On the Effects of Advection and Vortex Stretching
OOn the Effects of Advection and Vortex Stretching
Tarek Mohamed Elgindi and In-Jee JeongFebruary 7, 2017
Abstract
We prove finite-time singularity formation for De Gregorio’s model of the three-dimensionalvorticity equation in the class of L p ∩ C α ( R ) vorticities for some α > p < ∞ . We alsoprove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-timesingularity for certain infinite-energy solutions of the surface quasi-geostrophic equationwhich are C α -regular. One of the difficulties in the models we consider is that there arecompeting nonlocal stabilizing effects (advection) and destabilizing effects (vortex stretch-ing) which are of the same size in terms of scaling. Hence, it is difficult to establish thedomination of one effect over the other without having strong control of the solution.We conjecture that strong solutions to the De Gregorio model exhibit the following be-havior: for each 0 < α < ω ∈ C α ( R ) which is compactly supportedfor which the solution becomes singular in finite-time; on the other hand, solutions to DeGregorio’s equation are global whenever ω ∈ L p ∩ C ( R ) for some p < ∞ . Such a di-chotomy seems to be a genuinely non-linear effect which cannot be explained merely byscaling considerations since C α spaces are scaling subcritical for each α > Contents C -conjecture. . . . . . . . . 51.3.4 The models of Okamoto, Sakajo, and Wunsch . . . . . . . . . . . . . . . . 61.3.5 Connections with the SQG Equation . . . . . . . . . . . . . . . . . . . . . 71.4 The Setting of the Present Paper and the Main Results . . . . . . . . . . . . . . 71.5 A toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 The Self-Similar Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.1 The a = 0 case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.2 The case a >
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 The main steps in the proofs of Theorems 1 and 3 . . . . . . . . . . . . . . . . . 121.8 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 a r X i v : . [ m a t h . A P ] F e b Self-similar solutions for Constantin-Lax-Majda: Two Cases 13 < ω (cid:48) (0) < ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Case 2: ω (cid:48) (0) = + ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 a . . . . . . . . . . . . . . . . . . . . . . 173.1.1 The linear operator and its inverse . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Estimates for the inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 a and the linear operator . . . . . . . . . . . . . . . . . 334.2.2 Estimates for the inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.1 Properties of the Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Functional inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
In 1985 Constantin, Lax, and Majda introduced a one-dimensional equation which models thephenomenon of vortex-stretching in a three-dimensional incompressible fluid. They establishedfinite-time singularity formation in their model. Following in their footsteps, De Gregorio intro-duced a one-dimensional equation which models the effect of both transport and vortex stretch-ing. De Gregorio, aided by some numerical simulations, then conjectured that his introductionof velocity transport leads to global regularity. Thereafter, Okamoto, Sakajo, and Wunsch in-troduced a continuum of models which give different “weights” to the vortex stretching andtransport terms by introducing a real parameter a in front of the transport term. The fullmodel then became: ∂ t ω + a u∂ x ω = ω∂ x uu = ( − ∆) − / ω, with the Constantin-Lax-Majda and De Gregorio models being the cases a = 0 and a = 1respectively. They conjectured that when the transport term is strictly weaker than the vortexstretching term ( a < a ≥ a ≤ | a | < a for some a > | a | issmall. One of the interesting features of this construction is that the solutions, while becomingsingular in finite time, remain uniformly bounded in a norm which is strictly stronger than thosepredicted by the scaling of the equation and available conservation laws – a phenomenon which isconjectured to happen when a = − a ∈ R ) in classes of H¨older continuous solutions. Furthermore,for each (cid:15) > , we establish that such self-similar solutions lead to singularities in finite timewhich keep the velocity field bounded in the C − (cid:15) norm near the singular point though thevelocity’s C norm is becoming infinite. This relates to the C conjecture for the Cordoba-Cordoba-Fontelos model. Hence, it seems that the De Gregorio model exhibits two types ofbehavior for strong solutions: first there are “somewhat smooth” solutions for which there issingularity formation in finite time, then there are “very smooth” solutions and it is conjecturedthat these remain smooth for all time. Such a phenomenon is also neither captured by thescaling of the equation nor conservation of any critical quantity.Our interest in these models is two-fold: first, they give a simplified setting to understandthe complex interaction between transport and vortex stretching. Second, as the authors haveshown in [10], these models actually serve as the equations for the evolution of scale-invariantsolutions to the SQG equation and similar active scalar systems, which are locally well-posedin function spaces containing scale-invariant solutions (with some symmetry assumption on theinitial data). The main difference between the two-dimensional incompressible Euler equation and its three-dimensional counterpart is the presence of the so-called vortex-stretching term in three dimen-sions. The two-dimensional incompressible Euler equation can be written as: ∂ t ω + u · ∇ ω = 0 ,u = ∇ ⊥ ( − ∆) − ω, while the three-dimensional Euler equation can be written as: ∂ t ω + u · ∇ ω = ω · ∇ u,u = ∇ × ( − ∆) − ω. In two dimensions, ω is a scalar quantity and is advected by a velocity vector field which itinduces. In three dimensions, ω is a vector quantity. The term u · ∇ ω is a transport term and,since u is divergence free, it cannot cause any growth of the vorticity. The term ω ·∇ u is called thevortex stretching term and can lead to amplification of the vorticity. As is shown in the classicalwork of Beale, Kato, and Majda [1], finite-time singularity formation for strong solutions ofthe incompressible Euler equation happens if and only if the sup-norm of the vorticity becomesinfinite. 3he absence of vortex stretching in two dimensions allows the two dimensional Euler equationto enjoy infinitely many coercive conserved quantities, notably all L p norms of the vorticity for1 ≤ p ≤ ∞ . The presence of vortex stretching makes the three-dimensional Euler equationunstable with respect to all L p norms which were conserved in the two dimensional case andleads to point-wise exponential or double-exponential growth of the vorticity in time (see, forexample, [15],[11]). Despite the clear destabilizing properties of the vortex stretching term, theglobal regularity question of the three-dimensional Euler equation is wide open. In our view,there are three principle reasons why the regularity question is difficult:1. The quadratic and non-local nature of the vortex stretching term.2. The presence of both the vortex stretching and the transport terms.3. The 3D Euler equation is a system rather than a scalar equation.The importance of the second difficulty is discussed in great detail in [13]. In view of thesemany difficulties, experts in the field have devised simpler model equations that share some ofthe above properties for which more can be proven. The first attempt to derive a simplified 1D equation which models the effects of vortex stretchingseems to be that of Constantin, Lax, and Majda [4] where they wished to consider a non-linearevolution equation whose non-linearity is quadratic and non-local just like the vortex stretchingterm ω · ∇ u . Considering the exact relation between u and ω given by u = ∇ × ( − ∆) − ω , onesees that ∇ u is just a matrix of singular integral operators applied to ω . Hence, one has therelation: ω · ∇ u = R ( ω ) ω where R is some matrix of singular integrals. To extract a simple model from this, Constantin,Lax, and Majda dropped the transport term, made the 3D Euler system into a one-dimensionalscalar equation, and replaced R by the Hilbert transform. This led to: ∂ t ω = ωH ( ω )where H is the Hilbert transform. This equation can then be solved exactly using some specialproperties of the Hilbert transform – namely, the Tricomi identity (see Appendix A.1). Thereader should notice that in our above breakdown of the difficulties associated with the 3DEuler equation, the Constantin-Lax-Majda equation only takes into account the first of thedifficulties: the vortex stretching term is quadratic and non-local. One of the unfavorable properties of the reduction of Constantin, Lax, and Majda is that it doesnot behave well with the presence of diffusion. In fact, as Schochet has shown [19], the viscousCLM equation can experience finite time singularity formation before its inviscid counterpart4ven for small viscosity. More than this, the model does not take into account the effects ofthe transport term. To fix this problem, De Gregorio ([7],[8]) introduced a variant of the CLMequation that is much closer to the 3D Euler equation, which even has a large set of spatiallyperiodic steady states as the 3D Euler equation does. De Gregorio’s model is: ∂ t ω + u∂ x ω = ω∂ x u∂ x u = H ( ω ) . Notice that De Gregorio kept the same vortex stretching term but added a transport term aswell. It is important to note that any function of the form a n sin( nx ) + b n cos( nx ) is a stationarysolution to De-Gregorio’s equation while there are no non-trivial stationary states to the CLMequation . Moreover, if one considers the CLM equation with data ω = sin( x ), the solutionbecomes singular in finite time – developing a singularity like x near x = 0 as t → − (seenext section), but if one considers the De Gregorio equation with the same data, there is nosingularity formation (sin( x ) is stationary!). This already indicates that the transport term canhave a stabilizing effect. Numerical simulations done by De Gregorio [8], Okamoto, Sakajo, andWunsch [18], and others indicate, in fact, that singularity formation formation for De Gregorio’smodel is impossible from smooth data. In this work, we prove that singularity formation for DeGregorio’s model is possible in spaces of H¨older-continuous functions. This seems to be the firstrigorous large-data result for De Gregorio’s model. C -conjecture. In the spirit of Constantin, Lax, and Majda’s idea of getting a simple 1D model of the 3DEuler equation, Cordoba, Cordoba, and Fontelos found a 1D model for the so-called surfacequasi-geostrophic (SQG) equation. Recall the surface quasi-geostrophic equation: ∂ t θ + u · ∇ θ = 0 ,u = R ⊥ θ, with R ⊥ = ( R , − R ) where R i are the Riesz transforms on R . Cordoba, Cordoba, and Fontelosmodeled this by the following 1D equation: ∂ t θ + u∂ x θ = 0 ,u = H ( θ ) . Upon differentiating this equation and setting ∂ x θ = ω we get: ∂ t ω + u∂ x ω + ω∂ x u = 0 This is because any mean-zero periodic steady state ω would have to satisfy: ωH ( ω ) = 0 and ω − H ( ω ) = 0which would imply that ω ≡
0. When ω has non-zero mean, consider ω = ˜ ω + c with ˜ ω mean-zero and so˜ ωH (˜ ω ) + cH (˜ ω ) = 0 and H (˜ ω ) − ˜ ω = − c ˜ ω. Notice that if A is 0 somewhere, B is also 0 there. In other regions,we could have B = − c but since B must be mean-zero and B only takes on the values − c and 0, B ≡ . Thisrules out the existence of any non-constant periodic L stationary solution of the CLM equation. To our knowledge, the only existing results were local well-posedness results (see [18]). The SQG equation was introduced in the works of Constantin, Majda, and Tabak [6, 5] as a mathematicalmodel for geophysical atmospheric flows, and also as a model for the vorticity dynamics in 3D Euler equations –see for instance the book of Majda and Bertozzi [17]. x u = H ( ω )which is the same as De Gregorio’s model except that the vortex stretching term has the oppositesign.Cordoba, Cordoba, and Fontelos proved in [3] finite time singularity formation for thismodel, again, using some special properties of the Hilbert transform. Unlike the case withthe Constantin-Lax-Majda model, they were not able to solve the equation exactly but wereable to prove that certain kinds of smooth initial data cannot have global-in-time solutions byproving that ∂ t X ≥ cX for some positive quantity X which satisfies X ≤ (cid:107) ∂ x θ (cid:107) L ∞ and some constant c >
0. Thereafter,authors tried to push this blow-up result to prove that θ actually develops a shock at the time ofblow-up. However, numerics indicate that a different phenomenon than shock formation is takingplace. In fact, it seems that θ remains uniformly bounded in C for some unapparent reason.Proving that this is actually the case would be very interesting since it indicates that simply bychoosing the velocity field u to be H ( θ ) rather than θ , a phenomenon which “breaks the scaling”of the equation takes place. In other words, the dynamics of the equation would then not bedictated by clearly conserved quantities or scaling, since the strongest known conserved quantityis the W , norm of θ and the scaling critical quantity is the W , ∞ norm. The C conjecture isthen simply to prove that solutions to the Cordoba-Cordoba-Fontelos model remain uniformlybounded in C up to the time of blow-up (see [9],[16],[20],[14],[3]). We close by mentioning arecent work by Hoang and Radosz [12] which rigorously established cusp formation in a non-localmodel, which is a slight variant of the CCF model. The culmination of all of these developments is the introduction of one continuum of modelsunder which one can place all of those previously stated. This was done by Okamoto, Sakajo,and Wunsch [18]. They introduced the following continuum of models: ∂ t ω + au∂ x ω = ω∂ x u,∂ x u = H ( ω ) , with a ∈ R a parameter. The case a = − a = 0 corresponds to the Constantin-Lax-Majda model, and the case a = 1 correspondsto De Gregorio’s model. While it may seem that different values of a can lead to totally differentphenomena which would bring into question the efficacy of such a continuum, we will argue inthis work that there seems to be a deeper connection between all of these models as well as thequestions mentioned above: particularly the global regularity question for De Gregorio’s modeland the C conjecture for the Cordoba-Cordoba-Fontelos model. This connection is most visiblewhen one considers self-similar solutions.Returning to the issue of finite time singularities, it is intuitively clear that when a <
0, thetransport effect would work together with the stretching effect to develop a singularity. Indeed,Castro and C´ordoba [2] managed to prove that there is finite time blow up of smooth solutionsin the case of negative a . The question of finite time singularity/global regularity was left open6or a >
0, and one of our main results shows that when a is small and C ∞ data is considered,the blow up indeed occurs, and it may occur even in a self-similar way. Moreover, for each a > C α ( a ) for some α ( a ) > a large, which leads toa finite-time singularity. It is interesting to note that both the De Gregorio and Okamoto-Sakajo-Wunsch systems havedirect connections with the SQG equation discussed in the above. Indeed, in the work of Castroand C´ordoba [2], it was observed that with the so-called stagnation point similitude ansatz θ ( t, x , x ) = x f ( t, x ) , if f solves the De Gregorio system, then one obtains a solution to the SQG equation. Moreover,in our recent work [10], we have established that one can uniquely solve the SQG equation(locally in time) with the radial homogeneity ansatz, which is in polar coordinates θ ( t, r, s ) = r · h ( t, s ) , s ∈ [ − π, π )and then h ( t, · ) on the circle solves a system which is essentially the Okamoto-Sakajo-Wunschsystem with a = 2, up to a very smooth term in the relation connecting the scalar h with thevelocity u . The importance of this ansatz (as opposed to the stagnation point ansatz) is thatsingularity formation from smooth data for radially homogeneous solutions (which have infiniteenergy) implies finite-time singularity formation from compactly supported (finite energy) datain a suitable local well-posedness class, as shown in [10]. In this paper, we consider the Okamoto-Sakajo-Wunsch systems in the following form: ∂ t ω + a u∂ x ω = 2 ω∂ x u, u = − Λ − ω (1.1)where a ∈ R is a parameter and u ( t, · ) and ω ( t, · ) are functions of one spatial variable x ∈ R .Our first main result shows that when | a | is small, there exists a smooth initial data which blowsup in finite time in a self-similar manner. This is to say that Theorem 1.
There exists some absolute constant a > such that for | a | < a , there is an oddinitial data ω = ω a ∈ H ( R ) , depending analytically on a , which blows up in finite time in aself-similar manner.More precisely, for | a | < a , there exists some λ = λ ( a ) depending analytically on a andwith λ (0) = 0 , such that the unique local solution of (1.1) with initial data ω := ω a in C ([0 , H ( R )) is given by ω ( t, x ) = 11 − t ω (cid:18) x (1 − t ) λ (cid:19) , which blows up precisely at t = 1 . Note that the form of the equation is slightly different from that in the introduction and in the literature[2, 4, 7, 18], but it is equivalent up to a rescaling of time. Our convention makes some of the formulas a bitsimpler. emark 1.1. Even though we have chosen to work with a finite regularity space H ( R ), astraightforward modification of our arguments actually gives that the profile ω belongs to any H s ( R ) for s ≥ Theorem 2.
Let ω a ( x, t ) be the self-similar solutions constructed in Theorem 1. Then, ω a remains uniformly smooth up to the blow-up time away from x = 0 . That is, given b > , sup ( x,t ) ∈ ( R \ ( − b,b )) × [0 , | ω a ( x, t ) | ≤ C ( b ) . Moreover, ω a remains in the space of weak L λ ( a ) functions uniformly up to the blow-up time.That is, ω a ∈ L ∞ ([0 , L λ ( a ) , ∞ ) . Finally, sup t ∈ [0 , (cid:90) − | ω a ( x, t ) | p dx < ∞ for all p < λ ( a ) . Remark 1.2.
As a part of our construction, we shall see that λ (cid:48) (0) = − <
0. Hence, wededuce that for small a < , ω belongs to L p ([ − , ≤ p < λ ( a ) . Corollary 1.3.
Let ω a be the self-similar solutions constructed in Theorem 1 and suppose that a < is sufficiently small. Then, since u a = − Λ − ω a , sup t ∈ [0 , (cid:107) u a (cid:107) C α ( a ) < ∞ with α ( a ) = 1 − λ ( a ) . Remark 1.4.
Notice that with our convention the De Gregorio model is the case a = 2 andthe Cordoba-Cordoba-Fontelos model is the case a = −
2. Hence, to confirm the regularityconjecture for the De Gregorio model, one would need to show that λ (2) = − C conjecture for the CCF model, one would need to show that λ ( −
2) = 1 – assuming that λ can be continued to | a | ≤ all the Okamoto-Sakajo-Wunsch models – including De Gregorio’s model – by taking solutions with sufficiently lowH¨older continuity index. Theorem 3.
There exists some absolute constant c > such that for α ∈ { /n, n ∈ N } and | a | < c /α , there exists some value λ ( α ) ( a ) satisfying λ ( α ) (0) = 0 , λ ( α ) ( a ) > − , and an oddinitial data ω ( α )0 ( a ) ∈ L p, ∞ ∩ C α ( R ) for p = λ α ( a ) α which blows up in a self-similar way.More precisely, the unique local in time solution of (1.1) with initial data ω in C ([0 , T ); C α ( R )) is given by ω ( t, x ) = 11 − t ω ( α )0 x (1 − t ) λ ( α )( a ) α , which blows up at t = 1 . orollary 1.5. For a = − , the Cordoba-Cordoba-Fontelos model, there exists a family ofsolutions ω α ( x, t ) initially belonging to the class L λ ( α ) α , ∞ ∩ C α ( R ) for all | α | < α a smallconstant and such that λ ( α ) ≤ C α as α → which blows up at t = 1 in a neighborhoodof x = 0 . Moreover, the associated velocity field u α remains uniformly bounded in the space C − λ ( α ) ([ − , up to the time of blow-up. Notice that < − λ ( α ) for α sufficiently small. This is to say that for each (cid:15) > C − (cid:15) norm of the velocity under control up to the time of singularitywhere the C norm of the velocity becomes infinite. The main idea of our work is that if the transport term is “weaker” than the vortex stretchingterm, then singularities will form in finite time. The weakening of the transport term, in oursetting, can happen in two ways:1. By putting a small parameter in front of the transport term.2. By taking merely C α vorticity with α > We illustrate these ideas using an explicitly solvable toy model. Consider the following modelwhere one simply replaces ∂ x u in (1.1) by its value at zero ∂ x u (0) = − Hω (0): ∂ t ω − aHω (0) x · ∂ x ω + 2 Hω (0) · ω = 0 , (1.2)where we assume the initial data ω to be odd, nonnegative on R + , and smooth away from theorigin. Then, solving along the characteristics, ω ( t, x ) = e − (cid:82) t Hω ( s, ds · ω ( xe a (cid:82) t Hω ( s, ds ) , and using this representation for ω ( t, · ) we compute Hω ( t,
0) = − π (cid:90) ∞ e − (cid:82) t Hω ( s, ds ω ( xe a (cid:82) t Hω ( s, ) dxx = − e − (cid:82) t Hω ( s, ds c , where c = − Hω (0) >
0, unless ω is identically zero. This in turn determines Hω ( t,
0) = − c − − t , and inserting this back into the formula for ω ( t, · ), we conclude ω ( t, x ) = 11 − c t ω (cid:18) x (1 − c t ) − a/ (cid:19) . We observe that: We would like to mention a relevant work of Zlatoˇs [21] who obtained exponential in time growth of thevorticity gradient in L ∞ for the 2D Euler equation on T . This was done by using initial vorticity whose gradientat the origin is only C α -regular. For a <
2, the solution develops a singularity at time t ∗ = (2 c ) − , for any smooth profile ω with ω (cid:48) (0) (cid:54) = 0. • In the case a = 2, if the initial data is C or better, then the solution stays smooth up tothe time moment (2 c ) − and can be continued as a smooth solution for all time; indeed,using the mean value theorem with ω (0) = 0, (cid:12)(cid:12)(cid:12)(cid:12) ω ( x (1 − c t ))1 − c t (cid:12)(cid:12)(cid:12)(cid:12) ≤ x (cid:107) ω (cid:48) (cid:107) L ∞ . On the other hand, if initially ω ( x ) ≈ C sgn( x ) | x | α near the origin for some 0 < α < C >
0, then the solution blowsup in L ∞ at time (2 c ) − . In the case of a = 0, the system (1.1) is just the Constantin-Lax-Majda equation, which has asimple and explicit self-similar solution ω ( t, x ) = 11 − t F (cid:18) x − t (cid:19) , F ( z ) = z z , see Figure 1.The scaling symmetry of the model permits one to work in the self-similar variable z = x/ (1 − t ) λ for any λ ∈ R , and it is easy to see that a profile F gives a self-similar solution to(1.1) if and only if F solves the differential equation F ( z ) + (cid:0) (1 + λ ) z − a Λ − F ( z ) (cid:1) F (cid:48) ( z ) + 2 F ( z ) HF ( z ) = 0 . Then we proceed to show existence of a solution by essentially “linearizing” around the specialsolution which corresponds to a = 0 , λ = 0, and F = F . Proving the main theorem boils downto establishing that the inverse of the associated linear operator is bounded. a = 0 case. In the special case of a = 0, the system (1.1) reduces to the Constantin-Lax-Majda equation [4]on R : ∂ t ω + 2 H ( ω ) ω = 0 . (1.3)A particularly nice feature of this model is that it can be integrated: one may check thatthe pair ω ( t, x ) = ω ( x )(1 + t · Hω ( x )) + ( t · ω ( x )) , Hω ( t, x ) = ( Hω )( x )(1 + t · Hω ( x )) + t · ω ( x )(1 + t · Hω ( x )) + ( t · ω ( x )) (1.4)10 - - - Figure 1: Evolution of the data ω ( x ) = x/ (1 + x ) at time moments t = 0 . , . , . , Z = { x : ω ( x ) = 0 , Hω ( x ) < } is nonempty, the formula (1.4) explicitly shows that ω ( t, · ) blows up in L ∞ ( R ) at t ∗ = (sup x ∈ Z − Hω ( x )) − > , at the point of Z which attains the supremum. In particular, when the initial data ω ( x ) is anodd function of x and strictly positive on (0 , ∞ ), the solution becomes singular at the originexactly at t ∗ = − Hω (0) − > ω ( x ) = x x , whose Hilbert transform equals Hω ( x ) = −
11 + x . Plugging these identities into (1.4), one can see that ω ( t, x ) = xx +1 (cid:16) − tx +1 (cid:17) + t · (cid:16) xx +1 (cid:17) = 11 − t · x − t (cid:16) x − t (cid:17) = 11 − t · ω (cid:18) x − t (cid:19) , (1.5)that is, the solution is self-similar and blows up at time t = 1; see Figure 1. Note that the“vorticity stretching” factor Hω ( t, · ) is always maximal at the origin, and therefore it drags themaximum point of ω ( t, · ) towards the origin, and at the same time gets intensified.11 .6.2 The case a > . Moving on to the case of a >
0, let us stick to the situation where ω ( t, · ) is odd and positiveon (0 , ∞ ). Then, the velocity u ( t, · ) is again odd and always directed away from the origin, andtherefore it counteracts the stretching term which tries to form a sharp gradient by transportingthe maximum point towards the origin. It is far from clear whether the stretching term canovercome the transport effect and still result in a finite-time blow up; what seems to happen isthat for some critical value of a (likely a = 2), the transport term takes over and prevents thesingularity formation. We conjecture that for smooth data there will be global regularity when a ≥ a <
2. Theorem 1 confirms this for all a < a for someconstant a >
0. For merely C α data we will show that there can be singularity formation forany value of a so long as α is taken small enough and this is the content of Theorem 3. We close the introduction by giving a succinct explanation of how we go about proving Theorems1 and 3.1.
Construction of self-similar solutions when a = 0 which become singular in finite time. Todo this, one studies carefully the solution formula for the solutions to the a = 0 equation.In the smooth case, one uses Taylor’s theorem to extract a universal self-similar profile.In the C α case, it turns out that we can do a similar procedure using an interesting factabout the Hilbert transform (Lemma 2.2). This lemma says that if g ( z ) = sgn( z ) k ( | z | α )for some 0 < α < k ∈ C ∞ and k (0) = 0 then lim z → H ( g )( z ) − H ( g )(0) | z | α is well definedand is determined only by k (cid:48) (0). One can then check that the asymptotic profiles derivedthis way are actually self-similar solutions. This leaves us with one smooth self similarsolution when a = 0 as well as a family of merely C α self-similar solutions. In the smoothcase, this solution is just of the form ω ( x, t ) = − t F ( x − t ) with F ( z ) = z z . Analysis of the linearization of the equation satisfied by self-similar solutions around the a = 0 solution. Normally, when one considers a functional equation N ( f, a ) = 0 alongwith a known solution ( f , f a , a ) for a small can be done using the Implicit Function Theorem applied in the right functionalsetting. Applying the Implicit Function Theorem, in turn, requires invertibility of thelinear operator ∂N∂f ( f , . Unfortunately, this linear operator is neither onto nor one-to-one in our case. In fact, for an H ( R ) function to be in the range of this linear operator, itmust satisfy a consistency condition relating its Hilbert transform at 0 and its derivativeat 0. However, since this consistency condition is one-dimensional, this problem can beremedied by introducing one other parameter which we call λ ( a ). In particular, insteadof looking merely for a solution of the form − t F a ( x − t ) , we now look for a solution ofthe form − t F a ( x (1 − t ) λ ( a ) ) . A judicious choice of λ ( a ) allows us to carry out an iterationprocedure to formally continue the a = 0 solution and the iteration procedure converges solong as certain operators can be bounded on the spaces we are working in. These boundsbecome very delicate in the C α case, especially since we need to be careful to get boundswith explicit dependence on α . 12. Technical lemmas related to the Hilbert transform and related operators.
To prove bound-edness of the inverse of the modified linearized operator discussed above, we must useseveral important technical lemmas, some of which are classical and some of which seemto be new. We often make use of the Tricomi identity (A.1) and the classical Hardy in-equality (A.14). In fact, these are all we need in the smooth case. In the C α case we need toprove (sharp) bounds for Hilbert transform-like operators which describe how the Hilberttransform acts on functions of | x | n for n ∈ N . If for some n ∈ N , f ( x ) = ˜ f ( | x | n ) with ˜ f aneven L function then we define the operators ˜ H ( n ) by ˜ H ( n ) ( ˜ f )( w ) := sgn( w ) H ( f )( | w | n ) . In particular, it can be shown that the operators ˜ H ( n ) are bounded on the space of L even functions with operator norms Cn for some universal constant C > n ∈ N .We also need the Hardy-type inequalities (A.15). In Section 2 we discuss self-similar blow-up solutions for the CLM equation and how blow-up forCLM is generically self-similar. We discuss two cases therein: smooth self-similar profiles andmerely H¨older continuous self similar profiles. In Section 3, we prove that the self-similar profilesfor the CLM equation in the smooth case can be continued to self-similar blow-up profiles for theOkamoto-Sakajo-Wunsch equation with | a | small which is the content of Theorem 1. In Section4, we do the same for the merely H¨older continuous profiles for the CLM equation but there | a | can be taken to be on the order of α − where α is the H¨older continuity index of the CLMsolution, which is the content of Theorem 3. Lastly, we discuss an approach which could leadto showing finite-time singularity formation for the De Gregorio system on the circle in Section5. Several useful identities regarding the Hilbert transform and some functional inequalities arecollected in the Appendix. Notation
As it is usual, we use letters
C, c , and so on to denote absolute constants whose values mayvary from a line to another. Moreover, we write X (cid:46) Y when there is some absolute constant c > X ≤ cY . When a constant or a function depends on a parameter, we write out thedependence using either a subscript or a superscript. In particular, for 0 < α <
1, the expression F ( α )0 is reserved for the odd and self-similar profile for the Constantin-Lax-Majda equation; see(4.2). In Section 4, we use tildes to denote the transform which acts on functions on R + definedvia ˜ f ( w ) := f ( w α )with some given value of 0 < α < In this section we completely characterize the blow-up profile for solutions to the Constantin-Lax-Majda equation which are odd and positive on (0 , ∞ ). It shouldn’t be too surprising that we For clarity of exposition in the proof, we will often write these operators simply as ˜ H since n is always keptfixed. ω near x = 0. Moreover, the blow-up is of self-similar type with different scaling rates depending uponwhether ω (cid:48) (0) = 0 , < ω (cid:48) (0) < ∞ , or ω (cid:48) (0) = + ∞ . By “of self-similar type” we mean that thesolution consists of purely self similar function plus a function which is uniformly bounded upto the time of blow-up. We now go into more detail on only the second and third cases sincewe shall not need the third case. Recall the solution formula for the Constantin-Lax-Majdaequation: ω ( x, t ) = ω ( x )(1 + tHω ( x )) + t ω ( x ) . < ω (cid:48) (0) < ∞ Let’s first consider the case where ω (cid:48) (0) = 1. Also assume that Hω (0) = −
1. Then, ω ( x, t ) = x (1 + tHω ( x )) + t ω ( x ) + ω ( x ) − x (1 + tHω ( x )) + t ω ( x ) . Notice that the second term is uniformly bounded for ≤ t ≤ | ω ( x ) − x | (cid:46) | x | and | ω ( x ) | ≥ | x | . Hence, ω ( x, t ) ≈ x (1 + tHω ( x )) + t ω ( x ) , where the ≈ sign means equal up to a term uniformly bounded up to t = 1. Now, | Hω ( x ) + 1 | (cid:46) | x | since Hω is even. Hence, ω ( x, t ) ≈ x (1 − t ) + t ω ( x ) + (cid:104) x (1 + tHω ( x )) + t ω ( x ) − x (1 − t ) + t ω ( x ) (cid:105) x (1 + tHω ( x )) + t ω ( x ) − x (1 − t ) + t ω ( x ) = x [(1 − t ) − (1 + tHω ( x )) ][(1 − t ) + t ω ( x ) ][(1 + tHω ( x )) + t ω ( x ) ]Now, | (1 − t ) − (1 + tHω ( x )) | = | t (1 + Hω ( x ))(2 − t + tHω ( x )) | (cid:46) (1 − t ) x + | x | Therefore, ω ( x, t ) ≈ x (1 − t ) + t ω ( x ) . We remark that Hω (0) = − π (cid:82) ∞−∞ ω ( x ) x dx so that Hω (0) ≤ ω ≡ ω (cid:48) (0) = 1 and Hω (0) = − ω ( x, t ) ≈ x (1 − t ) + t x = 11 − t F ( x − t )with F ( z ) = zz +1 . As mentioned in the footnote, a simple scaling argument will then imply thatwhenever 0 < ω (cid:48) (0) < ∞ , and ω ∈ C , ω is asymptotically equal to a self similar profile. Infact, this is the only such self-similar profile.It is straightforward to check that with the self-similar ansatz ω ( t, x ) = 11 − t F (cid:18) x − t (cid:19) , the Constantin-Lax-Madja equation (1.3) reduces to the following: F + zF (cid:48) ( z ) + 2 F H ( F ) = 0 . (2.1)We have already seen that F ( z ) = Cz/ (1 + C z ) is a solution to this equation. We also have: Proposition 2.1.
Assuming that the profile F ( z ) in (2.1) is odd, smooth, and decays at infinity,we have F ( z ) = z z , up to a scaling of the variable z .Proof. Taking the Hilbert transform of both sides of the equation (2.1), we see: H ( F ) + zH ( F ) (cid:48) − F + H ( F ) = 0 . (2.2)Now let F + iH ( F ) = V. Then, since2 F H ( F ) + i ( H ( F ) − F ) = − i ( F + iH ( F )) = − iV , the system of equations (2.1), (2.2) can be rewritten as V + zV (cid:48) − iV = 0 . Integrating the system gives V = 1 i + Cz , and it is easy to see that the real and imaginary parts of V are odd and even respectivelyprecisely when C is a real constant. Thus, V = Cz C z + i C z and in particular, F = Cz C z . .2 Case 2: ω (cid:48) (0) = + ∞ Let’s assume that ω ( x ) = sgn( x ) | x | α Ω ( x ) for some 0 < α < a sufficientlysmooth rapidly decaying even function with Ω (0) = 1 and Ω > R . In this case, it is clearthat Hω ( x ) = C + | x | α Ω ( x ) with C some constant and Ω ( x ) a smooth even decaying functionon R . We actually need the following interesting lemma:
Lemma 2.2.
Suppose g ( z ) = sgn( z ) k ( | z | α ) with k smooth and k (0) = 0 . Then, lim z → | z | α [ H ( g )( z ) − H ( g )(0)] = cot (cid:16) απ (cid:17) k (cid:48) (0) . Proof.
Assume 0 < x < .
1. Note that (cid:90) ∞−∞ sgn( z ) k ( | z | α ) x − z dz − (cid:90) ∞−∞ sgn( z ) k ( | z | α ) z dz = x (cid:90) ∞−∞ sgn( z ) k ( | z | α )( x − z ) z dz. Since we are only concerned about the limit as z → x α , we can cutout the large z part. So we see that the quantity we want is equal tolim x → x α x (cid:90) − sgn( z ) | z | α ( x − z ) z dz. After a rescaling, it is equal tolim x → p.v. (cid:90) /x − /x dz (1 − z ) | z | − α = p.v. (cid:90) ∞−∞ − z ) | z | − α dz. A direct numerical check gives us that p.v. (cid:90) ∞−∞ − z ) | z | − α dz = π cot (cid:16) απ (cid:17) , and therefore multiplying by π , we are done.Using the lemma, we see that Ω (0) = cot( απ ) . It is clear from the formula ω ( x, t ) = ω ( x )(1 + tHω ( x )) + t ω ( x ) that singularity formation happens at x = 0 and t = − C − . Note that if Ω is fixed, C ≈ − α as α →
0. This can be seen by direct computation or by observing that H (sgn( x )) has a logarithmicsingularity at the origin. Using arguments similar to the above, we see: ω ( x, t ) ≈ sgn( x ) | x | α Ω (0)(1 − C − ( C + | x | α Ω (0))) + C − | x | α Ω (0) . Note that Ω (0) = 1 and Ω (0) = cot( απ ). ω ( x, t ) ≈ sgn( x ) | x | α (1 + t ( C + | x | α cot( απ ))) + t | x | α .
16 sgn( x ) | x | α (1 + tC ) + 2(1 + tC ) | x | α cot( απ ) + t | x | α csc ( απ )To make things easier, let’s assume C = −
1. So, ω ( x, t ) ≈ sgn( x ) | x | α (1 − t ) + 2(1 − t ) | x | α cot( απ ) + | x | α csc ( απ )Hence, we see that the behavior near the time of singularity ( t = 1) is self similar: ω ( x, t ) ≈ − t F ( x (1 − t ) α )with F ( z ) = sgn( z ) | z | α | z | α cot( απ ) + | z | α csc ( απ ) . This coincides with the exact self-similar solution (4.2) up to a multiplicative constant and aterm of order | z | α . In this section we prove Theorem 1. In particular, we consider the Okamoto-Sakajo-Wunschequation and prove the existence of self-similar solutions which become singular in finite timefor | a | small enough. This is done using a continuation argument since we already have sucha solution in the case when a = 0 . This cannot actually be done directly since normally thisrequires invertibility of a certain linear operator and we will find the operator to be neitherone-to-one nor onto. This will be fixed by introducing another parameter λ ( a ) which changesthe scaling of the self-similar solution. Once this λ ( a ) is chosen properly, the relevant linearoperator becomes onto and once we make a further restriction the linear operator becomesinvertible. Introducing this parameter λ ( a ) is likely necessary to construct smooth solutions ofthe Okamoto-Sakajo-Wunsch equation because, as is alluded to in the corollaries to Theorem 1,this seems to be related to the C conjecture for the case a = −
2. Namely, it seems to be thatthe velocity fields associated to solutions of the Okamoto-Sakajo-Wunsch equation with a < C α ( a ) estimate for some α ( a ) > a This time, we consider the following ansatz ω ( t, x ) = 11 − t F (cid:18) x (1 − t ) λ ( a ) (cid:19) , λ (0) = 0for the system (1.1), where F is an odd and smooth function which decays at infinity. Then, interms of F , (1.1) takes the following form: F ( z ) + (cid:0) (1 + λ ( a )) z − a Λ − ( F )( z ) (cid:1) F (cid:48) ( z ) + 2 F ( z ) HF ( z ) = 0 . (3.1)Our previous computations show that when a = 0, then F = F (with λ (0) = 0) solves (3.1).17e consider the following expansions in a around ( F , F ( z ) = F ( z ) + ∞ (cid:88) n =1 a n F n ( z ) , λ ( a ) = ∞ (cid:88) n =1 a n λ n (3.2)where F n and λ n for each n ≥ F n are C ∞ ( R ) functions with suitable decay at infinity. For simplicity, wewill just work in the space H ( R ): (cid:107) f (cid:107) H ( R ) := (cid:88) k =0 (cid:107) ∂ kz f (cid:107) L ( R ) , and show that the series (3.2) converges in H ( R ). Note that for f ∈ H , the expressions suchas f (cid:48) (0), f (cid:48)(cid:48) (0), and Hf (cid:48) (0) make sense, and in particular F will solve (3.1) pointwise, once weshow that the series for F converges in H .Let us briefly outline how F n and λ n are going to be determined, given F , · · · , F n − and λ , · · · , λ n − . We shall work with a linear operator L defined on the space of odd functions,which has a one-dimensional kernel and the image of codimension 1. Then at each step of theiteration, we will be able to determine F n by inverting L in an equation of the form L ( F n )( z ) = G n ( z ) − λ n zF (cid:48) ( z ) , with some G n = G n ( F , · · · , F n − ; λ , · · · , λ n − ) where λ n is the unique number which gives asolution F n , up to a one-dimensional kernel. Any non-trivial element in the kernel has a nonzeroderivative at the origin, and therefore we may define L − on the image of L by forcing thederivative of F n to be zero at the origin. Then we proceed to show that, using simple normestimates, both series in (3.2) are convergent. Let us insert the expansion (3.2) directly into (3.1). Then one obtains the sequence of identities,for each power of a : F n + zF (cid:48) n + 2 HF F n + 2 F HF n = n − (cid:88) j =0 Λ − ( F j ) F (cid:48) n − − j − z n (cid:88) j =1 λ j F (cid:48) n − j − n − (cid:88) j =1 F j HF n − j , (3.3)for each n ≥
1. We note that, by considering the linear operator L defined by Lf := f + zf (cid:48) + 2 HF f + 2 F Hf = f + zf (cid:48) −
21 + z f + 2 z z Hf, (3.4)the above set of equations can be rewritten as L ( F n ) = G n ( z ) − λ n zF (cid:48) ( z ) = G n ( z ) − λ n z (1 − z )(1 + z ) (3.5)18here G n = G n ( F , · · · , F n − ; λ , · · · , λ n − ) is simply the right hand side of (3.3) without theterm involving λ n . The following lemma will guarantee that we can always pick λ n in a (unique)way that (3.5) is solvable. Then, we can define F n := L − (cid:18) G n ( z ) − λ n z (1 − z )(1 + z ) (cid:19) by an explicit integral formula; see below (3.6).The purpose of the following lemma is mainly to extract a representation formula for the L − , supposing that the right hand side is sufficiently smooth and decays fast at infinity. Thenin the next lemma, we carry out precise norm estimates for F n . Lemma 3.1.
Consider the linear equation Lf = g where g ∈ C ∞ c ( R ) is a given odd function. Then a solution f ∈ H ( R ) (which is necessarilyodd) exists if and only if g (cid:48) (0) + 2 Hg (0) = 0 and once we further require that f (cid:48) (0) = 0 , it isuniquely determined by the expression ( L − g )( z ) = z (1 − z )(1 + z ) · (cid:90) z − w w ˆ g ( w ) + 2ˆ h ( w ) dw + − z (1 + z ) · (cid:90) z − g ( w ) + 1 − w w ˆ h ( w ) dw, (3.6) with ˆ g ( z ) := g ( z ) z −
11 + z g (cid:48) (0) , ˆ h ( z ) := Hg ( z ) − Hg (0) z + 2 z z Hg (0) . Proof.
Assuming that we are given a solution f ∈ H ( R ) to Lf = g , one may take the Hilberttransform of both sides to obtain Hf + z ( Hf ) (cid:48) − z z f −
21 + z Hf = Hg, (3.7)where we have used the Tricomi identity. Setting d = Hf for simplicity, we observe that asolution f of Lf = g must solve the following linear system of ODEs (cid:18) fd (cid:19) + z (cid:18) fd (cid:19) (cid:48) + 21 + z (cid:18) − z − z − (cid:19) (cid:18) fd (cid:19) = (cid:18) gh (cid:19) , (3.8)with d = Hf and h = Hg . Since we are interested in the case where f is odd (and therefore d is necessarily even), it suffices to solve the initial value problem with given f (0) and d (0) on thepositive reals. Due to the presence of the coefficient z , it is not clear whether the initial valueproblem is well-posed, and the condition g (cid:48) (0) + 2 h (0) = 0 comes out naturally in this setting.Indeed, evaluating both sides of the second component of (3.8) at z = 0, one obtains d (0) − d (0) = h (0) , z and taking the limit z → + ,0 = lim z → + (cid:18) f ( z ) z + f (cid:48) ( z ) −
21 + z f ( z ) z + 21 + z d ( z ) − g ( z ) z (cid:19) → d (0) − g (cid:48) (0) . Therefore we need d (0) = − h (0) = 12 g (cid:48) (0)for the ODE (3.8) to have a C solution on [0 , ∞ ). In particular we see that the system (3.8)selects a unique pair of initial conditions ( f (0) , d (0)) = (0 , − h (0)) = (0 , g (cid:48) (0) /
2) which has achance of having a smooth solution.Introducing for convenience ˆ d ( z ) := d ( z ) − d (0) and using the above condition, one cansimply rewrite the system as (cid:18) f ˆ d (cid:19) (cid:48) + 1 z (1 + z ) (cid:18) − (1 − z ) 2 z − z − (1 − z ) (cid:19) (cid:18) f ˆ d (cid:19) = (cid:32) g ( z ) z − z g (cid:48) (0) h ( z ) − h (0) z + z z h (0) (cid:33) . (3.9)Using complex notation V ( z ) = F ( z ) + iH ( F )( z ) = z − i z = 1 z + i ,U ( z ) = f ( z ) + id ( z ) , and ˆ G ( z ) = ˆ g ( z ) + i ˆ h ( z ) , the above vector system can be simply re-written as − z · iz − iz ( U ( z ) − U (0)) + ( U ( z ) − U (0)) (cid:48) = ˆ G ( z ) z . With this complex notation, it is easy to directly integrate (3.9) to obtain the formula U ( z ) = U (0) + z ( − z + i (1 − z ))(1 + z ) · (cid:32) c + (cid:90) z ( − − i − w w ) ˆ G ( w ) w dw (cid:33) , (3.10)with some (complex) constant of integration c , and converting back to the real notation, this is (cid:18) f ˆ d (cid:19) = z (1 + z ) (cid:18) − z − z z − z (cid:19) (cid:18)(cid:18) c f c d (cid:19) + (cid:90) z (cid:18) (1 − w ) /w − − w ) /w (cid:19) (cid:18) ˆ g ( w )ˆ h ( w ) (cid:19) dw (cid:19) with some constants c f and c d . In the expression inside the large brackets, the integral term isof order O ( z ) for | z | small, so that c f and c d are precisely the derivatives of f and ˆ d evaluated at z = 0. The latter is zero since ˆ d is even and, and c f is zero if we assume further that f (cid:48) (0) = 0.In particular, this shows that the solution f is unique.To actually conclude that f given by the above formula provides a solution of Lf = g , itneeds to be argued that Hf = d . This follows simply by observing that the functions V, U ,and ˆ G can be extended as holomorphic functions on the upper half-plane, and that the formula(3.10) actually defines a holomorphic function of z on the upper half-plane, which has an oddreal part and even imaginary part once restricted onto the real axis.20 .1.2 Estimates for the inverse We need an estimate for the inverse L − in H ( R ). Let us begin by breaking L − into pieces:for σ ∈ {− , , } , define the operators for odd functions g ∈ H ( R ) via T ,σ ( g ) := z (1 − z )(1 + z ) (cid:90) z w σ (cid:18) g ( w ) w −
11 + w g (cid:48) (0) (cid:19) dwT ,σ ( g ) := 2 z (1 + z ) (cid:90) z w σ (cid:18) g ( w ) w −
11 + w g (cid:48) (0) (cid:19) dw and S ,σ ( g ) := z (1 − z )(1 + z ) (cid:90) z w σ (cid:18) Hg ( w ) − Hg (0) w + 2 w w ( Hg )(0) (cid:19) dwS ,σ ( g ) := 2 z (1 + z ) (cid:90) z w σ (cid:18) Hg ( w ) − Hg (0) w + 2 w w ( Hg )(0) (cid:19) dw. Note that L − in (3.6) is a linear combination of twelve operators { T l,σ , S l (cid:48) ,σ (cid:48) } with l, l (cid:48) ∈ { , } and σ, σ (cid:48) ∈ {− , , } .Recall that λ n and F n were chosen to satisfy F n = L − (cid:18) G n ( z ) − λ n z (1 − z )(1 + z ) (cid:19) , and since G n involves derivatives of F j for j < n , G n belongs to H ( R ) and no better. It turnsout that unfortunately the operators T , − and S , − are not bounded from H ( R ) to H ( R )(although naively one would expect that these operators gain one derivative), and to actuallydeduce that F n ∈ H ( R ), we shall need to use the specific form of G n .Let us begin with the L estimates. Lemma 3.2 ( L -bounds) . Assume that f and g are odd functions on R . Then we have thefollowing estimates: (cid:107) T l,σ g (cid:107) L ( R ) + (cid:107) S l,σ g (cid:107) L ( R ) ≤ C (cid:107) g (cid:107) H ( R ) , (3.11) (cid:107) T l,σ ( zg (cid:48) ) (cid:107) L ( R ) + (cid:107) S l,σ ( zg (cid:48) ) (cid:107) L ( R ) ≤ C (cid:107) g (cid:107) H ( R ) , (3.12) and (cid:107) T l,σ (Λ − ( f ) · g (cid:48) ) (cid:107) L ( R ) + (cid:107) S l,σ (Λ − ( f ) · g (cid:48) ) (cid:107) L ( R ) ≤ C (cid:107) f (cid:107) H ( R ) · (cid:107) g (cid:107) H ( R ) (3.13) Proof.
We shall first establish all the inequalities in the case l = 1. During the course of theargument, it will become clear that the case l = 2 can be treated in the same way and is onlysimpler.Let us first show (3.11). We consider the most difficult case of σ = − T , − ( g ) = z (1 − z )(1 + z ) (cid:90) z g ( w ) − wg (cid:48) (0) w + w w g (cid:48) (0) dw (cid:107) z (1 − z )(1 + z ) · g (cid:48) (0) · (cid:90) z w w dw (cid:107) L ≤ C (cid:107) g (cid:107) H · (cid:107) z (1 − z ) ln(1 + z )(1 + z ) (cid:107) L ≤ C (cid:107) g (cid:107) H . For the other term, we obtain (cid:107) z (1 − z )(1 + z ) (cid:90) z g ( w ) − wg (cid:48) (0) w dw (cid:107) L ≤ (cid:107) z (1 − z )(1 + z ) (cid:107) L ∞ · (cid:107) z (cid:90) z g ( w ) − wg (cid:48) (0) w dw (cid:107) L ≤ C (cid:107) g ( z ) − zg (cid:48) (0) z (cid:107) L = C (cid:107) − ∂ z (cid:18) (cid:82) z g (cid:48) ( w ) dwz (cid:19) + 1 z (cid:90) z g (cid:48)(cid:48) ( w ) dw (cid:107) L ≤ C (cid:107) g (cid:107) H , via applications of the Hardy inequality (A.14). The argument for the case of S , − ( g ) is strictlyanalogous; we have S , − ( g ) = z (1 − z )(1 + z ) (cid:90) z Hg ( w ) − Hg (0) w + 21 + w Hg (0) dw, and as before, the latter term can be estimated directly in L . For the first term, we may rewritethe integral as (cid:90) z Hg ( w ) − Hg (0) w dw = (cid:90) z (cid:18) − ∂ w (cid:18) w (cid:90) w Hg (cid:48) ( x ) dx (cid:19) + 1 w (cid:90) w Hg (cid:48)(cid:48) ( x ) dx (cid:19) dw and then proceed exactly as before.We consider the cases σ = 0 ,
1. When σ = 0, T , ( g ) = z (1 − z )(1 + z ) (cid:90) z g ( w ) w −
11 + w g (cid:48) (0) dw and note that the term involving g (cid:48) (0) is certainly bounded in L . For the other term, we simplyuse Hardy inequalities to obtain (cid:107) z (1 − z )(1 + z ) (cid:90) z g ( w ) w dw (cid:107) L ≤ (cid:107) z (1 − z )(1 + z ) (cid:107) L ∞ · (cid:107) z (cid:90) z (cid:18) w (cid:90) w g (cid:48) ( x ) dx (cid:19) dw (cid:107) L ≤ C (cid:107) z (cid:90) z g (cid:48) ( w ) dw (cid:107) L ≤ C (cid:107) g (cid:48) (cid:107) L . In the case σ = 1, T , ( g ) = z (1 − z )(1 + z ) (cid:90) z g ( w ) − w w g (cid:48) (0) dw, and again, the term involving g (cid:48) (0) can be separately estimated in L . For the other term, (cid:107) z (1 − z )(1 + z ) (cid:90) z g ( w ) dw (cid:107) L ≤ (cid:107) z (1 − z )(1 + z ) (cid:107) L ∞ · (cid:107) z (cid:90) z g ( w ) dw (cid:107) L ≤ C (cid:107) g (cid:107) L . For each case of σ = 0 ,
1, the arguments for the S ,σ is again analogous and result in the sameestimates. 22ext, we deal with (3.12). First, for T ,σ , we have T ,σ ( zg (cid:48) ) = z (1 − z )(1 + z ) (cid:90) z w σ (cid:18) g (cid:48) ( w ) − g (cid:48) (0) + w w g (cid:48) (0) (cid:19) dw. In the case σ = −
1, one can directly evaluate the integral in the second term, which resultsin the bound (cid:107) z (1 − z )2(1 + z ) ln(1 + z ) g (cid:48) (0) (cid:107) L ≤ C (cid:107) g (cid:107) H . Regarding the first term, we rewrite it to bound (cid:107) z (1 − z )(1 + z ) · z (cid:90) z (cid:18) w (cid:90) w g (cid:48)(cid:48) ( x ) dx (cid:19) dw (cid:107) L ≤ (cid:107) z (1 − z )(1 + z ) (cid:107) L ∞ (cid:107) z (cid:90) z (cid:18) w (cid:90) w g (cid:48)(cid:48) ( x ) dx (cid:19) dw (cid:107) L ≤ C (cid:107) g (cid:107) H via a repeated application of the Hardy inequality.We treat the cases σ = 0 , T ,σ ( zg (cid:48) ) as z (1 − z )(1 + z ) (cid:90) z w σ (cid:18) g (cid:48) ( w ) −
11 + w g (cid:48) (0) (cid:19) dw and note that the latter term can be directly estimated in L by (cid:107) g (cid:107) H in both cases. When σ = 0, after integrating by parts we are left with simply (cid:107) z (1 − z )(1 + z ) g ( z ) (cid:107) L ≤ C (cid:107) g (cid:107) L , and when σ = 1, we have (cid:107) z (1 − z )(1 + z ) (cid:18) zg ( z ) − (cid:90) z g ( w ) dw (cid:19) (cid:107) L ≤ (cid:107) z (1 − z )(1 + z ) · g ( z ) (cid:107) L + (cid:107) z (1 − z )(1 + z ) · z (cid:90) z g ( w ) dw (cid:107) L ≤ C (cid:107) g (cid:107) L . We now turn to S ,σ ( zg (cid:48) ). We need to estimate in L the following: S ,σ ( zg (cid:48) ) = z (1 − z )(1 + z ) (cid:90) z w σ Hg (cid:48) ( w ) dw since the Hilbert transform of zg (cid:48) is zH ( g ) (cid:48) , which vanishes at the origin. In the case σ = − Hg (cid:48) (0) = 0) S , − ( zg (cid:48) ) = z (1 − z )(1 + z ) · z (cid:90) z (cid:18) w (cid:90) w Hg (cid:48)(cid:48) ( x ) dx (cid:19) dw L by C (cid:107) g (cid:107) H by the Hardy inequality (A.14). When σ = 0, S , ( zg (cid:48) ) = z (1 − z )(1 + z ) · z (cid:90) z ( Hg ) (cid:48) ( w ) dw, and this time, (cid:107) S , ( zg (cid:48) ) (cid:107) L is bounded by (cid:107) g (cid:107) H . Finally, in the remaining case of σ = 1, onemay integrate by parts to obtain S , ( zg (cid:48) ) = z (1 − z )(1 + z ) (cid:18) − (cid:90) z Hg ( w ) dw + zHg ( z ) (cid:19) , and note that each term is bounded by a constant multiple of (cid:107) g (cid:107) L in L .Turning to (3.13), we first consider the case of T , − (Λ − f · g (cid:48) ). The derivative of Λ − f · g (cid:48) at the origin is Hf (0) · g (cid:48) (0). We need to bound z (1 − z )(1 + z ) (cid:18)(cid:90) z (cid:82) w Hf ( x ) dx · g (cid:48) ( w ) − wHf (0) g (cid:48) (0) w dw + Hf (0) g (cid:48) (0) · (cid:90) z w w dw (cid:19) in L , and it is straightforward to estimate the second term. Regarding the first term, it sufficesby an application of the Hardy inequality to estimate1 w (cid:18)(cid:90) w Hf ( x ) dx · g (cid:48) ( w ) − wHf (0) g (cid:48) (0) (cid:19) in L , and we first rewrite it as1 w (cid:18)(cid:90) w Hf ( x ) dx − wHf (0) (cid:19) · g (cid:48) ( w ) + 1 w Hf (0) · (cid:0) g (cid:48) ( w ) − g (cid:48) (0) (cid:1) . Then, it is clear that the latter is bounded in L by C (cid:107) f (cid:107) H · (cid:107) g (cid:107) H . Next, the former can befurther re-written as g (cid:48) ( w ) multiplied with − ∂ w (cid:18) w (cid:90) w Hf ( x ) dx (cid:19) + 1 w ( Hf ( w ) − Hf (0)) = − ∂ w (cid:18) w (cid:90) w Hf ( x ) dx (cid:19) + 1 w (cid:90) w Hf (cid:48) ( x ) dx, each of which is bounded by C (cid:107) f (cid:107) H . We omit the argument for the (simpler) cases of T ,σ (Λ − f · g (cid:48) ) with σ = 0 , S , − (Λ − f · g (cid:48) ), we need an L bound on z (1 − z )(1 + z ) (cid:90) z H (Λ − f · g (cid:48) )( w ) − H (Λ − f · g (cid:48) )(0) w + 2 w w H (Λ − f · g (cid:48) )(0) dw. As before, the latter term can be estimated directly by evaluating the integral; (cid:107) z (1 − z )(1 + z ) ln(1 + z ) H (Λ − f · g (cid:48) )(0) (cid:107) L ≤ C | H (Λ − f · g (cid:48) )(0) | ≤ C (cid:107) f (cid:107) L (cid:107) g (cid:107) H , where we have used (cid:12)(cid:12) H (Λ − ( f ) · g (cid:48) )(0) (cid:12)(cid:12) = 2 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ z Λ − ( f )( z ) · g (cid:48) ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:107) f (cid:107) L · (cid:107) g (cid:107) H . (3.14)24or the former, we first use the identity (A.3) to rewrite it as z (1 − z )(1 + z ) (cid:90) z H (Λ − f · g (cid:48) )( w ) − H (Λ − f · g (cid:48) )(0) w dw = z (1 − z )(1 + z ) · z (cid:90) z w H (cid:18) Λ − f ( x ) x · g (cid:48) ( x ) (cid:19) ( w ) dw. Then it suffices to estimate in L the following function:1 w H (cid:18) Λ − f ( x ) x · g (cid:48) ( x ) (cid:19) ( w ) = 1 w (cid:90) w H (cid:18) Λ − f ( x ) x · g (cid:48) ( x ) (cid:19) (cid:48) ( s ) ds, where we have used that the function Λ − f ( x ) x · g (cid:48) ( x )is even, so that its Hilbert transform is odd. At this point, we note that (cid:107) w (cid:90) w H (cid:18) Λ − f ( x ) x · g (cid:48) ( x ) (cid:19) (cid:48) ( s ) ds (cid:107) L ≤ C (cid:107) Λ − f ( x ) x · g (cid:48) ( x ) (cid:107) H ≤ C (cid:107) f (cid:107) H · (cid:107) g (cid:107) H . As in the previous cases, the arguments for S ,σ with σ = 0 , l = 2. However, note that the prefactor in thiscase equals 2 z (1 + z ) , which decays faster as | z | → ∞ and also has one more factor of z when | z | →
0. Therefore, thearguments from the previous case simply carries over.We now state and prove the necessary H -bounds. Lemma 3.3 ( H -bounds) . Let E and F be odd functions in H ( R ) . For each l ∈ { , } and σ ∈ {− , , } , we have the estimates (cid:107) S l,σ ((Λ − E ) · F (cid:48) ) (cid:107) H ( R ) + (cid:107) T l,σ ((Λ − E ) · F (cid:48) ) (cid:107) H ( R ) ≤ C (cid:107) E (cid:107) H ( R ) (cid:107) F (cid:107) H ( R ) , (3.15) (cid:107) S l,σ ( zF (cid:48) ) (cid:107) H ( R ) + (cid:107) T l,σ ( zF (cid:48) ) (cid:107) H ( R ) ≤ C (cid:107) F (cid:107) H ( R ) , (3.16) and (cid:107) S l,σ ( E · H ( F )) (cid:107) H ( R ) + (cid:107) T l,σ ( E · H ( F )) (cid:107) H ( R ) ≤ C (cid:107) E (cid:107) H ( R ) (cid:107) F (cid:107) H ( R ) . (3.17) Proof.
Let us give a brief outline of the proof. We first establish the inequality (3.15), and thenwe proceed to the proof of (3.16), which can be done in a similar way. After that we prove(3.17). We fix l = 1, since as in the previous Lemma, the l = 2 case is only simpler. (i) proof of (3.15) 25ake σ = −
1, and let us establish (3.15) starting with the term involving T , − . Noting that ∂ z (Λ − E · F (cid:48) )(0) = (Λ − E ) (cid:48) (0) F (cid:48) (0) = H ( E )(0) F (0), we have to bound (cid:107) ∂ z (cid:18) z (1 − z )(1 + z ) (cid:90) z Λ − E ( w ) · F (cid:48) ( w ) − wH ( E )(0) F (cid:48) (0) w + w w H ( E )(0) F (cid:48) (0) dw (cid:19) (cid:107) L . To begin with, the second term is clearly bounded in H by C (cid:107) E (cid:107) H · (cid:107) F (cid:107) H . Regarding thefirst term, from the previous L -bound and an interpolation, it suffices to treat the case whereall three derivatives fall on the integral term. Taking one derivative, we get1 − z (1 + z ) · Λ − E ( z ) · F (cid:48) ( z ) − zH ( E )(0) F (cid:48) (0) z = 1 − z (1 + z ) (cid:18) Λ − E ( z ) z F (cid:48) ( z ) − H ( E )(0) F (cid:48) (0) (cid:19) and then using the algebra property (cid:107) f g (cid:107) H ≤ C (cid:107) f (cid:107) H (cid:107) g (cid:107) H gives (cid:107) − z (1 + z ) · z (cid:90) z HE ( w ) dw · F (cid:48) ( z ) (cid:107) H ≤ C (cid:107) z (cid:90) z HE ( w ) dw (cid:107) H · (cid:107) F (cid:48) (cid:107) H ≤ C (cid:107) E (cid:107) H (cid:107) F (cid:107) H with the Hardy inequality (A.14).Next, for S , − , we need a bound on (cid:107) z (1 − z )(1 + z ) (cid:90) z H (Λ − E · F (cid:48) )( w ) − H (Λ − E · F (cid:48) )(0) w + 2 w w H (Λ − E · F (cid:48) )(0) dw (cid:107) H . The latter term can be estimated directly; integrating in z gives (cid:107) z (1 − z )(1 + z ) ln(1 + z ) H (Λ − E · F (cid:48) )(0) (cid:107) H ≤ C | H (Λ − E · F (cid:48) )(0) | ≤ C (cid:107) E (cid:107) H (cid:107) F (cid:107) H , where we have used (3.14). For the other term, we first use the identity (A.3) to first rewrite itas z (1 − z )(1 + z ) (cid:90) z H (Λ − E · F (cid:48) )( w ) − H (Λ − E · F (cid:48) )(0) w dw = z (1 − z )(1 + z ) (cid:90) z w H (cid:18) Λ − E ( x ) x · F (cid:48) ( x ) (cid:19) ( w ) dw. As before, we only need to consider the case when the derivatives fall on the integral term, andtaking one derivative, we obtain1 − z (1 + z ) H (cid:18) Λ − E ( w ) w · F (cid:48) ( w ) (cid:19) ( z ) , and since (cid:107) − z (1 + z ) H (cid:18) Λ − E ( w ) w · F (cid:48) ( w ) (cid:19) ( z ) (cid:107) H ≤ C (cid:107) Λ − E ( z ) z · F (cid:48) ( z ) (cid:107) H ,
26e conclude the desired bound as in the above case of T , − . This concludes the estimate (3.15)in the case σ = − σ = 0. We need to bound z (1 − z )(1 + z ) (cid:90) z Λ − E ( w ) w · F (cid:48) ( w ) −
11 + w H ( E )(0) F (cid:48) (0) dw in H , and again we only worry about the case when derivatives fall on the integral. Taking onederivative, (1 − z )(1 + z ) (cid:18) Λ − E ( z ) · F (cid:48) ( z ) − z z H ( E )(0) F (cid:48) (0) (cid:19) and it is direct to see that (cid:107) (1 − z )(1 + z ) · z z H ( E )(0) F (cid:48) (0) (cid:107) H ≤ C (cid:107) E (cid:107) H · (cid:107) F (cid:107) H . Regarding the other term, we rewrite it as z (1 − z )(1 + z ) · F (cid:48) ( z ) · z (cid:90) z H ( E )( w ) dw, and then we have (cid:107) z (1 − z )(1 + z ) · F (cid:48) ( z ) · z (cid:90) z H ( E )( w ) dw (cid:107) H ≤ C (cid:107) F (cid:107) H · (cid:107) E (cid:107) H , using the Hardy inequality. Now the remaining case σ = 1 can be handled similarly; this timewe have z (1 − z )(1 + z ) (cid:90) z Λ − E ( w ) · F (cid:48) ( w ) − w w H ( E )(0) F (cid:48) (0) dw, and note that the latter term is bounded in H . The other term, when a derivative falls on theintegral, becomes z (1 − z )(1 + z ) Λ − E ( z ) z · F (cid:48) ( z ) = z (1 − z )(1 + z ) z (cid:90) z H ( E )( w ) dw · F (cid:48) ( z )and again using the Hardy inequality, (cid:107) z (1 − z )(1 + z ) z (cid:90) z H ( E )( w ) dw · F (cid:48) ( z ) (cid:107) H ≤ C (cid:107) E (cid:107) H · (cid:107) F (cid:107) H . Lastly we deal with S , and S , . In these cases, for simplicity set g = Λ − E · F (cid:48) and weneed to bound z (1 − z )(1 + z ) (cid:90) z w σ (cid:18) Hg ( w ) w + (cid:18) w w − w (cid:19) Hg (0) (cid:19) dw σ = 0 , − z (1 + z ) (cid:18) z σ Hg ( z ) + z σ (cid:18) z z − (cid:19) Hg (0) (cid:19) , and it is straightforward to see that when σ = 0 ,
1, both terms are bounded in H by C (cid:107) F (cid:107) H ·(cid:107) E (cid:107) H . This establishes (3.15). (ii) proof of (3.16)In this case, H ( zF (cid:48) ) = zH ( F ) (cid:48) and hence the corresponding bound for S l,σ follows similarlyfrom the bound for T l,σ . We first take σ = −
1, and proceed to show that (cid:107) ∂ z (cid:18) z (1 − z )(1 + z ) (cid:90) z wF (cid:48) ( w ) − wF (cid:48) (0) w + w w F (cid:48) (0) (cid:19) (cid:107) L ≤ C (cid:107) F (cid:107) H holds. Taking one derivative on the integral term gives1 − z (1 + z ) · (cid:18) F (cid:48) ( z ) + (cid:18) z z − (cid:19) F (cid:48) (0) (cid:19) , and it is straightforward to see that both terms are bounded by C (cid:107) F (cid:107) H .On the other hand, when σ = 1, we need a bound on (cid:107) ∂ z (cid:18) z (1 − z )(1 + z ) (cid:18)(cid:90) z wF (cid:48) ( w ) − wF (cid:48) (0) + w w F (cid:48) (0) dw (cid:19)(cid:19) (cid:107) H , and taking one derivative on the integral term gives z (1 − z )(1 + z ) (cid:18) zF (cid:48) ( z ) + (cid:18) z z − z (cid:19) F (cid:48) (0) (cid:19) = z (1 − z )(1 + z ) (cid:18) F (cid:48) ( z ) −
11 + z F (cid:48) (0) (cid:19) , so that (cid:107) z (1 − z )(1 + z ) (cid:18) F (cid:48) ( z ) −
11 + z F (cid:48) (0) (cid:19) (cid:107) H ≤ C (cid:107) F (cid:48) ( z ) −
11 + z F (cid:48) (0) (cid:107) H ≤ C (cid:107) F (cid:107) H . We omit the proof for the intermediate case of σ = 0, which is simpler. (iii) proof of (3.17)Noting that the function G := H ( E ) · F is again odd, it suffices to show that (cid:107) T l,σ ( G ) (cid:107) H + (cid:107) S l,σ ( G ) (cid:107) H ≤ C (cid:107) G (cid:107) H .Taking the derivative of the integral in T , − ( G ) = z (1 − z )(1 + z ) (cid:90) z w (cid:18) G ( w ) w −
11 + w G (cid:48) (0) (cid:19) dw, we obtain 1 − z (1 + z ) (cid:18) z (cid:90) z G (cid:48) ( w ) dw −
11 + z G (cid:48) (0) (cid:19) , H by C (cid:107) G (cid:107) H .Similarly, differentiating the integral term in S , − ( G ) = z (1 − z )(1 + z ) (cid:90) z w (cid:18) H ( G )( w ) − H ( G )(0) w + 2 w w H ( G )(0) (cid:19) dw, we get 1 − z (1 + z ) (cid:18) z (cid:90) z H ( G ) (cid:48) ( w ) dw + 2 z z H ( G )(0) (cid:19) , which is again bounded in H by C (cid:107) G (cid:107) H .As before, the cases σ = 0 , | a | small, which completes the proof of Theorem 1. Proof of Theorem 1.
It suffices to show that there exists an absolute constant r >
0, such thatfor all n ≥
1, we have (cid:107) F n (cid:107) H ( R ) ≤ r n , | λ n | ≤ r n . Then we may pick a = 1 /r , and take a smaller if necessary to guarantee that λ ( a ) > − | a | < a . For simplicity we set (cid:107) F n (cid:107) H =: µ n and let us translate the estimates from Lemma 3.2and Lemma 3.3 in terms of sequences { µ n } , { λ n } .Let us begin by recalling that L ( F n ) = G n − λ n z (1 − z )(1 + z ) , with G n ( z ) = n − (cid:88) j =0 Λ − ( F j )( z ) · F (cid:48) n − − j ( z ) − n − (cid:88) j =1 zλ j F (cid:48) n − j ( z ) − n − (cid:88) j =1 H ( F n − j )( z ) F j ( z ) , and λ n = G (cid:48) n (0) + 2( HG n )(0) . Noting that G (cid:48) n (0) = n − (cid:88) j =0 HF j (0) F (cid:48) n − − j (0) − n − (cid:88) j =1 λ j F (cid:48) n − j (0) − n − (cid:88) j =1 F (cid:48) j (0) HF n − j (0) HG n (0) = n − (cid:88) j =0 H (Λ − ( F j ) F (cid:48) n − − j )(0) − n − (cid:88) j =1 ( − F j (0) F n − j (0) + HF j (0) HF n − j (0))29nd simply using crude bounds (cid:107) H ( F j ) (cid:107) L ∞ , (cid:107) F (cid:48) j (cid:107) L ∞ ≤ C (cid:107) F j (cid:107) H as well as (3.14) we deducethat | λ n | ≤ Cµ µ n − + C n − (cid:88) j =1 ( µ j µ n − − j + | λ j | µ n − j + µ j µ n − j ) . (3.18)Given λ n , we first write L − ( G n − λ n z (1 − z )(1+ z ) ) as a linear combination (with constant coeffi-cients) of twelve operators T l,σ , S l (cid:48) ,σ (cid:48) with l, l (cid:48) ∈ { , } , σ, σ (cid:48) ∈ {− , , } , and applying Lemmas3.2, 3.3 to each term of G n , we deduce that µ n ≤ C | λ n | + Cµ µ n − + C n − (cid:88) j =1 ( µ j µ n − − j + | λ j | µ n − j + µ j µ n − j ) . (3.19)Therefore, combining (3.18) and (3.19), there exists some absolute constant C > ζ n := µ n + C ( | λ n | + µ n − ) for n ≥ ζ n ≤ C n − (cid:88) j =1 ζ j ζ n − j (3.20)for all n ≥
1. Then, denoting ¯ ζ n to be the solution of the quadratic recursion¯ ζ n := n − (cid:88) j =1 ¯ ζ j ¯ ζ n − j , n ≥ ζ := Cζ (here C is the absolute constant from (3.20)), we deduce the desired statement,simply because the sequence ¯ ζ n is precisely the sequence of coefficients of the Taylor expansionof f satisfying f ( z ) − f ( z ) = ¯ ζ z around z = 0, so that in particular we have, for some r > | ζ n | ≤ C n | ¯ ζ n | ≤ r n where C is the same constant from (3.20). This finishes the proof. Remark 3.4.
As we have mentioned in the introduction, a similar argument goes through for H s with any value of s ≥
3, which in particular concludes that the functions F , F , · · · , as wellas the profile F are C ∞ ( R )-smooth. It is likely that these functions are indeed real analytic,and one way to establish such a result would be to carefully carry out the H s -estimates, keepingtrack of the dependence of multiplicative constants in terms of s . Proof of Theorem 2.
We simply use the equation (3.1) to obtain the desired decay statementfor the self-similar profile F . Multiplying both sides of (3.1) by z λ − and then integrating,we get: z λ F ( z ) = 11 + λ (cid:90) z s λ − (cid:0) F ( s ) · H ( F )( s ) − a Λ − ( F )( s ) · F (cid:48) ( s ) (cid:1) ds. a is small, let’s assume λ > −
1. Then the function s λ − is locally integrable. Weconsider two cases: 1 / (1 + λ ) − ≥ <
0. In the latter case, the right hand side is simplybounded in absolute value in the limit z → + ∞ , which gives the desired decay of z − λ . To seethis, just note that the integral in the region 0 ≤ s ≤ (cid:107) F · H ( F ) − a Λ − ( F ) · F (cid:48) (cid:107) L ∞ , and for s > L bound on the first term (cid:107) F H ( F ) (cid:107) L ≤ (cid:107) F (cid:107) L · (cid:107) H ( F ) (cid:107) L . For the transport term we integrate by parts to get az λ − Λ − ( F )( z ) · F ( z ) − a (cid:90) z s λ − H ( F ) F + ( 11 + λ − s λ − · s Λ − ( F ) · F ds, and since | Λ − ( F )( z ) | ≤ z due to the L estimate on H ( F ), the boundary term can be subsumedinto the term on the left hand side for z large, which is z λ F . This takes care of the boundaryterm. Now, the two integral terms are handled just as before noting that s − Λ − F ∈ L usingHardy’s inequality.In the other case of 1 / (1 + λ ) ≥
1, running the above argument gives instead the decay | F ( z ) | (cid:46) z − , and in particular F and H ( F ) belongs to L p for all p >
1. Once one has this, were-insert this information above to get better decay on F until we see that F actually decayslike z − λ . In this section, our goal consists of establishing Theorem 3, which states the self-similar blowup of merely C α solutions to (1.1), over a range of a scaling as 1 /α for α → + . This is donefirst by constructing a family of self-similar solutions to the CLM equation which are merely C α ; in fact, they are smooth functions of | z | α . Then the proof will be similar to the smoothcase except that we will need to take great care in that we are dealing with smooth functionsof | z | α rather than simply smooth functions. This leads us to define the operators ˜ H ( n ) whichdescribe how the Hilbert transform acts on L functions of | z | α . We effectively prove that L functions of | z | α are mapped to L functions of | z | α with an L operator norm on the order of Cα , though we only do this for α = n with n ∈ N , as all we actually need is a sequence of C α solutions with α →
0. Other than the extra technical machinery needed to deal with this caseas well as the careful checking of the dependence of constants on α as α →
0, the main idea issimilar that in the preceding section.
As in the case of smooth data, the starting point is to find self-similar solutions to the Constantin-Lax-Majda equation which is only H¨older continuous. More precisely, for each 0 < α < C α -profile F ( α )0 such that ω ( t, x ) = 11 − t F ( α )0 (cid:32) x (1 − t ) α (cid:33) provides a solution to the Constantin-Lax-Majda equation. This reduces to the following differ-ential equation for F ( α )0 : F ( α )0 + 1 α zF ( α ) (cid:48) + 2 F ( α )0 · H ( F ( α )0 ) = 0 , (4.1)and it can be checked explicitly that the following pair of functions F ( α )0 ( z ) = sin (cid:0) απ (cid:1) sgn( z ) | z | α (cid:0) απ (cid:1) | z | α + | z | α (4.2)and H ( F ( α )0 )( z ) = − (cid:0) απ (cid:1) | z | α (cid:0) απ (cid:1) | z | α + | z | α . (4.3)provides a solution to (4.1) (see Figure 2). In the case α = 1, we obtain our familiar pairof functions ( F , H ( F )). To argue that (4.3) is indeed the Hilbert transform of (4.2), we usecomplex notation V ( z ) := F ( α )0 ( z ) + iH ( F ( α )0 )( z )and then taking the Hilbert transform of (4.1), one obtains the ODE system V + 1 α zV (cid:48) − iV = 0 . Then, integrating the system, one obtains that V is necessarily in the following form with some(complex) constant of integration C = C ( α ) : V ( z ) = 1 i + Cz α , where we define z α := r α e iαθ for z = re iθ (0 ≤ θ ≤ π ) as a holomorphic function on the upperhalf-plane. It is easy to check that C ( α ) = sin (cid:16) απ (cid:17) + i cos (cid:16) απ (cid:17) is the unique number (up to a scaling in R + ) which makes V ( z ) holomorphic in the upperhalf-plane with an odd real part when restricted to the real axis. With this value of C ( α ) , onehas F ( α )0 ( z ) = (cid:60) ( V ) ( z ) , H ( F ( α )0 )( z ) = (cid:61) ( V ) ( z )which in particular concludes that (4.3) is the Hilbert transform of (4.2). We have proved thefollowing proposition: Proposition 4.1.
For each < α ≤ , F ( α ) is the only odd data which is a smooth function of z α on R + and extends as holomorphically to the upper half-plane, which defines a self-similarsolution to the Constantin-Lax-Majda equation (1.3) . - - - - - - Figure 2: The functions F (1 / (odd) and H ( F (1 / ) (even) plotted on [ − , a and the linear operator To seek C α self-similar solutions to the De Gregorio equation, we now take the ansatz ω ( t, x ) = 11 − t F (cid:32) x (1 − t ) λ ( a ) α (cid:33) , λ (0) = 0 , as in the smooth case. In terms of F , we obtain F ( z ) + (cid:18) λ ( a ) α z + a · u F ( z ) (cid:19) F (cid:48) ( z ) + 2 F ( z ) HF ( z ) = 0 . (4.4)We expand F and λ again as F ( z ) = F ( α )0 ( z ) + ∞ (cid:88) n =1 a n F n ( z ) , λ ( a ) = ∞ (cid:88) n =1 a n λ n (4.5)Each term in the expansions depend on α but we suppress from writing out the dependence.Inserting the expansion (4.5) into (4.4), for each 0 < α <
1, we obtain with F = F ( α )0 F n + 1 α zF (cid:48) n + 2 H ( F ) F n + 2 F H ( F n ) = n − (cid:88) j =0 Λ − ( F j ) F (cid:48) n − − j − α n (cid:88) j =1 λ j zF (cid:48) n − j − n − (cid:88) j =1 F j H ( F n − j )(4.6)for each n ≥
1. Just as in the case of smooth solutions, we write (4.6) in the form L ( F n ) = G n − α λ n zF (cid:48) , G n = G n ( F , · · · , F n − ; λ , · · · , λ n − )33ith the linear operator L = L ( α ) defined by Lf = f + 1 α zf (cid:48) + 2 H ( F ) f + 2 F H ( f ) . For each n ≥
1, there is a unique λ n which makes the linear system solvable, and this in turndefines F n .We shall take advantage of the fact that the functions F , H ( F ), and then Λ − ( F ) F (cid:48) aresmooth (actually, analytic) functions of w := z α on the positive real axis R + . First, we writedown an explicit formula for the inverse L − in terms of the new variable w . Then, we willsimply estimate the functions F , F , and so on in H ( R + ) with respect to w . From now on, letus use tildes to denote functions of w : given a function f on R + , we set ˜ f ( w ) := f ( w /α ).In particular, we write˜ F ( w ) = sin (cid:0) απ (cid:1) w (cid:0) απ (cid:1) w + w , ˜ H ( F )( w ) = − (cid:0) απ (cid:1) w (cid:0) απ (cid:1) w + w and the linear system may be written as˜ L ˜ f := ˜ f + w ˜ f (cid:48) + 2 ˜ H ( F ) · ˜ f + 2 ˜ F · ˜ H ( f ) = ˜ g. Taking the Hilbert transform of both sides, and using complex notation V ( w ) = ˜ F ( w ) + i ˜ H ( F )( w ) = 1 i + C ( α ) w , and U ( w ) = ˜ f ( w ) + i ˜ H ( f )( w ) , G ( w ) = ˜ g ( w ) + i ˜ H ( g )( w ) , the linear system takes the form U + wU (cid:48) − iV U = G. (4.7)We view the variable w as varying on the strip S ( α ) := { w ∈ C : 0 ≤ arg( w ) ≤ πα } , as the right hand side G as well as V are holomorphic on S ( α ) with continuous extension up tothe boundary. In this section, for simplicity we use the word “holomorphic” to describe suchfunctions. Lemma 4.2.
Consider the differential equation (4.7) on the sector S ( α ) , where G = G ( w ) isholomorphic. Then, we have the following statements: • There exists a holomorphic solution U if and only if G satisfies G (cid:48) (0) − i (cid:16) sin (cid:16) απ (cid:17) + i cos (cid:16) απ (cid:17)(cid:17) G (0) = 0 . (4.8) Any such solution satisfies U (0) = − G (0) . There is a unique such U once we require in addition that U (cid:48) (0) = 0 . • This unique solution is given explicitly by the integral formula U ( w ) − U (0) = w (1 + 2 w cos (cid:0) απ (cid:1) + w ) · (cid:16) − (1 + w ) cos (cid:16) απ (cid:17) − w + i (1 − w ) sin (cid:16) απ (cid:17)(cid:17) × (cid:32)(cid:90) w (cid:18) − s s cos (cid:16) απ (cid:17) − − i − s s sin (cid:16) απ (cid:17)(cid:19) ˆ G ( s ) s ds (cid:33) , (4.9) where ˆ G ( w ) := G ( w ) − C ( α ) iw − C ( α ) iw G (0) , C ( α ) = sin (cid:16) απ (cid:17) + i cos (cid:16) απ (cid:17) . Proof.
The proof is strictly analogous to the corresponding lemma from the smooth case.To begin with, the consistency condition (4.8) simply implies that the holomorphic functionˆ G ( s ) /s vanishes at s = 0, therefore canceling the singularity 1 /s inside the integral in (4.9). Itis then clear that the formula (4.9) defines a holomorphic function of w , and a straightforwardcomputation shows that it provides a solution to the equation (4.7).This explicit formula is easy to derive: first re-write (4.7) as(1 − iV ) U + wU (cid:48) = G, and subtracting the value at 0, we obtain − w · C ( α ) iw − C ( α ) iw ( U − U (0)) + ( U − U (0)) (cid:48) = 1 w (cid:32) G ( w ) − C ( α ) iw − C ( α ) iw G (0) (cid:33) =: ˆ G ( w ) w . The integrating factor can be computed explicitly:exp (cid:32)(cid:90) w s · C ( α ) is − C ( α ) is ds (cid:33) = w (1 + 2 w cos (cid:0) απ (cid:1) + w ) · (cid:16) − (1 + w ) cos (cid:16) απ (cid:17) − w + i (1 − w ) sin (cid:16) απ (cid:17)(cid:17) , with the inverseexp (cid:32) − (cid:90) w s · C ( α ) is − C ( α ) is ds (cid:33) = − w w · cos (cid:16) απ (cid:17) − − i − w w · sin (cid:16) απ (cid:17) . This establishes (4.9), together with the fact that there is a one (complex) dimensional kernelspanned by the function w (1 + 2 w cos (cid:0) απ (cid:1) + w ) · (cid:16) − (1 + w ) cos (cid:16) απ (cid:17) − w + i (1 − w ) sin (cid:16) απ (cid:17)(cid:17) . w = 0 directly in (4.7) to deduce uniqueness.Lastly, the statement G (0) = − U (0) simply follows from evaluating (4.7) at w = 0. Next,subtracting the values at 0 from both sides of (4.7), dividing both sides by 1 /w , and then takingthe limit w →
0, one sees that the condition (4.8) is necessary to have a smooth solution.This finishes the proof.
Remark 4.3.
Assuming further that (cid:60) ( G (0)) = 0 and the function G ( z α ) is odd in z whenrestricted onto the real axis, we have seen in Lemma 2.2 that the condition (4.8) reduces to justhaving (cid:60) ( G (cid:48) (0)) + 2 sin (cid:16) απ (cid:17) (cid:61) ( G (0)) = 0 . (4.10)Note that in the case α = 1, this simply reduces back to (cid:60) ( G (cid:48) (0)) + 2 (cid:61) ( G (0)) = 0. In this subsection, we shall restrict the variable w onto the positive real axis R + and obtainprecise norm estimates for the real and imaginary parts of the solution of the linear operator(4.7) given in (4.9).It will be convenient to define for each 0 < α ≤ H and ˜Λ − forfunctions defined on R + in a way that˜ H ( ˜ f )( w ) := ( Hf )( w /α ) , ( ˜Λ − ˜ f )( w ) := (Λ − f )( w /α ) . As an example, we have˜ H (cid:32) sin (cid:0) απ (cid:1) w (cid:0) απ (cid:1) w + w (cid:33) = − (cid:0) απ (cid:1) w (cid:0) απ (cid:1) w + w . Let us write out an explicit integral representation for the operator ˜ H . From the definition,˜ H ( ˜ f )( w ) = 1 π p.v. (cid:90) R + tf ( t ) w /α − t dt = 1 π p.v. (cid:90) R + t ˜ f ( t α ) w /α − t dt = 1 π p.v. (cid:90) R + s /α − w /α − s /α ˜ f ( s ) ds. In the case α = 1 /n for some integer n ≥
1, notice that the kernel is a rational function of w and s . From this it is easy to obtain an L estimate for the operator ˜ H = ˜ H ( n ) , see Lemma A.5in the Appendix. From now on, we restrict ourselves to such values of α . Next, note that theoperator ˜Λ − has the representation( ˜Λ − ˜ f )( w ) := Λ − f ( w /α ) = (cid:90) w /α Hf ( s ) ds = (cid:90) w ˜ H ˜ f ( t ) · α t − αα dt. H estimates for the inverse ˜ L − , let us first write out the realpart of U ( w ) from (4.9). Denoting the real part by ˜ L − (˜ g ), we have:˜ L − (˜ g ) = ˜ L − ,I (˜ g ) + ˜ L − ,II (˜ g )with˜ L − ,I (˜ g ) := w (cid:0) − (1 + w ) cos (cid:0) απ (cid:1) − w (cid:1)(cid:0) (cid:0) απ (cid:1) w + w (cid:1) × (cid:34)(cid:90) w (cid:18) − s s cos (cid:16) απ (cid:17) − (cid:19) · s (cid:32) ˜ g ( s ) − s (cid:0) απ (cid:1) s + s ˜ g (cid:48) (0) (cid:33) ds + (cid:90) w (cid:18) − s s sin (cid:16) απ (cid:17)(cid:19) · s (cid:32) ˜ H ˜ g ( s ) − − s (cid:0) απ (cid:1) s + s ˜ H ˜ g (0) (cid:33) ds (cid:35) , and ˜ L − ,II (˜ g ) := w (1 − w ) sin (cid:0) απ (cid:1)(cid:0) (cid:0) απ (cid:1) w + w (cid:1) × (cid:34)(cid:90) w (cid:18) s s cos (cid:16) απ (cid:17) + 2 (cid:19) · s (cid:32) ˜ H ˜ g ( s ) − − s (cid:0) απ (cid:1) s + s ˜ H ˜ g (0) (cid:33) ds + (cid:90) w (cid:18) − s s sin (cid:16) απ (cid:17)(cid:19) · s (cid:32) ˜ g ( s ) − s (cid:0) απ (cid:1) s + s ˜ g (cid:48) (0) (cid:33) ds (cid:35) , where we have used the consistency condition (4.10).We now split the above inverse operator into several pieces: we define for σ ∈ {− , , } T σ (˜ g )( w ) := P ( w ) · (cid:90) w s σ (cid:32) ˜ g ( s ) s −
11 + 2 cos (cid:0) απ (cid:1) s + s ˜ g (cid:48) (0) (cid:33) ds (4.11)and S σ (˜ g )( w ) := P ( w ) · (cid:90) w s σ (cid:32) ˜ H ˜ g ( s ) s − − s s (1 + 2 cos (cid:0) απ (cid:1) s + s ) ˜ H ˜ g (0) (cid:33) ds = P ( w ) · (cid:90) w s σ (cid:32) ˜ H ˜ g ( s ) − ˜ H ˜ g (0) s + 2 cos (cid:0) απ (cid:1) + 2 s (cid:0) απ (cid:1) s + s ˜ H ˜ g (0) (cid:33) ds (4.12)where the prefactor P ( w ) equals P ( w ) = w · (cid:0) − (1 + w ) cos (cid:0) απ (cid:1) − w (cid:1)(cid:0) (cid:0) απ (cid:1) w + w (cid:1) . Lemma 4.4.
Assume that ˜ E and ˜ F vanish at zero. Then we have the H estimates of the form (cid:107) T σ ( ˜ F ) (cid:107) H + α · (cid:107) S σ ( ˜ F ) (cid:107) H ≤ c (cid:107) ˜ F (cid:107) H , (4.13)37 T σ ( w ˜ F (cid:48) ) (cid:107) H + α · (cid:107) S σ ( w ˜ F (cid:48) ) (cid:107) H ≤ c (cid:107) ˜ F (cid:107) H , (4.14) and (cid:107) T σ (cid:16) (Λ − E · F (cid:48) ) (cid:101) (cid:17) (cid:107) H + α · (cid:107) S σ (cid:16) (Λ − E · F (cid:48) ) (cid:101) (cid:17) (cid:107) H ≤ c (cid:107) ˜ E (cid:107) H · (cid:107) ˜ F (cid:107) H , (4.15) for each σ ∈ {− , , } , where c > is some absolute constant independent on < α ≤ .Proof. The proof is completely analogous to those of the estimates in Lemmas 3.2 and 3.3. (i) proof of (4.13)We begin with the proof of the inequality (4.13). Consider first the function T σ ( ˜ F ) in themost difficult case σ = −
1: after a simple rewriting, we have that T − ( ˜ F )( w ) = P ( w ) · (cid:90) w s (cid:32) ˜ F ( s ) s − ˜ F (cid:48) (0) (cid:33) + 2 cos (cid:0) απ (cid:1) + s (cid:0) απ (cid:1) s + s ˜ F (cid:48) (0) ds. We first deal with the second term: P ( w ) ˜ F (cid:48) (0) · (cid:90) w (cid:0) απ (cid:1) + s (cid:0) απ (cid:1) s + s ds. In L , this is bounded by C (cid:107) ˜ F (cid:107) H · (cid:107) P ( w ) · (cid:90) w (cid:0) απ (cid:1) + s (cid:0) απ (cid:1) s + s ds (cid:107) L ∞ ≤ C (cid:107) ˜ F (cid:107) H · (cid:107) P ( w ) ln(1 + w ) (cid:107) L ∞ ≤ C (cid:107) ˜ F (cid:107) H , and it is straightforward to show that this is bounded in H again by a constant multiple of (cid:107) ˜ F (cid:107) H . Next, the first term of T − ( ˜ F ) is bounded in L by (cid:107) wP ( w ) (cid:107) L ∞ · (cid:107) w (cid:90) w s (cid:32) ˜ F ( s ) s − ˜ F (cid:48) (0) (cid:33) ds (cid:107) L ≤ C (cid:107) ˜ F (cid:107) H just as in the smooth case, using the Hardy inequality. To obtain the H bound, we may assumethat a derivative falls on the integral to get P ( w ) w · (cid:32) ˜ F ( w ) w − ˜ F (cid:48) (0) (cid:33) , which is bounded in H by a constant multiple of (cid:107) ˜ F (cid:107) H again using the Hardy inequality.The cases σ = 0 , T σ ( ˜ F ) into (cid:18) P ( w ) (cid:90) w s σ − ˜ F ( s ) ds (cid:19) − (cid:32) P ( w ) ˜ F (cid:48) (0) · (cid:90) w s σ (cid:0) απ (cid:1) s + s ds (cid:33) , and then clearly both terms can be bonded in H by C (cid:107) ˜ F (cid:107) H .38he corresponding estimate for the function S σ ( ˜ F ) can be carried out in a similar way. Letus only point out that we lose by a factor of 1 /α simply because of the loss in H -estimate fromLemma A.5: (cid:107) ˜ H ˜ F (cid:107) H ≤ Cα (cid:107) ˜ F (cid:107) H . (ii) proof of (4.14)We proceed to the proof of (4.14), starting with the case of T − ( w ˜ F (cid:48) ). We need to estimate: P ( w ) (cid:90) w s (cid:32) ˜ F (cid:48) ( s ) −
11 + 2 cos (cid:0) απ (cid:1) s + s ˜ F (cid:48) (0) (cid:33) ds = P ( w ) · (cid:32)(cid:90) w ˜ F (cid:48) ( s ) − ˜ F (cid:48) (0) s ds + ˜ F (cid:48) (0) · (cid:90) w (cid:0) απ (cid:1) + s (cid:0) απ (cid:1) s + s ds (cid:33) , but the corresponding proof from the smooth case carries over to this one, which bounds theabove in H by a constant multiple of ˜ F in H . The cases σ = 0 , T σ ( w ˜ F (cid:48) ), the corresponding result for S σ ( w ˜ F (cid:48) ) similarly follows,using the convenient fact that ˜ H (cid:16) w ˜ F (cid:48) (cid:17) = w ( ˜ H ( ˜ F )) (cid:48) , see (A.11) in the Appendix. (iii) proof of (4.15)To begin with, it is necessary to note that(Λ − E · F (cid:48) ) (cid:101) ( w ) = ( 1 z (Λ − E )( z ) · zF (cid:48) ( z )) (cid:101) ( w ) = w − α ˜Λ − ( ˜ E )( w ) · αw ˜ F (cid:48) ( w )and that its derivative at w = 0 equals α · ˜ H ( ˜ E )(0) · ˜ F (cid:48) (0) . Let us restrict ourselves to obtaining an H estimate for T − ((Λ − E · F (cid:48) ) (cid:101) ). We need tobound T − ( ˜Λ − ( ˜ E ) · ˜ F (cid:48) )( w ) = P ( w ) · α (cid:90) w s (cid:16) s − α ˜Λ − ( ˜ E )( s ) · ˜ F (cid:48) ( s ) − ˜ H ( ˜ E )(0) · ˜ F (cid:48) (0) (cid:17) ds + P ( w ) · α ˜ H ( ˜ E )(0) · ˜ F (cid:48) (0) · (cid:90) w (cid:0) απ (cid:1) + s (cid:0) απ (cid:1) s + s ds. To estimate the latter term, it suffices to observe that˜ H (cid:0) Λ − ( E ) · F (cid:48) (cid:1) ˜ (0) = H (cid:0) Λ − ( E ) · F (cid:48) (cid:1) (0)= − π (cid:90) ∞ z (cid:18)(cid:90) z H ( E )( s ) ds (cid:19) F (cid:48) ( z ) dz = − π (cid:90) ∞ (cid:18) w − α (cid:90) w ˜ H ( ˜ E )( t ) · α t − αα dt (cid:19) · ˜ F (cid:48) ( w ) dw α · (cid:12)(cid:12)(cid:12) ˜ H (cid:0) Λ − ( E ) · F (cid:48) (cid:1) ˜ (0) (cid:12)(cid:12)(cid:12) ≤ Cα (cid:107) ˜ H ( ˜ E ) (cid:107) L · (cid:107) ˜ F (cid:107) H ≤ C (cid:107) ˜ E (cid:107) H · (cid:107) ˜ F (cid:107) H (4.16)using the Hardy-type inequality (A.15). The former term can be estimated in H by followingalong the lines of the corresponding proof from Lemmas 3.2 and 3.3, using the Hardy-typeinequality (A.15) instead of the usual Hardy inequality.Regarding S − ((Λ − E · F (cid:48) ) (cid:101) ), one again just needs to follow along the arguments given inthe smooth case. The only difference is that one needs first to decompose the operator ˜ H into (cid:80) n − j = − n +1 ˜ H j where n = 1 /α (see (A.5),(A.6) in the Appendix for the definition of the pieces˜ H j ), and then use the identity (A.8) which is a convenient substitute for (A.3).Given the above estimates, let us conclude the proof of Theorem 3. Proof of Theorem 3.
Let us first proceed to show that, with some absolute constants c > c > < α ≤
1, we have the bounds (cid:107) ˜ F n (cid:107) H ≤ cα ( c α ) n , | λ n | ≤ c ( c α ) n . for all n ≥
0. Note that in the case n = 0, these estimates trivially hold.We set (cid:107) ˜ F n (cid:107) H =: α n +1 µ n , | λ n | =: α n l n and let us write down the set of inequalities for thesequences { µ n } and { l n } .To begin with, we recall that in terms of the original variable z , L ( F n ) = G n − α λ n zF (cid:48) , with G n ( z ) = n − (cid:88) j =0 Λ − ( F j ) · F (cid:48) n − − j − α n − (cid:88) j =1 λ j zF (cid:48) n − j − n − (cid:88) j =1 H ( F n − j ) · F j and H ( G n )( z ) = n − (cid:88) j =0 H (cid:0) Λ − ( F j ) · F (cid:48) n − − j (cid:1) − α n − (cid:88) j =1 λ j zH ( F n − j ) (cid:48) + n − (cid:88) j =1 ( F n − j · F j − H ( F n − j ) · H ( F j )) . Also recall that λ n is defined by λ n = 1sin (cid:0) απ (cid:1) ˜ G (cid:48) n (0) + 2 ˜ H ( ˜ G n )(0) . A straightforward computation shows that ˜ G (cid:48) n (0) equals n − (cid:88) j =0 ˜ H ( ˜ F j )(0) · α ˜ F (cid:48) n − − j (0) − n − (cid:88) j =1 λ j ˜ F (cid:48) n − j (0) − n − (cid:88) j =1 ˜ H ( ˜ F n − j )(0) · ˜ F (cid:48) n − j (0) . (cid:12)(cid:12)(cid:12) ˜ H (cid:0) Λ − ( F j ) · F (cid:48) n − − j (cid:1) ˜ (0) (cid:12)(cid:12)(cid:12) ≤ C (cid:107) ˜ H ( ˜ F j ) (cid:107) L · (cid:107) ˜ F n − − j (cid:107) H from (4.16). Since ˜ H ( ˜ G n )(0) equals n − (cid:88) j =0 ˜ H (cid:0) Λ − ( F j ) · F (cid:48) n − − j (cid:1) ˜ (0) − n − (cid:88) j =1 ˜ H ( ˜ F n − j )(0) · ˜ H ( ˜ F j )(0) , we finally deduce that | λ n | ≤ Cα n − (cid:88) j =0 µ j α j · α · µ n − − j α n − j + n − (cid:88) j =1 l j α j · µ n − j α n − j +1 + n − (cid:88) j =1 µ n − j α n − j · µ j α j +1 + C n − (cid:88) j =0 µ j α j · µ n − − j α n − j + n − (cid:88) j =1 µ n − j α n − j · µ j α j , using the H ( R ) ⊂ L ∞ ( R ) embedding together with the estimate (cid:107) ˜ H ( ˜ F j ) (cid:107) H ≤ Cα (cid:107) ˜ F j (cid:107) H . Equivalently, l n ≤ C µ µ n − + n − (cid:88) j =1 ( µ j µ n − − j + l j µ n − j + µ n − j µ j ) (4.17)with some absolute constant C > < α ≤ λ n , ˜ F n := ˜ L − (cid:16) ˜ G n − λ n w ˜ F (cid:48) (cid:17) can be written as a linear combination of T σ , S σ (cid:48) with σ, σ (cid:48) ∈ {− , , } , and it is importantto notice that whenever we use the operator S σ (cid:48) , its coefficient in the expansion of ˜ L − comeswith a factor of α . Then, applying Lemma 4.4 to each term in ˜ G n , we deduce that µ n α n +1 ≤ C ( l n α n · α ) + Cα n +1 n − (cid:88) j =1 ( µ j µ n − − j + l j µ n − j + µ j µ n − j ) , or equivalently, µ n ≤ C l n + µ µ n − + n − (cid:88) j =1 ( µ j µ n − − j + l j µ n − j + µ j µ n − j ) . { l n } , { µ n } are equivalent with that of {| λ n |} , { µ n } from the smooth case, we can run the exact same argument to deduce the desiredbounds.This shows that the series F ( α ) ( z ) = F ( α )0 ( z ) + ∞ (cid:88) n =1 a n F ( α ) n ( z ) , λ ( α ) ( a ) = ∞ (cid:88) n =1 a n λ ( α ) n are convergent for some interval a ∈ ( − / ( c α ) , / ( c α )) on which it can be also guaranteedthat λ ( α ) ( a ) > − F ( α ) indeed belongs to C α ( R ): it sufficesto observe that ˜ F ( α ) belongs to ˙ C ∩ L ∞ . To show decay of the function F ( α ) ( z ), we simplyargue using the equation (4.4) just as we did in the proof of Theorem 2. Indeed, note that if F solves F + 1 + λα zF (cid:48) + 2 F H ( F ) + a Λ − ( F ) F (cid:48) = 0and if F = ˜ F ( z α ) with ˜ F ∈ H (as we have shown above) then ˜ F solves˜ F ( w ) + (1 + λ ) w ˜ F (cid:48) ( w ) + 2 ˜ F · ˜ H ( ˜ F ) + aw − α ˜Λ − ( ˜ F ) · w ˜ F (cid:48) ( w ) = 0 . Proceeding as in the proof of Theorem 2, we obtain that ˜ F decays like w − λ , and therefore F decays like z − α λ . Now we are done.
To close the paper, we would like to mention another method which likely could lead to finite-timesingularity formation for C α solutions to the De Gregorio model which are periodic. ConsiderDe Gregorio’s model: ∂ t ω + 2 u∂ x ω = 2 ∂ x uω. Assume that the data is odd in x , 2 π periodic, and positive in (0 , π ). Suppose further that | ω ( x ) | ≥ C (sin( x )) α on (0 , π ) for some α < C >
0. Now define f ( x ) := (cid:90) x dyω ( y ) . Since ω is bounded from below as above and since α < f is well-defined. In general, if wedefine f ( t, x ) = (cid:90) x dyω ( t, y )we see that f is a 2 π periodic even function satisfying a transport equation: ∂ t f + 2 u∂ x f = 0 , = − Λ − ( 1 ∂ x f ) , with f ( x ) ≈ | sin( x ) | α near x = 0. To prove that ω becomes singular in finite time, it sufficessimply to prove that there is a time where f ceases to have a cusp at x = 0 in the sense thatthe quantity A ( t ) := lim x → | f ( x, t ) || x | α hits zero in finite time. Since the velocity field u is directed away from the origin, A ( t ) iscertainly a decreasing function. However, to prove that A ( t ) actually hits zero in finite time, itis necessary to understand the operator f → Λ − ( ∂ x f ) more precisely.This reformulation of the problem allows us to clearly see how the case of α < α ≥
1, since f cannot even be defined from ω if ω is C and vanishes.This leads one to conjecture that the C case and the C α case ( α <
1) are actually quite differ-ent. Finally, for the Okamoto-Sakajo-Wunsch models one can define f similarly, and the rightdefinition for f in that case is f ( t, x ) = (cid:90) x dyω ( t, y ) a/ . For a < f can be defined even when ω is C ∞ . This leads one to further conjecture thatsingularity formation will occur in the full range a < The authors would like to thank Tej-Eddine Ghoul, Vu Hoang, Hao Jia, Andrew Majda, NaderMasmoudi, Huy Nguyen, Vladimir Sverak, and Vlad Vicol for helpful remarks and stimulatingdiscussions. T. M. Elgindi acknowledges funding from NSF grant DMS-1402357.
A Appendix
A.1 Properties of the Hilbert transform
We collect a few simple properties of the Hilbert transform. It will be implicitly assumed thatthe functions f and g are in the domain of the Hilbert transform. Lemma A.1 (The Tricomi identity) . Given two functions f and g , H ( f g ) = H ( f ) g + f H ( g ) + H ( Hf Hg ) . (A.1) Proof.
There are complex analytic functions u and v on the upper half plane whose restrictionson the real line equals u = f + iHf,v = g + iHg. Then, it suffices to note that uv = f g − Hf Hg + i ( f Hg + gHf ) , H ( f g − Hf Hg ) = f Hg + gHf holds.The following identities are very well-known: Lemma A.2.
Assuming that zf ( z ) ∈ L ( R ) , the Hilbert transform of zf ( z ) is related to theHilbert transform of f via H ( wf ( w )) ( z ) = zHf ( z ) − π (cid:90) R f ( w ) dw. (A.2) Moreover, if one assumes that ( f ( z ) − f (0)) /z ∈ L ( R ) , then we have H (cid:18) f ( w ) − f (0) w (cid:19) ( z ) = Hf ( z ) − Hf (0) z . (A.3) Proof.
Note that H ( wf ( w )) ( z ) := p.v. π (cid:90) wf ( w ) z − w dw = 1 π (cid:90) ( w − z ) f ( w ) z − w dw + z (cid:18) p.v. π (cid:90) f ( w ) z − w dw (cid:19) . The proof of the second identity is strictly analogous.We state and prove a few elementary facts regarding the transforms H ( α ) of Hilbert-typewhich appear in Section 4. We first recall the definition of these transforms. Given some positiveinteger n , and a function f defined on R + , we consider the transformation ˜ H ( n ) for w > H ( n ) ( f )( w ) = 1 π p.v. (cid:90) R + nt n − w n − t n f ( t ) dt. (A.4)First, we may decompose the kernel as follows: Lemma A.3.
We have ˜ H ( n ) ( f )( w ) = 1 π n (cid:88) j =1 (cid:90) R + − ζ j n w − ζ j n t f ( t ) dt = 1 π p.v. (cid:90) R + tw − t dt + n (cid:88) j =1 π (cid:90) R + t − (cid:60) ( ζ j n ) ww − (cid:60) ( ζ j n ) wt + t f ( t ) dt where ζ n := exp ( iπ/n ) is the primitive n -th root of unity.Proof. This is a purely algebraic statement which can be checked directly.44e set ˜ H ( n )0 ( f )( w ) = 1 π p.v. (cid:90) R + tw − t f ( t ) dt (A.5)and for j ∈ {± , · · · , ± ( n − } ˜ H ( n ) j ( f )( w ) = 1 π (cid:90) R + − ζ j n w − ζ j n t f ( t ) dt. (A.6)Note that ˜ H ( n )0 is exactly the Hilbert transform of the odd function f ( | w | )sgn( w ) restricted onto R + . In particular, (cid:107) ˜ H ( n )0 ( f ) (cid:107) L ≤ (cid:107) f (cid:107) L . The following lemma follows directly from the definition of the operators ˜ H ( n ) j . Lemma A.4.
For ≤ | j | < n , we have the identities ˜ H ( n ) j ( tf ( t )) ( w ) = − ζ j n · w ˜ H ( n ) j ( f )( w ) − π (cid:90) R + f ( t ) dt, (A.7)˜ H ( n ) j (cid:18) f ( t ) − f (0) t (cid:19) ( w ) = ζ j n · ˜ H ( n ) j f ( w ) − ˜ H ( n ) j f (0) w , (A.8) and ˜ H ( n ) j ( f (cid:48) )( w ) = − ζ j n · ˜ H ( n ) j ( f ) (cid:48) . (A.9) As a consequence, we obtain the identity ˜ H ( n ) j ( tf (cid:48) ( t ))( w ) = w ( ˜ H ( n ) j f ) (cid:48) ( w ) , (A.10) which in particular implies that ˜ H ( n ) ( tf (cid:48) ( t ))( w ) = w ( ˜ H ( n ) f ) (cid:48) ( w ) . (A.11) Lemma A.5.
For each ≤ | j | ≤ n , we have the estimate (cid:107) ˜ H ( n ) j ( f ) (cid:107) L ≤ C ln (cid:16) jπn (cid:17) (cid:107) f (cid:107) L with some absolute constant C > . In particular, for each n ≥ , (cid:107) ˜ H ( n ) ( f ) (cid:107) L ≤ Cn (cid:107) f (cid:107) L . (A.12) Similarly, for any integer m ≥ , we have (cid:107) ˜ H ( n ) ( f ) (cid:107) H m ≤ C m n (cid:107) f (cid:107) H m . (A.13)45 roof. Without loss of generality, we assume that f is real and consider the real and the imag-inary parts of ˜ H ( n ) j separately.Regarding the real part, we need to bound (cid:107) π (cid:90) R + −(cid:60) ( ζ j n ) w + tw − (cid:60) ( ζ j n ) wt + t f ( t ) dt (cid:107) L w , and note that (cid:60) ( ζ j n ) = cos (cid:16) jπn (cid:17) . First making the change of variable t = ws and then usingthe Minkowski inequality, the above quantity is bounded by1 π (cid:90) R + (cid:12)(cid:12)(cid:12) s − cos (cid:16) jπn (cid:17)(cid:12)(cid:12)(cid:12) s − (cid:16) jπn (cid:17) s + 1 (cid:107) f ( ws ) (cid:107) L w ds ≤ C (cid:107) f (cid:107) L · (cid:90) R + (cid:12)(cid:12)(cid:12) s − cos (cid:16) jπn (cid:17)(cid:12)(cid:12)(cid:12) s − (cid:16) jπn (cid:17) s + 1 · s / ds. Then it suffices to show that (cid:90) R + (cid:12)(cid:12)(cid:12) s − cos (cid:16) jπn (cid:17)(cid:12)(cid:12)(cid:12) s − (cid:16) jπn (cid:17) s + 1 · s / ds ≤ C ln (cid:16) jπn (cid:17) for some universal C >
0. To see this, we consider the case when cos( jπ/n ) ≥ jπ/n ) → + , it is easy to see thatthe integrand is uniformly bounded in j, n by an integrable function and therefore the integralsare uniformly bounded as well. Hence, we may assume that cos( jπ/n ) ≥ /
4, and in this case,we split the integral as (cid:34)(cid:90) ≤ s< cos ( jπn ) + (cid:90) cos ( jπn ) ≤ s< cos ( jπn ) + (cid:90) cos ( jπn ) ≤ s (cid:35) (cid:12)(cid:12)(cid:12) s − cos (cid:16) jπn (cid:17)(cid:12)(cid:12)(cid:12)(cid:16) s − cos (cid:16) jπn (cid:17)(cid:17) + sin (cid:16) jπn (cid:17) · s / ds, and in the first and third regions, the integrals are uniformly bounded whenever cos( jπ/n ) ≥ / C (cid:90) ≤ s ≤ cos ( jπn ) ss + sin (cid:16) jπn (cid:17) · (cid:16) s + cos (cid:16) jπn (cid:17)(cid:17) / ds ≤ C (cid:48) (cid:90) ≤ s ≤ ss + sin (cid:16) jπn (cid:17) ds ≤ C ln (cid:16) jπn (cid:17) , which is the bound we wanted. The imaginary part can be treated in a similar way and resultsin a better estimate, without the logarithmic factor.Given this L -bound for ˜ H ( n ) j , the bound (A.12) follows simply from summing the estimatesover 1 ≤ | j | ≤ n : it only suffices to observe that (cid:88) ≤| j |≤ n ln (cid:16) jπn (cid:17) = ln (cid:89) ≤| j |≤ n (cid:16) jπn (cid:17) ≤ C ln n (cid:89) j =1 (cid:18) nj (cid:19) ≤ C (cid:48) n, n (cid:89) j =1 (cid:18) nj (cid:19) ≤ (2 n )!( n !) ≤ C exp( Cn )for some constant C > H m -bound follows similarly, using the identity (A.9). A.2 Functional inequalities
We state and prove the Hardy inequalities.
Lemma A.6 (Hardy inequalities) . For any f ∈ H s ( R ) with s ≥ , we have (cid:107) z (cid:90) z f ( w ) dw (cid:107) H s ( R ) ≤ C s (cid:107) f (cid:107) H s ( R ) . More precisely, for each ≤ σ ≤ s , we have (cid:107) ∂ σz (cid:18) z (cid:90) z f ( w ) dw (cid:19) (cid:107) L ( R ) < σ + 1 (cid:107) ∂ σz f (cid:107) L . (A.14) Proof.
In the proof, without loss of generality we shall assume that f ∈ C ∞ c ([0 , ∞ )).We first consider the case s = 0: we have (cid:107) z (cid:90) z f ( w ) dw (cid:107) L (0 , ∞ ) = (cid:90) ∞ − ∂ z z − (cid:18)(cid:90) z f ( w ) dw (cid:19) dz = 2 (cid:90) ∞ f ( z ) · (cid:18) z (cid:90) z f ( w ) dw (cid:19) dz< (cid:107) z (cid:90) z f ( w ) dw (cid:107) L (0 , ∞ ) · (cid:107) f (cid:107) L (0 , ∞ ) , which establishes the statement in this case, noting that the strict equality forces f ( z ) = c z c for some constants c , c , which never belongs to L ( R ). Next, take s = σ = 1, and from ∂ z (cid:18) z − (cid:90) z f ( w ) dw (cid:19) = z − (cid:18) zf ( z ) − (cid:90) z f ( w ) dw (cid:19) , we proceed similarly as before: (cid:107) ∂ z (cid:18) z (cid:90) z f ( w ) dw (cid:19) (cid:107) L (0 , ∞ ) = − (cid:90) ∞ ∂ z z − (cid:18) zf ( z ) − (cid:90) z f ( w ) dw (cid:19) dz< (cid:107) ∂ z (cid:18) z (cid:90) z f ( w ) dw (cid:19) (cid:107) L (0 , ∞ ) · (cid:107) ∂ z f (cid:107) L (0 , ∞ ) . The argument for the case s > emma A.7 (Hardy-type inequalities) . For any f ∈ H s ( R + ) with s ≥ and < α ≤ , wehave (cid:107) z − α (cid:90) z f ( w ) · α w − αα dw (cid:107) H s ( R + ) ≤ C s (cid:107) f (cid:107) H s ( R + ) with a constant uniform for < α ≤ . More precisely, for each ≤ σ ≤ s , we have (cid:107) ∂ σz (cid:18) z − α (cid:90) z f ( w ) · α w − αα dw (cid:19) (cid:107) L ( R + ) < σ + 1 (cid:107) ∂ σz f (cid:107) L ( R + ) . (A.15) Proof.
The proof is strictly analogous to that of the usual Hardy inequalities. Let us restrictourselves to the case σ = 0. Then, we simply write out (cid:107) z − α (cid:90) z f ( w ) · α w − αα dw (cid:107) L ( R + ) = 11 − /α (cid:90) ∞ ∂ z (cid:16) z − α (cid:17) · (cid:18) z − α (cid:90) z f ( w ) · α w − αα dw (cid:19) dz = − − /α (cid:90) ∞ (cid:18) z − α (cid:90) z f ( w ) · α w − αα dw (cid:19) · α f ( z ) dz = 22 − α (cid:90) ∞ (cid:18) z − α (cid:90) z f ( w ) · α w − αα dw (cid:19) · f ( z ) dz and then applying the Cauchy-Schwartz inequality finishes the proof. References [1] J. T. Beale, T. Kato, and A. Majda. Remarks on the breakdown of smooth solutions forthe 3-D Euler equations.
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