energy levels e / R, [1] , ( k ) ∼ Z − k/ as in (9.26), weobtain H¨older regularity up to d +1) as stated in Theorem 1.5.In the next Sections 4-8, we give the proof of the Main Lemma. In the following Sections 9-11, we thenexplain how the Main Lemma can be used to deduce Theorems 1.1-1.2. The following Lemma will be used heavily in the construction in order to control the output of a convolutionoperator applied to a highly oscillatory input. The Lemma allows us to show that, to leading order, a convolutionoperator simply behaves like a multiplication operator when it is applied to a high frequency input with anonlinear phase function.In all of our applications, the kernel K ( h ) below will be a Schwartz function essentially supported on lengthscales of order | h | ∼ λ − for large λ . We normalize the Fourier transform of a function K : R → C to be ˆ K ( ξ ) = Z R e − iξ · h K ( h ) dh Lemma 4.1 (Microlocal Lemma) . Suppose that T [Θ]( x ) = Z R Θ( x − h ) K ( h ) dh is a convolution operator acting on functions Θ : T → C , with a kernel K : R → C in the Schwartz class.Let ξ : T → R and θ : T → C be smooth, periodic functions and λ ∈ Z be an integer. Then for any input ofthe form Θ = e iλξ ( x ) θ ( x ) we have the formula T [Θ]( x ) = e iλξ ( x ) (cid:16) θ ( x ) ˆ K ( λ ∇ ξ ( x )) + δ [ T Θ]( x ) (cid:17) (4.1) where the error in the amplitude term has the explicit form δ [ T Θ]( x ) = Z dr ddr Z R e − iλ ∇ ξ ( x ) · h e iZ ( r,x,h ) θ ( x − rh ) K ( h ) dhZ ( r, x, h ) = rλ Z h a h b ∂ a ∂ b ξ ( x − sh )(1 − s ) ds (4.2) Proof.
Observe that e − iλξ ( x ) T [Θ]( x ) = Z R e iλ ( ξ ( x − h ) − ξ ( x )) θ ( x − h ) K ( h ) dh (4.3)By Taylor expanding, we express ξ ( x − h ) − ξ ( x ) = −∇ ξ ( x ) · h + Z h a h b ∂ a ∂ b ξ ( x − sh )(1 − s ) ds (4.4)In our applications, the kernel K is localized to small values of | h | ∼ λ − for large λ , so we view the secondterm in (4.4) as a small error. Similarly, we think of θ ( x − h ) as a perturbation of θ ( x ) , which motivates us toexpress the right hand side of (4.3) as e − iλξ ( x ) T [Θ]( x ) = Z R e − iλ ∇ ξ ( x ) · h θ ( x ) K ( h ) dh + δ [ T Θ]( x ) , (4.5)19here δ [ T Θ]( x ) is expressed in (4.2). The proof concludes by recognizing that θ ( x ) can be factored out of theintegral in (4.5), which gives formula (4.1). Remark . We remark that the same method applied here to prove Lemma 4.1 can also be iterated to obtain ahigher order expansion of T [Θ]( x ) involving only the functions θ ( x ) , ∇ ξ ( x ) and their derivatives evaluated atthe point x δ [ T θ ]( x ) = − i ∂ a θ ( x ) ∂ a ˆ K ( λ ∇ ξ ( x )) − iλ θ ( x ) ∂ a ∂ b ξ ( x ) ∂ a ∂ b ˆ K ( λ ∇ ξ ( x )) + . . . (4.6)To obtain this further expansion, one modifies the function Z defined in (4.2) to have an additional factor of r in the argument of the phase function Z ( r, x, h ) = rλ Z h a h b ∂ a ∂ b ξ ( x − rsh )(1 − s ) ds The expansion (4.6) is then obtained by Taylor expansion in the variable r via integration by parts. We donot take this approach here because it does not improve our estimates, and results in some more complicatedformulas. We now give a detailed description of the construction. We start by obtaining a complete list of the error terms.Suppose that we are in the setting of Lemma 3.1. Thus, we have a solution ( θ, u, c A , R J ) to the compoundscalar-stress equation with vector A l = m l ( ξ (1) ) + m l ( − ξ (1) ) as in (2.14) ( ∂ t θ + ∂ l ( θu l ) = ∂ l ( c A A l + R lJ ) u l = T l ( θ ) (5.1)whose frequency-energy levels are below (Ξ , e v , e R , e J ) . After adding a correction Θ to the scalar field, thecorrected scalar θ = θ + Θ and drift velocity u l = u l + U l , U l = T l [Θ] satisfy the system ∂ t θ + ∂ l ( θ u l ) = ∂ t Θ + ∂ l ( u l Θ) + ∂ l ( θU l ) + ∂ l (Θ U l + c A A l + R lJ ) (5.2)As a preliminary step, it is necessary to define suitable regularizations ( θ ǫ , u ǫ , ˜ c A , R ǫ ) of the given ( θ, u, c A , R J ) .The purpose of these regularizations is to ensure that only the “low frequency parts” of the given solutions ( θ, u, c A , R J ) will influence the building blocks of the construction. These mollifications give rise to an errorterm R lM = ( u l − u lǫ )Θ + ( θ − θ ǫ ) U l + ( c A − ˜ c A ) A l + ( R lJ − R lǫ ) (5.3)Our goal is to design a correction Θ for the scalar field θ so that the corrected scalar θ = θ + Θ and driftvelocity u l = u l + U l satisfy the compound scalar-stress equation with vector B l = m l ( ξ (2) ) + m l ( − ξ (2) ) asin (2.14) ( ∂ t θ + ∂ l ( θ u l ) = ∂ l ( c B B l + R l ) u l = T l [ θ ] (5.4)whose compound frequency energy levels are bounded as in Lemma 3.1.20 .1 The shape of the corrections Our correction is a sum of individual waves
Θ = X I Θ I (5.5) Θ I = e iλξ I ( θ I + δθ I ) (5.6)where we are free to specify the amplitudes θ I and the phase function ξ I . The parameter λ is a large frequencyparameter of the form λ = B λ N Ξ (5.7)where B λ is a very large constant associated to λ which is chosen at the end of the argument. (For technicalreasons, we will require that λ ∈ Z + is a positive integer, so B λ will really have some dependence on N Ξ , butwill nonetheless be bounded, and should be thought of as a constant.) The term δθ I in (5.6) is a small correctionterm which is present to ensure that the wave Θ I has compact support in frequency space. We will specify δθ I later, but it is important to remark that k δθ I k C → , as λ → ∞ Each wave Θ I has a conjugate wave Θ ¯ I = Θ I with an opposite phase function ξ ¯ I = − ξ I and amplitude θ ¯ I = ¯ θ I so that the overall correction is real-valued. We will choose the amplitudes θ I = ¯ θ I to be real-valued as well.The index I for the wave Θ I consists of two parts I = ( k, f ) ∈ Z × {±} . The discrete index k ∈ Z specifiesthe support of the wave Θ I = Θ ( k,f ) in time. Specifically, the support of Θ ( k,f ) will be contained in the timeinterval [( k − ) τ, ( k + ) τ ] where τ is a time scale parameter that will be chosen during the iteration. Theindex f ∈ {±} is a sign which specifies the direction of oscillation of the wave Θ ( k,f ) .The phase functions ξ I are solutions to the transport equation ( ∂ t + u lǫ ∂ l ) ξ I = 0 ξ I ( t ( I ) , x ) = ˆ ξ I ( x ) (5.8)The amplitudes θ I will be supported on a small time interval during which the phase functions remain close totheir initial data. The initial data ˆ ξ I for the phase function ξ I = ξ ( k,f ) is chosen at the time t ( I ) = kτ dependingon the index I = ( k, f ) ˆ ξ I ( x ) = ˆ ξ ( k, ± ) ( x ) = ± [ k ] ξ (1) · x (5.9)where [ k ] ∈ { , } is equal to when k is even and is equal to when k is odd. In particular, we have ∇ ˆ ξ I = ± [ k ] ξ (1) , [ k ] ∈ { , } Our individual waves are localized in frequency and take the form Θ I = P I ≈ λ [ e iλξ I θ I ] (5.10)The operators P I ≈ λ in (5.10) restrict to frequencies of order λ in a neighborhood of λ ∇ ˆ ξ I . To be explicit, let ˆ η ≈ ( ξ ) be a bump function supported on frequencies ˆ η ≈ ( ξ ) ∈ C ∞ c (cid:16) B | ξ (1) | / ( ξ (1) ) (cid:17) which has the property that ˆ η ≈ ( ξ ) = 1 , if | ξ − ξ (1) | ≤ | ξ (1) |
21e then define a frequency cutoff supported on high frequencies of order λ by rescaling and reflection ˆ η I ≈ λ ( ξ ) = ˆ η ≈ ( ± − [ k ] λ − ξ ) . Then P I ≈ λ is given explicitly by a Fourier multiplier \ P I ≈ λ F ( ξ ) = ˆ η I ≈ λ ( ξ ) ˆ F ( ξ ) . Including this “projection operator” P I ≈ λ guarantees that all the corrections (5.10) have frequency support in theball | ξ − ( λ ∇ ˆ ξ I ) | ≤ λ |∇ ˆ ξ I | , and in particular have integral . Having compact support in frequency space willallow us to easily control the resulting increment to the velocity field, which is obtained by applying anotherFourier multiplier.By the Microlocal Lemma 4.1, it is possible to write the wave (5.10) in the form (5.6) with an explicitremainder δθ I , since we have Θ I = P I ≈ λ [ e iλξ I θ I ] = e iλξ I ( θ I ˆ η I ≈ λ ( λ ∇ ξ I ) + δθ I )= e iλξ I ( θ I + δθ I ) (5.11)provided that the phase gradient is sufficiently close to its initial value |∇ ξ I − ∇ ˆ ξ I | E |∇ ˆ ξ I | (5.12)We will verify that inequality (5.12) is satisfied when the parameter lifespan parameter τ is chosen.Applying the Microlocal Lemma 4.1 again, we can also calculate the resulting correction to the drift veloc-ity. U lI := T l Θ I (5.13) T l P I ≈ λ [ e iλξ I θ I ] = e iλξ I ( θ I ˆ K l ( λ ∇ ξ I )ˆ η I ≈ λ ( λ ∇ ξ I ) + δu lI ) (5.14) U lI = e iλξ I ( θ I m l ( ∇ ξ I ) + δu lI ) (5.15)Therefore, once we have verified (5.12), we have U lI = e iλξ I ( u lI + δu lI ) (5.16) u lI = θ I m l ( ∇ ξ I ) (5.17)with an explicit error term δu lI given by Lemma 4.1. According to Section 5.1, we can now decompose the remaining error terms in Equation (5.2) as follows ∂ t θ + ∂ l ( θ u l ) = ∂ t Θ + ∂ l ( u lǫ Θ) + ∂ l ( θ ǫ U l ) (5.18) + ∂ l [ X I Θ I U l ¯ I + ˜ c A A l + R lǫ ] (5.19) + X J =¯ I ∂ l (Θ I U lJ ) (5.20) + ∂ l R lM (5.21)22he term R lM comes from the regularizations in Equation (5.3).The first objective of the correction is to eliminate the term (5.19), which is the only low frequency termthat arises. However, since we consider oscillations in essentially only one direction ∇ ξ I ≈ ± [ k ] ξ (1) , we willonly able to eliminate the A l component of (5.19).We begin by expanding the low frequency part of the interactions in line (5.19) as X I (Θ I U l ¯ I ) = 12 X I (Θ I U l ¯ I + Θ ¯ I U lI )= X I ∈I + θ I ¯ θ I ( m l ( −∇ ξ I ) + m l ( ∇ ξ I )) + Lower Order Terms (5.22) = X I ∈I + | θ I | ( m l ( −∇ ˆ ξ I ) + m l ( ∇ ˆ ξ I )) + Lower Order Terms (5.23) = X I ∈I + | θ I | A l + Lower Order Terms (5.24)We will give a complete list of the lower order terms below after we have chosen the amplitudes θ I .We wish to choose the amplitudes θ I so that the main term in (5.24) cancels with the A l component of theother terms in line (5.19). We achieve this cancellation in two steps. First, we decompose R ǫ into components R lǫ = c J A l + c B B l (5.25)We also subtract a constant vector field ∂ l ( e ( t ) A l ) = 0 from line (5.19), which leads us to impose to an equation X I ∈I + | θ I | A l = e ( t ) A l + ˜ c A A l + c J A l (5.26) = e ( t )(1 + ε ) A l (5.27) ε = ˜ c A + c J e ( t ) (5.28)for the amplitudes θ I . In this way, the amplitudes θ I are chosen to eliminate the A l component of the lowfrequency part of the stress R lǫ .It will be important for our construction that the term ε is smaller than the constant in the (1 + ε ) term in(5.27). From the lower bound e ( t ) ≥ Ke R assumed in (3.10), we can obtain an upper bound k ε k C ≤ ZK (5.29)on the size of the term (5.28), where Z is a constant depending only on the vectors A l and B l . Now, provided K ≥ K = 2 Z , we have k ε k C ≤ (5.30)A subtle point here is that the bound (5.29) does not follow immediately from (3.10). Namely, we must alsocheck that the same lower bound remains true on the set e ( t ) ≥ Ke R for all ( t, x ) ∈ supp (˜ c A + c J ) , (5.31)which is slightly larger than the supports of the given R J and c A due to a regularization in time in the definitionsof ˜ c A , c J . Thus, the estimates (5.29)-(5.30) are guaranteed only after (5.31) has been verified, which is accom-plished in Line (6.17) below when we choose the mollifying parameters. We now assume that (5.29)-(5.30)hold in order to finish defining the construction. 23rom Equation (5.27), we are led to choose amplitudes of the form θ I = e / ( t ) η k ( t ) γ, I = ( k, f ) (5.32) γ = (1 + ε ) / (5.33)The functions η k ( t ) = η (cid:18) t − kττ (cid:19) are elements of a rescaled partition of unity in time X u ∈ Z η ( t − u ) = 1 which we use to patch together local solutions of Equation (5.27). Our choice of η k ensures that each amplitude θ ( k,f ) has support in a time interval [ kτ − τ , kτ + τ ] of duration τ . The coefficient γ ensures that (5.27) issatisfied, and γ is assured to be well-defined by the bound (5.30).To express the remaining error terms in a compact way, let us introduce the notation ˜ θ I = θ I + δθ I , ˜ u lI = u lI + δu lI Thus, Θ I = e iλξ I ˜ θ I and U lI = e iλξ I ˜ u lI .Having chosen θ I , we can now expand the error term in (5.19) as follows X I Θ I U l ¯ I + ˜ c A A l + R lǫ = c B B l + R lS (5.34) R lS = X I (Θ I U l ¯ I ) − X I ∈I + | θ I | A l (5.35)We now expand (Θ I U l ¯ I + Θ ¯ I U lI ) = ˜ θ I ˜ u l ¯ I + ˜ θ ¯ I ˜ u lI = | θ I | ( m l ( −∇ ˆ ξ I ) + m l ( ∇ ˆ ξ I )) + R lS, + R lS, R lSI, = | θ I | [( m l ( −∇ ξ I ) − m l ( −∇ ˆ ξ I )) + ( m l ( ∇ ξ I ) − m l ( ∇ ˆ ξ I ))] R lSI, = δθ I ˜ u l ¯ I + ˜ θ I δu l ¯ I − δθ I δu l ¯ I + δθ ¯ I ˜ u lI + ˜ θ ¯ I δu lI − δθ ¯ I δu lI which gives R lS = X I ( R lSI, + R lSI, ) (5.36)Note that, at any given time t , at most four indices I contribute to the sum in (5.36). In Sections 5.1-5.2 we have defined the construction up to the specification of a few parameters. Our result isthat the corrected field θ = θ + Θ and drift velocity u l = u l + U l satisfy Equation (5.4) with c B defined inline (5.25), and R = R T + R L + R H + R M + R S (5.37)24he terms R M R S are defined in (5.3) and (5.36). We now rewrite the remaining terms in Equations (5.18)-(5.20) using the fact that the velocity fields appearing in these equations are divergence free.The transport term R T is obtained by solving ∂ l R lT = ( ∂ t + u lǫ ∂ l )Θ (5.38) = X I e iλξ I ( ∂ t + u lǫ ∂ l )˜ θ I (5.39)Here the term where the derivative hits the phase functions vanishes according to equation (5.8). Formula (5.39)suggests that the transport term has frequency λ , so we expect to gain a factor λ − in solving equation (5.38). Infact, we will choose our mollification u ǫ to be a frequency-localized version of u so that together with (5.10), theterm (5.38) is literally supported on frequencies of size λ ≤ | ξ | ≤ λ . Hence, there is a frequency localizingoperator P ≈ λ satisfying ( ∂ t + u lǫ ∂ l )Θ = P ≈ λ [( ∂ t + u ǫ · ∇ )Θ] This frequency localization property allows us to simply define R lT = ∂ l ∆ − P ≈ λ [( ∂ t + u ǫ · ∇ )Θ] (5.40)In particular, we obtain the bound k R T k C ≤ λ − k ( ∂ t + u ǫ · ∇ )Θ k C (5.41)The terms remaining from (5.18) and (5.20) are the High-Low term ∂ l R lL = U l ∂ l θ ǫ (5.42) = X I e iλξ I ˜ u jI ∂ j θ ǫ (5.43)and the high frequency interference terms ∂ l R lH = X J =¯ I U lJ ∂ l Θ I (5.44)The frequency cutoffs in our definitions of θ ǫ , U I and Θ I ensure both of these terms have Fourier support infrequencies λ ≤ | ξ | ≤ λ . Here it is important that we have localized the frequency support each Θ I and U lI to a limited range of angles. As a consequence, U l ∂ l θ ǫ = P ≈ λ [ U l ∂ l θ ǫ ] U lJ ∂ l Θ I = P ≈ λ [ U lJ ∂ l Θ I ] for some frequency projection operator P ≈ λ , and we can define R lL = ∂ l ∆ − P ≈ λ [ U j ∂ j θ ǫ ] (5.45) R lH = X J =¯ I ∂ l ∆ − P ≈ λ [ U lJ ∂ l Θ I ] (5.46)Now that we have written down the error terms (5.37), we must observe that each of these terms can be madesmall. For the transport term R lT , the estimate (5.41) ensures that R T is small once λ is chosen sufficiently large,and the same type of estimate can be used to control the High-Low term R L . The high-frequency interferenceterms require a more careful treatment. 25et us focus on an individual term in the sum. U lJ ∂ l Θ I = ( iλ ) e iλ ( ξ I + ξ J ) ˜ u lJ ∂ l ˜ θ I (5.47)We expand this term as U lJ ∂ l Θ I = ( iλ ) e iλ ( ξ I + ξ J ) ( θ J m l ( ∇ ξ J ) + δu lJ ) ∂ l ξ I ˜ θ I (5.48)If we regard the phase gradients ∇ ξ I ≈ ∇ ˆ ξ I as perturbations of their initial values, the main term in (5.48)vanishes m l ( ∇ ˆ ξ J ) ∂ l ˆ ξ I = m l ( ±∇ ˆ ξ I ) ∂ l ˆ ξ I = 0 from the degree zero homogeneity of m ( ξ ) and the identity m ( ξ ) · ξ = 0 .The terms which remain are all lower order iλ ) U lJ ∂ l Θ I = e iλ ( ξ I + ξ J ) θ J ˜ θ I ( m l ( ∇ ξ J ) − m l ( ∇ ˆ ξ J )) ∂ l ξ I (5.49) + e iλ ( ξ I + ξ J ) θ I ˜ θ J m l ( ∇ ˆ ξ J )( ∂ l ξ I − ∂ l ˆ ξ I ) (5.50) + e iλ ( ξ I + ξ J ) δu lJ ∂ l ξ I ˜ θ I (5.51)The terms (5.49), (5.50) are made small by choosing the lifespan parameter τ to be small, while the term (5.51)is made small once λ is chosen sufficiently large (see Section 6.1). The high-frequency term R H itself is thencontrolled by the estimate k R H k C ≤ Cλ k X J =¯ I U lJ ∂ l Θ I k C from the formula (5.46). This calculation concludes our list of the error terms (5.37). What remains is to specifythe parameters in the construction, prove estimates for the elements of the construction and finally to check thatthe estimates stated in Lemma 3.1 are satisfied. To initialize the argument, we must specify how we regularize the given solution ( θ, u, c A , R l ) to the compoundscalar-stress equation. In this section, we specify how these regularizations are defined. Because the flow mapof the regularized velocity is used to define the regularizations of c A and R l , it is necessary to start with thedefininition of the regularized velocity. After the regularizations of c A and R l are defined, we are able to verifythe lower bound (5.31) which had been assumed previously to guarantee a well-defined construction.To obtain the regularized scalar field θ ǫ and drift velocity u ǫ , we take low frequency projections in the spatialvariables with length scale parameters ǫ θ and ǫ u θ ǫ = P ≤ q θ, where − q = ǫ θ (6.1) u ǫ = P ≤ q u, where − q = ǫ u (6.2)The reason for the double mollification in equation (6.2) will become apparent during the commutator estimatesof Section 7. The operator is given by rescaling a Fourier multiplier \ P ≤ χ F ( ξ ) = ˆ η (cid:18) ξ χ (cid:19) ˆ F ( ξ ) where ˆ η ( ξ ) is a smooth function with compact support in | ξ | ≤ that is equal to on | ξ | ≤ .26y well-known estimates for convolutions with mollifiers satisfying vanishing moment conditions (see [31]Section 5.1), we have k θ − θ ǫ k C ≤ C L ǫ Lθ k∇ L θ k C (6.3) k u − u ǫ k C ≤ C L ǫ Lu k∇ L u k C (6.4)We want to choose the length scales ǫ θ and ǫ u as large as possible while ensuring that the mollification term R lM in (5.3) is acceptably small. The main terms in (5.3) where these mollification errors appear are R lM,θ = X I ( θ − θ ǫ ) e iλξ I u lI (6.5) R lM,u = X I e iλξ I θ I ( u l − u lǫ ) (6.6)Logically, the terms (6.5) and (6.6) are not well-defined until we have specified how to define u ǫ . However,from the expressions (5.32) and (5.17) and the bound (3.11) we have an a priori estimate k R M,θ k C + k R M,u k C ≤ Ae / R ( k θ − θ ǫ k C + k u − u ǫ k C ) (6.7)for A a constant depending only on the parameter M in Lemma 3.1.Using (6.3)-(6.4) for a = L and the bound (3.3), we can choose parameters of the form ǫ θ = ǫ u = 1 B N /L Ξ (6.8)Here B is some large constant depending on A in (6.7) chosen to assure that k R M,θ k C + k R M,u k C ≤ e / v e / R N (6.9)The estimate (6.9) is stronger than what we require for Lemma 3.1. Rather, estimate (6.9) is the type of boundone requires to obtain solutions with regularity / − (see [31] Section 4.6).Observe that the parameter choice (6.8) is exactly the choice of parameter taken in [31] Section 5.2 up to aconstant. We will therefore in many cases be able to refer to the estimates of [31] without repeating the proofs.Having defined θ ǫ and u ǫ , we can now define our regularizations ˜ c A and R lǫ of c A and R lJ .Following [31], we define these regularizations using the coarse scale flow Φ s associated to ∂ t + u ǫ · ∇ ,whose definition we now recall. Definition 6.1.
We define the coarse scale flow Φ s ( t, x ) : R × R × T → R × T to be the unique solution tothe ODE dds Φ s ( t, x ) = 1 dds Φ js ( t, x ) = u jǫ (Φ s ( t, x )) , j = 1 , ( t, x ) = ( t, x ) We can now define our regularizations for c A and R J . First, we mollify both c A and R lJ in space to define c A,ǫ x = η ǫ x ∗ c A (6.10) R ǫ x = η ǫ x ∗ R J (6.11)27e then use the coarse scale flow Φ s and a smooth function η ǫ t ( s ) supported in | s | ≤ ǫ t with integral R η ǫ t ( s ) ds = 1 to average in time and form ˜ c A ( t, x ) = Z c A,ǫ x (Φ s ( t, x )) η ǫ t ( s ) ds (6.12) R lǫ ( t, x ) = Z R ǫ x (Φ s ( t, x )) η ǫ t ( s ) ds (6.13)Thus, the values of R ǫ ( t, x ) and ˜ c A ( t, x ) are obtained by averaging c A and R lJ over an ǫ x -neighborhood of atime ǫ t flow line through ( t, x ) .To estimate ˜ c A and R lǫ , we recall that both c A and R J satisfy the estimates ||∇ k c A || C + (cid:18) e R e J (cid:19) k∇ k R J k ≤ k e R k = 0 , . . . , L (6.14) ||∇ k ( ∂ t + u · ∇ ) c A || C + (cid:18) e R e J (cid:19) ||∇ k ( ∂ t + u · ∇ ) R J || C ≤ k +1 e / v e R k = 0 , . . . , L − (6.15)coming from the compound frequency-energy levels of Definition 3.2.Since the bounds (6.14)-(6.15) coincide with the bounds for the tensor R jl in [31], we can draw from theresults of that paper to control R lǫ and ˜ c A .Following Section 5.5 of [31], we choose length and time scales of the form ǫ x = 1 B N /L Ξ , ǫ t = 1 B ( N Ξ) − e − / R (6.16)We choose B ≥ large enough to bound the terms k (˜ c A − c A ) A l k C + k R lJ − R lǫ k C ≤ e / v e / R N (6.17)which appear in the list of error terms from mollification of Equation (5.3).Note that the choice of parameters (6.16) is the same as the choice made in [31, Section 18.3], and thereforeleads to the same bounds k∇ k (cid:18) ¯ D∂t (cid:19) r ˜ c A k C ≤ C k Ξ k e R (Ξ e / v ) ( r ≥ ( N Ξ e / R ) ( r ≥ N ( k +1 − L ) + /L k∇ k (cid:18) ¯ D∂t (cid:19) r R J k C + k∇ k (cid:18) ¯ D∂t (cid:19) r c J k C ≤ C k Ξ k e J (Ξ e / v ) ( r ≥ ( N Ξ e / R ) ( r ≥ N ( k +1 − L ) + /L (6.18)where we use the notation ( r ≥ m ) = χ [ m, ∞ ) ( r ) . The fact we are using here is that c A obeys the same estimatesas the stress R jl in [31], and the terms R J and c J satisfy even better bounds. The details of the proof are carriedout in [31, Section 18].A crucial point here is that (6.18) contains estimates on second order advective derivatives, even though ourassumed bounds (3.6), (3.8) on c A and R J contain only information regarding first order advective derivatives.The ability to obtain this estimate comes from the fact that the advective derivative ¯ D∂t commutes with its ownflow Φ s , and thus commutes with the averaging along the flow. This observation allows us to integrate by partsin ¯ D∂t R lǫ ( t, x ) = Z ¯ D∂t R ǫ x (Φ s ( t, x )) η ǫ t ( s ) ds (6.19) = − Z R ǫ x (Φ s ( t, x )) η ′ ǫ t ( s ) ds (6.20)28his computation explains why the cost of the second advective derivative in (6.18) is exactly a factor of ǫ − t for the choice of parameter (6.16). We refer to [31, Section 18.6.1] and to [33, Section 12.1] for two differentproofs of this identity.Having defined ˜ c A and R ǫ , we are now able to verify the lower bound (5.31) on the energy profile, whichhad been assumed previously in many of the formulas in our construction. From the assumption (3.9) that supp c A ∪ supp R J ⊆ I × T , we have by construction that supp ˜ c A ∪ supp R ǫ ∪ supp c J ⊆ I ± ǫ t × T Since we assumed the lower bound (3.10) for e ( t ) on the interval I ± Ξ − e − / v , it suffices to check that ǫ t < Ξ − e − / v . This inequality follows from the definition (6.16) of ǫ t and the inequality N ≥ (cid:16) e v e R (cid:17) / ,which follows from condition (3.12).At this point, the only term that remains to be estimated in the mollification term R lM = R lM,θ + R lM,u + ( c A − ˜ c A ) A l + ( R lJ − R lǫ ) + R lM ′ (6.21)is given by R lM ′ = X I e iλξ I ( u l − u lǫ ) δθ I + X I e iλξ I ( θ − θ ǫ ) δu lI (6.22)This term will be estimated when we choose the parameter λ at the end of the argument. The present Section is devoted to choosing the lifespan parameter τ . Here we motivate the choice of τ bycomparing the estimates that will be satisfied by the main error terms and optimizing. However, we warn thereader that the estimates stated in this Section have not yet been established, but will follow from the bounds ofSection 7 below.The lifespan parameter τ determines the length of time during which an amplitude θ I = e / ( t ) η (cid:18) t − t ( I ) τ (cid:19) γ (6.23)is allowed to remain nonzero. The parameter τ is chosen to be small so that the gradients of the phase functions,which satisfy the transport equation ( ∂ t + u jǫ ∂ j ) ∂ l ξ I = − ∂ l u jǫ ∂ j ξ I , (6.24)remain close to their initial values ∇ ξ I ≈ ∇ ˆ ξ I . More precisely, equation (6.24) with initial data ξ I ( t ( I ) , x ) =ˆ ξ I ( x ) leads to a bound of the form |∇ ξ I (Φ s ( t ( I ) , x )) − ∇ ˆ ξ I ( x ) | ≤ Ae A Ξ e / v τ (Ξ e / v ) τ, | s | ≤ τ (6.25)where Ξ e / v is an upper bound on k∇ u ǫ k C ≤ Ξ e / v , cf. Lemma 7.3 below.In our case, we require that τ ≤ Ξ − e − / v , so that the estimate (6.25) becomes k∇ ξ I − ∇ ˆ ξ I k C ≤ A (Ξ e / v ) τ (6.26)Here, there are two main error terms which require the choice of a sharp time cutoff in order to control. Thefirst such term, which is familiar from the case of the Euler equations, is the set of high-frequency interferenceterms in (5.46) R lH = X J =¯ I ∂ l ∆ − P ≈ λ [ U jJ ∂ j Θ I ] (6.27)29ecall from (5.49)-(5.51) that each term in the series (6.27) can be expressed to leading order as iλ ) U lJ ∂ l Θ I = e iλ ( ξ I + ξ J ) θ J ˜ θ I ( m l ( ∇ ξ J ) − m l ( ∇ ˆ ξ J )) ∂ l ξ I (6.28) + e iλ ( ξ I + ξ J ) θ I ˜ θ J m l ( ∇ ˆ ξ J )( ∂ l ξ I − ∂ l ˆ ξ I ) (6.29) + lower order terms (6.30)Formula (6.27) leads to the bound k R H k C ≤ Aλ k X J =¯ I U jJ ∂ j Θ I k C ≤ A max I k θ I k C ( k m l ( ∇ ξ I ) − m l ( ∇ ˆ ξ I ) k C + k∇ ξ I − ∇ ˆ ξ I k C ) + Lower order terms ≤ Ae R max I k∇ ξ I − ∇ ˆ ξ I k C + Lower order terms (6.31) ≤ Ae R (Ξ e / v τ ) + Lower order terms (6.32)where the constant A changes from line to line. The error term (6.32) is made small by choosing the lifespanparameter τ to rhe small compared to the natural time scale Ξ − e − / v of the coarse scale velocity u ǫ . The otherterms in (6.32) are lower order in the sense that they can be made small by a suitable choice of λ .The price we pay by introducing sharp cutoffs is a worse bound on the transport term. R lT = ∂ l ∆ − P ≈ λ [( ∂ t + u ǫ · ∇ )Θ] (6.33) ( ∂ t + u ǫ · ∇ )Θ = X I e iλξ I ( ∂ t + u ǫ · ∇ ) θ I + Lower order terms (6.34)The time cutoffs appear in the formula (6.23) for the amplitude, and give rise to a term of size k ( ∂ t + u ǫ · ∇ )Θ k C = Aτ − e / R + Lower order terms (6.35)which leads in turn from the definition (5.7) of λ . to a bound on the transport term k R T k C ≤ Aλ − k ( ∂ t + u ǫ · ∇ )Θ k C (6.36) ≤ AB − λ ( N Ξ) − τ − e / R + Lower order terms (6.37)We therefore choose τ = B − / λ e / v e / R N ! / Ξ − e − / v (6.38)in order to optimize between the estimates for the leading term in (6.32) and (6.37). This choice leads to the C estimate k R k C E e / v e / R N ! / e / v e / R N (6.39)stated in Lemma 3.1, and ultimately to the regularity / − .Unlike the case of the Euler equations, there is also a second error term which requires sharp time cutoffs tomake small in our present scheme, namely the Stress term R S appearing in (5.36). It turns out that this term alsosatisfies the same estimate (6.32), and consequently will be among the largest error terms, having size (6.39)30fter the above choice of τ . The reason that we see this extra term compared to the case of Euler is that themethod we have used here to solve the quadratic equation (5.26) requires the phase gradients ∇ ξ I to remainvery close to their initial values ∇ ξ I ≈ ∇ ˆ ξ I to within an error much smaller than O (1) . In the case of Euler,the equation analogous to (2.15) can be solved using nonlinear phase functions in a way which allows for thephase gradients to depart from their initial values by an error of size k∇ ξ I − ∇ ˆ ξ I k C = O (1) (see [31, Section7.3]). Ideally, one would hope to solve equation (2.15) in a similar manner to avoid generating error terms suchas these which require sharp time cutoffs to treat as above.We now turn our attention to obtaining estimates for the terms in the construction. In particular, we needto establish the estimates (6.32) and (6.37) precisely, and also to estimate the other error terms. The proof isconcluded by choosing the constant B λ in (5.7) to be sufficiently large so that the inequality (6.39) holds asstated, without any implicit constant factor. Lemma 7.1 (Coarse Scale Flow Estimates) . Let L ≥ be an integeras in Lemma 3.1. The mollified velocityfield u ǫ defined in (6.2) obeys the estimates k∇ k u ǫ k C ≤ C k Ξ k e / v N ( k − L ) + /L , k ≥ , (7.1) k∇ k ( ∂ t + u ǫ · ∇ ) u ǫ k C ≤ C k Ξ k +1 e v N ( k +1 − L ) + /L , k ≥ (7.2) for some universal positive constants C k .Proof. For k ≤ L , we see that (7.1) holds in view of the iterative assumption (3.3). For k > L , there is anadditional cost of ǫ ( k − L ) + u = ( B − N /L Ξ) ( k − L ) + , where we have used the choice of ǫ u in (6.8).In order to prove (7.2), we recall that u jǫ = P ≤ q u j , where − q = ǫ u = B − N /L Ξ . We then have P ≤ q ( ∂ t u + u · ∇ u ) = ( ∂ t u ǫ + u ǫ · ∇ u ǫ ) − Q ǫ ( u, u ) , (7.3)where Q jǫ ( u, u ) = u iǫ ∂ i u jǫ − P ≤ q ( u i ∂ i u j ) (7.4) = [ P ≤ q u i ∂ i , P ≤ q ]( P ≤ q u j ) + P ≤ q (cid:16) [ u i ∂ i , P ≤ q ] u j (cid:17) − P ≤ q (cid:16) ( u i − P ≤ q u i ) ∂ i ( P ≤ q u j ) (cid:17) . (7.5)The estimate k Q ǫ ( u, u ) k C ≤ Cǫ u Ξ e v ≤ CN − /L Ξ e v (7.6)follows from (7.5) precisely as in [31, Section 16], by appealing to (3.3). The decomposition (7.5) of thequadratic commutator term is convenient since it allows one to estimate without additional complications thehigher order derivatives ∇ k Q ǫ ( u, u ) . Derivatives up to order L − each introduce a factor of Ξ , while pastthat order the derivatives fall on the mollifier P ≤ q and the cost per derivative is a constant multiple of Ξ N /L .Combining with k∇ k P ≤ q ( ∂ t u j + u i ∂ i u j ) k C ≤ C k Ξ k +1 e v N ( k +1 − L ) + /L (7.7)which follows from the definition of q and (3.4), the proof of (7.2) is completed. Lemma 7.2 (Commutator Estimates) . Let D ≥ and let Q be a convolution operator Qf ( x ) = Z R f ( x − h ) q ( h ) dh hose kernel q satisfies the estimates k |∇ k q | ( h ) k L ( R ) + k| h | |∇ k q | ( h ) k L ( R ) ≤ λ k for some λ ≥ N Ξ , and all ≤ | k | ≤ D . Then the commutator h ¯ D∂t , Q i = [ ∂ t + u ǫ · ∇ , Q ] satisfies theestimates (cid:13)(cid:13)(cid:13)(cid:13) ∇ k (cid:20) ¯ D∂t , Q (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C k Ξ e / v λ k , ≤ k ≤ D − as a bounded operator on C ( R × T ) . In fact, the above lemma will only be applied to operators Q for which λ is given as in (5.7). Lemma 7.3 (Transport Estimates) . Let L ≥ , and denote by ¯ D∂t the convective derivative associated to theflow u ǫ . The phase gradients ∇ ξ I obey the bound k∇ k (cid:18) ¯ D∂t (cid:19) r ∇ ξ I k C ≤ C k Ξ k (Ξ e / v ) r N ( k +( r − + +1 − L ) + /L (7.8) for all k ≥ and r ∈ { , , } . Moreover, the same bound holds if ∇ k ( ¯ D∂t ) r is replaced by D ( k,r ) , where thelatter is defined by D ( k,r ) = ∇ k (cid:18) ¯ D∂t (cid:19) r ∇ k (cid:18) ¯ D∂t (cid:19) r ∇ k , (7.9) with k + k + k = k , k i ≥ , r + r = r , and r i ≥ .We also have the estimate |∇ ξ I (Φ s ( t, x )) − ∇ ˆ ξ I ( x ) | ≤ Cb (7.10) b = B − / λ e / v e / R N ! / = τ Ξ e / v where Φ s is the coarse scale flow defined in 6.1, and τ is specified as in line (6.38) .Proof. In order to establish (7.8) for r = 0 , one appeals to (6.24), and obtains ( ∂ t + u jǫ ∂ j ) ∇ k ∂ l ξ I = −∇ k ( ∂ l u jǫ ∂ j ξ I ) + [ u jǫ ∂ j , ∇ k ] ∂ l ξ I . (7.11)The bound for r = 0 then follows from the Gr¨onwall inequality in the above identity, estimate (7.1), and thechoice for τ in (6.38). Similarly, from ∇ k ( ∂ t + u jǫ ∂ j ) ∂ l ξ I = −∇ k ( ∂ l u jǫ ∂ j ξ I ) (7.12)and (3.3) we obtain the estimate (7.8) with r = 1 .Lastly, in order to obtain the desired estimate when r = 2 we note that (cid:18) ¯ D∂t (cid:19) ∂ l ξ I = ( ∂ t + u iǫ ∂ i ) ∂ l ξ I = − ( ∂ t + u iǫ ∂ i )( ∂ l u jǫ ∂ j ξ I ) (7.13) = − ∂ j ξ I ( ∂ t + u iǫ ∂ i )( ∂ l u jǫ ) − ∂ l u jǫ ( ∂ t + u iǫ ∂ i )( ∂ j ξ I ) (7.14) = − ∂ j ξ I ∂ l ( ∂ t + u iǫ ∂ i ) u jǫ + 2 ∂ l u jǫ ∂ j u iǫ ∂ i ξ I . (7.15)32n particular, it is important that the second convective derivative of ∇ ξ I only depends on a single convectivederivative of u ǫ . By appealing to Lemma 7.1, from (7.15) we obtain that k (cid:18) ¯ D∂t (cid:19) ∇ ξ I k C ≤ C Ξ e v . (7.16)The bound (7.8) with r = 2 and k ≥ , similarly follows from (7.15), the Leibniz rule, Lemma 7.1, and estimate(7.8) with r = 0 .The estimate (7.10) follows from the bound (7.8) with k = 0 and r = 1 , from the calculation |∇ ξ I (Φ s ( t, x )) − ∇ ˆ ξ I ( x ) | ≤ Z s (cid:12)(cid:12)(cid:12) ¯ D∂t ∇ ξ I (Φ σ ( t, x )) (cid:12)(cid:12)(cid:12) dσ ≤ C | s | Ξ e / v ≤ Cb, if | s | ≤ τ Lemma 7.4 (Principal Amplitude estimates) . Let L ≥ and τ be defined in (6.38) . Then the principal parts ofthe scalar amplitude θ I , and the velocity amplitude u I , obey the bounds k D ( k,r ) θ I k C + k D ( k,r ) u I k C ≤ C k Ξ k e / R τ − r N ( k +1 − L ) + /L (7.17) for all k ≥ and r ∈ { , , } , for some suitable universal constants C k > .Proof. First, we note that in view of (5.17) we have u lI = θ I m l ( ∇ ξ I ) . Since the multiplier m is smooth outsidethe origin and in view of Lemma 7.3 we have bounds for the derivatives of ∇ ξ I , the bound on u I follows fromthat on θ I , up to possibly increasing the constant C K by a constant factor.From (5.28) and (5.32) we recall that θ I = η (cid:18) t − kττ (cid:19) e ( t ) / γ = η (cid:18) t − kττ (cid:19) e ( t ) / (1 + ε ) / (7.18)where ε = (˜ c A + c J ) /e ( t ) . Using (6.18), the lower bound e ( t ) ≥ K e R and (5.30), we obtain the followingestimates for ε and γ = (1 + ε ) / k∇ k (cid:18) ¯ D∂t (cid:19) r ε k C + k∇ k (cid:18) ¯ D∂t (cid:19) r γ k C ≤ C k Ξ k e R (Ξ e / v ) ( r ≥ ( N Ξ e / R ) ( r ≥ N ( k +1 − L ) + /L . (7.19)The bounds for spatial derivatives of θ I now follow from (7.19) since the other terms η (cid:0) t − kττ (cid:1) and e / ( t ) donot depend on x . Lemma 7.4 requires us also to show that that each advective derivative up to order costs atmost Cτ − per derivative. For the time cutoff and the function e / ( t ) in (7.18), the cost of τ − follows bydefinition for the time cutoff and by (3.11) for e / ( t ) using the inequality Ξ e / v ≤ τ − from the choice of τ inSection 6.1. For the other terms, the estimate (7.19) tells us that the first advective derivative costs Ξ e / v ≤ τ − ,and taking two advective derivatives costs (cid:12)(cid:12)(cid:12)(cid:12) ¯ D ∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (Ξ e / v )( N Ξ e / R ) = CN Ξ e / v e / R = Cτ − from the choice of τ in (6.38). The bounds for the spatial derivatives then follow from the pattern in (7.19). Lemma 7.5 (Amplitude correction estimates) . Under the hypotheses of Lemma 7.4, the corrections δθ I and δu lI to the scalar amplitude and the velocity amplitude obey the bounds k D ( k,r ) δθ I k C + k D ( k,r ) δu I k C ≤ C k B − λ N − Ξ k e / R τ − r N ( k +2 − L ) + /L (7.20) for all k ≥ and r ∈ { , , } , for some suitable universal constants C k > . roof. These estimates are obtained by explicitly differentiating the formulas for δθ I and δu lI coming from theMicrolocal Lemma, Lemma 4.1. Here we carry out the calculation for the case of δθ I , since the term δu lI canbe treated in the same way. Recall that Θ I = P I ≈ λ ( e iλξ I θ I ) = e iλξ I ( θ I + δθ I ) Applying Lemma 4.1 with K ( h ) = η I ≈ λ ( h ) , we have the following formula for δθ I δθ I = δθ I, − δθ I, δθ I, = Z dr Z e − iλ ∇ ξ I ( x ) · h e iZ ( r,x,h ) ( iλ ) (cid:20)Z h a h b ∂ a ∂ b ξ I ( x − sh )(1 − s ) ds (cid:21) θ I ( x − rh ) η I ≈ λ ( h ) dh (7.21) δθ I, = Z dr Z e − iλ ∇ ξ I ( x ) · h e iZ ( r,x,h ) ∂ a θ I ( x − rh ) h a η I ≈ λ ( h ) dh (7.22)with Z ( r, x, h ) = rλ R h a h b ∂ a ∂ b ξ ( x − sh )(1 − s ) ds and where η I ≈ λ is defined after line (5.10). In particular,recall that the kernel η I ≈ λ ( h ) = 10 k ] λ η ≈ ( ± [ k ] λh ) is constructed by rescaling a Schwartz kernel by afactor λ , and therefore satisfies the estimates k| h | m η I ≈ λ k L h ≤ C m λ − m (7.23)Combining the estimate (7.23) with the bounds of Lemma 7.4 and Lemma 7.3 gives the C estimate for ∇ k δθ I .Proving estimates for advective derivatives of δθ I is tedious, but straightforward. To ease notation let uswrite Z ( r, x, h ) = rλh a h b Z ab where Z ab = Z ab ( t, x, h ) = Z ∂ a ∂ b ξ I ( x − sh )(1 − s ) ds We will sketch one example and estimate the advective derivative of the term in (7.21). ( ∂ t + u ǫ · ∇ ) δθ I, ( t, x ) = − iT (1) + iT (2) + T (3) + T (4) (7.24) T (1) = Z dr Z e − iλ ∇ ξ I ( x ) · h e iZ ( iλ ) h a h b Z ab θ I ( x − rh ) ¯ D∂t ∂ c ξ I ( x ) λh c η I ≈ λ ( h ) dh (7.25) T (2) = Z dr Z e − iλ ∇ ξ I ( x ) · h e iZ ( iλ ) h a h b Z ab θ I ( x − rh ) r (cid:18) ∂ t + u iǫ ( x ) ∂∂x i (cid:19) Z ab λh a h b η I ≈ λ ( h ) dh (7.26) T (3) = Z dr Z e − iλ ∇ ξ I ( x ) · h e iZ ( iλ ) h a h b (cid:18) ∂ t + u iǫ ( x ) ∂∂x i (cid:19) Z ab θ I ( x − rh ) η I ≈ λ ( h ) dh (7.27) T (4) = Z dr Z e − iλ ∇ ξ I ( x ) · h e iZ ( iλ ) h a h b Z ab (cid:18) ∂ t + u iǫ ( x ) ∂∂x i (cid:19) θ I ( x − rh ) η I ≈ λ ( h ) dh (7.28)The pattern we observe in (7.25)-(7.28) is that the cost of the first advective derivative is given by Ξ e / v forevery term. This cost is most clear for the term (7.25). The advective derivative brings down one term of size k ¯ D∂t ∇ ξ I k C ≤ C Ξ e / v and also introduces the factor λh c . The λ and the h cancel out in terms of the estimate, since we gain a λ − when we apply the bound k h a h b h c η I ≈ λ ( h ) k L h ≤ Cλ − u iǫ ( x ) with the nearbypoint in the integral. For example, for the term (cid:0) ∂ t + u iǫ ( x ) ∂∂x i (cid:1) θ I ( x − rh ) in (7.28) we write (cid:18) ∂ t + u iǫ ( x ) ∂∂x i (cid:19) θ I ( x − rh ) = ¯ D∂t θ I ( x − rh ) + ( u iǫ ( x ) − u iǫ ( x − rh )) ∂ i θ I ( x − rh ) (7.29)The cost of Ξ e / v for the advective derivative on θ I follows from Lemma 7.4. For the latter term, we write ( u iǫ ( x ) − u iǫ ( x − rh )) ∂ i θ I ( x − rh ) = − r Z ∂ c u iǫ ( x − σrh ) dσ ∂ i θ I ( x − rh ) h c (7.30)The term where ∂ c u iǫ appears accounts for the cost of k∇ u ǫ k C ≤ Ξ e / v . The derivative hitting θ I costs afactor of Ξ , but this factor is regained by the factor h c that has appeared, which gains a λ − when combinedwith the kernel as usual. Repeating this observation many times for each one of (7.26)-(7.28), one obtains thefirst advective derivative bound in (7.20). We omit the details.One also has to take a second advective derivative in order to prove (7.20), giving rise to another long seriesof terms which obey the correct bounds. We omit the proof of this estimate also, but we remark that one canavoid using these bounds during the course of the proof. The only applications of these bounds are in Section 8.3for a lower order part of the advective derivative of the transport term, and in this case one can substitute secondorder commutator estimates as in Lemma 7.2, which are somewhat less tedious to write down. Corollary 7.1.
The bounds (7.17) of Lemma 7.4 hold also for ˜ θ I = θ I + δθ I and for ˜ u lI = u lI + δu lI . In this Subection, we obtain estimates for the corrections Θ and U l = T l [Θ] to the scalar field and driftvelocity. These bounds confirm that the estimates (3.15)-(3.20) of Lemma 3.1 are satisfied. As with the previousLemmas 7.1-7.5 and our choices of parameters, the results we obtain in this section are familiar from [31,Section 22.1]. In our setting, these estimates turn out to be a bit easier to check thanks to our use of frequencylocalizing projections. Proposition 7.1.
Under the hypotheses of Lemma 7.4, the corrections Θ I and U lI to the scalar field and thedrift velocity satisfy k D ( k,r ) Θ I k C + k D ( k,r ) U I k C ≤ C k ( B λ N Ξ) k τ − r e / R (7.31) for ≤ r ≤ .Proof. We outline the proof of (7.31) for Θ I , as the velocity field U I can be treated in the same way. Here werecall again that Θ I = P I ≈ λ [ e iλξ I θ I ] = e iλξ I ˜ θ I For r = 0 , the estimates for ∇ k Θ I follow from the bound k θ I k C ≤ Ce / R , and the definition of λ . To estimatethe advective derivatives, we write ¯ D∂t Θ I = e iλξ I ¯ D∂t ˜ θ I (7.32) ¯ D ∂t Θ I = e iλξ I ¯ D ∂t ˜ θ I (7.33)35he bounds (7.31) now follow from the bounds of Lemma 7.3 and Corollary 7.1. The main terms in the estimatesfor spatial derivatives arise in every case when the derivatives fall on e iλξ I . Alternatively, one can obtain thesame bounds using commutator estimates such as those of Lemma 7.2 extended to second order commutators.Note that this latter approach avoids using the second advective derivative estimates proven in Lemma 7.5.Lemma 3.1 also requires bounds on a vector field W l satisfying div W = Θ . To define W l , first recall thatthe corrections Θ I = P I ≈ λ ( e iλξ I θ I ) are frequency localized, which allows us to invert the divergence using the standard Helmholtz solution W lI = ∂ l ∆ − P I ≈ λ ( e iλξ I θ I ) (7.34)With this definition, we have Θ = div W for W l = P I W lI . The bounds (3.18)-(3.20) of Lemma 3.1 nowfollow from Lemma 7.2 and Lemma 7.4 by writing ( ∂ t + u ǫ · ∇ ) W I = (cid:20) ¯ D∂t , ∂ l ∆ − P I ≈ λ (cid:21) ( e iλξ I θ I ) + ∂ l ∆ − P I ≈ λ ( e iλξ I ¯ D∂t θ I ) (7.35)and differentiating in space. We conclude this Section by verifying the estimates (3.21) and (3.22) for prescribing the energy increment. Toobtain the estimate (3.21), let t ∈ R and write Z T | Θ | ( t, x ) dx = X I,J Z Θ I · Θ J ( t, x ) dx (7.36)For indices J = ¯ I which are not conjugate to each other, the product Θ I · Θ J is localized at frequency ≈ λ , andin particular has integral . The only remaining terms are Z T | Θ | ( t, x ) dx = X I Z | Θ I | ( t, x ) dx | Θ I | = | θ I + δθ I | = | θ I | + 2 δθ I θ I + δθ I The terms involving δθ I can all be estimated using Lemma 7.5 and Lemma 7.4. X I (cid:12)(cid:12)(cid:12) Z T θ I δθ I + ( δθ I ) dx (cid:12)(cid:12)(cid:12) ≤ C e R B λ N The main terms are then given by X I Z T | θ I | ( t, x ) dx = X I Z η k ( t ) e ( t ) γ dx = 2 Z e ( t ) γ dx = 2 Z T e ( t )(1 + ε ) dx The bound (3.21) now follows from (5.30) provided B λ is sufficiently large.36n order to obtain the estimate (3.22), we differentiate (7.36) with respect to t , and use the fact that the coarsescale velocity field u ǫ is divergence free ddt Z T | Θ | ( t, x ) dx = X I,J Z T ( ∂ t + u ǫ · ∇ )Θ I · Θ J ( t, x ) dx At this point, we again observe that the terms ( ∂ t + u ǫ ·∇ )Θ I · Θ J are localized in frequency space at frequenciesof order λ for all nonconjugate indices J = ¯ I . These terms therefore integrate to and we are left with ddt Z T | Θ | ( t, x ) dx = X I Z T ( ∂ t + u ǫ · ∇ ) | Θ I | ( t, x ) dx = X I Z T ( ∂ t + u ǫ · ∇ ) | ˜ θ I | dx The bound (3.22) now follows from Corollary 7.1.
The statement (3.13) in Lemma 3.1 requires us to prove that the new scalar field and drift velocity θ = θ + Θ , u l = u l + U l satisfy the bounds (3.3)-(3.4) for the new compound frequency energy levels (Ξ ′ , e ′ v , e ′ R , e ′ J ) =( CN Ξ , e R , K e J , e ′ J ) with e ′ J = e / v e / R N ! / e R The bounds in (3.3) already follow from the arguments in [31, Section 22], as the scalar field θ and drift velocity u l both share the same estimates as the coarse scale velocity v l in that paper, and the corrections Θ and U l bothshare the same estimates at the corrections V l in that paper. The only new point here is how we establish thebound k ( ∂ t u + u · ∇ ) u k C E (Ξ ′ e ′ v ) = CN Ξ e R (7.37)This estimate, which is quadratic in the velocity, is analogous to the bound for the pressure gradient in the caseof Euler.The idea is to use the assumed bound (3.4) and write ( ∂ t u + u · ∇ ) u = ( ∂ t u + u · ∇ u ) + U · ∇ u + ( ∂ t + u · ∇ ) U (7.38)In the case of Euler, the first term ( ∂ t u + u · ∇ u ) can be bounded using the Euler-Reynolds equations. In ourcase, though, the bound (7.37) on the advective derivative cannot be obtained from commuting the operator T l with the compound scalar stress equation due to the lack of C boundedness of T l , and arguments involvingfrequency truncations still give logarithmic losses.The idea is that we have already assumed the bound k ( ∂ t u + u · ∇ u ) k C ≤ Ξ e v , so that (7.37) follows fromthe condition N ≥ (cid:16) e v e R (cid:17) . Also, further advective derivatives can be estimated at a cost smaller than N Ξ perderivative up to order L − , giving (3.4) for this term. The proof of (3.4) for the other two terms is the same asin [31, Section 22]. The main idea is to write ( ∂ t + u · ∇ ) = ( ∂ t + u ǫ · ∇ ) + ( u − u ǫ ) · ∇ , and then to applythe relevant bounds established earlier on in Sections 6-7.37 Estimates for the New Stress
In this Section, we conclude the proof of Lemma 3.1 by establishing estimates for the error terms contributingto the new stress field which were derived in Section 5.3. Recall from that section that the new scalar field θ = θ + Θ and the new drift velocity u l = u l + U l satisfy the compound scalar stress equation ( ∂ t θ + ∂ l ( θ u l ) = ∂ l ( c B B l + R l ) u l = T l [ θ ] (8.1)The function c B is defined in (5.25), and the new stress field has the form R l = R T + R L + R H + R M + R S (8.2)as in (5.37). For these error terms, the Main Lemma requires us to show that the bounds of Definition 3.2 aresatisfied for the compound frequency energy levels (Ξ ′ , e ′ v , e ′ R , e ′ J ) specified in (3.13). Our starting point willbe to prove the C estimates k c B k C E K e J (8.3) k R k C E e ′ J (8.4) e ′ J = e / v e / R N ! / e R (8.5)We will obtain these bounds in Section 8.1, at which point we will finally specify the large constant B λ appearingin line (5.7) where λ is defined.Once the C estimates are established and B λ is chosen, the bounds on spatial derivatives k∇ k c B k C E C ( N Ξ) k K e J k = 0 , . . . , L (8.6) k∇ k R k C E C ( N Ξ) k e ′ J , k = 0 , . . . , L (8.7)will be clear, and we will also need to verify the estimates for the advective derivatives k∇ k ( ∂ t + u · ∇ ) c B k C E C ( N Ξ) k ( N Ξ e / R ) K e J (8.8) k∇ k ( ∂ t + u · ∇ ) R k C E C ( N Ξ) k ( N Ξ e / R ) e ′ J (8.9) k = 0 , . . . , L − These bounds will be checked in Sections 8.2 and 8.3, which will conclude the proof of Lemma 3.1. C bounds In this Section, we establish the C bounds (8.3)-(8.4). The bound (8.4) will be obtained only after the constant B λ of line (5.7) is chosen sufficiently large.First, observe that the estimate (8.3) for c B follows immediately from line (5.25) where c B is defined, andthe bound k R ǫ k C ≤ k R J k C . Note that the constant K depends only on the operator T l .It now remains to estimate the stress terms appearing in (8.2). We estimate each of these in turn. The mollification term R lM . We recall from (5.3) and (6.21) that R lM = ( u l − u lǫ )Θ + ( θ − θ ǫ ) U l + ( c A − ˜ c A ) A l + ( R lJ − R lǫ ) (8.10) = R lM,θ + R lM,u + ( c A − ˜ c A ) A l + ( R lJ − R lǫ ) + R lM ′ . (8.11)38ote that by the choice of B , from (6.9) and (6.17) we have that k R M,θ k C + k R M,u k C + k ( c A − ˜ c A ) A l k C + k ( R lJ − R lǫ ) k C ≤ e / v e / R N = 150 e / v e / R N ! e R (8.12)The factor (cid:18) e / v e / R N (cid:19) is less than , so this estimate is more than enough to achieve the bound (8.4).To estimate the remaining term R lM ′ = X I e iλξ I ( u l − u lǫ ) δθ I + X I e iλξ I ( θ − θ ǫ ) δu lI (8.13)recall the estimates k ( u l − u lǫ ) k C + k ( θ − θ ǫ ) k C ≤ e / v N ≤ e / v (8.14) k δθ I k C + k δu lI k C ≤ C e / R B λ N (8.15)from Section 6 and Lemma 7.5 (in fact the estimates for the terms ( θ − θ ǫ ) and ( u − u ǫ ) are even better). Notealso that, at any given time t , at most four indices I contribute to the sum in (8.13).For sufficiently large values of B λ , we therefore obtain k R lM ′ k C ≤ e / v e / R N which is sufficient for (8.4). The Stress term R S . To estimate R S , let us recall from (5.36) that we can express R lS = X I ( R lSI, + R lSI, ) (8.16) R lSI, = | θ I | [( m l ( −∇ ξ I ) − m l ( −∇ ˆ ξ I )) + ( m l ( ∇ ξ I ) − m l ( ∇ ˆ ξ I ))] (8.17) R lSI, = δθ I ˜ u l ¯ I + ˜ θ I δu l ¯ I − δθ I δu l ¯ I + δθ ¯ I ˜ u lI + ˜ θ ¯ I δu lI − δθ ¯ I δu lI (8.18)The estimates of Lemma 7.4, Lemma 7.5 and Corollary 7.1 give k R SI, k C ≤ CB λ N e R At any given time t , as most terms of the form R SI, are nonzero, which allows us to obtain the estimate k X I | R SI, |k C ≤ e / v e / R N which is sufficient for (8.4), by taking the value of B λ sufficiently large.We estimate the terms in (8.17) using (7.10) and Lemma 7.4, in order to obtain the bound k R lSI, k C ≤ CB / λ e / v e / R N ! / e R
39y choosing the constant B λ sufficiently large, we obtain the bound k X I | R lSI, |k C ≤ e ′ J (8.19)where e ′ J , as defined in (8.4), is our goal for the size of the new stress term R l .For the next stress terms, R L and R T , we use that they are frequency localized between two constantmultiples of λ , and thus we can appeal to the estimate k∇ ∆ − P ≈ λ k C → C ≤ Cλ − = CN B λ . (8.20) The High-Low term R L . We recall from (5.45) that R lL = ∂ l ∆ − P ≈ λ [ U j ∂ j θ ǫ ] and thus k R L k C ≤ k∇ ∆ − P ≈ λ k C → C k U j k C k ∂ j θ ǫ k C ≤ CN B λ e / R e / v holds, upon appealing to (8.20). Choosing B λ sufficiently large, we see that k R L k C ≤ e / v e / R N which is sufficient for (8.4) to be satisfied. The Transport term R T . We use (5.40) to recall that R T = ∂ l ∆ − P ≈ λ (cid:20) ¯ D∂t Θ (cid:21) . Thus, from (8.20) and (7.31) we obtain k R T k C ≤ Cλ τ − e / R = CB λ N Ξ B / λ e / v e / R N ! − / Ξ e / v e / R = CB / λ e / v e / R N ! / e R in view of the choice of τ in (6.38). Choosing B λ sufficiently large immediately shows that k R T k C ≤ e ′ J holds. 40 he High-Frequency Interference term R H . To conclude the C stress estimates we recall from (5.46) and(5.49)–(5.51) that R H = X J =¯ I iλ∂ l ∆ − P ≈ λ h e iλ ( ξ I + ξ J ) (cid:16) θ J ˜ θ I ( m l ( ∇ ξ J ) − m l ( ∇ ˆ ξ J )) ∂ l ξ I + θ I ˜ θ J m l ( ∇ ˆ ξ J )( ∂ l ξ I − ∂ l ˆ ξ I ) + δu lJ ∂ l ξ I ˜ θ I (cid:17) i . From (8.20) we thus obtain k R H k C ≤ CAe R (Ξ e / v τ ) + CB λ N e R ≤ CAB / λ e / v e / R N ! / e R + CB λ N e R . For all sufficiently large choices of B λ , we finally have the estimate k R H k C ≤ e ′ J This error term is the last one, so the estimate (8.4) will finally be satisfied for any sufficiently large choice of B λ ≥ B λ . The only restriction now is that λ = B λ N Ξ in (5.7) must be a positive integer. Since we assume Ξ ≥ in Definition 3.2 and N ≥ , an appropriate choice of B λ exists in the interval B λ ∈ [ B λ , B λ ] . Ourconstruction is now fully specified once such a value is chosen. First we claim that k∇ k c B k C ≤ C k ( N Ξ) k K e J For this purpose, recall the definition (5.25) and the bound (6.18). This above estimate holds since we havealready verified the C estimate (8.3), and each spatial derivatives costs no more than a factor |∇| ≤ CN Ξ . The stress terms R T , R L , and R H each are contain a frequency localizing operator P ≈ λ , so that again, eachspatial derivative costs at most CN Ξ , since the constant B λ has now been fixed, in the previous subsection.The term R lM is treated in the same fashion as the mollified stress term in [31, Section 25.1]. The mainideas is that comparing the bound k u − u ǫ k C ≤ e / v N which had been used to establish (8.12) in Section 6, to the estimate k∇ u k C + k∇ u ǫ k C ≤ C Ξ e / v we notice the cost is at most CN Ξ upon taking a spatial derivative.The R lS stress is treated by writing R lS = P I ( R lSI, + R lSI, ) . The estimate for R SI, follows from thebounds established in Lemmas 7.4–(7.5). To treat R SI, we need to observe that the spatial derivative costs atmost N Ξ when it is applied to the difference ∇ ξ I − ∇ ˆ ξ I . Comparing the bounds of Lemma 7.3 and (7.10) k∇ ξ I − ∇ ˆ ξ I k C ≤ e / v e / R N ! / k∇ ξ I k C ≤ C Ξ |∇| ≤ CN / Ξ , which is smaller than the threshold N Ξ . All further derivatives of this term costat most CN / Ξ according to Lemma 7.3. We now proceed to establish the advective derivative bounds (8.8)-(8.9) for the new frequency energy levels,which is more subtle than the spatial derivative estimates due to the improved regularity of the advective deriva-tive. As observed in [31, Proposition 24.1 and Proposition 24.2], note that it suffices to check the bounds forthe coarse scale advective derivative ¯ D∂t = ∂ t + u ǫ · ∇ after we write ∂ t + u · ∇ = ( ∂ t + u ǫ · ∇ ) + ( u − u ǫ ) · ∇ + U · ∇ . Having established spatial derivative estimates on all our error terms, the the bounds for the two error spatialderivative terms follow from the results of Section 8.2, the already established estimates on spatial derivativesfor k∇ k ( u − u ǫ ) k C which follow from (6.4), and the bounds on k∇ k U k C , which follow from Propositon 7.1.Since each term has been estimated already in C by the energy level e ′ J , our goal at this point is to checkthat the advective derivative never costs any more than (cid:12)(cid:12)(cid:12) ¯ D∂t (cid:12)(cid:12)(cid:12) E CN Ξ e / R (8.21)compared the estimates that were used to obtain the C bound.For most terms, the advective derivative costs τ − , so it is useful to observe that our goal is also implied bya bound of the type (cid:12)(cid:12)(cid:12) ¯ D∂t (cid:12)(cid:12)(cid:12) ≤ Cτ − (8.22)from the fact that τ − ≤ N Ξ e / R . For terms involving the difference between the phase gradients and theirinitial values, the following Lemma stating the cost of differentiating ∇ ξ I − ∇ ˆ ξ I is helpful Lemma 8.1.
For k ≥ and ≤ r ≤ , we have the following bounds k∇ k (cid:18) ¯ D∂t (cid:19) r ( ∇ ξ I − ∇ ˆ ξ I ) k C ≤ C k ( N Ξ) k τ − r b (8.23) b = e / v e / R N ! / (8.24)Lemma 8.1 follows from Lemma 7.3 after checking the relationships of the parameters using the condition N ≥ e v e R . Corollary 8.1.
The bounds in Lemma 8.1 hold also for m l ( ∇ ξ I ) − m l ( ∇ ˆ ξ I ) = ( ∇ ξ I − ∇ ˆ ξ I ) Z ∂ a m l (cid:16) (1 − σ ) ∇ ˆ ξ I + σ ∇ ξ I (cid:17) dσ With these bounds in hand, we can now quickly verify (8.21).
The term c B . The term c B inherits the estimates for R ǫ from its definition in (5.25). These bounds are noworse than the bounds stated for R J in (6.18) as long as one takes no more than advective derivatives and nomore than L total spatial or advective derivatives (see [31, Section 18]). As a consequence, we obtain (8.8).42 he mollification term R lM . The mollification term (8.11) is handled in the same way as in [31, Sections25.1, 25.2]. Among these estimates, the most subtle are the terms ( u − u ǫ )Θ + ( θ − θ ǫ ) U For the purposes of proving our result and the main theorem in [31], these terms can be estimated separately as (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ¯ D∂t u − ¯ D∂t u ǫ (cid:19) Θ (cid:13)(cid:13)(cid:13)(cid:13) C ≤ C (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) ¯ D∂t u (cid:13)(cid:13)(cid:13)(cid:13) C + (cid:13)(cid:13)(cid:13)(cid:13) ¯ D∂t u ǫ (cid:13)(cid:13)(cid:13)(cid:13) C (cid:19) k Θ k C at the cost of requiring the condition N ≥ (cid:16) e v e R (cid:17) / . However, as discussed in [31, Section 25.1], it appearsthat a scheme aimed at proving / regularity might require this term to be estimated more delicately. A moredelicate commutator estimate would allow us to require instead that N ≥ (cid:16) e v e R (cid:17) . The stress term R S For the term R S , the cost (8.21) is obtained for every term appearing in (8.16) using theestimates of Lemmas 7.4-7.5 and Corollary 7.1 for the amplitudes, and using Lemma 8.1 and Corollary 8.1 forthe terms involving differences of phase gradients. The terms R T , R L and R H The commutator estimates of Lemma 7.2 and the use of frequency localizedwaves make it especially simple to estimate the terms obtained by solving the divergence equation. We listthese terms here. ¯ D∂t R lT = (cid:20) ¯ D∂t , ∂ l ∆ − P ≈ λ (cid:21) (cid:20) ¯ D∂t Θ (cid:21) + ∂ l ∆ − P ≈ λ (cid:20) ¯ D ∂t Θ (cid:21) (8.25) ¯ D∂t R lL = (cid:20) ¯ D∂t , ∂ l ∆ − P ≈ λ (cid:21) (cid:2) U j ∂ j θ ǫ (cid:3) + ∂ l ∆ − P ≈ λ ¯ D∂t (cid:2) U j ∂ j θ ǫ (cid:3) (8.26) ¯ D∂t R lH = X J =¯ I (cid:20) ¯ D∂t , ∂ l ∆ − P ≈ λ (cid:21) r H,IJ + ∂ l ∆ − P ≈ λ ¯ D∂t r
H,IJ (8.27) r H,IJ = ( iλ ) e iλ ( ξ I + ξ J ) (cid:16) θ J ˜ θ I ( m l ( ∇ ξ J ) − m l ( ∇ ˆ ξ J )) ∂ l ξ I (cid:17) (8.28) + ( iλ ) e iλ ( ξ I + ξ J ) (cid:16) θ I ˜ θ J m l ( ∇ ˆ ξ J )( ∂ l ξ I − ∂ l ˆ ξ I ) + δu lJ ∂ l ξ I ˜ θ I (cid:17) . (8.29)Combining Lemma 7.2 with Corollary 8.1 and all the bounds of Section 7, we obtain a cost of (8.22) (andtherefore (8.21)) for the advective derivative. Further spatial derivatives cost at most CN Ξ as all the terms arein fact localized to frequencies of order λ .This estimate concludes the proof of the Main Lemma. In this Section, we explain how Theorem 1.1 can be deduced from the Main Lemma, Lemma 3.1. Morespecifically, the Theorem we establish directly is the following:
Theorem 9.1.
As in the hypotheses of Theorem 1.1, let α < / , let ǫ > be given, and let f : R × T → R be any smooth function of compact support supp f ⊆ I × T or which the integral Z T f ( t, x ) dx = 0 , t ∈ R (9.1) remains constant in time. Then there exists a function θ : R × T → R with the following properties:1. θ satisfies the Active Scalar Equation (1.1) in the sense of distributions.2. The scalar field θ and the drift velocity u l = T l [ θ ] both belong to the H¨older class θ, u l ∈ C αt,x θ is supported in the time interval supp θ ⊆ I ǫ × T , (9.2) where I ǫ = [ a − ǫ, b + ǫ ] is an ǫ -neighborhood of the interval I = [ a , b ] θ satisfies a uniform estimate k θ k C ≤ C (9.3) with C depending only on f .5. For any smooth function φ : R × T → C , we have (cid:12)(cid:12)(cid:12)(cid:12)Z R × T ( θ − f ) φ dtdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ k∇ φ k L t,x ( I ǫ × T ) (9.4) for some constant C depending on f . Theorem 1.1 follows from Theorem 9.1 by a straightforward argument that is implicit in Section 10 below.Our starting point is the observation that the function f can be viewed as a solution to the scalar-stressequation ∂ t f + ∂ l ( f u l ) = ∂ l R l (9.5) u l = T l [ f ] R j = ∂ j ∆ − [ ∂ t f + ∂ l ( f u l )] (9.6)thanks to the condition Z T ∂ t f dx = ddt Z T f ( t, x ) dx = 0 . Furthermore, the functions ( f, u l , R l ) in (9.5) are all smooth functions on R × T with support contained ina finite time interval I × T . In particular, the scalar function θ (0) = f can be viewed as part of a smooth,compactly supported solution ( f, u l , c A , R l ) to the compound scalar stress equation (3.2) with c A = 0 .Our proof of Theorem 9.1 will be completed once we prove the following Claim. Claim 9.1.
Under the assumptions of Theorem 9.1, there exists a sequence sequence of scalar-stress fields ( θ ( k ) , u l ( k ) , c A, ( k ) , R lJ, ( k ) ) satisfying the following properties.1. For even indices k = 0 , , , . . . , ( θ ( k ) , u l ( k ) , c A, ( k ) , R lJ, ( k ) ) solves the Compound Scalar-Stress Equa-tion (3.2) with vector A l , whereas for odd indices k = 1 , , , . . . , ( θ ( k ) , u l ( k ) , c A, ( k ) , R lJ, ( k ) ) solves theCompound Scalar-Stress Equation (3.2) with vector B l . Here A l and B l are defined as in (3.1) . . We have k c A, ( k ) k C + k R J, ( k ) k C → as k → ∞
3. The sequences θ ( k ) , u l ( k ) are Cauchy in C t,x with uniform bounds on k θ ( k ) k C , k u ( k ) k C depending onlyon f .4. The sequences θ ( k ) , u l ( k ) are also Cauchy in C αt,x .5. We have supp θ ( k ) ⊆ I ǫ × T for all k .6. The estimate (9.4) holds for θ ( k ) uniformly in k . The scalar stress fields described in Claim 9.1 will be constructed by iteration of Lemma 3.1.
To initialize the construction, we set θ (0) = f , u l (0) = T l [ f ] , c A, (0) = 0 , and R J, (0) as in (9.6). We define I (0) to be the smallest closed inerval such that supp f ⊆ I (0) × T . We set e J, (0) = k R J, (0) k C . In order to be consistent with the iteration rules (9.8)-(9.11) below and to maintain the inequality e v ≥ e R ≥ e J during the iteration, we take e v, (0) = e R, (0) = K e J, (0) where K is the constant in Lemma 3.1.Now let Ξ be a sufficiently large constant such that the bounds ofDefinition 3.2 hold with L = 2 for the frequency energy levels (Ξ , e v , e R , e J ) = (Ξ , e J, (0) , e J, (0) , e J, (0) ) .We will choose our initial frequency level Ξ (0) to be even larger than the parameter Ξ . More specifically, Ξ (0) will take the form Ξ (0) = Y Ξ (9.7)Here Y ≥ is a large parameter whose purpose will ultimately be to make sure that the time interval containingthe support of the solution will be small without disturbing the required C estimate. In terms of the con-struction, choosing the parameter Y to be large will imply that we perform the iteration with a large frequencyparameter λ and a small lifespan parameter τ when we iterate the Main Lemma. k ≥ . We will proceed with the proof by iteration of the Main Lemma, which requires us to specify a sequenceof frequency energy levels (Ξ ( k ) , e v, ( k ) , e R, ( k ) , e J, ( k ) ) , a sequence of functions e ( k ) ( t ) prescribing the energyincrement, a sequence of intervals I ( k ) containing the support of the compound scalar-stress fields, and a se-quence of frequency growth factors N ( k ) ≥ . The present section is devoted to choosing these parameters, andstudying how these parameters grow or decay during the iteration.We will choose our frequency energy levels so that the following iteration rules hold for all k ≥ : e v, ( k +1) = e R, ( k ) (9.8) e R, ( k +1) = K e J, ( k ) (9.9) e J, ( k +1) = e J, ( k ) Z (9.10) Ξ ( k +1) = C N ( k ) Ξ ( k ) (9.11)45he parameter Z will be chosen in the proof to be a large constant satisfying Z ≥ K ≥ . From (9.8)-(9.10)and the choices of Section 9.1, the energy levels decay exponentially according to the following pattern: ( e v , e R , e J ) (0) = ( K e J, (0) , K e J, (0) , e J, (0) )( e v , e R , e J ) (1) = (cid:18) K e J, (0) , K e J, (0) , Z e J, (0) (cid:19) ( e v , e R , e J ) ( k ) = (cid:18) K Z k − e J, (0) , K Z k − e J, (0) , Z k e J, (0) (cid:19) , k ≥ (9.12)The constant C in (9.11) is the constant C appearing in line (3.13) of the Main Lemma. Thus, C will dependon how we construct our energy increment functions e ( k ) ( t ) , which will be specified momentarily. Accordingto the bound of line (3.13), we have that e J, ( k +1) = e / v, ( k ) e / R, ( k ) N ( k ) / e R, ( k ) (9.13)The iteration rules (9.8)-(9.10) are therefore achieved by taking N ( k ) = (cid:18) e v, ( k ) e R, ( k ) (cid:19) / (cid:18) e R, ( k ) e J, ( k ) (cid:19) Z (9.14)More specifically, recalling (9.12), N (0) = K Z , N (1) = K Z , N ( k ) = K Z / , k ≥ . (9.15)As we always have (cid:16) e v, ( k ) e R, ( k ) (cid:17) / ≤ Z / ≤ N , the assumption of line (3.12) is always satisfied, so this choiceof N ( k ) is admissible. With this choice, iteration of (9.11) results in exponential growth of the frequency levels Z k Ξ (0) ≤ Ξ ( k ) ≤ C k K k Z (9 / k Ξ (0) (9.16)for all k ≥ .We will now specify how our sequence of energy functions e ( k ) ( t ) and time intervals I ( k ) will be chosen,beginning with stage k = 0 . Define ˆ τ ( k ) = Ξ − k ) e − / v, ( k ) (9.17)to be the natural time scale associated to these frequency energy levels. Let I (0) be the time interval containingthe support of the initial scalar-stress field. Let η ǫ ( t ) be a standard, non-negative mollifying kernel in onevariable, with support in | t | ≤ ǫ . The initial energy function e (0) ( t ) is required to satisfy the lower bound e (0) ( t ) ≥ K e R, (0) on the time interval I (0) ± ˆ τ (0) according to (3.10), and must have a square root e / ( t ) which satisfies bounds of the form (3.11). We construct e (0) ( t ) by mollifying the characteristic function of I (0) according to the formula e / ( t ) = (2 K ) / e / J, (0) η ˆ τ ∗ χ I (0) ± τ ( t ) (9.18)With this choice, the lower bound (3.10) and the bounds (3.11) are satisfied with K = K and with M beingsome absolute constant which arises from differentiating the mollifier. Having constructed e (0) ( t ) , we can applyLemma 3.1 to obtain a solution ( θ (1) , u l (1) , c A, (1) , R lJ, (1) ) to the Compound Scalar-Stress equation with vector B l with support in the interval I (1) × T , I (1) = I (0) ± τ (0) .46e now iterate this procedure to form a sequence of scalar stress fields ( θ ( k ) , u l ( k ) , c A, ( k ) , R lJ, ( k ) ) whosecompound frequency energy levels obey the rules -(9.8)-(9.11) by choosing N ( k ) = Z / according to (9.15).We define e / k ) ( t ) = (2 K ) / e / J, ( k ) η ˆ τ ∗ χ I ( k ) ± τ ( t ) so that the bounds on e / k ) are consistent with the bounds on (9.18), and Lemma 3.1 applies with the sameconstant M . According to Lemma 3.1, the time intervals containing the support of the scalar stress fieldssupport grow according to the rule I ( k +1) = I ( k ) ± (4ˆ τ k ) (9.19)In (9.19) and below, we use the notation I ± δ to denote the δ -neighborhood of an interval I . In other words, I ± δ = [ a − δ, b + δ ] if I = [ a, b ] . During this iteration, the vector in the scalar-stress equation alternatesbetween A l and B l as in Property 1 of Claim 9.1.We have now defined our iteration up to the choice of the parameters Y and Z . We will choose theseparameters in the following Subsection to ensure that the properties listed in Claim 9.1 are all satisfied. We now verify that the properties in Claim 9.1 hold for sufficiently large values of Y and Z . Property 1
This property follows immediately from the construction.
Property 2
To check that the error terms converge uniformly to , we observe that k R J, ( k ) k C ≤ e J, ( k ) = Z − k e J, (0) from (9.12), and the same type of estimate holds for k c A, ( k +1) k C . Thus, both terms composing the stress errorconverge uniformly to . Property 3
Here we verify that the sequence θ ( n ) , u l ( n ) is Cauchy in C . Recall that, for n ≥ we have θ ( n ) = θ (0) + n − X k =0 Θ ( k ) , u l ( n ) = u l (0) + n − X k =0 U ( k ) (9.20)where the properties of Θ ( k ) and U l ( k ) are as described in Lemma 3.1. The functions θ (0) and u l (0) are smoothwith compact support, and are therefore uniformly bounded. Our estimates for the corrections have the form k Θ ( k ) k C + k U ( k ) k C ≤ Ce / R, ( k ) (9.21)From (9.12), the bounds e R, ( k ) decays exponentially in k for any choice of Z ≥ , and both series in (9.20)therefore converge uniformly. In particular, as θ (0) = f , we have k θ ( k ) k C + k u ( k ) k C ≤ k f k C + Ce / R, (0) , k ≥ (9.22)where C is proportional to the constant in Lemma 3.1. In particular, the bound (9.22) depends only on f , anddoes not depend on our choices of parameters Y and Z . Property 4
We now verify that the series (9.20) also converges in C αt,x once Z is chosen large enough. Weclaim that the bounds from Lemma 3.1 give k∇ Θ ( k ) k C + k∇ U ( k ) k C + k ∂ t Θ ( k ) k C + k ∂ t U ( k ) k C ≤ CN ( k ) Ξ ( k ) e / R, ( k ) (9.23)47he bounds on k∇ Θ ( k ) k C + k∇ U ( k ) k C follow directly from Lemma 3.1. We obtain the same bound for thetime derivatives by writing ∂ t Θ ( k ) = − u ( k ) · ∇ Θ ( k ) + ( ∂ t + u ( k ) · ∇ )Θ ( k ) (9.24)and similarly for U ( k ) . As we have shown in (9.22) that the sequence k u ( k ) k C is uniformly bounded by someconstant, we have that the terms − u ( k ) · ∇ Θ ( k ) and − u ( k ) · ∇ U ( k ) both obey the estimate (9.23). Lemma 3.1also supplies the following bound on the advective derivative: k ( ∂ t + u ( k ) · ∇ )Θ ( k ) k C + k ( ∂ t + u ( k ) · ∇ ) U ( k ) k C ≤ C b − / k ) Ξ / k ) e / v, ( k ) e / R, ( k ) b ( k ) = N − k ) ( e / v, ( k ) /e / R, ( k ) ) Note that the parameter N ( k ) = K Z / and the ratio e / v, ( k ) e / R, ( k ) = Z / are both independent of k once k ≥ ,while e v, ( k ) = K Z − ( k − e J, (0) decays to exponentially. Thus, the estimate for the advective derivative iseven better than the bound (9.23). From (9.24) we now conclude that (9.23) holds for the time derivative aswell.Interpolation of (9.23) and (9.21) gives k Θ ( k ) k C αt,x + k U ( k ) k C αt,x ≤ C [ N ( k ) Ξ ( k ) ] α e / R, ( k ) (9.25)Applying (9.16) and (9.15), we have k Θ ( k ) k C αt,x + k U ( k ) k C αt,x ≤ C Z,K ,C (cid:16) C α K α Z ( α − ) (cid:17) k (cid:16) Ξ α (0) e / R, (0) (cid:17) (9.26)As we have assumed α < / , we can take Z large enough depending on α , K and C so that Z ( α − ) < K α C α (9.27)Under this assumption, the right hand side of (9.26) tends to exponentially fast as k → ∞ , and it follows thatthe series (9.20) converges in C αt,x . Property 5
To bound the support of θ ( k ) , recall that supp θ ( k ) ⊆ I ( k ) , where I (0) is the smallest time intervalcontaining the support of f , and the intervals I ( k ) grow according to the rule (9.19). As a consequence, we have(in terms of the notation introduced in (9.19)) I ( k ) ⊆ I (0) ± T, k = 0 , , , . . .T = 4 ∞ X k =0 ˆ τ ( k ) = 4 ∞ X k =0 Ξ − k ) e − / v, ( k ) We recall now that e v, (0) = e v, (1) = e v, (2) = K e J, (0) while e v, ( k ) decays exponentially as (9.12) for k ≥ .We also recall the lower bound in (9.16) to obtain T ≤ − e − / J, (0) (cid:18) − ( C Z / ) − (cid:19) (9.28)Recalling the definition (9.7) of Ξ (0) , and noting that C ≥ and Z ≥ , we have T ≤ Y − Ξ − e − / J, (0) (9.29)Property 5 is satisfied for the ǫ > given in (5) once Y is chosen sufficiently large to ensure T < ǫ .48 roperty 6
For a smooth test function φ with compact support, we have Z R × T ( θ − f ) φ ( t, x ) dtdx = Z R × T ( θ − θ (0) ) φ ( t, x ) dtdx (9.30) = ∞ X k =0 Z Θ ( k ) φ ( t, x ) dtdx (9.31)According to Lemma 3.1, we can write the corrections in divergence form Θ ( k ) = ∂ l W l ( k ) for some vector fields W l ( k ) obeying the estimates (3.18)-(3.20). Integrating by parts, we have ∞ X k =0 Z Θ ( k ) φ ( t, x ) dtdx = − ∞ X k =0 Z W l ( k ) ∂ l φ ( t, x ) dtdx (9.32)Recalling (9.12), (9.16) and the definition (9.7) of Ξ (0) , we obtain (cid:12)(cid:12)(cid:12) Z R × T ( θ − f ) φ ( t, x ) dtdx (cid:12)(cid:12)(cid:12) ≤ ∞ X k =0 k W ( k ) k C k∇ φ k L t,x ( I ǫ × T ) (9.33) ≤ C ∞ X k =0 N ( k ) Ξ ( k ) e / R, ( k ) ! k∇ φ k L t,x ( I ǫ × T ) (9.34) = ∞ X k =0 C Ξ ( k +1) e / R, ( k ) ! k∇ φ k L t,x ( I ǫ × T ) (9.35) ≤ C C Ξ (1) e / R, (0) k∇ φ k L t,x ( I ǫ × T ) (9.36) ≤ C Z Ξ − e / J, (0) k∇ φ k L t,x ( I ǫ × T ) (9.37) ≤ C Z Y Ξ − e / J, (0) k∇ φ k L t,x ( I ǫ × T ) (9.38)where C above denotes the constant in the Main Lemma. Taking Y (or alternatively Z ) to be large enoughdepending on C , Ξ and e J, (0) , we obtain (9.4). This choice concludes the proof of Theorem 9.1.
10 Proof of Corollary 1.1
In this Section, we explain how Theorem 1.1 (or alternatively Theorem 9.1) can be applied to prove Corollary1.1, which characterizes the closure of compactly supported solutions to the active scalar equations in the space L ∞ endowed with the weak-* topology. Proof of Corollary 1.1.
As in the statement of Theorem 1.1, consider an Active Scalar Equation (1.1) with asmooth multiplier that is not odd. Let I ⊆ R be a nonempty, finite, open interval. Let α < / and let S ⊆ L ∞ denote the set of all weak solutions ( θ, u l ) to the Active Scalar equation (1.1) which have compact support in I × T , and which belong to the H¨older class ( θ, u l ) ∈ C αt,x . Let S denote the closure of S in L ∞ with respectto the weak-* topology. Corollary 1.1 asserts that S is equal to the space of f ∈ L ∞ ( I × T ) which satisfy theconservation law R f ( t, x ) dx = 0 as a distribution in the variable t . In other words, we assume that for everysmooth test function η ( t ) : I → R with compact support, we have Z I × T η ( t ) f ( t, x ) dtdx = 0 (10.1)49irst, observe that any f ∈ L ∞ which belongs to S must satisfy (10.1) for all such η ( t ) , since the integrationagainst η ( t ) is continuous with respect to the weak-* topology, and because equality (10.1) is satisfied by all ofthe elements ( θ, u l ) ∈ S . This conservation law is proven for each ( θ, u l ) ∈ S by writing the test function in(10.1) as η = ˜ η + ( η − ˜ η ) , where ˜ η is a smooth function whose support is disjoint support from that of ( θ, u l ) that satisfies Z I ˜ η ( t ) dt = Z I η ( t ) dt This condition allows us to write η − ˜ η = ddt h ( t ) , where h ( t ) is smooth and compactly supported in I . Thedefinition of weak solution for (1.1) then implies Z I × T η ( t ) θ ( t, x ) dtdx = Z I × T ˜ η ( t ) θ ( t, x ) dtdx + Z I × T ( η − ˜ η )( t ) θ ( t, x ) dtdx = Z ∂∂t h ( t ) θ ( t, x ) dtdx = Z u l ∂∂x l h ( t ) θ ( t, x ) dtdx = 0 We now show conversely that every f ∈ L ∞ satisfying (10.1) belongs to S . Let us assume by contradiction that f / ∈ S . By definition of the weak-* topology, there exists a finite collection { η , . . . , η m } ⊆ L ( I × T ) and δ > such that for all θ ∈ S the lower bound (cid:12)(cid:12)(cid:12) Z ( f ( t, x ) − θ ( t, x )) η j ( t, x ) dtdx (cid:12)(cid:12)(cid:12) ≥ δ (10.2)holds for at least one η j ∈ { η , . . . , η m } .Let ˜ f ∈ L ∞ ( I × T ) be a smooth function of compact support with k ˜ f k L ∞ ≤ k f k L ∞ such that property(10.1) holds for ˜ f and for all such η j , and we have the bound (cid:12)(cid:12)(cid:12) Z ( f ( t, x ) − ˜ f ( t, x )) η j ( t, x ) dtdx (cid:12)(cid:12)(cid:12) ≤ δ/ (10.3)Such a function ˜ f can be constructed by first applying a smooth cutoff in time to restrict to a compact subsetof I × T , and then convolving with a mollifier in time and space, noting that both operations preserve theproperty (10.1) without enlarging the L ∞ norm. Inequality (10.3) is established by duality, as the adjoint cutoffand mollifier operations converge strongly in L when applied to each η j . We choose the mollification in such away that the support of ˜ f remains inside a time interval ˜ I strictly smaller than I with ˜ I ± τ ⊆ I for some τ > .Now apply Theorem 9.1 for the function ˜ f with ǫ = 1 /n to obtain a sequence ( θ n , u ln ) ∈ S such that thebound k θ n k L ∞ ≤ A holds uniformly, and (9.4) holds for ˜ f with ǫ = 1 /n . We assume here that n ≥ τ − islarge enough to ensure ( θ n , u ln ) have compact support in I × T thanks to the compact support of ˜ f and (9.2).Now let ˜ η j be smooth functions of compact support in I × T with k η j − ˜ η j k L ( I × T ) ≤ δ k f k L ∞ + A ) . Weobtain an upper bound on the left hand side of (10.2) (cid:12)(cid:12)(cid:12) Z ( f ( t, x ) − θ n ( t, x )) η j ( t, x ) dtdx (cid:12)(cid:12)(cid:12) ≤ δ (cid:12)(cid:12)(cid:12) Z ( ˜ f − θ n ) η j dtdx (cid:12)(cid:12)(cid:12) ≤ δ δ (cid:12)(cid:12)(cid:12) Z ( ˜ f − θ n )˜ η j dtdx (cid:12)(cid:12)(cid:12) ≤ δ δ n k∇ ˜ η j k L t,x Taking n large enough contradicts inequality (10.2) and concludes the proof. At this point, Theorem 1.1 would also suffice. In this section, we outline how Lemma 3.1 can also be applied to yield Theorem 1.2. The proof follows an ideaof [31, Section 12]. The same argument below also shows that one can glue any two solutions which have thesame integral .Let θ be a smooth solution of (1.1) on ( − T, T ) × T , with multiplier m l which is not odd. Let ¯ θ = | T | R T θ (0 , x ) dx be the average value of θ , which is conserved by θ along the flow. Let ψ ( t ) be a smoothcutoff function, equal to on | t | ≤ T and equal to for | t | ≥ T = T .Now consider the scalar field θ (0) ( t, x ) = ψ ( t ) θ ( t, x ) + (1 − ψ ( t ))¯ θ . Then θ (0) is an integral-conservingscalar field (i.e. R T ∂ t θ (0) dx = ddt R T θ (0) dx = 0 ), and therefore solves the scalar stress equation ∂ t θ (0) + ∂ l ( θ (0) T l [ θ (0) ]) = ∂ l R l (11.1) R j = ∂ j ∆ − [ ∂ t θ (0) + ∂ l ( θ (0) T l [ θ (0) ])] (11.2)Note also that, because both θ and ¯ θ are solutions to (1.1), the support of R l is contained in the support of ψ ′ ( t ) ,namely supp R l ( t, x ) ⊆ { T ≤ | t | ≤ T } × T Repeating the argument of Sections 9.1-9.3, we can now iterate Lemma 3.1 to obtain a sequence of solutions θ ( k ) to the compound scalar stress equation, such that supp ( θ ( k ) − θ (0) ) ⊆ { T ≤ | t | ≤ T } × T for all indices k ≥ , and such that θ ( k ) → ˜ θ converge in C αt,x to a solution of (1.1). At this point, themain difference in the argument is that we choose energy functions e ( k ) ( t ) which are supported within pairs ofintervals containing a small neighborhood of { T ≤ | t | ≤ T } . (In fact, the argument is simpler at this pointbecause we do not need to achieve a weak approximation, and hence there is no need to introduce the parameter Y .) As we can take this intervals of support to form an arbitrarily small neighborhood of { T ≤ | t | ≤ T } , wecan keep the support of the iteration contained within { T ≤ | t | ≤ T } , and thereby obtain Theorem 1.2.
12 Proof of Weak Rigidity for Odd Active Scalars
In this section we give the proof of Theorem 1.3. Let θ ∈ { θ n } n ≥ be a weak solution to (1.1), with associatedvelocity field u l = T l [ θ ] . Also let φ ∈ D ( I × T ) be a fixed test function. The proof of the theorem is based onthe following computation. For each fixed time t , let N t [ θ, φ ] = Z T θ ( t, x ) u l ( t, x ) ∂ l φ ( t, x ) dx (12.1) This observation is due to Sung-Jin Oh [41]. t slice. Since T l is given by a Fourier multiplier, it commuteswith differentiation, and upon integrating by parts several times we obtain N t [ θ, φ ] = Z T ∂ k ∆ − ∂ k θ ∂ j T l [∆ − ∂ j θ ] ∂ l φ dx = − Z T ∆ − ∂ k θ ∂ k ∂ j T l [∆ − ∂ j θ ] ∂ l φ dx − Z T ∆ − ∂ k θ ∂ j T l [∆ − ∂ j θ ] ∂ k ∂ l φ dx = Z T ∂ k ∆ − ∂ j θ ∂ k T l [∆ − ∂ j θ ] ∂ l φ dx + Z T ∆ − ∂ k θ ∂ k T l [∆ − ∂ j θ ] ∂ j ∂ l φ dx − Z T ∆ − ∂ k θ ∂ j T l [∆ − ∂ j θ ] ∂ k ∂ l φ dx = − Z T ∆ − ∂ j θ ∂ j T l [ θ ] ∂ l φ dx − Z T ∆ − ∂ j θ ∂ k T l [∆ − ∂ j θ ] ∂ k ∂ l φ dx + Z T ∆ − ∂ k θ ∂ k T l [∆ − ∂ j θ ] ∂ j ∂ l φ dx − Z T ∆ − ∂ k θ ∂ j T l [∆ − ∂ j θ ] ∂ k ∂ l φ dx. (12.2)At this stage we use that the Fourier multiplier m l ( ξ ) is odd in ξ , which implies that ∂ j T l , given by the Fouriermultiplier iξ j m l ( ξ ) which is even in ξ , is self-adjoint in L ( T ) . We may thus write Z T ∆ − ∂ j θ ∂ j T l [ θ ] ∂ l φ dx = Z T θ ∂ j T l [∆ − ∂ j θ ∂ l φ ] dx = Z T θ ∂ j T l [∆ − ∂ j θ ] ∂ l φ dx + Z T θ (cid:0) ∂ j T l [∆ − ∂ j θ ∂ l φ ] − ∂ j T l [∆ − ∂ j θ ] ∂ l φ (cid:1) dx = N t [ θ, φ ] + Z T θ (cid:2) ∂ j T l , ∂ l φ (cid:3) ∆ − ∂ j θ dx. (12.3)Combining (12.2) and (12.3) we arrive at N t [ θ, φ ] = − Z T θ (cid:2) ∂ j T l , ∂ l φ (cid:3) ∆ − ∂ j θ dx − Z T ∆ − ∂ j θ ∂ k T l [∆ − ∂ j θ ] ∂ k ∂ l φ dx + Z T ∆ − ∂ k θ ∂ k T l [∆ − ∂ j θ ] ∂ j ∂ l φ dx − Z T ∆ − ∂ k θ ∂ j T l [∆ − ∂ j θ ] ∂ k ∂ l φ dx. (12.4)From the H¨older inequality, and the bounds k∇ T ∆ − ∇ η k L ≤ C k η k L , for η ∈ L ( T ) (12.5) k [ ∇ T l , ∂ l φ ] η k L ≤ C k η k L k φ k H ǫ , for η ∈ ˙ H ( T ) (12.6)we thus obtain from (12.4) that | N t [ θ, φ ] | ≤ C k θ ( t, · ) k L k ∆ − ∇ θ ( t, · ) k L k φ ( t, · ) k H ǫ (12.7)for any ǫ > . The above estimate is a manifestation of the compactness inherent in N t in the spatial variables.Since we have only assumed θ ∈ L p ( I ; L ( T )) , compactness in the time variable must come from theactive scalar equation. Below we give two essentially equivalent approaches to obtaining this compactness. Thefirst proof is based on a variant of the Arzel`a-Ascoli principle due to Aubin-Lions. The second proof is a moredirect argument in the spirit of [32], using Littlewood-Paley theory to extract regularity in time.52 ime compactness via Aubin-Lions compactness lemma. At this stage we notice that for any weak solution θ ∈ L p ( I ; L ( T )) of (1.1), and any index j , we have we have that ∂ t (∆ − ∂ j θ ) = ∆ − ∂ j ∂ l ( θ T l [ θ ]) (12.8)holds in the sense of distributions, and thus k ∂ t (∆ − ∂ j θ ) k L p/ ( I ; H − ( T )) = k ∆ − ∂ j ∂ l ( θ T l [ θ ]) k L p/ ( I ; H − ( T )) ≤ C k ∆ − ∂ j ∂ l ( θ T l [ θ ]) k L p/ ( I ; L ( T )) ≤ C k θ k L p ( I ; L ( T )) (12.9)in view of the compact embedding of L ( T ) ⊂ W , ( T ) .Now assume that θ n ⇀ f ∈ L p ( I ; L ( T )) (12.10)for some p > . The convergence of the mean Z T θ n dx → Z T f dx in D ′ ( I ) is automatic. In view of the Sobolev embedding and (12.9), by (12.10) we have that ∆ − ∇ θ n is uniformly bounded in L p ( I ; H ( T )) (12.11) ∂ t (∆ − ∇ θ n ) is uniformly bounded in L p/ ( I ; H − ( T )) . (12.12)Therefore, applying the Aubin-Lions compactness lemma (see e.g. [16, Lemma 8.4]), we obtain that there is asubsequence { θ n j } such that ∆ − ∇ θ n j → ∆ − ∇ f ∈ L p ( I ; L ( T )) , (12.13)i.e. the convergence is strong. To conclude, we integrate (12.4) in time, use (12.7) and (12.13), and obtain that Z I Z T θ n j T l [ θ n j ] ∂ l φdxdt → Z I Z T f T l [ f ] ∂ l φdxdt for any test function φ , since the product of a strong and a weak limit is a weak limit. The convergence holds infact along any subsequence n j → ∞ , and therefore holds also along the original sequence. Time compactness via Littlewood-Paley theory.
We now give a more direct proof which illustrates theusefulness of Littlewood-Paley theory in extracting time regularity.Let us use the notation P ≤ I θ , P I θ and P [ a,b ] θ denote the standard, Littlewood-Paley projection operators.Thus, [ P ≤ I θ ( ξ ) = η (2 − I ξ )ˆ θ ( ξ ) , I = 0 , , , . . . is a truncation of ˆ θ to frequencies of order supp [ P ≤ I θ ⊆ {| ξ | ≤ I +1 } , η is a smooth cutoff supported in | ξ | ≤ with η ( ξ ) = 1 for | ξ | ≤ / . We let P I = P ≤ I − P ≤ I − denote the Littlewood-Paley piece which occupiesfrequencies supp d P I θ ⊆ { I − ≤ | ξ | ≤ I +1 } . We use the notation P [ a,b ] = P a ≤ I ≤ b P I .Now let φ ∈ C ∞ ( I × T ) be a smooth test function, and let θ n be a sequence of solutions to (1.1) convergingweakly to θ n ⇀ f in L p ( I ; L ( T )) for some p > as in (12.10). Let N = R R N t [ θ, φ ] dt = R R R T θu l ∂ l φ dxdt denote the full nonlinear term. 53e claim that N [ θ n , φ ] → N [ f, φ ] . To simplify the calculation, a simple approximation argument allowsus to assume that that ˆ φ has compact support in supp ˆ φ ⊆ {| ξ | ≤ r − } for some r ≥ . In this case, for all θ ∈ { θ n } n ≥ , we decompose the nonlinear term (12.1) into dyadic frequency increments N [ θ, φ ] = N [ P ≤ r θ, φ ] + ∞ X I = r +1 δN I [ θ, φ ] (12.14) δN I [ θ, φ ] = N [ P ≤ I +1 θ, φ ] − N [ P ≤ I θ, φ ] (12.15) = Z R Z T P I +1 θP ≤ I +1 u l ∂ l φdxdt + Z R Z T P ≤ I θP I +1 u l ∂ l φdxdt (12.16) = Z P I +1 θP [ I − r,I + r ] u l ∂ l φdx + Z P [ I − r,I + r ] θP I +1 u l ∂ l φdx. (12.17)In the last line we take advantage of the compact support of ˆ φ for convenience. Using the commutator formula-tion (12.4), each δN I decomposes into several terms of the type δN I [ θ, φ ] = Z R Z T ∆ − ∂ k P I +1 θ ∂ k T l [∆ − ∂ j P [ I − r,I + r ] θ ] ∂ j ∂ l φ dxdt + other similar termsFrom (12.6) and k ∆ − ∇ P I θ k L x ≤ C − I k θ k L x , each δN I is bounded by | δN I [ θ, φ ] | ≤ C φ − I k θ k L t,x (12.18)for some constant C φ depending on φ .We now show that the weak convergence (12.10) can in fact be upgraded to uniform convergence for eachdyadic piece P I θ n → P I f , which implies the convergence of each term δN I [ θ n , φ ] → δN I [ f, φ ] . The uniformconvergence is obtained by compactness. We start with the bounds k P I θ n k L pt L ∞ x + k∇ P I θ n k L pt L ∞ x ≤ C I k θ n k L pt L x Applying P I to (1.1), the equation ∂ t P I θ = − ∂ l P I [ θu l ] gives regularity in time k ∂ t P I θ n k L p/ t L ∞ x ≤ C I k θ n k L pt L x As we have assumed p > and uniform in n bounds on k θ n k L pt L x from (12.10), it follows by Sobolev em-bedding that the sequence P I θ n for each I is H¨older continuous in time and space, uniformly in n . By Arzel`a-Ascoli, there exists a uniformly convergent subsequence P I θ n j for each I . From the weak convergence (12.10),we have uniform convergence of P I θ n → P I f on any subsequence, which implies that the original sequence P I θ n → P I f converges uniformly.It now follows that δN I [ θ n , φ ] → δN I [ f, φ ] for each index I and that N [ P ≤ r θ n , φ ] → N [ P ≤ r f, φ ] . We alsohave the estimate (12.18), so the convergence of N [ θ n , φ ] → N [ f, φ ] follows from the dominated convergencetheorem applied to (12.14).We remark that the same two arguments can be upgraded to prove compactness of solutions when we onlyassume weak convergence in L pt H sx for some p > and s > − / . The main difference involves using thecommutator formulation (12.4) to obtain an estimate for the time derivative from the lower regularity in space.
13 Conservation of the Hamiltonian for Odd Active Scalars
In this Section we give the proof of Theorem 1.4. Recall that the symbol of the Fourier multiplier L defined in(1.3), which defines the Hamiltonian is given by ˆ L ( ξ ) = | ξ | − ( iξ m ( ξ ) − iξ m ( ξ )) (13.1)54ith the convention that ˆ L (0) = 0 . Since we are in two spatial dimensions and ξ · m ( ξ ) = 0 for all nonzerovectors ξ , automatically we must have that m ( ξ ) = iξ ⊥ | ξ | − ℓ ( ξ ) (13.2)for some even , zero-order homogenous, smooth on the unit sphere, real-valued scalar function ℓ ( ξ ) . The factthat ℓ ( ξ ) ∈ R follows from the fact that ℓ ( ξ ) = ℓ ( − ξ ) = ℓ ( ξ ) . In the case of the SQG equation, ℓ ( ξ ) = 1 .In summary, we have that ˆ L ( ξ ) = | ξ | − ℓ ( ξ ) (13.3)which reiterates that L is a self-adjoint operator, which is smoothing of degree − when ℓ ( ξ ) is nonvanishingon the unit sphere. The Hamiltonian then is H ( t ) = Z T θ ( t, x ) Lθ ( t, x ) dx (13.4)or equivalently, in view of Plancherel’s theorem, H ( t ) = X k ∈ Z ∗ | ˆ θ ( t, k ) | | k | − ℓ ( k ) . (13.5)Let φ ǫ = ǫ − φ ( x/ǫ ) be a standard mollifier on T , and denote · ǫ = · ∗ φ ǫ , · ǫ,ǫ = · ∗ φ ǫ ∗ φ ǫ . The conservation of the Hamiltonian H for solutions of (1.1) is implied by establishing that ddt H ( t ) = 0 asa distribution in t . Namely, we show that lim ǫ → Z R Z T η ′ ( t ) θ ǫ ( t, x ) Lθ ǫ ( t, x ) dxdt = 0 (13.6)holds for every smooth function η ( t ) which is supported in I . Note that since mollification ∗ φ ǫ is given by aFourier multiplier, it commutes with L .Considering the test function η ( t ) Lθ ǫ,ǫ in the weak formulation of (1.1), we arrive at Z R Z T θ∂ t ( ηLθ ǫ,ǫ ) dxdt + Z R Z T θu l ∂ l ( ηLθ ǫ,ǫ ) dxdt = 0 , (13.7)for every ǫ > . Strictly speaking, the above test function is not smooth in time, but this restriction can beignored after a time mollification argument, as in the proof of [33, Theorem 2.2]. The first term in (13.7) maynow be rewritten as Z R Z T θ∂ t ( ηLθ ǫ,ǫ ) dxdt = Z R Z T θ ǫ ∂ t ( ηLθ ǫ ) dxdt = Z R Z T θ ǫ η ′ Lθ ǫ dxdt + Z R Z T θ ǫ ηL∂ t θ ǫ dxdt = Z R Z T θ ǫ η ′ Lθ ǫ dxdt + Z R Z T Lθ ǫ η∂ t θ ǫ dxdt = Z R Z T θ ǫ η ′ Lθ ǫ dxdt − Z R Z T ∂ t ( Lθ ǫ,ǫ η ) θdxdt. (13.8)55ombining the above with (13.7) we see that establishing (13.6) is equivalent to establishing lim ǫ → Z R η Z T ( θu l ) ǫ ∂ l Lθ ǫ dxdt = 0 . (13.9)for u l = T l [ θ ] .Up to this point, we have presented the proof of conservation of H ( t ) analogously to the proof of energyconservation for Euler in the Onsager critical Besov space L t B / ,c ( N ) of [10], but the remaining analysis turnsout to be less subtle. In particular, there is no need for a quadratic commutator estimate as in [15] (and themollification above could also be simpler).To proceed, we view the cubic term on the left hand side of (13.9) as the diagonal part of a family of trilinearoperators Q ǫ [ θ (1) , θ (2) , θ (3) ] = Z R η Z T ( θ (1) T l [ θ (2) ]) ǫ ( ∂ l Lθ (3) ) ǫ dxdt (13.10)In this notation, equation (13.9) asks to show lim ǫ → Q ǫ [ θ, θ, θ ] = 0 for all θ ∈ L t,x . Observe first that theoperators Q ǫ satisfy the bound | Q ǫ [ θ (1) , θ (2) , θ (3) ] | ≤ k θ (1) k L t,x · k T [ θ (2) ] k L t,x · k∇ L [ θ (3) ] k L t,x ≤ C k θ (1) k L t,x · k θ (2) k L t,x · k θ (3) k L t,x (13.11)This bound follows from the fact that both operators T and ∇ L = ∇ ( − ∆) − / ( R T − R T ) (13.12)are bounded as operators from L x to itself, thanks to the smoothness and degree homogeneity of m .Because the operators Q ǫ are trilinear, and the bound (13.11) they satisfy is uniform in ǫ , it suffices to prove(13.9) under the additional assumption that θ is smooth with compact support by the density of such functionsin L t,x . Assuming now that θ is smooth, we may pass ǫ to in (13.9), and it remains to show that Z R η Z T θT l [ θ ] ∂ l Lθdxdt = 0 . At this stage we recall that u l = T l [ θ ] = ( ∂ l ) ⊥ Lθ, (13.13)which may be seen on the Fourier side from (13.2) and (13.3). As a result, we have Z T θu l ∂ l Lθdx = Z T θ ( ∂ l ) ⊥ Lθ ∂ l Lθdx = 0 (13.14)which concludes the proof.
14 Constraints on Weak Limits of Degenerate Active Scalars in HigherDimensions
In this Section, we give a proof of Theorem 1.6, which shows that the nondegeneracy condition in Theorem 1.5is necessary for the weak limit statement of Theorem 1.1.56n this section, we assume that there is a nonzero frequency ξ (0) ∈ c T n \ { } = Z n ∗ in the dual lattice suchthat the image of the even part of the multiplier is contained in { m ( ξ ) + m ( − ξ ) | ξ ∈ c R n } ⊆ h ξ (0) i ⊥ (14.1)In this case, we have the following restriction on weak limits of solutions to the active scalar equation, whichbears resemblance to a new conservation law. Lemma 14.1.
Consider the active scalar equation (1.1) on I × T n and suppose that the image of the even partof the multiplier is contained in the hyperplane (14.1) . Let T l denote the Fourier multiplier with symbol [ T l [ θ ]( ξ ) = 12 ( m l ( ξ ) − m l ( − ξ ))ˆ θ ( ξ ) Suppose that φ ∈ C ∞ ( I × T n ) has the property that its spatial gradient takes values in the direction ξ (0) ∇ φ ( t, x ) ∈ h ξ (0) i (14.2) Suppose that f ∈ L ∞ ( I × T n ) can be realized as a weak-* limit θ ( k ) ⇀ f in L ∞ of some sequence of solutions θ ( k ) to (1.1) . Then Z I × T n f ∂ t φ + f T l [ f ] ∂ l φdxdt = 0 (14.3) Proof.
Consider the sequence of solutions θ ( k ) to (1.1) converging to f in the L ∞ weak-* topology. Decomposethe operator T l as T l = T l + T le , where the term T le of the operator is the Fourier multiplier with symbol [ T le [ θ ]( ξ ) = 12 ( m l ( ξ ) + m l ( − ξ ))ˆ θ ( ξ ) (14.4)By equation (1.1), we have for all indices k that Z I × T n ( θ ( k ) ∂ t φ + θ ( k ) T l [ θ ( k ) ] ∂ l φ ) dxdt = − Z I × T n θ ( k ) T le [ θ ( k ) ] ∂ l φdxdt = 0 by the condition (14.1). By the proof of Theorem 1.3, the nonlinear term is continuous with respect to weak-*limits in L ∞ when restricted to active scalar fields, giving (14.3). To make this conclusion, it is important to notethat, in the proof of compactness for the nonlinear term, it was not important that the operator in the nonlinearterm was identical to the operator appearing in the active scalar equation. The proof used only the oddness ofthe multiplier in the nonlinear term, and certain time regularity estimates from the active scalar equation comingfrom the boundedness properties of the operator in the equation.Assuming that the hyperplane containing the image of the even part of m is in the dual lattice ξ (0) ∈ c T n , itis now not so hard to design a test function φ obeying (14.2) and an integral-conserving function f which failsto satisfy (14.3). As a first attempt, we can let ζ ( t ) be a smooth cutoff in time, and take φ ( t, x ) = ζ ( t ) cos( ξ (0) · x ) (14.5) f ( t, x ) = ζ ′ ( t ) cos( ξ (0) · x ) (14.6)Then (14.2) is satisfied, and we also have Z f ( t, x ) ∂ t φ ( t, x ) dxdt = Z ( ζ ′ ( t )) cos ( ξ (0) · x ) dxdt > (14.7)is strictly positive.The positivity of (14.7) does not necessarily imply the failure of (14.3). However, if the equality (14.3)holds for this function f , then (14.3) cannot hold for the function f , because the linear term (which is positiveby (14.7)) scales linearly, whereas the quadratic term scales quadratically. Thus, at least one of f or f fails tosatisfy (14.3), and we have Theorem 1.6. 57 Active scalar equations arise naturally in fluid dynamics in several asymptotic regimes, and as model equationsfor the full fluid systems. The problem of constructing active scalar fields for which the energy k θ k L x fails to beconserved is a natural generalization of Onsager’s conjecture for the Euler equations. This problem, however,encounters several additional difficulties when compared to Euler. Most importantly, a suitable analogue ofBeltrami flows, which provide an essential ingredient for obtaining regularity up to / in the case of Euler, areunavailable.For active scalars with multipliers that are not odd, we obtain nonuniqueness of weak solutions and even h -principles among integral-conserving functions for weak solutions with H¨older regularity up to / (Theo-rem 1.1, Theorem 1.2, and Corollary 1.1). Our proof is based on the observation that the interference termswhich arise due to self-interactions between individual waves must vanish to leading order. This observationallows for an approach in the spirit of the isometric embedding equations, where we eliminate one componentof the error in each stage of the iteration using one-dimensional oscillations. Our observation is general, andapplies in arbitrary dimensions even to the case of the Euler equations, giving a new approach to solutions inthat case as well. However, our inability to remove more than one component of the error leads to further lossesin regularity.These results however should not be expected for multipliers which are odd. For odd symbols, the Hamil-tonian is conserved at the level of θ ∈ L t,x (Theorem 1.4), and the nonlinearity exhibits a weak rigidity whichmakes it impossible to obtain an h -principle type result (Theorem 1.3). In higher dimensions, the presence ofconservation laws and other rigidity properties of weak solutions can even be sensitive to more subtle alge-braic properties of the multiplier, and our method applies in a generality which is essentially optimal (Theo-rems 1.5, 1.6).Several related questions remain open. Part of our proof does not apply to the nonperiodic setting andsome new idea could be required to produce nonperiodic solutions (currently even L ∞ t,x solutions have not beenconstructed in this case). Other significant questions include1. In the case of SQG, exhibit a weak solution θ ∈ L pt L x , that does not conserve energy.2. In the case of SQG, exhibit a weak solution θ ∈ C x,t that does not conserve energy, but does conservethe Hamiltonian.3. In the case of IPM, or more generally for not odd symbols, exhibit weak solutions θ ∈ C αt,x , with α ∈ (1 / , / that do not conserve energy.We believe that answering these questions may shed some light into the field of two dimensional turbulence.Finally, further sharpening approaches which do not rely on the use of Beltrami flows may be found usefulin resolving Onsager’s conjecture. The current approaches introduce anomalous time scales in the constructionwhich are incompatible with the time regularity bounds held by more regular solutions. With this obstruction,it seems unlikely at this time that an approach based on stationary solutions alone will go beyond the exponent / even for L -based function spaces. 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