Estimate on the dimension of the singular set of the supercritical surface quasigeostrophic equation
aa r X i v : . [ m a t h . A P ] D ec ESTIMATE ON THE DIMENSION OF THE SINGULAR SET OF THESUPERCRITICAL SURFACE QUASIGEOSTROPHIC EQUATION
MARIA COLOMBO AND SILJA HAFFTER
Abstract.
We consider the SQG equation with dissipation given by a fractional Laplacian oforder α < . We introduce a notion of suitable weak solution, which exists for every L initialdatum, and we prove that for such solution the singular set is contained in a compact set inspacetime of Hausdorff dimension at most α (cid:0) αα (1 − α ) + 2 (cid:1) . Contents
1. Introduction 12. Preliminaries 53. The local energy inequalities 94. Decay of the excess 175. Iteration of the excess decay 226. ε -regularity results and proof of Theorem 1.3 317. The singular set and proof of Theorem 1.1 398. Stability of the singular set 41Appendix A. Local spacetime regularity of the fractional heat equation 44Appendix B. C δ -H¨older continuous solutions are classical for δ > − α Introduction
For α ∈ (cid:0) , (cid:3) we consider the following fractional drift-diffusion equation ( ∂ t θ + u · ∇ θ = − ( − ∆) α θ div u = 0 , (1)where θ : R × [0 , ∞ ) → R is an active scalar, u : R × [0 , ∞ ) → R is the velocity field and ( − ∆) α corresponds to the Fourier multiplier with symbol | ξ | α . The system is usually complementedwith the initial condition θ ( · ,
0) = θ . (2)We will be particularly interested in the surface quasigeostrophic (SQG) equation where thevelocity field u is determined from θ by the Riesz-transform R on R . More precisely, we require u = ∇ ⊥ ( − ∆) − θ = R ⊥ θ . (3) There is a natural scaling invariance associated to the system: whenever ( θ, u ) solves (1), thenso does the pair θ r ( x, t ) := r α − θ ( rx, r α t ) u r ( x, t ) = r α − u ( rx, r α t ) . (4)1.1. Main result.
Our main result shows that for every L initial datum and every α ∈ (cid:2) , (cid:1) , there exists an almost everywhere smooth solution of the SQG equation and, more precisely, itprovides a bound on the box-counting and Hausdorff dimension of the closed set of its singularpoints. Theorem 1.1.
Let α := √ . For any α ∈ (cid:0) α , (cid:1) and any initial datum θ ∈ L ( R ) thereis a Leray–Hopf weak solution ( θ, u ) of (1) – (3) (see Definition 3.1) and a relatively closed set Sing θ ⊂ R × (0 , ∞ ) such that • θ ∈ C ∞ (cid:0) [ R × (0 , ∞ )] \ Sing θ (cid:1) , • Sing θ ∩ [ R × [ t, ∞ )] is compact with box-counting dimension at most α (cid:0) αα (1 − α ) + 2 (cid:1) for any t > , • the Hausdorff dimension of Sing θ does not exceed α (cid:0) αα (1 − α ) + 2 (cid:1) . Remark . We will in fact prove a slightly stronger statement, namely that all suitable weaksolutions θ of (1)–(3) on R × (0 , ∞ ) (see Definition 3.4) satisfy the estimate on the dimension ofthe spacetime singular set Sing θ ; in particular, they are smooth almost everywhere in spacetime.Moreover, the set Sing θ is compact as soon as the initial datum is regular enough to guaranteelocal smooth existence.The regularity issue for the equation (1)–(3) is fully understood only in the subcritical andcritical regime, namely for α ≥ . The critical case (without bounderies) is now well-understoodthanks to Kiselev, Nazarov and Volberg [19] and Caffarelli and Vasseur [4] (see also [9]) and oneeven has a description of the long time behaviour of the system [8, 6]. On bounded domains,the critical case has been well-studied in a series of works initiated by [7]. In the supercriticalrange α < , the global regularity of Leray-Hopf weak solutions to the SQG equation is anopen problem related to the problem of global existence of classical solutions: in fact, it iswell-known that Leray-Hopf weak solutions coincide with classical solutions as long as the latterexist. Constantin and Wu [10, 11] obtained partial results by extending the program of [4] tothe supercritical regime. In [4] the technique of De Giorgi for uniformly elliptic equations withmeasurable coefficients is adapted to prove the smoothness of Leray-Hopf weak solutions in threesteps: the local boundedness of L solutions, the H¨older continuity of L ∞ solutions, and thesmoothness of H¨older solutions. While the L ∞ -bound for Leray-Hopf weak solutions still worksin the supercritical case [10], only conditional regularity results are kown regarding the secondand third step of the scheme. For instance, H¨older solutions in C δ are smooth for δ > − α ,while for δ < − α this is left open [11]. On the negative side, [1] established non-uniqueness ofa class of (very) weak-solution for the system (1)–(3), even for subcritical dissipations. In thiscontext Theorem 1.1 is, to our knowledge, the first a.e. smoothness / partial regularity result.1.2. An ε -regularity theorem. The estimate on the dimension of the singular set in Theorem1.1 follows from a simple covering argument and a so-called ε -regularity result: in order to fixthe main ideas, we present the latter in a simplified version in Theorem 1.3 below. In whatfollows we denote by θ ∗ the Caffarelli-Silvestre extension of θ and by M the maximal function with respect to the space variable (see Sections 2.4 and 2.6); K q , as defined in (51), is a constantdepending on the local-in-time L ∞ t L qx ( R ) estimate of θ (recalled in Section 3.1). Theorem 1.3.
Let α ∈ (cid:2) α , (cid:1) , q ≥ and p := αα + q . There exists a universal ε = ε ( α ) > such that the following holds: Let ( θ, u ) be a suitable weak solution of (1) – (3) on R × (0 , T ) (see Definition 3.4) satisfying k θ k p − L ∞ ( R × [ t − r α ,t + r α ]) r p (1 − α )+2 Z C ∗ ( x,t ; r ) y b |∇ θ ∗ | dz ds dy + Z C ( x,t ; r ) M (cid:0) ( D α, θ ) (cid:1) dz ds ! ≤ ε , (5) where C ( x, t ; r ) := B K q r α − /q ( x ) × ( t − r α , t + r α ) , C ∗ ( x, t ; r ) := [0 , r ) × C ( x, t ; r ) and ( D α, θ )( z, s ) := (cid:18)Z R | θ ( z, s ) − θ ( z ′ , s ) | | z − z ′ | α dz ′ (cid:19) . (6) Then θ is smooth on B r/ ( x ) × ( t − r α / , t + r α / . The integral quantities present in (5) are two non-equivalent localized versions of the dissi-pative part of the energy, i.e. the L ((0 , T ) , W α, ( R ))-norm of θ , and are globally controlledthrough the latter. At this point, the careful reader will object that Theorem 1.3 cannot beused in a covering argument since the maximal function is not bounded on L . This issue rep-resents a mere technical difficulty though: it is resolved by introducing a suitable variant of thesharp maximal function which leads to the more involved ε -regularity criterion of Corollary 6.6.Theorem 1.3 is a consequence of the ε -regularity Theorem 5.3 (which holds for every α ∈ ( , ))whose smallness requirement features an L p -based excess quantity and can be met at some smallscale by requiring (5). Theorem 5.3 on the other hand is obtained via an excess decay resultand a linearization argument, in analogy with [23] for the classical Navier-Stokes equations andwith [5] for the hyperdissipative Navier-Stokes equations. Nevertheless there are some noveltiesin our approach with respect to the corresponding results for Navier-Stokes: • Our ε -regularity result relies on the crucial observation (previously used in [4, 11]) thatthe equation (1) is invariant under a change of variables which sets the space average of u to zero. Indeed, the scaling (4), in contrast to the analogous situation for the Navier-Stokes equations, does not guarantee any control on the average of u on B r in termsof the average of the rescaled solution u r on B as r →
0. The lack of control on theaverages introduces a challenge to iterate the excess decay, since at each step we need tocorrect for this change of variable, in a similar spirit to [11]. • As a second ingredient we introduce a new notion of suitable weak solution which enablesus to perform energy estimates of nonlinear type controlling a potentially large power of θ . Such nonlinear energy estimates exploit the boundedness of Leray-Hopf weak solutionsin an essential way and are not available for the Navier-Stokes equations. The freedom ofchoosing a suitable nonlinear power on the other hand is crucial in the context of the SQGequation: Indeed, the classical (local) energy controls naturally θ ∈ L ∞ ((0 , T ) , L ( R )) ∩ L ((0 , T ) , W α, ( R )) and hence, by interpolation, θ ∈ L α ) ( R × (0 , T )). Yet, since2(1 + α ) < α < , this is not enough to conclude a strong enough Caccioppoli-typeinequality which accounts for the cubic nonlinearity in the local energy. • On one side Theorem 1.3 may be seen as an analogue of Scheffer’s result [26] for Navier-Stokes, providing ε -regularity criterion at a fixed scale. On the other side, in order to give COLOMBO AND HAFFTER an estimate on the dimension of the singular set, the smallness (5) must be required interms of differential quantities of θ , as it happens in the more refined result by Caffarelli,Kohn and Nirenberg for Navier-Stokes [2]. In the context of the SQG equation, theeasier Corollary 6.1 below may be seen as the full analogue of Scheffer’s result. AlthoughCorollary 6.1 still establishes the compactness of the singular set, it does, in contrast toNavier-Stokes, not yield any estimate on the dimension of the singular set.Using the “continuity” of the aforementioned ε -regularity Theorem 5.3 under strong conver-gence in L p , an immediate consequence is the stability of the singular set in the fractional order α ∈ ( , ] which in particular recovers the following result of [12]. Corollary 1.4 (Gobal regularity for slightly supercritical SQG) . Let θ ∈ H ( R ) with k θ k H ≤ R .
Then there exists ε = ε ( R ) > such that (1) – (3) has a unique smooth solution θ ∈ L ∞ loc ([0 , ∞ ) , H ( R )) ∩ L loc ((0 , ∞ ) , H α ( R )) for all fractional orders α ∈ (cid:2) − ε, (cid:3) . Remark . The corollary could be set in any H δ ( R ) for δ > α close to and there admits a (quantified) short-time existence of smooth solutions. In [12]the assumption k θ k H ≤ R is replaced by the scaling invariant assumption k θ k αL k θ k − α ˙ H ≤ R .
The latter statement can be reduced to ours by applying a first rescaling which renormalizes the L -norm of the initial datum to 1.Moreover, by the decay of the L ∞ -norm of solutions (see Theorem 3.2 below), the ε -regularitycriterion is verified for large times and we recover the eventual regularization of suitable weaksolutions from L -initial data for α ∈ ( , ) previously established for Leray–Hopf solutions in[29] for α close to and in [13, 20] for any α ∈ (0 , ).1.3. A conjecture on the optimal dimension estimate.
Theorem 1.1 leaves open the ques-tion of whether or not the estimate on the dimension of the singular set, as well as the range of α for which it is valid, is optimal. We believe that a natural conjecture for an optimal estimateof the dimension of the spacetime singular set isdim P (Sing θ ) ≤ α (4 − α ) , (7)and dim H (Sing θ ) ≤ α (cid:0) − α (cid:1) . (8)In (7), P is the parabolic Hausdorff measure that is, for α < , the Hausdorff measure resultingfrom restricting the class F of admissible covering sets to the spacetime cylinders ˜ Q r ( x, t ) = B r / (2 α ) ( x ) × ( t − r, t ] . We refer for instance to [5] for its construction for α > . The cylinders˜ Q r ( x, t ) are the natural choice for α < because their diameter is less than 4 r , at differencefrom the classical parabolic cylinders B r ( x ) × ( t − r α , t ] whose diameter is of the order of r α .The conjecture (7) is based on a dimensional analysis of the equation: We may assign a“dimension” to any function f ( θ ) of θ via the exponent β of the rescaling factor 1 /r β whichmakes the spacetime integral of f ( θ ) on ˜ Q r dimensionless, i.e. scaling-invariant with respectto (4). The number appearing on the right-hand side of (7) corresponds then to dimension ofthe energy, whose dissipative part is the globally controlled quantity in the form of a spacetimeintegral which scales best. This would correspond to the result of Caffarelli, Kohn and Nirenberg[2] ) for the Navier-Stokes system (see [32, 5, 21] for fractional dissipations of order α ∈ [ , )) who proved that suitable weak solutions of the latter are smooth outside a closed set of dimension1 . In fact, for the Navier-Stokes system this bound on the dimension of the singular set is whatthe scaling of the equations and boundedness of the energy suggest. Notice that the right-handside of both (7) and (8) does not converge to 0 as α → : this is due to the fact that the quantitythat dictates the scaling-criticality of the equation, namely the L ∞ -norm of θ , is not of integraltype and hence cannot be used in a covering argument of the type that we do in the proof ofTheorem 1.1. In turn, this covering argument finds his pivotal quantity in the dissipative partof the energy, which has a worse scaling than the L ∞ -norm of θ .In the proof of Theorem 1.1, it is natural to consider the classical Hausdorff measure, sincethe tilting effect of the change of variables, which sets the space average of u to zero, forces usto work on balls in spacetime (rather than parabolic cylinders, see Section 6.4 and in particularStep 3 of the proof of Corollary 6.6). This effect of the change of variables constitutes a seriousobstacle for any parabolic Hausdorff dimension estimate. However, our estimate is nonoptimal:to obtain the optimal estimate, one should replace αα by 2 in the estimate of the dimensionof the singular set in Theorem 1.1; however, the integrability exponent αα represents the leastpossible exponent for which we are able to use a “nonlinear” localized energy inequality in anexcess decay argument (cf. Lemma 3.8). An analogous difficulty appears for the ipodissipativeNavier-Stokes equations for low fractional orders α < where the Caccioppoli-type inequalityas described before fails to be strong enough to control the cubic nonlinearity and indeed noestimate of the dimension of the singular set is known.1.4. Structure of the paper.
The paper is structured as follows. After recalling some technicalpreliminaries in Section 2, we discuss in Section 3 the global and local energy inequalities ofthe SQG equation and we define the notion of suitable weak solutions. The key compactnessproperty of the latter is proven in Section 3.5 and leads to an excess decay result established inSection 4. The iteration of the excess decay on all scales is performed in Section 5 and requiresto introduce a change of variables which sets to 0 the average of the velocity u on suitable balls.This excess decay yields the basis for several ε -regularity results, in particular Theorem 1.3,which are deduced in Section 6. The proof of Theorem 1.1 is given in Section 7. In Section 8,we discuss the stability of the singular set with respect to variations of the fractional order ofdissipation. 2. Preliminaries
Notation.
We use the following notation for space(time) averages of functions or vectorfields f defined on R × [0 , ∞ ): For bounded sets E ⊆ R × [0 , ∞ ) and F ⊆ R , we define( f ) E := − Z E f ( x, t ) dx dt and [ f ( t )] F := − Z F f ( x, t ) dx . We introduce the spacetime cylinder adapted to the parabolic scaling (4) of the equation Q r ( x, t ) := B r ( x ) × ( t − r α , t ] . In the upper half-space R we define B ∗ r ( x ) := B r ( x ) × [0 , r ) and we define the extended cylinder Q ∗ r ( x, t ) := B ∗ r ( x ) × ( t − r α , t ] . We will omit the center of the cylinders whenever ( x, t ) = (0 , . Moreover, we use the followingconvention to describe spacetime H¨older spaces: For α, β ∈ (0 ,
1) and Q ⊂ R × R we denote COLOMBO AND HAFFTER by C α,β ( Q ) the functions which are α - and β -H¨older continuous in space and time respectively,namely such that the following semi-norm is finite k θ k C α,β ( Q ) = sup (cid:26) | θ ( x, t ) − θ ( y, s ) || x − y | α + | s − t | β : ( x, t ) , ( y, s ) ∈ Q with ( x, t ) = ( y, s ) (cid:27) . Whenever α = β , we denote the above space just by C α ( Q ) . Furthermore, we will also workwith spatial Sobolev spaces of fractional order: For Ω ⊆ R n , s ∈ (0 ,
1) and 1 ≤ p < ∞ , wedenote by W s,p (Ω) := ( f ∈ L p (Ω) : | f ( x ) − f ( y ) || x − y | np + s ∈ L p (Ω × Ω) ) . Correspondingly, we define for f ∈ W s,p (Ω) the Gagliardo semi-norm by[ f ] W s,p (Ω) := (cid:18)Z Ω Z Ω | f ( x ) − f ( y ) | p | x − y | n + sp dx dy (cid:19) p . In the special p = 2, we will sometimes denote W α, by H α and we recall that for Ω = R n theGagliardo semi-norm co¨ıncides, up to a universal constant, with the semi-norms (9). Finally,we will consider the Bochner spaces L q ((0 , T ) , X ) for 1 ≤ q ≤ ∞ and for some Banach space X (here: X = L p ( R ) or X = W α, ( R )). Whenever we work on a parabolic cube Q r ( x, t ), we willuse the short-hand notation L W α, ( Q r ) := L q (( t − r α , t ) , W α, ( B r ( x ))) . Singular points.
We call a point ( x, t ) ∈ R × (0 , ∞ ) a regular point of a Leray-Hopfweak solution θ of (1)–(3) (see Definition 3.1) if there exists a neighbourhood of ( x, t ) where θ is smooth. We denote by Reg θ the open set of regular points in spacetime. Correspondingly,we define the spacetime singular set Sing θ := [ R × (0 , ∞ )] \ Reg θ .
Riesz-transform.
We recall that the Riesz-transforms admit a singular integral represen-tation. Indeed, for f : R → [0 , ∞ ) and i = 1 , R i f ( x ) = c p . v . Z R x i − z i | x − z | f ( z ) dz . By Calderon-Zygmund they are bounded operators on L p for 1 < p < ∞ and from L ∞ to BM O .
Caffarelli-Silvestre extension.
We recall the following extension problem. We use thenotation ∇ , ∆ for differential operators defined on the upper half-space R n +1+ . Theorem 2.1 (Caffarelli–Silvestre [3]) . Let θ ∈ H α ( R n ) with α ∈ (0 , and set b := 1 − α .Then there is a unique “extension” θ ∗ of θ in the weighted space H ( R n +1+ , y b ) which satisfies ∆ b θ ∗ ( x, y ) := ∆ θ ∗ + by ∂ y θ ∗ = 1 y b div (cid:0) y b ∇ θ ∗ (cid:1) = 0 and the boundary condition θ ∗ ( x,
0) = θ ( x ) . Moreover, there exists a constant c n,α , depending only on n and α , with the following properties: (a) The fractional Laplacian ( − ∆) α w is given by the formula ( − ∆) α θ ( x ) = c n,α lim y → y b ∂ y θ ∗ ( x, y ) . (b) The following energy identity holds Z R n | ( − ∆) α θ | dx = Z R n | ξ | α | b θ ( ξ ) | dξ = c n,α Z R n +1+ y b |∇ θ ∗ | dx dy . (9)(c) The following inequality holds for every extension η ∈ H ( R n +1+ , y b ) of θ : Z R n +1+ y b |∇ θ ∗ | dx dy ≤ Z R n +1+ y b |∇ η | dx dy . Poincar´e inequalities.
Let α ∈ (0 , ≤ p < nα and p ∗ := pnn − pα . There exists a universalconstant C = C ( α, n, p ) such that for every f ∈ W α,p ( R n ), q ∈ [ p, p ∗ ] , x ∈ R n and r > Z B r ( x ) | f ( z ) − [ f ] B r ( x ) | q dz ! q ≤ Cr α − n ( p − q ) [ f ] W α,p ( B r ( x )) . (10)We will also need a weighted Poincar´e inequality in the spirit of the classical work [16] for α = 1(where on the other side much more general weights are admissible). Let ω ∈ C ∞ c ( R n ) be aradial, non-increasing weight such that ω ≡ B r/ ( x ), ω ≡ B r ( x ) and |∇ ω | ≤ Cr pointwise. We introduce the weighted average[ f ] ω,B r ( x ) := (cid:18) Z B r ( x ) ω ( z ) dz (cid:19) − Z B r ( x ) f ( z ) ω ( z ) dz . The following weighted Poincar´e inequality is classical for α = 1 (see [22, Lemma 6.12]) and itis established for q = p in [15, Proposition 4]: Their proof extends to the other endpoint q = p ∗ and hence to the range q ∈ [ p, p ∗ ] by interpolation. Lemma 2.2.
Under the above assumptions, we have the weighted Poincar´e inequality (cid:18) Z B r ( x ) | f ( z ) − [ f ] ω,B r ( x ) | q ω ( z ) dz (cid:19) q ≤ ˜ Cr α − n ( p − q ) [ f ] W α,p ( B r ( x )) (11) where ˜ C = 2 − α + n/p C .
In the case p = 2, we can rewrite the right-hand side of (10) and (11) in terms of the extensionas follows. Lemma 2.3.
Let n ≥ , α ∈ (0 , , < r < s , f ∈ W α, ( R n ) and g ∈ C ( R ) . Then thereexists C = C ( n, α ) such that [ g ◦ f ] W α, ( B r ) ≤ C (cid:18) Z B ∗ s y b |∇ [ g ( f ∗ )] | dx dy + ( s − r ) − Z B ∗ s \ B ∗ r y b ( g ( f ∗ )) dx dy (cid:19) (12) and for any ≤ q ≤ nn − α k g ◦ f k L q ( B r ) ≤ C (cid:18) Z B ∗ s y b |∇ [ g ( f ∗ )] | dx dy + ( s − r ) − Z B ∗ s \ B ∗ r y b ( g ( f ∗ )) dx dy + Z B r f dx (cid:19) . (13) COLOMBO AND HAFFTER
In particular, for any x ∈ R n [ f ] W α, ( B r ( x )) ≤ C (cid:18) Z r Z B r ( x ) y b |∇ f ∗ | ( z, y ) dz dy (cid:19) . (14) Proof.
Let n ≥ α ∈ (0 , < r < s and g ∈ C ( R ). By approximation we may assume that f is Schwartz. Fix a smooth cut-off function ϕ ∈ C ∞ c ( R n +1+ ) such that 0 ≤ ϕ ≤ ϕ ≡ B ∗ r , supp ϕ ⊆ B ∗ s and |∇ ϕ | ≤ C ( s − r ) − . For α ∈ (0 ,
1) we use the minimizing property of theextension to write[ g ◦ f ] W α, ( B r ) = Z B r Z B r | g ( f ( x )) − g ( f ( z )) | | x − z | n +2 α dx dz ≤ Z R n Z R n | (( g ◦ f ) ϕ | y =0 )( x ) − (( g ◦ f ) ϕ | y =0 )( z ) | | x − z | n +2 α dx dz = c n,α Z R n +1+ y b |∇ (( g ◦ f ) ϕ | y =0 ) ∗ | dx dy ≤ c n,α Z R n +1+ y b |∇ ( g ( f ∗ ) ϕ ) | dx dy ≤ c n,α Z R n +1+ y b |∇ [ g ( f ∗ )] | ϕ dx dy + 2 c n,α Z R n +1+ y b ( g ( f ∗ )) |∇ ϕ | dx dy and thus (12) follows. The estimate (13) follows from (12) via Sobolev embedding and interpo-lation. As for (14) , we may assume x = 0 and denoting by c the weighted average of f ∗ withrespect to the weight y b on B r/ × [0 , r/ ω ( x, y ) = y b ∈ A on R ) r − Z r Z B r y b | f ∗ − c | dx dy . Z r Z B r y b |∇ f ∗ | dx dy . (15)The estimate (14) follows then from (12) (applied to f − c and s = 4 r/
3) and (15). (cid:3) (Sharp) maximal function.
For a function f : R × [0 , ∞ ) → R we introduce themaximal function (in space) M f ( x, t ) := sup r> − Z B r ( x ) | f ( z, t ) | dz as well as the sharp fractional maximal function (in space) f α ( x, t ) := sup r> r − α − Z B r ( x ) | f ( z, t ) − [ f ( t )] B r ( x ) | dz and for q > √ f α,q ( x, t ) := sup r> r − α (1 − /q ) − Z B r ( x ) | f ( z, t ) − [ f ( t )] B r ( x ) | − /q ) dz . (16)In order to use a spacetime integral of θ α,q in a covering argument, we need to know that it isglobally controlled; to guarantee the latter, we are forced to choose q < ∞ . Lemma 2.4.
Let α ∈ (0 , , f ∈ W α, ( R n ) and q ∈ ( √ , ∞ ) . Then there exists a constant C = C ( n, q ) ≥ (which is uniformly bounded for q bounded away from ∞ ) such that k f α k L ( R n ) + k f α,q k / ( q − L / ( q − ( R n ) ≤ C [ f ] W α, ( R n ) . For α = 1 , the equivalent of Lemma 2.4 is a simple consequence of the Poincar´e inequality andthe maximal function estimate. Indeed, by Poincar´e we have almost everywhere the pointwiseestimate f ( x ) . M ( |∇ f | )( x ) f ,q ( x ) . M (cid:0) |∇ f | − /q ) (cid:1) ( x )for f ∈ W , loc and f ∈ W , − /q ) loc respectively. Integrating in x and using the boundedness ofthe maximal function on L and L / ( q − , we obtain the equivalent of Lemma 2.4. Proof.
We give the proof for f α,q . We estimate the quantity in the supremum in (16) − Z B r ( x ) (cid:12)(cid:12)(cid:12)(cid:12) − Z B r ( x ) f ( z ) − f ( y ) r α dy (cid:12)(cid:12)(cid:12)(cid:12) − /q ) dz ≤ − Z B r ( x ) (cid:18) − Z B r ( x ) | f ( z ) − f ( y ) | r α dy (cid:19) − /q dz ≤ C − Z B r ( x ) (cid:18) Z B r ( x ) | f ( z ) − f ( y ) | | z − y | n +2 α dy (cid:19) − /q dz ≤ C − Z B r ( x ) ( D α, f ( z )) − /q ) dz , (17)where D α, f is the n -dimensional version of (6), i.e. for z ∈ R n ( D α, f )( z ) := (cid:18)Z R n | f ( z ) − f ( z ′ ) | | z − z ′ | n +2 α dz ′ (cid:19) . By taking the supremum over r > , we deduce from (17) that for almost every xf α,q ( x ) ≤ C M (cid:0) ( D α, f ) − /q ) (cid:1) ( x ) (18)and hence by the maximal function estimate on L / ( q − k f α,q k / ( q − L / ( q − ( R n ) ≤ C k ( D α, f ) − /q ) k / ( q − L / ( q − ( R n ) = C k D α, f k L ( R n ) = C [ f ] W α, ( R n ) . (cid:3) The local energy inequalities
Leray-Hopf weak solutions.
We recall the notion of Leray–Hopf weak solutions.
Definition 3.1.
Let θ ∈ L ( R ) . A pair ( θ, u ) is a Leray–Hopf weak solution of (1) – (2) on R × (0 , T ) if: (a) θ ∈ L ∞ ((0 , T ) , L ( R )) ∩ L ((0 , T ) , W α, ( R )) ; (b) θ solves (1) – (2) in the sense of distributions, namely div u = 0 and Z (cid:16) ∂ t ϕθ + uθ · ∇ φ − ( − ∆) α ϕθ (cid:17) dx dt = − Z θ ( x ) · ϕ (0 , x ) dx (19) for any ϕ ∈ C ∞ c ( R × R ) . (c) The following inequalities hold for every t ∈ (0 , T ) and for almost every s ∈ (0 , T ) andevery t ∈ ( s, T ) respectively: Z θ ( x, t ) dx + Z t Z | ( − ∆) α θ | ( x, τ ) dx dτ ≤ Z | θ | ( x ) dx (20)12 Z | θ | ( x, t ) dx + Z ts Z | ( − ∆) α θ | ( x, τ ) dx dτ ≤ Z | θ | ( x, s ) dx (21) Correspondingly, we say that θ is a Leray–Hopf weak solution of (1) – (3) if additionally (3) holds. Observe that from the weak formulation (19) it follows that for all ϕ ∈ C ∞ c ( R × [0 , T )) Z θ ( x, t ) ϕ ( x, , t ) dx − Z θ ( x, s ) ϕ ( x, , s ) dx = Z ts Z (cid:16) θ∂ τ ϕ | y =0 + ( uθ ) · ∇ ϕ | y =0 − θ ( − ∆) α ϕ | y =0 (cid:17) dx dτ = Z ts Z (cid:16) θ∂ τ ϕ | y =0 + ( uθ ) · ∇ ϕ | y =0 (cid:17) dx dτ − c α Z ts Z R y b ∇ θ ∗ · ∇ ϕ dx dy dτ (22)for s = 0 and almost every t ∈ (0 , T ) and for almost every 0 < s < t < T (with c α given byTheorem 2.1). Indeed, the equality between the last term of the second line and the last term of(22) holds for every θ ∈ L ((0 , T ) , W α, ( R )); in the smooth case this equality is a consequenceof Theorem 2.1 which one recovers for general θ through regularization.We recall that any Leray-Hopf weak solution is actually in L ∞ for t > Theorem 3.2 ([11] Theorem 2.1) . Let θ ∈ L ( R ) and let ( θ, u ) be a Leray-Hopf weak solutionof (1) – (2) . Then there exist a universal constant, independent on u , such that for any t > x ∈ R | θ ( x, t ) | ≤ C k θ k L t − α . (23) In the particular case (3) , where u = R ⊥ θ , we obtain as a consequence that for any t > k u ( · , t ) k BMO ( R ) ≤ C k θ k L t − α . (24) Remark . In [11] Theorem 3.2 is proven for Leray–Hopf weak solutions of the coupled system(1)–(3). However, in the proof of (23) only the energy inequality on level sets together with theassumption div u = 0 is used; the structure (3) is only used to deduce (24) from (23).3.2. Suitable weak solutions.
We are now ready to give our definition of suitable weak so-lution. Both this notion and the one of Leray-Hopf solution are given without requiring thecoupling (3), since, in the proof of Theorem 1.3, we will need to work on a larger class of equa-tions, where u is obtained from θ by means of the Riesz transform and a temporal translation. Definition 3.4.
A Leray–Hopf weak solution ( θ, u ) of (1) – (2) on R × (0 , T ) is a suitable weaksolution if the following two inequalities hold for almost every t ∈ (0 , T ) , all nonnegative testfunctions ϕ ∈ C ∞ c ( R × (0 , T )) with ∂ y ϕ ( · , , · ) = 0 in R × (0 , T ) , for all q ≥ and every That is, the function ϕ vanishes when | x | + y + | t | is large enough and if t is sufficiently close to 0, but it canbe nonzero on some regions of { ( x, y, t ) : y = 0 } . linear transformation of the form η := ( θ − M ) /L with scalar L > and shift M ∈ R : Z R ϕ ( x, , t ) η ( x, t ) dx + 2 c α Z t Z R y b |∇ η ∗ | ϕ dx dy ds (25) ≤ Z t Z R ( η ∂ t ϕ | y =0 + uη · ∇ ϕ | y =0 ) dx ds + c α Z t Z R y b ( η ∗ ) ∆ b ϕ dx dy ds , Z R ϕ ( x, , t ) | η | q ( x, t ) dx + 4 (cid:0) − q (cid:1) c α Z t Z R y b |∇| η ∗ | q | ϕ dx dy ds (26) ≤ Z t Z R ( | η | q ∂ t ϕ | y =0 + u | η | q · ∇ ϕ | y =0 ) dx ds + c α Z t Z R y b | η ∗ | q ∆ b ϕ dx dy ds , where the constant c α depends only on α and comes from Theorem 2.1.Correspondingly, we say that θ is a suitable weak solution of (1) – (3) if additionally (3) holds.Remark . In the classical notion of suitable weak solutions for the (hyperdissipative) Navier-Stokes equations, the local energy inequality (25) is asked to hold only for θ and not for everylinear transformation η := ( θ − M ) /L . However, it can be proved (see for instance [5]) thatthe class of suitable weak solutions is stable under this transformation. Here on the otherhand, since we use a “nonlinear” energy inequality (26), it is no longer obvious that the classof suitable weak solutions is stable under linear transformations; hence we require it already inthe definition. The class of suitable weak solutions contains smooth solutions (see Section 3.3)and is non-empty (see Section 3.4) for any L initial datum.3.3. Local energy equality for smooth solutions.
It is not difficult to see that (25) and(26) hold with an equality for every smooth solution of (1)–(2). Indeed, let f ∈ C ( R ). Wemultiply (1) by f ′ ( θ ) ϕ | y =0 and integrate in space to obtain for t ∈ [0 , T ] Z R f ( θ )( x, t ) ϕ | y =0 ( x, t ) dx − Z t Z R [ f ( θ ) ∂ t ϕ | y =0 + uf ( θ ) · ∇ ϕ | y =0 ] dx ds = − Z t Z R ( − ∆) α θf ′ ( θ ) ϕ | y =0 dx ds . By means of the divergence theorem, we compute for fixed time t Z R ( − ∆) α θ ( x, t ) f ′ ( θ )( x, t ) ϕ | y =0 ( x, t ) dx = c α lim y → + Z R y b ∂ y θ ∗ ( x, y, t ) (cid:0) f ′ ( θ ∗ ) ϕ (cid:1) ( x, y, t ) dx = c α Z R div( y b ∇ θ ∗ f ′ ( θ ∗ ) ϕ ) dx dy = c α Z R y b |∇ θ ∗ | f ′′ ( θ ∗ ) ϕ dx dy − c α Z R y b f ( θ ∗ )∆ b ϕ dx dy , where we integrated by parts in the third equality and used that the boundary terms vanish dueto the hypothesis ∂ y ϕ ( · , , · ) = 0 . We obtain that for f ∈ C ( R ) Z R f ( θ )( x, t ) ϕ | y =0 ( x, t ) dx + c α Z t Z R y b |∇ θ ∗ | f ′′ ( θ ∗ ) ϕ dx dy ds = Z t Z R [ f ( θ ) ∂ t ϕ | y =0 + uf ( θ ) · ∇ ϕ y =0 ] dx ds + c α Z t Z R y b f ( θ ∗ )∆ b ϕ dx dy ds . Observe that if f is moreover convex and nonnegative, both the left- and the right-hand side ofthe above equality have a sign. In particular, we obtain (25) with an equality when choosing f ( x ) = (cid:0) x − ML (cid:1) (since f ′′ ≡ L − ) and (26) when choosing f ( x ) = (cid:12)(cid:12) x − ML (cid:12)(cid:12) q for q ≥ . Existence of suitable weak solutions.
For any α ∈ (0 , ) the existence of suitable weaksolutions can be established from any initial datum θ ∈ L ( R ) by adding a vanishing viscosityterm ǫ ∆ θ on the right-hand side and letting ǫ → . The key argument is a classical Aubin-Lionstype compactness argument that we sketch in Appendix C.
Theorem 3.6.
For any θ ∈ L ( R ) there is a suitable weak solution of (1) – (3) on R × (0 , ∞ ) . Compactness.
We establish the compactness of a sequence of suitable weak solutions withvanishing excess. Let ( θ, u ) be a solution of (1)–(2) on R × (0 , T ) . For r > Q r ( x, t ) ⊆ R × (0 , T ), we define the excess as E ( θ, u ; x, t, r ) := E S ( θ ; x, t, r ) + E V ( u ; x, t, r ) + E NL ( θ ; x, t, r )where E S ( θ ; x, t, r ) := (cid:18) − Z Q r ( x,t ) | θ ( z, s ) − ( θ ) Q r ( x,t ) | p dz ds (cid:19) p E V ( u ; x, t, r ) := (cid:18) − Z Q r ( x,t ) | u ( z, s ) − [ u ( s )] B r ( x ) | p dz ds (cid:19) p E NL ( θ ; x, t, r ) := (cid:18) − Z tt − r α sup R ≥ r (cid:16) rR (cid:17) σp (cid:18) − Z B R ( x ) | θ ( z, s ) − [ θ ( s )] B r ( x ) | dz (cid:19) p ds (cid:19) p , for p ∈ (3 , ∞ ) and σ ∈ (0 , α ) yet to be chosen. Observe that both parameters serve as (hidden)parameter for now and will be chosen in the very end to close the main ε -regularity Theorem(see also Remark 5.4). Whenever ( x, t ) = (0 , E ( θ, u ; r ) . Remark . The excess behaves nicely under the natural rescaling (4).Indeed, for r > E ( θ, u ; x, t, r ) = r − α E ( θ r , u r ; x, t, . Lemma 3.8 (Compactness) . Let α ∈ (0 , ] , σ ∈ (0 , α ) and p > αα . Let ( θ k , u k ) be a sequenceof suitable weak solutions of (1) – (2) on R × [ − , with • lim k →∞ E ( θ k , u k ; 1) = 0 • and [ u k ( s )] B = 0 for all s ∈ [ − , . Set E k := E ( θ k , u k ; 1) and define η k := ( θ k − ( θ k ) Q ) /E k . Then there exists η ∈ L / loc ( R × [ − , such that, up to subsequences, η k ⇀ η weakly in L / loc ( R × [ − , . Moreover, η k → η strongly in L p ( Q / ) and η solves ∂ t η + ( − ∆) α η = 0 on Q / with E S ( η ; 1) + E NL ( η ; 1) ≤ . We will need the following auxiliary Lemma.
Lemma 3.9 (Tail estimate) . Let α ∈ (0 , ] , σ ∈ (0 , α ) and < p < ∞ . Then there exists auniversal constant C = C ( α, σ, p ) ≥ such that for every θ ∈ L p ( B ) with sup R ≥ R σp (cid:18) − Z B R | θ ( x ) | dx (cid:19) p < + ∞ , we have the estimate Z B ∗ y b | θ ∗ ( x, y ) | p dx dy ≤ C Z B | θ ( x ) | p dx + sup R ≥ R σp (cid:18) − Z B R | θ ( x ) | dx (cid:19) p ! . Proof.
We set θ := θ B and θ i +1 := θ ( B i +1 − B i ) for i ≥ y > θ ∗ ( x, y ) = ( P ( · , y ) ∗ θ )( x ) for P ( x, y ) = y α / ( | x | + y ) α . We estimate Z B ∗ y b | θ ∗ | p dx dy . Z B ∗ y b | θ ∗ | p dx dy + Z y b (cid:16) X i> k θ ∗ i ( · , y ) k L ∞ ( B ) (cid:17) p dy . The first term is estimated using Young and the fact that k P ( · , y ) k L ( R ) = k P ( · , k L ( R ) = C α is a universal constant (see for instance the appendix of [5]). Indeed, Z B ∗ y b | θ ∗ | p dx dy = Z y b k θ ∗ P ( · , y ) k pL p ( B ) dy ≤ Z y b k P ( · , y ) k pL ( R ) k θ k pL p ( R ) dy ≤ C pα Z B | θ | p dx . For i ≥
1, we estimate, using the fact that for x ∈ B and z ∈ B i +1 \ B i we have | x − z | ≥ i − and thus P ( x − z, y ) ≤ P (2 i − , y ) uniformly in z , k θ ∗ i +1 ( · , y ) k L ∞ ( B ) ≤ P (2 i − , y ) Z B i +1 \ B i | θ | dx ≤ Z B i +1 \ B i y α | θ | α )( i − dx , so that X i> k θ ∗ i ( · , y ) k L ∞ ( B ) ≤ X i ≥ y α (2 α − σ )( i − Z B i +1 \ B i | θ | (2+ σ )( i − dx ≤ Cy α (cid:16) X i ≥ (2 α − σ )( i − (cid:17) sup R ≥ R σ − Z B R | θ | dx . We obtain the claim by raising the previous inequality to the power p . (cid:3) Proof of Lemma 3.8.
Observe that u k → L p ( Q ) and thus, we may assume thatsup k ≥ k u k k L p ( Q ) ≤ . Moreover, by construction the pair ( η k , u k ) is a distributional solution to ∂ t η k + u k · ∇ η k + ( − ∆) α η k = 0with E ( η k , u k E k ; 1) = 1 . Step 1: We prove the uniform boundedness of η k in (cid:0) L ∞ L ∩ L W α, ∩ L ( p − α ) (cid:1) ( Q / ) . Fix a test function ϕ ∈ C ∞ c ( R × (0 , ∞ )) such that 0 ≤ ϕ ≤
1, supp ϕ ⊂ Q ∗ / and ϕ ≡ Q ∗ / . Moreover, we assume that ϕ is constant in y for small y , that is ∂ y ϕ = 0 for { y < } . From the local energy inequality (25) we deduce that for t ∈ [ − (13 / α , Z B / η k ( x, t ) dx + 2 c α Z t − ( ) α Z B ∗ / y b |∇ η ∗ k | ( x, y, s ) dx dy ds ≤ Z B / η k ( x, t ) ϕ ( x, , t ) dx + 2 c α Z t − ( ) α Z B ∗ / y b |∇ η ∗ k | ( x, y, s ) ϕ ( x, y, s ) dx dy ds ≤ Z t − ( ) α Z B / ( η k ∂ t ϕ | y =0 + u k η k · ∇ ϕ | y =0 ) dx ds + c α Z t − ( ) α Z B ∗ / y b ( η ∗ k ) ∆ b ϕ dx dy ds . Z Q / ( η k + | η k | pp − + | u k | p ) dx ds + Z Q ∗ / y b ( η ∗ k ) dx dy ds . Using Lemma 2.3, the previous inequality and Lemma 3.9, we deduce that for t ∈ [ − (13 / α , Z B / η k ( x, t ) dx + Z t − ( ) α [ η k ( s )] W α, ( B / ) ds . Z B / η k ( x, t ) dx + 2 c α Z t − ( ) α Z B ∗ / y b (cid:0) |∇ η ∗ k | + ( η ∗ k ) (cid:1) dx dy ds . Z Q ( η k + | η k | pp − + | u k | p ) dx ds + Z − sup R ≥ R σ (cid:18) − Z B R | η k ( x, s ) − [ η k ( s )] B | dx (cid:19) ds . , where we used in the last inequality that pp − ≤ p together with the fact E ( η k , u k E k ; 1) = 1 . Taking the supremum over t ∈ [ − (13 / α , , we deduce that uniformly in k ≥ t ∈ [ − (13 / α , Z B / η k ( x, t ) dx + Z − ( ) α [ η k ( s )] W α, ( B / ) ds ≤ C .
We now consider ψ k := | η k | p − . Using Lemma 2.3 applied with r = , s = , g ( x ) = | x | p − ,the local energy inequality (26) for ψ k and proceeding as before, using also that p >
3, we thus have for any t ∈ [ − (13 / α ,
0] that Z B / ψ k ( x, t ) dx + Z t − ( ) α [ ψ k ( s )] W α, ( B / ) ds . Z B / | η k | p − ( x, t ) dx + Z t − ( ) α Z B ∗ / y b |∇| η ∗ k | p − | dx dy ds + Z Q ∗ / y b | η ∗ k | p − dx dy ds . Z Q / (cid:0) | η k | p − + | η k | p + | u k | p (cid:1) dx ds + Z Q ∗ / y b | η ∗ k | p − dx dy ds . Z Q (cid:0) | η k | p − + | η k | p + | u k | p (cid:1) dx ds + Z − sup R ≥ R ( p − σ (cid:18) − Z B R | η k ( x, s ) − [ η k ( s )] | dx (cid:19) p − ds . . Taking the supremum over t ∈ [ − (13 / α , k boundsup t ∈ [ − (13 / α , Z B / ψ k ( x, t ) dx + Z − ( ) α [ ψ k ( s )] W α, ( B / ) ds ≤ C .
From Sobolev embedding ˙ W α, ֒ → L − α , we obtain by interpolation that k ψ k k L α ) ( Q / ) ≤ C uniformly in k ≥ k ≥ k η k k L ( p − α ) ( Q / ) ≤ C .
Step 2: We use an Aubin-Lions type compactness argument to deduce strong convergence of η k in L q ( Q / ) for every ≤ q < ( p − α ) . Since ( p − α ) > p by hypothesis, wededuce in particular that η k → η strongly in L p ( Q / ) . We may assume q ∈ [2 , ( p − α )). Since the excess uniformly bounds the L / loc -norm of η k , there exists by Banach-Alaoglu a limit η ∈ L / loc ( R × [ − , η k ⇀ η weakly in L / loc ( R × [ − , η k ⇀ η weakly in L q ( Q / ). We now claim that the latter convergence is in fact strong on the slightly smallercube Q / . Indeed, fix ε > { φ δ } δ> of mollifiers in the space variable. For k, j ≥ k η k − η j k L q ( Q / ) ≤ k η k − η k ∗ φ δ k L q ( Q / ) + k η j − η j ∗ φ δ k L q ( Q / ) + k ( η k − η j ) ∗ φ δ k L q ( Q / ) . We claim that the first two contributions converge to 0 as δ →
0, uniformly in k and j . Indeed,we compute for δ small enough by H¨older and the uniform boundedness of η k in L W α, ( Q / ) k η k − η k ∗ φ δ k L ( Q / ) = Z − ( ) α Z B / (cid:12)(cid:12)(cid:12)(cid:12) Z ( η k ( x ) − η k ( y )) φ δ ( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) dx dt ≤ Z − ( ) α Z B / (cid:18) Z | η k ( x ) − η k ( y ) | | x − y | α | x − y |≤ δ dy (cid:19)(cid:18) Z φ δ ( x − y ) | x − y | α dy (cid:19) dx dt ≤ πδ α k φ k L ∞ Z − ( ) α Z B / δ Z B / δ | η k ( x ) − η k ( y ) | | x − y | α dy dx dt ≤ πδ α k φ k L ∞ Z − ( ) α [ η k ( t )] W α, ( B / ) dt ≤ Cδ α , (27)where C does not depend on k ≥ η k is uniformly bounded in L (1+ α )( p − ( Q / ),we have by interpolation for some ϑ ∈ (0 ,
1] and C ≥ k η k − η k ∗ φ δ k L q ( Q / ) ≤ Cδ αϑ . We now fix δ small enough, independently of k , such that this contribution does not exceed ε . As for the third term, we consider for fixed δ > { t η k ∗ φ δ } k ≥ .From the equation, we have the identity ∂ t ( η k ∗ φ δ ) = − ( u k · ∇ η k ) ∗ φ δ − ( − ∆) α η k ∗ φ δ . Observe that u k · ∇ η k = div( u k η k ) so that k ( u k · ∇ η k ) ∗ φ δ k L pqp + q ([ − (3 / α , ,W , ∞ ( B / )) ≤ k u k k L p ( Q / ) k η k k L q ( Q / ) k φ δ k W , pqpq − p − q ( B / ) . As for the last term, we have that for x ∈ B / | ( − ∆) α φ δ ( x ) | = c α (cid:12)(cid:12)(cid:12)(cid:12)Z φ δ ( x ) − φ δ ( y ) | x − y | α dy (cid:12)(cid:12)(cid:12)(cid:12) . k φ δ k C Z B dy | y | α + k φ δ k L ∞ Z B c dy | x − y | α ≤ C ( δ )1 + | x | α . Analogously, | ( − ∆) α ∇ φ δ ( x ) | ≤ C ( δ )1+ | x | α . We estimate the convolution on dyadic balls for fixedtime. We set η k,i := η k ( B i +1 − B i ) for i ≥ k ( − ∆) α φ δ ∗ η k k W , ∞ ( B / ) ≤ C ( δ ) (cid:18) Z B | η k | + X i ≥ k ( − ∆) α φ δ ∗ η k,i k W , ∞ ( B / ) (cid:19) . For i ≥ x ∈ B / and z ∈ B i +1 \ B i we have | x − z | ≥ i − , so that X i ≥ k ( − ∆) α φ δ ∗ η k,i k W , ∞ ( B / ) ≤ C ( δ ) X i ≥ ( i − α ) Z B i +1 \ B i | η k | dy ≤ C ( δ ) X i ≥ ( i − α − σ ) Z B i +1 \ B i | η k | (2+ σ )( i − dy ≤ C ( δ ) (cid:18) sup R ≥ R σ − Z B R | η k − [ η k ] | dy + Z B | η k | dy (cid:19) . We conclude by integrating in time that k ( − ∆) α φ δ ∗ η k k L p ([ − (3 / α , ,W , ∞ ( B / )) ≤ C ( δ ) (cid:0) E S ( η k ; 1) + E NL ( η k ; 1) (cid:1) . Summarizing, we have shown that k ∂ t ( η k ∗ φ δ ) k L pqp + q ([ − (3 / α , ,W , ∞ ( B / )) ≤ C ( δ )uniformly in k ≥
1. Hence the family of curves { t η k ∗ φ δ } k ≥ is an equicontinuous sequencewith values in a bounded subset of W , ∞ ( B / ). By Arzela-Ascoli there exists a uniformlyconvergent subsequence (which we don’t relabel), and in particular, there exists N = N ( δ ) > k, j ≥ N we have k ( η k − η j ) ∗ φ δ k L q ( Q / ) ≤ ε , hence k η k − η j k L q ( Q / ) ≤ ε for all k, j ≥ N which proves the claim. Step 3: Conclusion.
By Step 2 we can pass to the limit in the equation in Q / and deduce that η ∈ L p ( Q / ) is adistributional solution of ∂ t η + ( − ∆) α η = 0 in Q / . Moreover, by weak lower semicontinuity E S ( η ; 1) + E NL ( η ; 1) ≤ . (cid:3) Decay of the excess
In this section, we prove the self-improving property of the excess, namely that if the excessis small at any given Q r , there exists a small, fixed scale µ ∈ (0 , ), independent of r , at whichthe excess decays between Q r and Q µ r - provided that the velocity field has zero average on B r . This requirement is crucial to guarantee the decay of the excess related to the non-localpart of the velocity (see E V ( v k ; µ ) in the proof of Proposition 4.1). More generally, one couldprove this excess decay at scale µ under the weaker assumption that the average of the rescaledvelocity u r on B is bounded uniformly in r for r ∈ (0 , L p -norms aresupercritical with respect to the scaling (4) of the equation, we will not be able to guaranteesuch an assumption. In this section, we will also for the first time make use of the structure ofthe velocity field (3). Similar arguments should apply for velocity fields determined from θ byother singular integral operators. Proposition 4.1 (Excess decay) . Let α ∈ (0 , ) , σ ∈ (0 , α ) and p > max (cid:8) αα , ασ (cid:9) . Forany c > and any γ ∈ (0 , σ − αp ) there exist universal ε = ε ( α, σ, p, c, γ ) ∈ (0 , ) and µ = µ ( α, σ, p, c, γ ) ∈ (cid:0) , (cid:1) such that the following holds: Let Q r ( x, t ) ⊆ R × (0 , ∞ ) and let ( θ, u ) be a suitable weak solution to (1) – (2) . We assume that the velocity field satisfies [ u ( s )] B r ( x ) = 0 for all s ∈ [ t − r α , t ] and is obtained from θ by u ( y, s ) = R ⊥ θ ( y, s ) + f ( s ) (28) for some f ∈ L ([ t − r α , t ]) . Then, if E ( θ, u ; x, t, r ) ≤ r − α ε , the excess decays at scale µ ,that is E ( θ, u ; x, t, µ r ) ≤ cµ γ E ( θ, u ; x, t, r ) . Remark . If in (28) f = 0 we recover simply the SQG equation. We will need the freedomto subtract a function of time f from the velocity field u in order to satisfy the zero-averageassumption (see Lemma 5.1).We will need the following auxiliary Lemma. Lemma 4.3.
Assume θ ∈ L loc ( R ) with supp θ ⊆ (cid:0) B / (cid:1) c and that for some σ ∈ (0 , we have sup R ≥ R σ − Z B R | θ ( x ) | dx < + ∞ . Then R ⊥ θ ∈ C ∞ ( B / ) and there exists a universal C = C ( σ ) > such that k D R ⊥ θ k L ∞ ( B / ) ≤ C sup R ≥ R σ − Z B R | θ ( x ) | dx . Proof of Lemma 4.3.
Observe that from ( − ∆) R ⊥ θ = 0 on B / , we infer that R ⊥ θ ∈ C ∞ ( B / ) . Moreover for i, j = 1 , x ∈ B / , we notice that the integral representation is no longersingular and we can compute by integration by parts ∂ j R i θ ( x ) = Z | z |≥ x i − z i | x − z | ∂ j θ ( z ) dz = − Z | z |≥ ∂ j (cid:18) x i − z i | x − z | (cid:19) θ ( z ) dz , where we used that the boundary terms at {| z | = } and at infinity vanish. Observe that for x ∈ B / and z ∈ B i +1 \ B i we have | x − z | ≥ i − ≥ i − for i ≥ − . Thus | ∂ j R i θ ( x ) | ≤ Z ≤| z |≤ (cid:12)(cid:12)(cid:12) ∂ j (cid:16) x i − z i | x − z | (cid:17) θ ( z ) (cid:12)(cid:12)(cid:12) dz + X i ≥− Z B i +1 \ B i (cid:12)(cid:12)(cid:12) ∂ j (cid:16) x i − z i | x − z | (cid:17) θ ( z ) (cid:12)(cid:12)(cid:12) dz ≤ C (cid:18) Z B | θ ( z ) | dz + X i ≥− Z B i +1 \ B i | θ ( z ) | i − dz (cid:19) ≤ C (cid:16) X i ≥− ( i − − σ ) (cid:17)(cid:18) sup R ≥ R σ − Z B R | θ ( z ) | dz (cid:19) . (cid:3) Proof of Proposition 4.1.
By translation and scaling invariance, we may assume w.l.o.g. ( x, t ) =(0 ,
0) and r = 1 . We argue by contradiction. Then there exists a sequence ( θ k , u k ) of suitableweak solutions to (1)–(2) such that • E ( θ k , u k ; µ ) > cµ γ E ( θ k , u k ; 1) for all µ ∈ (0 , ) , • lim k →∞ E ( θ k , u k ; 1) = 0 , • [ u k ( s )] B = 0 for all s ∈ [ − ,
0] and for all k ≥ • u k ( y, s ) = R ⊥ θ ( y, s ) + f k ( s ) for some f k ∈ L ([ − , . We set E k := E ( θ k , u k ; 1) and M k := ( θ k ) Q . We will consider the rescaled and shifted sequence η k := θ k − M k E k and v k := u k E k . By construction, ( η k ) Q = 0 and E ( η k , v k ; 1) = 1 . In particular, we have for all µ ∈ (0 , ) that E ( η k , v k ; µ ) > cµ γ . (29)We will now take the limit k → ∞ and argue that (29) contradicts the excess decay dictatedby the linear limit equation. Indeed, by Lemma 3.8, the sequence η k converges weakly to η in L / loc ( R × [ − , η k → η strongly in L p ( Q / ) . Hence we have for µ ∈ (0 , ) E S ( η ; µ ) = lim k →∞ E S ( η k ; µ ) . We also know from Lemma 3.8 that η ∈ L p ( Q / ) solves the fractional heat equation ∂ t η +( − ∆) α η = 0 on Q / with E S ( η ; 1) + E NL ( η ; 1) ≤ . In particular, we deduce from Lemma A.1that η is smooth (in space) on Q / and that η ∈ C − /p ( Q / ) with the estimate k η k L ∞ ([ − (1 / α , ,C ( B / )) + k η k C − p ( Q / ) ≤ ¯ C ( E S ( η ; 1) + E NL ( η ; 1)) ≤ ¯ C . (30)In particular, we infer that for µ ∈ (0 , )lim k →∞ E S ( η k ; µ ) = E S ( η ; µ ) ≤ ¯ Cµ α (1 − p ) . (31)Let us now consider the non-local part of the excess. We split E NL ( η k ; µ ) ≤ (cid:18) − Z − µ α sup µ ≤ R< (cid:16) µR (cid:17) σp (cid:18) − Z B R | η k ( x, t ) − [ η k ( t )] B µ | dx (cid:19) p dt (cid:19) p + (cid:18) − Z − µ α sup R ≥ (cid:16) µR (cid:17) σp (cid:18) − Z B R | η k ( x, t ) − [ η k ( t )] B µ | dx (cid:19) p dt (cid:19) p . We estimate the second term by adding and subtracting [ η k ( t )] B for fixed time t . In the sequel C ′ = C ′ ( α, σ ) will denote a universal constant which may change line by line. Using that E NL ( η k ; 1) + E S ( η k ; 1) ≤ k ≥
1, we obtain that (cid:18) − Z − µ α sup R ≥ (cid:16) µR (cid:17) σp (cid:18) − Z B R | η k ( x, t ) − [ η k ( t )] B µ | dx (cid:19) p dt (cid:19) p ≤ µ σ − αp E ( η k ; 1) + (4 µ ) σ (cid:18) − Z − µ α | [ η k ( t )] B − [ η k ( t )] B µ | p dt (cid:19) p ≤ µ σ − αp + (4 µ ) σ (cid:18) − Z − µ α | [ η k ( t )] B − [ η k ( t )] B / | p dt (cid:19) p + (cid:18) − Z − µ α | [ η k ( t )] B / − [ η k ( t )] B µ | p dt (cid:19) p ! ≤ µ σ − αp + C ′ µ σ − αp E S ( η k ; 1) + (4 µ ) σ (cid:18) − Z − µ α | [ η k ( t )] B / − [ η k ( t )] B µ | p dt (cid:19) p ≤ C ′ µ σ − αp + (4 µ ) σ (cid:18) − Z − µ α | [ η k ( t )] B / − [ η k ( t )] B µ | p dt (cid:19) p . We infer, using the strong converge of η k → η in L p ( Q / ), thatlim inf k →∞ E NL ( η k ; µ ) ≤ C ′ µ σ − αp + (4 µ ) σ (cid:18) − Z − µ α | [ η ( t )] B / − [ η ( t )] B µ | p dt (cid:19) p + (cid:18) − Z − µ α sup µ ≤ R< (cid:16) µR (cid:17) σp (cid:18) − Z B R | η ( x, t ) − [ η ( t )] B µ | dx (cid:19) p dt (cid:19) p . Using (30) again, we obtain that | [ η ( t )] B µ − [ η ( t )] B / | ≤ ¯ C uniformly in time as well as − Z − µ α sup µ ≤ R< (cid:16) µR (cid:17) σp (cid:18) − Z B R | η ( x, t ) − [ η ( t )] B µ | (cid:19) p ! p ≤ ¯ C (cid:18) − Z − µ α sup µ ≤ R< (cid:16) µR (cid:17) σp R p (cid:19) p ≤ ¯ Cµ σ . We conclude that lim inf k →∞ E NL ( η k ; µ ) ≤ C ′ µ σ − αp . (32)Finally, let us consider the part of the excess which is related to the velocity v k . We observethat, using the structure of the velocity (28), E V ( v k ; µ ) = (cid:18) − Z Q µ (cid:12)(cid:12) R ⊥ (cid:0) E − k θ k (cid:1) ( x, t ) − (cid:2) R ⊥ (cid:0) E − k θ k (cid:1) ( t ) (cid:3) B µ (cid:12)(cid:12) p dx dt (cid:19) p . We write R ⊥ ( E − k θ k ) = v k + v k where we introduce v k := R ⊥ ( η k χ )for some cut-off χ between B / and B / . Correspondingly, we write E V ( v k ; µ ) ≤ (cid:18) − Z Q µ | v k − [ v k ( t )] B µ | p dx dt (cid:19) p + (cid:18) − Z Q µ | v k − [ v k ( t )] B µ | p dx dt (cid:19) p . By Calderon-Zygmund estimates, we infer that v k → R ⊥ ( ηχ ) =: v strongly in L p ( Q / ).Moreover by Schauder estimates [28, Proposition 2.8], we have for fixed time[ v ( t )] C − p ( R ) ≤ C ′ k η ( t ) χ k C − p ( R ) ≤ C ′ k η ( t ) k C − p ( Q ) ≤ C ′ ¯ C uniformly in t ∈ [ − (1 / α ,
0] by (30). We conclude thatlim k →∞ (cid:18) − Z Q µ | v k − [ v k ( t )] B µ | p dx dt (cid:19) p = (cid:18) − Z Q µ | v − [ v ( t )] B µ | p dx dt (cid:19) p ≤ C ′ µ − p . We now come to the excess related to v k . By construction, v k = R ⊥ (cid:18) θ k E k − η k χ (cid:19) = R ⊥ (cid:18) η k (1 − χ ) + M k E k (cid:19) . Correspondingly, we define w k,ρ := R ⊥ ( η k (1 − χ ) χ ρ ) and w k,ρ := R ⊥ (cid:16) M k E k χ ρ (cid:17) for some radiallysymmetric cut-off χ ρ between B ρ and B ρ +1 . By Calderon-Zygmund, we have for fixed time t that w k,ρ ( t ) + w k,ρ ( t ) → v k ( t ) as ρ → ∞ strongly in L p ( R ) . In particular, (cid:18) − Z Q µ | v k − [ v k ( t )] B µ | p dx dt (cid:19) p = lim ρ →∞ (cid:18) − Z Q µ | w k,ρ + w k,ρ − [( w k,ρ + w k,ρ )( t )] B µ | p dx dt (cid:19) p ≤ lim sup ρ →∞ (cid:18) − Z Q µ | w k,ρ − [ w k,ρ ( t )] B µ | p dx dt (cid:19) p + lim sup ρ →∞ (cid:18) − Z B µ | w k,ρ − [ w k,ρ ( t )] B µ | p dx dt (cid:19) p . Let us consider first w k,ρ . We apply Lemma 4.3 to w k,ρ to deduce that for fixed time t [ w k,ρ ( t )] Lip(B / ) ≤ C ′ sup R ≥ R σ − Z B R | η k | ( x, t ) dx ≤ C ′ (cid:18) sup R ≥ R σ − Z B R | η k ( x, t ) − [ η k ( t )] B | dx + Z B | η k | ( x, t ) dx (cid:19) . Integrating in time, we infer that uniformly in ρ ≥ k w k,ρ k L p ([ − , , Lip(B / )) ≤ C ′ (cid:0) E S ( η k ; 1) + E NL ( η k ; 1) (cid:1) . We deduce that for µ ∈ (0 , ) we havelim ρ →∞ (cid:18) − Z Q µ | w k,ρ ( x, t ) − [ w k,ρ ( t )] B µ | p dx dt (cid:19) p ≤ C ′ µ . (33)We now come to the contribution of w k,ρ . Observe that ( − ∆) / w k,ρ = 0 in B for ρ ≥ w k,ρ is smooth in the inside of B . Recall moreover, that we have the integral representation(in the principal value sense) (cid:0) w k,ρ (cid:1) ⊥ ( x ) = − c M k E k Z x − y | x − y | χ ρ ( y ) dy so that, by spherical symmetry of χ ρ , we immediately infer w k,ρ (0) = 0 . Moreover, for x ∈ B / we have | w k,ρ ( x ) | = c | M k | E k (cid:12)(cid:12)(cid:12)(cid:12) Z ρ< | x − y | <ρ +1 y | y | χ ρ ( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | M k | E k π (( ρ + 1) − ρ ) (cid:16) ρ (cid:17) − = C ′ | M k | E k ρ − . Thus for fixed k ≥ , we have lim ρ → k w k,ρ k L ∞ ( B / ) = 0 , so that excess associated to v k, is controlled by (33). Collecting the terms (31) and (32)and taking the limes inferior k → ∞ in (29) we have obtained, for a universal constant C ′ = C ′ ( α, σ ) > , that cµ γ ≤ C ′ µ σ − αp = µ γ C ′ µ σ − αp − γ . for all µ ∈ (0 , ) . We reach the desired contradiction for µ ≤ (cid:16) cC ′ (cid:17) σ − αp − γ . (cid:3) Iteration of the excess decay
In this section, we prove the decay of the excess on all scales. We iteratively define shiftedrescalings of ( θ, u ) verifying the zero average assumption of Proposition 4.1 as well as (28) andtherefore allowing the decay of the excess when passing at scale µ . From the decay of the excesson all scales, we deduce H¨older continuity by means of Campanato’s Theorem. In contrast toNavier-Stokes, we need our estimates to be quantitative, since it is not known whether localsmoothness for SQG follows from a mere L ∞ -bound; instead we need to prove spatial C δ -H¨oldercontinuity of the velocity for a δ > − α (see Lemma B.1). The main mechanism of theiteration is the invariance of the equation under the following change of variables realizing thezero average assumption on B / . The latter has been exploited previously in [4, 11].
Lemma 5.1 (Change of variables at unit scale) . Let α ∈ ( , ) and σ ∈ (0 , α ) . Let ( θ, u ) bea suitable weak solution of (1) on R × [ − α , . Fix ( x, t ) ∈ Q . Define θ ( y, s ) := θ ( y + x + x ( s ) , s + t ) and u ( y, s ) := u ( y + x + x ( s ) , s + t ) − ˙ x ( s ) , where ˙ x ( s ) = − Z B u ( y + x ( s ) + x, s + t ) dyx (0) = 0 . (34) Then ( θ , u ) is a suitable weak solution of (1) on R × [ − , with [ u ( s )] B / = 0 for s ∈ [ − , . Moreover, there exists universal ε = ε ( p, α ) ∈ (0 , ) and C ≥ such that if − Z Q ( x,t ) | u | p dy ds ! p ≤ ε , (35) then E ( θ , u ; 14 ) ≤ C E ( θ, u ; x, t, . Remark r ) . Under the hypothesis of Lemma 5.1, the rescaledpair ( θ r , u r ) is still a suitable weak solution for r ∈ (0 ,
1) (see (4)) and we can apply the changeof variables of Lemma 5.1 to it. More precisely, we define for ( x, t ) ∈ Q θ r, ( y, s ) := r α − θ ( r ( y + x ( s ) + x ) , r α ( s + t ))and u r, ( y, s ) = r α − u ( r ( y + x ( s ) + x ) , r α ( s + t )) − ˙ x ( s ) , where ˙ x ( s ) = r α − − R B / u ( r ( y + x ( s )+ x ) , r α ( s + t )) with x (0) = 0 . Observe that equivalently,by considering ˜ x ( s ) := r − α x ( s ), we can write θ r, ( y, s ) := r α − θ ( ry + r α ˜ x ( s ) + rx, r α ( s + t ))and u r, ( y, s ) = r α − (cid:0) u ( ry + r α ˜ x ( s ) + rx, r α ( s + t )) − ˙˜ x ( s ) (cid:1) . Proof.
By Peano, the ODE (34) admits a solution and we claim it is unique since the vectorfieldgenerating the flow is log-Lipschitz and hence satisifies the Osgood uniqueness criterion [33,Chapter II.7 and III.12.7-8]. Indeed, we know from Theorem 3.2 that θ ∈ L ∞ ( R × [ − , u ∈ L ∞ ([ − , , BM O ( R )) . We estimate for fixed time τ as long as | ξ − ζ | ≤ , using also thebound | B ( ξ )∆ B ( ζ ) | . | ξ − ζ | on the volume of the symmetric difference, (cid:12)(cid:12)(cid:12)(cid:12) − Z B ( ξ ) u ( y + x, τ ) dy − − Z B ( ζ ) u ( y + x, τ ) dy (cid:12)(cid:12)(cid:12)(cid:12) . | ξ − ζ | (cid:12)(cid:12)(cid:12)(cid:12) − Z B ( ξ ) \ B ( ζ ) u ( y + x, τ ) dy − − Z B ( ζ ) \ B ( ξ ) u ( y + x, τ ) dy (cid:12)(cid:12)(cid:12)(cid:12) . | ξ − ζ | (cid:18) − Z B ( ξ ) \ B ( ζ ) (cid:12)(cid:12) u ( y + x, τ ) − [ u ( · + x, τ )] B / ( ( ξ + ζ )) (cid:12)(cid:12) dy + − Z B ( ζ ) \ B ( ξ ) (cid:12)(cid:12) u ( y + x, τ ) − [ u ( · + x, τ )] B / ( ( ξ + ζ )) (cid:12)(cid:12) dy (cid:19) . Recall from the John-Nirenberg inequality [18] that BMO functions are exponentially integrable,that is for every f ∈ BM O ( R ) there exist constants c , c > B in R |{ x ∈ B : | f − [ f ] B | > λ }| ≤ c exp( − c λ k f k − BMO ) | B | . As an immediate consequence, we observe that for A ≥ C k f k BMO and any ball B in R sup B − Z B e | f − [ f ] B | A dx < + ∞ . We now estimate the last two contributions, setting z := x + ( ξ + ζ ) and using Jensen − Z B ( ξ ) \ B ( ζ ) (cid:12)(cid:12) u ( y + x, τ ) − [ u ( · + x, τ )] B / ( ( ξ + ζ )) (cid:12)(cid:12) dy . k u ( τ ) k BMO log (cid:18) | B ( ξ ) \ B ( ζ ) | − − Z B ( z ) e A | u ( y,τ ) − [ u ( τ )] B / z ) | dy (cid:19) . k u ( τ ) k BMO log( C | ξ − ζ | − ) . We infer that the function ξ − R B / ( ξ ) u ( y, s + t ) is log-Lipschitz in space for s ∈ [ − , x, t ) ∈ Q is log-Lipschitz. The functions θ and u as in the statementare now well-defined. We remark first that, in the sense of distributions, div u = 0 and ∂ s θ + u · ∇ y θ = ( ∂ t θ + ˙ x ( s ) · ∇ θ ) (cid:12)(cid:12) ( y + x + x ( s ) ,s + t ) + ( u · ∇ θ − ˙ x ( s ) · ∇ θ ) (cid:12)(cid:12) ( y + x + x ( s ) ,s + t ) = ( − ∆) α θ ( y + x + x ( s ) , s + t )= ( − ∆) α θ , so that ( θ , u ) is a distributional solution of (1). It is straightforward to check that ( θ , u ) isin fact a suitable weak solution. Moreover, u ( y, s ) = R ⊥ θ ( y, s ) − ˙ x ( s ) and[ u ( s )] B = − Z B u ( y + x + x ( s ) , s + t ) dy − ˙ x ( s ) = 0by construction. Assume now that (35) holds for an ε ∈ (0 , ) yet to be chosen small enough.As long as x ( s ) ∈ B / we estimate | ˙ x ( s ) | ≤ (cid:18) − Z B + x ( s ) | u | p ( x + y, s + t ) dy (cid:19) p ≤ p (cid:18) − Z B | u | p ( x + y, s + t ) dy (cid:19) p , so that for s ∈ ( − , | x ( s ) | ≤ k ˙ x k L p (( − , | s | − p ≤ p (cid:18) − Z Q ( x,t ) | u | p dy dτ (cid:19) p | s | − p . Choosing ε ≤ − p the assumption (35) guarantees that x ( s ) ∈ B / ⊂ B / for s ∈ [ − , . We then estimate, using again that B / + x ( s ) ⊆ B for s ∈ [ − , E S ( θ ; 14 ) ≤ (cid:18) − Z Q | θ ( y + x + x ( s ) , s + t ) − ( θ ) Q ( x,t ) | p dy ds (cid:19) p + | ( θ ) Q − ( θ ) Q ( x,t ) |≤ (cid:18) − Z − (1 / α − Z B + x ( s ) | θ ( x + y, s + t ) − ( θ ) Q ( x,t ) | p dy ds (cid:19) p ≤ (2+2 α ) p (cid:18) − Z Q ( x,t ) | θ − ( θ ) Q ( x,t ) | p dy ds (cid:19) p ≤ E S ( θ ; x, t, . Proceeding analogously, we have that E V ( u ; ) ≤ E V ( u ; x, t, . As for the non-local part ofthe excess, we observe that for R ≥ we have B R ( x ( s )) ⊆ B R +3 / ⊆ B R and hence E NL ( θ ; 14 ) ≤ (cid:18) − Z − (1 / α sup R ≥ (cid:26)(cid:16) R (cid:17) σp (cid:18) − Z B R | θ − [ θ ] B ( x ) | dy (cid:19) p + | [ θ ] B − [ θ ] B ( x ) | p (cid:27) ds (cid:19) p ≤ (cid:18) − Z − (1 / α sup R ≥ (cid:16) R (cid:17) σp (cid:18) − Z B R | θ ( y + x, s + t ) − [ θ ] B ( x ) | dy (cid:19) p ds (cid:19) p + 4 σ +1 E S ( θ ; x, t, ≤ (cid:0) E NL ( θ ; x, t,
1) + E S ( θ ; x, t, (cid:1) . (cid:3) Theorem 5.3.
Let α ∈ ( , ) , σ ∈ (0 , α ) , p > max (cid:8) αα , ασ (cid:9) and γ ∈ [1 − α, σ − αp ) . Thereexists ε ∈ (cid:0) , (cid:1) (depending only on α, σ, p and γ ) such that the following holds: Let ( θ, u ) bea suitable weak solution to (1) – (3) on R × ( − α , . Assume that for any ( x, t ) ∈ Q it holdsthat E ( θ , u ; 14 ) ≤ ε , (36) where ( θ , u ) is obtained from θ through the change of variables of Lemma 5.1. Then θ ∈ C δ, p − p δ ( Q ) where δ := γ − p − (cid:20) − α + 2 αp (cid:21) . (37) Remark . The parameter p is crucial since it determines the di-mension of the singular set (see proof of Theorem 1.1): the lower the power p, the better thedimension estimate. All the other parameters are of technical nature; yet, the range of admis-sible parameters is sufficiently large to allow us to conclude the desired estimate on the size ofthe singular set for all fractional orders for which the latter is meaningful (i.e. for α > α , seeRemark 6.7). We deliberately choose to leave all the parameters free to increase the readabilityof the paper; but one could also read the paper fixing the parameters as in the proof of Theorem1.3. Let us now comment on the role of the single parameters in more detail: • p cannot go below the threshold αα : This corresponds to spacetime integrability thatguarantees the compactness of the ( p − the crucial ingredient of the excess decay. The requirement p > ασ on the otherhand is purely technical and harmless for σ close to 2 α . • σ captures the decay at infinity of the non-local part of both the fractional Laplacianand the velocity and should be thought arbitrarily close to 2 α . • γ describes the decay of the excess when passing to the smaller scale µ . In order to applythe excess decay of Proposition 4.1 iteratively, we have to verify its smallness requirementalong a sequence of by µ := µ rescaled solutions which is possible only if the decay ratebeats the supercritical scaling of the excess (see Remark 3.7), i.e. γ ≥ − α . • The exponent of the local H¨older continuity in space, δ , is obtained from γ , but consid-erably worsened. This stems form the fact that in order to use the decay of the excess onall scales to deduce H¨older continuity via Campanato’s Theorem, we have to control the effect of the flow of Lemma 5.1. This loss in the H¨older continuity exponent is peculiarto SQG and is not observed in the similar results for Navier-Stokes.
Proof.
Let ε > C ≥ x, t ) ∈ Q . We obtain the suitableweak solution ( θ , u ) by applying Lemma 5.1 to ( θ, u ) at the point ( x, t ) . This first change ofvariables does two things: It translates ( x, t ) to the origin (0 ,
0) and it produces a new suitableweak solution whose velocity u has zero average on B / . Hereafter, the excess will always becentered in (0 , . Let µ := µ ∈ (0 , ) where µ , ε ∈ (0 , ) are given by Proposition 4.1 with c = (16 C ) − .For k ≥ θ k , u k ) which we obtain from( θ k − , u k − ) by first rescaling it at scale µ according to (4), i.e. we set θ k − ,µ ( y, s ) := µ α − θ k − ( µy, µ α s ) u k − ,µ ( y, s ) := µ α − u k − ( µy, µ α s ) , and second, by applying the change of variables of Lemma 5.1 to ( θ k − ,µ , u k − ,µ ) at the point(0 , . This change of variables produces a new suitable weak solution, which we call ( θ k , u k ) , that evolves along the flow ˜ x k and whose velocity u k has zero average on B / . Indeed, setting x k ( s ) := µ α − ˜ x k ( s ) (compare also with Remark 5.2), we define iteratively for k ≥ θ k ( y, s ) := θ k − ,µ ( y + ˜ x k ( s ) , s ) = µ α − θ k − ( µy + µ α x k ( s ) , µ α s ) (38)and u k ( y, s ) := u k − ,µ ( y + ˜ x k ( s ) , s ) − ˙˜ x k ( s ) = µ α − ( u k − ( µy + µ α x k ( s ) , µ α s ) − ˙ x k ( s )) , (39)where ˙ x k ( s ) = − Z µ α − x k ( s )+ B u k − ( µy, µ α s ) dyx k (0) = 0 . Observe that by Lemma 5.1 and scaling invariance, ( θ k , u k ) are suitable weak solutions of (1)for all k ≥ u k ( s )] B / = 0 for all s ∈ [ − , . Next, we want to deduce the H¨older continuity of θ assuming (36) is enforced. To this end,we break the parabolic scaling and we consider in Step 3 a new excess of θ made on modifiedcylinders. This in turn is helpful to get sharper estimates at the level of the change of variable,performed in Step 2, since the translation has less time to act. Finally, we rewrite this decay inStep 4 in terms of θ rather than θ , and we apply Campanato’s Theorem to deduce the H¨oldercontinuity of θ in Step 5. Step 1: excess decay on the sequence of solutions after the change of variable. Let α , σ , p and γ as in the statement. There exists a universal constant ¯ ε ∈ (0 , ) (depending only on α , σ , γ and p ) such that if ε ∈ (0 , ¯ ε ] and if ( θ, u ) is a suitable weak solution to (1) on R × ( − α , with (36) , then for every k ≥ the excess of ( θ k , u k ) (see (38) – (39) ), decays at scale µ : E ( θ k , u k ; µ ) ≤ C − µ γ ( k +1) µ (2 α − k ε (40) E ( θ k , u k ; 14 ) ≤ µ ( γ − (1 − α )) k ε , (41) where C is the universal constant from Lemma 5.1. We proceed by induction on k ≥ The case k = 0 . Let ε ∈ (0 , ¯ ε ] for some ¯ ε ∈ (0 , ) to be chosen later and assume that (36)holds. We only need to show (40). If ¯ ε ≤ α − ε , then by Proposition 4.1 and (36) E ( θ , u ; µ ) = E ( θ , u ; µ ≤ C − (cid:16) µ (cid:17) γ E ( θ , u ; 14 ) ≤ C − µ γ ε . The inductive step.
By the inductive hypothesis, we can assume that • E ( θ k − , u k − ; µ ) ≤ C − µ kγ µ ( k − α − ε , • E ( θ k − , u k − ; ) ≤ µ ( k − γ − (1 − α )) ε . We recall that ( θ k , u k ) is obtained by applying the change of variables of Lemma 5.1 to ( θ k − ,µ , u k − ,µ )at the point (0 , u k − ( s )] B / = 0 for s ∈ [ − , (cid:18) − Z Q | u k − ,µ | p dy ds (cid:19) p = µ α − (cid:18) − Z Q µ | u k − | p dy ds (cid:19) p ≤ µ α − (4 µ ) − (2+2 α ) p (cid:18) − Z Q | u k − | p dy ds (cid:19) p = µ α − (4 µ ) − (2+2 α ) p (cid:18) − Z Q | u k − ( y, s ) − [ u k − ( s )] B | p dy ds (cid:19) p = µ α − (4 µ ) − (2+2 α ) p E V ( u k − ; 14 ) ≤ µ α − (4 µ ) − (2+2 α ) p ε . Choosing ¯ ε even smaller, namely,¯ ε := min (cid:8) (4 α − ε , µ − α (4 µ ) (2+2 α ) p ε (cid:9) we have (cid:18) − Z Q | u k − ,µ | p dy ds (cid:19) p ≤ ε . By Lemma 5.1, Remark 3.7 and the inductive hypothesis, we deduce that E ( θ k , u k ; 14 ) ≤ C E ( θ k − ,µ , u k − ,µ ; 1) = C µ α − E ( θ k − , u k − ; µ ) ≤ µ k ( γ − (1 − α )) ε , showing the second inequality and, recalling the choice of ε ∈ (0 , ¯ ε ), showing in particular that E ( θ k , u k ; 14 ) ≤ α − ε . Since by construction [ u k ( s )] B / = 0 for s ∈ [ − , E ( θ k , u k ; µ ) ≤ C − µ γ E ( θ k , u k ; 14 ) ≤ C − µ γ µ k ( γ − (1 − α )) ε = C − µ ( k +1) γ µ k (2 α − ε . Step 2: bound on the translation in the change of variables. We observe that θ k is just ashifted and rescaled (by µ k , according to the natural scaling (4) ) version of θ . Indeed, noticethat by construction, one can verify inductively for k ≥ θ k ( y, s ) = θ ,µ k ( y + µ − k r k ( s ) , s ) , (42) where θ ,µ k ( y, s ) := µ k (2 α − θ ( µ k y, µ αk s ) and r k ( s ) := µ α − k X j =1 µ j x j ( µ α ( k − j ) s ) . We claim that the center of the cylinders don’t move too much, namely for s ∈ [ − , | r k ( s ) | ≤ Cε | s | − p µ αk (1 − p ) − (1 − α + p ) . Indeed, for j ≥ x j ( s ) ∈ B / | ˙ x j ( s ) | ≤ (cid:18) − Z µ α x j ( s )+ B µ | u j − ( y, µ α s ) | p dy (cid:19) p ≤ µ − p (cid:18) − Z B | u j − ( y, µ α s ) − [ u j − ( µ α s )] B | p dy (cid:19) p , where we used that [ u j − ( µ α s )] B / = 0 uniformly in time. In particular, k ˙ x j k L p (( − , ≤ µ − p − αp E V ( u j − ; 14 ) ≤ µ − p E ( θ j − , u j − ; 14 )and hence for s ∈ [ − ,
0] we have, using (41), | x j ( s ) | ≤ µ − p E ( θ j − , u j − ; 14 ) | s | − p ≤ ε | s | − p µ − p . Collecting terms, we have | r k ( s ) | ≤ ε | s | − p µ αk (1 − p ) µ α − µ − p k X j =1 µ (1 − α (1 − p )) j ≤ Cε | s | − p µ αk (1 − p ) − (1 − α + p ) . Step 3: Decay of a modified excess of θ . We claim that for every r ∈ (0 , µ ) (cid:18) − Z − r pp − − Z B r | θ − ( θ ) B r × ( − r pp − , | p dy ds (cid:19) p ≤ C − µ − r γ − h − αp − + αp ( p − i ε . (43)Observe that by the scaling of the excess µ α − E S ( θ k ; µ ) = E S ( θ k,µ ; 1) and by (42) θ k,µ ( y, s ) = θ ,µ k +1 ( y + µ − ( k +1) r k ( µ α s ) , s ) . We introduce the set I k +1 := (cid:16) − µ ( k +1)((1 − α ) pp − + αp − ) , i . If s ∈ I k +1 we can ensure, by an appropriate choice of ε , that r k ( µ α s ) ∈ B µ k +1 / . Indeed | r k ( µ α s ) | ≤ Cε µ α ( k +1)(1 − p ) − (1 − α + p ) µ ( k +1)(1 − α ) µ ( k +1) αp = Cε µ k +1 µ − (1 − α + p ) , for s ∈ I k +1 by Step 2. It is thus enough to choose ε (if necessary) even smaller, or moreprecisely, we set ε := min (cid:26) ¯ ε , C − µ (1 − α + p ) (cid:27) . We now estimate, by adding and substracting ( θ k,µ ) Q and H¨older (cid:18) − Z I k +1 − Z B | θ ,µ k +1 ( y, s ) − ( θ ,µ k +1 ) B × I k +1 | p (cid:19) p ≤ (cid:18) − Z I k +1 − Z B | θ ,µ k +1 ( y, s ) − ( θ k,µ ) Q | p dy ds (cid:19) p . Since µ − ( k +1) r k ( µ α s ) ∈ B / we have B / ⊆ µ − ( k +1) r k ( µ α s ) + B as long as s ∈ I k +1 , so that (cid:18) − Z I k +1 − Z B | θ ,µ k +1 ( y, s ) − ( θ k,µ ) Q | p (cid:19) p ≤ p (cid:18) − Z I k +1 − Z µ − ( k +1) r k ( µ α s )+ B | θ ,µ k +1 ( y, s ) − ( θ k,µ ) Q | p dy ds (cid:19) p = 4 p (cid:18) − Z I k +1 − Z B | θ ,µ k +1 ( y + µ − ( k +1) r k ( µ α s ) , s ) − ( θ k,µ ) Q | p dy ds (cid:19) p ≤ p | I k +1 | − p (cid:18) − Z Q | θ k,µ ( y, s ) − ( θ k,µ ) Q | p dy ds (cid:19) p = 4 p | I k +1 | − p E S ( θ k,µ ; 1) = 4 p | I k +1 | − p µ α − E S ( θ k ; µ ) . Combining the previous inequality with Step 1 and observing that µ ( k +1)2 α I k +1 = (cid:0) − µ ( k +1) pp − , (cid:3) ,we deduce that for k ≥ (cid:18) − Z µ ( k +1)2 α I k +1 − Z B µk +1 | θ ( y, s ) − ( θ ) B µk +1 × µ ( k +1)2 α I k +1 | p dy ds (cid:19) p = µ ( k +1)(1 − α ) (cid:18) − Z I k +1 − Z B | θ ,µ k +1 ( y, s ) − ( θ ,µ k +1 ) B × I k +1 | p (cid:19) p ≤ C − | I k +1 | − p µ ( k +1) γ ε = 4 C − µ ( k +1) (cid:16) γ − h − αp − + αp ( p − i(cid:17) ε . This gives (43) for r = µ k +1 for some k ≥ . For r ∈ ( µ k +2 , µ k +1 ) instead, we observe that (cid:18) − Z − r pp − − Z B r | θ − ( θ ) B r × ( − r pp − , | p dy ds (cid:19) p ≤ (cid:18) µ k +1 r (cid:19) p (2+ pp − ) (cid:18) − Z − µ ( k +1) pp − − Z B µk +1 | θ − ( θ ) B µk +1 × ( − µ ( k +1) pp − , | p dy ds (cid:19) p ≤ C − (cid:18) µ k +1 r (cid:19) p (2+ pp − ) µ ( k +1) (cid:16) γ − h − αp − + αp ( p − i(cid:17) ε ≤ C − µ − r (cid:16) γ − h − αp − + αp ( p − i(cid:17) ε . Step 4: Decay of a modified excess of θ . There exists a r = r ( k u k L p +1 ( Q / ) ) > such thatfor every r ∈ (0 , r ) and for every ( x, t ) ∈ Q (cid:18) − Z tt − r pp − − Z B r ( x ) | θ − ( θ ) B r ( x ) × ( t − r pp − ,t ] | p dy ds (cid:19) p ≤ µ − r γ − h − αp − + αp ( p − i ε . Since by Theorem 3.2 u ∈ L ∞ ([ − (3 / α , , BM O ( R )) , we have u ∈ L qloc ( R × [ − (3 / α , q ∈ [1 , ∞ ) . Fix ( x, t ) ∈ Q . As long as x ( s ) ∈ B / and | s | < , we have the estimate | x ( s ) | ≤ | s | − p +1 k u k L p +1 ( Q / ) . (44)In particular for 0 ≤ | s | pp +1 ≤ min n k u k − L p +1 ( Q / ) , − pp +1 o the estimate (44) holds. Let now r := min (cid:26) µ , (cid:18) k u k − L p +1 ( Q / ) (cid:19) − p , (cid:18) k u k − L p +1 ( Q / ) (cid:19) p − (cid:27) . Recalling that µ ≤ , we observe that for all r ∈ (0 , r ), ( x, t ) ∈ Q and s ∈ ( − r pp − ,
0] (44)holds and we have | x ( s ) | ≤ k u k L p +1 ( Q / ) r p p − ≤ r (cid:18) k u k L p +1 ( Q / ) r p − (cid:19) ≤ r . (45)Hence we can estimate by the triangular inequality and H¨older, by (45) and by Step 3 (cid:18) − Z tt − r pp − − Z B r ( x ) | θ − ( θ ) B r ( x ) × ( t − r pp − ,t ] | p dy ds (cid:19) p ≤ (cid:18) − Z − r pp − − Z B r ( x ) | θ ( y, s + t ) − ( θ ) B r × ( r pp − , | p dy ds (cid:19) p ≤ (cid:18) − Z − r pp − − Z B r ( x + x ( s )) | θ ( y, s + t ) − ( θ ) B r × ( r pp − , | p dy ds (cid:19) p ≤ µ − r γ − h − αp − + αp ( p − i ε . Step 5: By Campanato’s Theorem, we deduce that θ is H¨older continuous in Q . By a variant of Campanato’s Theorem [17, Theorem 2.9.], we deduce from Step 4 that (37) holds.Indeed, observe that the sets B r ( x ) × ( t − r p/ ( p − , t ] are nothing else but balls with respect tothe metric d (( x, t ) , ( y, s )) := max {| x − y | , | t − s | ( p − /p } on spacetime where in time, as usualfor parabolic equations, we only look at backward-in-time intervals. The proof of this versionof Campanato’s Theorem follows, for instance, line by line [27, Theorem 1] when replacing theparabolic metric by d . (cid:3) ε -regularity results and proof of Theorem 1.3 In this section we prove some ε -regularity results, including Theorem 1.3 and its more preciseversion in Corollary 6.6, by meeting the smallness requirement of Theorem 5.3. As a first result,we deduce in Corollary 6.1 an ε -regularity criterion in terms of a spacetime integral of θ and u that constitutes an analogue of Scheffer’s Theorem [26] for the Navier-Stokes system. As in thecase of Navier-Stokes, it implies that the singular set of suitable weak solutions is compact inspacetime (see Step 1 in the proof of Theorem 1.1). In the context of the SQG equation though,in contrast to Navier-Stokes, Corollary 6.1 cannot be used to obtain their almost everywheresmoothness (or any estimate on the dimension of the singular set): The fact that the L ∞ -normis a controlled quantity necessitates to rely on spacetime integrals of derivatives of θ to showlocal smoothness. In order to pass from Theorem 5.3 to an ε -regularity criterion involving onlyfractional space derivatives of θ , which are globally controlled through the energy, we need toovercome the following difficulties: • The excess E S related to θ involves the spacetime average of θ . In particular, in orderto use a standard Poincar´e inequality (10) to pass to a differential quantity, we wouldneed some fractional differentiability in time too. Using the parabolic structure of theequation, we will be able to circumvent this and to establish in Lemma 6.2 a Poincar´einequality which is nonlinear but involves only fractional space derivatives. • The ε -regularity criterion of Theorem 5.3 features the composition of θ with the flow x , so that we need some control on the tilting effect of the flow. We will see that atscale r , the flow shifts the center of the excess in space by at most r α − /q k u k L ∞ L q (see (55)). As a consequence, at scale r , all quantities related to the excess of θ will nolonger live on parabolic cylinders but rather on modified cylinders Q ( x, t ; r ) in spacetime,approximately of radius r α in time and r α − /q in space. Morally q = ∞ ; however, sincethe Riesz-transform is bounded from L ∞ → BM O and not from L ∞ → L ∞ , we introducethe parameter q which should be thought to be arbitrarily large. • We set the excess in L p for p > αα in order to gain the compactness of the ( p − L W α, -control given by the energy via the nonlinearPoincar´e inequality described in the first point, we are lacking some higher integrabilityin time. We bypass this issue by factoring out p − θ in L ∞ .6.1. An analogue of Scheffer’s theorem.
We provide a first ε -regularity result featuringspacetime integrals of θ and u . Observe that in agreement with the previous discussion, smoothsolutions of (1)–(3) do, in general, not verify the ε -regularity criterion (46) at any small scale. Corollary 6.1.
Given α ∈ ( , ) there exists ε = ε ( α ) > such that if p = p ( α ) := α − and θ is a suitable weak solution of (1) – (3) on R × ( t − (4 r ) α , t + r α / satisfying r (1 − α ) p +2+2 α Z Q r ( x,t + r α / (cid:0) M θ (cid:1) p ( z, s ) + | u | p ( z, s ) dz ds < ε , (46) then θ ∈ C (cid:0) Q r ( x, t + r α / (cid:1) . In particular, θ is smooth in the interior of Q r/ ( x, t + r α / ⊇ B r/ ( x ) × ( t − r α / , t + r α / and hence ( x, t ) is a regular point.Proof of Corollary 6.1. Let α and p as in the statement. By translation and scaling invariance,we can assume w.l.o.g. that ( x, t + r α /
4) = (0 ,
0) and r = 1 , so that we assume that Z Q ( M θ ) p ( x, t ) + | u | p ( x, t ) dx dt ≤ ε . for an ε yet to be chosen small enough. We observe that p > max { αα , ασ } for any σ > . Wedefine accordingly σ := α + and γ := α + . Since σ ∈ ( , α ) and γ ∈ [1 − α, α − αp ) , this is an admissible choice of the parameters of Theorem 5.3 and we infer from the latter that θ ∈ C δ, p − p δ ( Q ) , with δ given by (37), provided the smallness requirement (36) holds for any( x, t ) ∈ Q . Since for α ∈ ( , ) δ = 1 − α + (4 α − (cid:18) − α − α + 33(7 α − (cid:19) > − α , we deduce from Lemma B.1 that θ is smooth in the interior of Q / . We observe that for any( x, t ) ∈ Q we have Q ( x, t ) ⊆ Q and hence (cid:18) − Z Q ( x,t ) | u | p dz ds (cid:19) p ≤ | Q | − p (cid:18) Z Q | u | p dz ds (cid:19) p ≤ ε p . Requiring ε ≤ ε p , we thus deduce from Lemma 5.1 that E ( θ , u ; ) ≤ C E ( θ, u ; x, t,
1) andhence (36) is enforced for any ( x, t ) ∈ Q ifsup ( x,t ) ∈ Q E ( θ, u ; x, t, ≤ C − ε , (47)where ε > θ ≤ M θ pointwise almost everywhere, we have E S ( θ ; x, t,
1) + E V ( u ; x, t, ≤ (cid:18)Z Q | θ | p dz ds (cid:19) p + 2 (cid:18)Z Q | u | p dz ds (cid:19) p ≤ ε p . As for the non-local part of the excess, we estimate, using again θ ≤ M θ almost everywhere, E NL ( θ ; x, t, ≤ − Z tt − sup R ≥ R σp (cid:18) − Z B R ( x ) θ dz (cid:19) p ds ! p + C | ( θ ) Q ( x,t ) |≤ C − Z tt − sup R ≥ R σp (cid:18) Z B ( x ) − Z B R ( z ′ ) θ ( z, s ) dz dz ′ (cid:19) p ds ! p + C (cid:18) Z Q | θ | p dz ds (cid:19) p ≤ C − Z tt − sup R ≥ R σp (cid:18) Z B ( x ) M θ ( z ′ , s ) dz ′ (cid:19) p ds ! p + Cε p ≤ C NL ε p . Hence we reach (47) by choosing ε ≤ min (cid:8) ( ε ) p , (4 + C NL ) − C − ε ) p (cid:9) . (cid:3) Nonlinear Poincar´e inequality of parabolic type.
We introduce the following scaling-invariant quantity which should be understood as a localized version of the dissipative part ofthe energy: E ( θ ; x, t, r ) := 1 r − α )+2 Z Q ∗ r ( x,t ) y b |∇ θ ∗ | ( z, y, s ) dz dy ds . The following Lemma and its proof is inspired by [31], where a parabolic Poincar´e inequality isobtained for the classical, linear heat equation, and by [25], where a nonlinear Poincar´e inequalityof similar nature is also crucially used in a ε -regularity result. Lemma 6.2 (Nonlinear Poincar´e inequality of parabolic type) . Let α ∈ (0 , . There exists aconstant C = C ( α ) ≥ such that the following holds: Let Q r ( x, t ) ⊆ R × (0 , ∞ ) and let ( θ, u ) be a Leray-Hopf weak solution of (1) – (2) . We assume that the velocity field is obtained by u ( z, s ) = R ⊥ θ ( z, s ) + f ( s ) for some f ∈ L loc ( R ) and that it satisfies [ u ( s )] B r ( x ) = 0 for all s ∈ [ t − (2 r ) α , t ] . Then we havefor any q ∈ (cid:2) , − α (cid:3) that r (1 − α )+ q + α Z tt − r α (cid:18) Z B r ( x ) | θ ( z, s ) − ( θ ) Q r ( x,t ) | q dz (cid:19) q ds ! ≤ C (cid:0) E ( θ ; x, t, r ) + (cid:18) r − α )+2+2 α Z Q r ( x,t ) | u ( z, s ) − [ u ( s )] B r ( x ) | dz ds (cid:19) E ( θ ; x, t, r ) (cid:1) . Proof.
By translation and scaling invariance (with respect to (4)), we may assume ( x, t ) = (0 , r = 1 . Step 1: By means of the weighted Poincar´e inequality (11) , we reduce the Lemma to anestimate on weighted space averages computed at two different times. To this aim, let ω ∈ C ∞ c ( R ) be a weight such that ω | y =0 is a radial non-increasing function, ≤ ω ≤ and ω ≡ on B × [0 , and ω ≡ outside B × [0 , . We estimate for fixed time (cid:18)Z B | θ ( x, t ) − ( θ ) Q | q dx (cid:19) q ≤ (cid:18)Z B | θ ( x, t ) − [ θ ( t )] ω | y =0 ,B | q ω ( x, dx (cid:19) q + π q (cid:0)(cid:12)(cid:12) [ θ ( t )] ω | y =0 ,B − ( θ ) ω | y =0 ,Q (cid:12)(cid:12) + (cid:12)(cid:12) ( θ ) ω | y =0 ,Q − ( θ ) Q (cid:12)(cid:12)(cid:1) , where we used ω ( · , ≡ B . Reusing this fact and H¨older, we bound the last term by (cid:12)(cid:12) ( θ ) ω | y =0 ,Q − ( θ ) Q (cid:12)(cid:12) ≤ Z − (cid:18) − Z B | θ − ( θ ) ω | y =0 ,Q | q dx (cid:19) q dt ! ≤ π − q Z − (cid:18)Z B | θ − [ θ ( t )] ω | y =0 ,B | q ω ( x, dx (cid:19) q dt ! + (cid:18)Z − | [ θ ( t )] ω | y =0 ,B − ( θ ) ω | y =0 ,Q | dt (cid:19) , so that we deduce by the weighted Poincar´e inequality (11) Z − (cid:18)Z B | θ ( x, t ) − ( θ ) Q | q dx (cid:19) q dt ! ≤ C " (cid:18)Z − [ θ ( t )] W α, ( B ) dt (cid:19) + (cid:18)Z − | [ θ ( t )] ω | y =0 ,B − ( θ ) ω | y =0 ,Q | dt (cid:19) . The first term on the right-hand side can be expressed in terms of the extension by (14). Sincethe weight ω is independent of time, the second term can be estimated by (cid:18)Z − | [ θ ( t )] ω | y =0 ,B − ( θ ) ω | y =0 ,Q | dt (cid:19) ≤ (cid:18)Z − α Z − α | [ θ ( t )] ω | y =0 ,B − [ θ ( s )] ω | y =0 ,B | ds dt (cid:19) . Step 2: We use the equation to estimate the difference of two weighted space averages computedat different times.
We use the weak formulation (22) of the equation with time-independent test function ϕ ( x, y ) :=sgn([ θ ( t )] ω | y =0 ,B − [ θ ( s )] ω | y =0 ,B ) ω ( x, y ) . We estimate the right-hand side of (22) from below andthe left-hand side from above for s, t ∈ [ − α , Z ( θ ( x, t ) − θ ( x, s )) ϕ ( x, dx = (cid:12)(cid:12) [ θ ( t )] ω | y =0 ,B − [ θ ( s )] ω | y =0 ,B (cid:12)(cid:12) Z B ω ( x, dx ≥ π (cid:12)(cid:12) [ θ ( t )] ω | y =0 ,B − [ θ ( s )] ω | y =0 ,B (cid:12)(cid:12) since ω ( · , ≡ B . As for the right-hand side, we estimate by H¨older (cid:12)(cid:12)(cid:12)(cid:12) Z ts Z R y b ∇ θ ∗ · ∇ ϕ dx dy dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ − α k ω k C | Q | Z Q ∗ y b |∇ θ ∗ | dx dy dτ ! . Since u is divergence-free and [ u ( τ )] B = 0 for τ ∈ [ − α ,
0] by assumption, we can estimate thenonlinear term by H¨older and the Poincar´e inequality (10) combined with (14) (cid:12)(cid:12)(cid:12)(cid:12)Z ts Z uθ · ∇ ϕ | y =0 dx dτ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ts Z ( u − [ u ( τ )] B )( θ − [ θ ( τ )] B ) · ∇ ϕ | y =0 dx dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ω k C (cid:18)Z Q | u − [ u ( τ )] B | dx dτ (cid:19) (cid:18)Z Q | θ − [ θ ( τ )] B | dx dτ (cid:19) ≤ C (cid:18)Z Q | u − [ u ( τ )] B | dx dτ (cid:19) Z Q ∗ y b |∇ θ ∗ | dx dy dτ ! . Collecting terms, we have for almost every s, t ∈ [ − α ,
0] that (cid:12)(cid:12) [ θ ( t )] ω | y =0 ,B − [ θ ( s )] ω | y =0 ,B (cid:12)(cid:12) . Z Q ∗ y b |∇ θ ∗ | dx dy dτ ! (cid:18) Z Q | u − [ u ( τ )] B | dx dτ (cid:19) ! . Combining this estimate with Step 1, we conclude. (cid:3)
The non-local part of excess.
We recall from the proof of Proposition 4.1 that the excessrelated to the velocity can be estimated in terms of θ .
More precisely, we have the following:
Lemma 6.3.
Let α ∈ ( , ) , Q r ( x, t ) ⊆ R × (0 , ∞ ) and θ ∈ L p ( R × [ t − (3 r/ α , t ]) . Considera velocity field of the form u ( z, s ) = R ⊥ θ ( z, s ) + f ( s ) for some f ∈ L loc ( R ) . There exists C = C ( p ) ≥ such that E V ( u ; x, t, r ) ≤ C (cid:18) − Z Q r ( x,t ) | θ ( z, s ) − [ θ ( s )] B r ( x ) | p dz ds (cid:19) p + E NL ( θ ; x, t, r ) ! . After reducing the Lemma to r = 1 and ( x, t ) = (0 , E V in the proof of Proposition 4.1. We now bound the quantity E NL in terms of avariant of the sharp maximal function introduced in Section 2.6. Lemma 6.4.
Let α ∈ ( , ) , Q r ( x, t ) ⊆ R × (0 , ∞ ) and θ ∈ L ∞ ( R × [ t − r α , t ]) ∩ L ([ t − r α , t ] , W α, ( R )) . If ( p, q, σ ) ∈ ( αα , ∞ ) × [2 , ∞ ) × (0 , α ) satisfy the admissibility criterion σp − (2 + 2 α ) − q − ≥ , (48) then there exists a constant C = C ( p ) ≥ such that r p (1 − α ) E NL ( θ ; x, t, r ) p ≤ C k θ k p − L ∞ ([ t − r α ,t ] × R ) r p (1 − α )+2 Z tt − r α Z B r ( x ) | θ α,q ( z, s ) | q − dz ds . Proof.
By translation and scaling invariance, we may assume ( x, t ) = (0 ,
0) and r = 1. Byfactoring out p − L ∞ , by adding and subtracting [ θ ( s )] B / for fixed time s and radius R , and by reabsorbing | [ θ ( s )] B − [ θ ( s )] B / | in the supremum, we have E NL ( θ ; 1) p ≤ (2 k θ k L ∞ ( R × [ − , ) p − Z − sup R ≥ (cid:18) R (cid:19) σp (cid:18) − Z B R | θ ( x, s ) − [ θ ( s )] B | dx (cid:19) ds . k θ k p − L ∞ ( R × [ − , Z − sup R ≥ (cid:18) R (cid:19) σp (cid:18) − Z B R | θ ( x, s ) − [ θ ( s )] B | dx (cid:19) ds . We estimate the argument of the supremum for fixed time s and radius R ≥ by the triangularinequality and H¨older (cid:18) − Z B R | θ − [ θ ( s )] B | dx (cid:19) ≤ (cid:18) − Z B R | θ − [ θ ( s )] B R | − /q ) dx (cid:19) q − + | [ θ ( s )] B − [ θ ( s )] B R | ! ≤ (cid:18) (4 R ) − Z B R | θ − [ θ ( s )] B R | − /q ) dx (cid:19) q − . (49)For z ∈ B / it holds B R ⊆ B R ( z ), so that by the triangular inequality and by averaging over z ∈ B / , we have − Z B R | θ ( x, s ) − [ θ ( s )] B R | − /q ) dx ≤ − Z B − Z B R ( z ) | θ ( x, s ) − [ θ ( s )] B R ( z ) | − /q ) dx dz ≤ R α (1 − /q ) − Z B θ α,q ( z, s ) dz . (50)We combine (49)–(50) and use H¨older to bring the power 1 + q − inside the integral. We obtain E NL ( θ ; 1) ≤ C k θ k p − L ∞ ( R × [ − , Z − Z B | θ α,q ( z, s ) | q − dz ds , provided sup R ≥ (cid:18) R (cid:19) σp − (2+2 α ) − q − < + ∞ . Observe that this is ensured through (48) and that the supremum can be estimated from aboveby 4 p such that the constant of the Lemma depends only on p . (cid:3) Proof of the Theorem 1.3.
The following Corollary of Theorem 5.3 gives a differentversion of the ε -regularity criterion in terms of θ rather than its composition with the flow θ . Theorem 1.3 will be an immediate consequence. To this aim, we introduce the following modifiedballs and cylinders (backwards and centered in time respectively) which are enlarged in spacein accordance with the “intrinsic effect” of the flow (see (55)): B ( x ; r ) := B K q r α − /q ( x ) and B ∗ ( x ; r ) := B ( x ; r ) × [0 , r ) , Q ( x, t ; r ) := B ( x ; r ) × ( t − r α , t ] and Q ∗ ( x, t ; r ) := B ∗ ( x ; r ) × ( t − r α , t ] , C ( x, t ; r ) := B ( x ; r ) × ( t − r α , t + r α ) and C ∗ ( x, t ; r ) := B ∗ ( x ; r ) × ( t − r α , t + r α ) , where K q = K q ( u ; x, t, r ) := 2 max n k u k L ∞ ([ t − r α ,t + r α ] ,L q ( R )) , r − α +2 /q o . (51)To shorten notation, we will often omit the dependence of K q on u and r , and whenever( x, t ) = (0 , , we will omit to specify the center of the balls and cylinders. The followingremark justifies that one should really think of B ( x ; r ) as a enlarged balls of radius r α − /q . Remark K q ) . For 0 < r ≤ ( t/ α by Calderon-Zygmund, Theorem 3.2and the energy inequality (20) of Leray-Hopf weak solutions k u k L ∞ ([ t − r α ,t + r α ] ,L q ( R )) ≤ C k θ k L ∞ ([ t − r α ,t + r α ] ,L q ( R )) ≤ C k θ k L t − α (1 − /q ) such that for 0 < r ≤ r := min (cid:26) ( t/ α , (cid:16) C k θ k L t − α (1 − /q ) (cid:17) / (1 − α +2 /q ) (cid:27) and any x ∈ R K q ( u ; x, t, r ) ≤ C k θ k L t − α (1 − /q ) . Corollary 6.6.
Let α := √ , α ∈ [ α , ) , q ≥ and p = p ( q ) := αα + q . There exists auniversal ε = ε ( α ) ∈ (0 , such that the following holds: If ( θ, u ) is a suitable weak solution to (1) – (3) on R × [ t − (4 r ) α , t ] satisfying k θ k p − L ∞ ( R × [ t − (4 r ) α ,t ]) (4 r ) p (1 − α )+2 Z Q ∗ ( x,t ;4 r ) y b |∇ θ ∗ | dz dy ds + Z Q ( x,t ;4 r ) | θ α,q | q − dz ds ! ≤ ε , (52) then θ is smooth in the interior of Q r/ ( x, t ) . Remark α ) . α is the threshold until which both the smallness hypothesisof Corollary 6.6 is verified, at sufficiently small scale, for smooth solutions at any point ( x, t ) inspacetime and the dimension estimate of Theorem 1.1 is non-trivial. Indeed, for α > α it holds12 α (cid:16) αα (1 − α ) + 2 (cid:17) < . Before proceeding with the proof, let us show that Theorem 1.3 is an immediate consequenceof Corollary 6.6.
Proof of Theorem 1.3.
Let α , p and q as in the statement and assume that (5) holds. Observethat C ( x, t ; r ) ⊇ Q ( x, t + r α / r ) . By (18) and H¨older we deduce the pointwise estimate θ α,q ( z, s ) ≤ M (cid:0) ( D α, θ ) − /q ) (cid:1) ( z, s ) ≤ (cid:2) M (cid:0) ( D α, θ ) (cid:1)(cid:3) − /q ( z, s ) . We infer that θ satisfies (52) at the radius r/ x, t + r α / . We deduce fromCorollary 6.6 that θ is smooth in the interior of Q r/ ( x, t + r α /
16) which contains the open ball B r/ ( x ) × ( t − r α / , t + r α / . (cid:3) Proof of Corollary 6.6.
By translation and scaling invariance, we assume ( x, t ) = (0 ,
0) and r = 1 . Step 1: We tune the free parameters σ and γ . We define σ := 2 α − q . Observe that with this choice the triple ( σ, p, q ) satisfies the hypothesisof Lemma 6.4. Indeed, recalling that by assumption α ≥ α > , we have for q ≥ σp − (2 + 2 α ) − q − q (cid:18) α − q (cid:18) αα + 1 q + 2 q q − (cid:19)(cid:19) > q (cid:18) α − (cid:19) > . We introduce also γ := 2 α − α α ∈ [1 − α, σ − α/p ), so that the triple ( σ, p, γ ) satis-fies the hypothesis of Theorem 5.3. We conclude from the latter that θ ∈ C δ, (1 − /p ) δ ( Q r ) ⊆ L ∞ (( − , C δ ( B )) , with δ given by (37), providedsup ( x,t ) ∈ Q E ( θ , u ; 14 ) ≤ ε , (53)where for ( x, t ) ∈ Q fixed, we define θ ( z, s ) = θ ( z + x ( s ) + x, s + t ) , u ( z, s ) = u ( z + x ( s ) + x, s + t ) − ˙ x ( s ) and x is the flow given by Lemma 5.1. We have δ = γ − p − (cid:20) − α + 2 αp (cid:21) > α − α α − α (cid:20) − α + 2 α α (cid:21) , where the right-hand side exceeds 1 − α for α ≥ √ −
1, so in particular for α ≥ α . We deducefrom Lemma B.1 that θ is smooth in the interior of Q / . We are thus left to verify that (53)can be enforced by requiring (52).
Step 2: We bound the full excess sup ( x,t ) ∈ Q E ( θ , u ; ) . For ( x, t ) ∈ Q fixed, we estimate by factoring out p − L ∞ , by Lemma 6.2, by H¨olderand Young E S ( θ ; 14 ) p . k θ k p − L ∞ ( Q ) E ( θ ; 34 ) + k θ k p − L ∞ ( Q ) E ( θ ; 34 ) E V ( u ; 12 ) . k θ k p − L ∞ ( R × [ − , E ( θ ; 1) + (cid:16) k θ k p − L ∞ ( R × [ − , E ( θ ; 1) (cid:17) p − + E V ( u ; 12 ) p . Moreover, by Lemma 6.3 and (10) combined with (14), recalling that u = R ⊥ θ − ˙ x ( s ), wehave E V ( u ; 12 ) p . Z Q | θ − [ θ ( s )] B | p dz ds + E NL ( θ ; 34 ) p . k θ k p − L ∞ ( Q ) E ( θ ; 1) + E NL ( θ ; 34 ) p . Collecting terms and applying Lemma 6.4, we deduce that for every ( x, t ) ∈ Q with a constant C = C ( α ) ≥ p cannot exceed αα + 1) E ( θ , u ; 14 ) p ≤ C (cid:18) k θ k p − L ∞ ( R × [ − , E ( θ ; 1) + (cid:16) k θ k p − L ∞ ( R × [ − , E ( θ ; 1) (cid:17) p − + k θ k p − L ∞ ( R × [ − , Z − Z B | ( θ ) α,q | q − dz ds (cid:19) . (54) Step 3: We estimate the tilting effect of the flow. To this aim, introduce a parameter q ≥ (to ensure that α − /q ≥ / > ). For ( x, t ) ∈ Q and s ∈ [ − ,
0] we have by the definition of the flow (34) | ˙ x ( s ) | ≤ | B | − q k u k L ∞ ([ t − ,t ] ,L q ( R )) ≤ k u k L ∞ ([ − α , ,L q ( R )) ≤ K q ( u ; 4) . Hence the center of the excess in space can be shifted by at mostsup ( x,t ) ∈ Q sup s ∈ [ − , | x ( s ) | ≤ K q ( u ; 4) . (55)Since θ is just a spatial translation of θ , we estimate k θ k L ∞ ( R × [ − , ≤ k θ k L ∞ ( R × [ − α , for( x, t ) ∈ Q . Recall that the extension is obtained by convolution with a Poisson kernel. Since translation and convolution commute, we have ( θ ) ∗ ( z, y, s ) = θ ∗ ( z + x ( s ) + x, y, s + t ) andhence E ( θ ; 1) = Z Q ∗ y b |∇ θ ∗ | ( z + x ( s ) + x, y, s + t ) dz dy ds ≤ Z Q ∗ (4) y b |∇ θ ∗ | ( z, y, s ) dz dy ds . (56)We used in the last inequality that for ( x, t ) ∈ Q it holds t − ≥ − α and B ( x ) + x ( s ) ⊆ B + B K q ( u ;4) ⊆ B (4) + 14 α − /q B (4) ⊆ B (4) (57)for any s ∈ [ − ,
0] by (55). As for the remaining term in (54), we observe that ( θ ) α,q ( z, s ) = θ α,q ( z + x ( s ) + x, s + t ) and we reuse (57) to estimate Z − Z B | ( θ ) α,q | q − dz ds ≤ Z Q (4) | θ α,q | q − dz ds . (58)Combining (54), (56) and (58), we reach (53) by requiring (52) . (cid:3) The singular set and proof of Theorem 1.1
We recall the box-counting dimension of a (compact) set
S ⊆ R : For every δ ∈ (0 ,
1) wedenote by N ( δ ) the minimal number of sets of diameter δ needed to cover S . We then definedim b ( S ) := lim sup δ → − log δ ( N ( δ )) . It is well-known that the box-counting dimension controls the Hausdorff dimension dim H , i.e.dim H S ≤ dim b S . Proof of Theorem 1.1. Step 1: Fix t > and define S := Sing θ ∩ (cid:2) R × [ t, ∞ ) (cid:3) . We claim that S is a compact set in spacetime. From the definition it is clear that S is closed and we claim that it is also bounded. Indeed,let p be as in Corollary 6.1 and let p ′ ≥ p . From the maximal function estimate and Calderon-Zygmund Z ∞ t Z ( M θ ) p ′ + | u | p ′ dx ds . Z ∞ t Z | θ | p ′ dx ds . k θ k p ′ − α ) L ∞ ( R × [ t, ∞ )) k θ k α ) L α ) ( R × [ t, ∞ )) ≤ C ( k θ k L ) t − p ′− α )2 α , by Theorem 3.2 and the global energy inequality (20) of Leray-Hopf weak solutions. By theabsolute continuity of the integral we deduce that for every ε > M = M ( θ, ε ) > T ∗ = T ∗ ( k θ k L , ε ) > Z ∞ t Z R \ B M ( M θ ) p + | u | p dx ds + Z ∞ T ∗ Z R ( M θ ) p + | u | p dx ds < ε , which, by choosing ε as in Corollary 6.1, means that S ⊂ B M +4 (0) × [0 , T ∗ + 4 α ] . Step 2: Let α ∈ ( α , ) (otherwise the dimension estimate is trivial by Remark 6.7). We showthat for every q ≥ we have dim b S ≤ α − q (cid:18)(cid:18) αα + 1 q (cid:19) (1 − α ) + 2 (cid:19) =: β ( q ) . Indeed, fix q ≥ p q := αα + q . From Corollary 6.6, we know that if ( x, s ) ∈ S , then for every r ∈ (0 , ( t/ α ) it holds1 r p q (1 − α )+2 Z C ∗ ( x,s ; r ) y b |∇ θ ∗ | dz dy dτ + Z C ( x,s ; r ) | θ α,q | q − dz dτ > ε k θ k − ( p − L ∞ ( R × [ t/ , ∞ )) =: 2 ε , where ε = ε ( α ) > r . By Theorem 3.2, thethreshold ε depends on t > k θ k L only. Following Remark 6.5, we observe that with thenotation of Remark 6.5 K q ( u ; x, s, r ) ≤ max n C k θ k L t − α (1 − /q ) , o =: L q for r ∈ (0 , δ ] , where δ := min n ( t/ α , ( L q / / (1 − α +2 /q ) , o . Observe that L q depends only on k θ k L and t > . For δ ∈ (0 , δ ) , we define the collection Γ δ containing balls B √ L q δ α − /q ( x, s ) centered at some point ( x, s ) ∈ R × [ t, ∞ ) satisfying Z C ∗ ( x,s ; δ ) y b |∇ θ ∗ | dz dy dτ + Z C ( x,s ; δ ) | θ α,q | q − dz dτ ≥ ε δ p q (1 − α )+2 . Observe that { Γ δ } δ is a family of coverings of S consisting of Euclidean balls in spacetime. Bythe Vitali covering Lemma, there exists for every δ a countable, disjoint family { B i } i ∈ I suchthat S ⊆ [ i ∈ I B i and B i ∈ Γ δ , in particular B i = B √ L q δ α − /q ( x i , s i ) for some ( x i , s i ) ∈ R × [ t, ∞ ) . Observethat by Lemma 2.4, Theorem 2.1 and the global energy inequality (20) for Leray-Hopf weaksolutions, we have the global control Z ∞ Z R y b |∇ θ ∗ | dz dy ds + Z ∞ Z R | θ α,q | q − dz ds ≤ C Z ∞ [ θ ( s )] W α, ( R ) ds ≤ C k θ k L . Therefore, setting η := 2 √ L q δ α − /q and using the disjointness of { B i } i ∈ I , we can estimatethe minimal number N ( η ) of sets of diameter η needed to cover S by N ( η ) ≤ H ( I ) ≤ C k θ k L ε δ p q (1 − α )+2 = C k θ k L ( √ L q ) pq (1 − α )+22 α − /q ε η pq (1 − α )+22 α − /q . We conclude that dim b S = lim sup η → − log η N ( η ) ≤ p q (1 − α ) + 22 α − /q . Step 3: Conclusion.
By taking the limit q → ∞ in Step 2, we conclude thatdim b (Sing θ ∩ [ t, ∞ )) ≤ α (cid:18) αα (1 − α ) + 2 (cid:19) for every t >
0. Writing Sing θ = S n ≥ Sing θ ∩ (cid:2) R × (cid:2) n , ∞ (cid:1)(cid:3) , we deduce thatdim H (Sing θ ) ≤ sup n ≥ dim H (cid:0) Sing θ ∩ (cid:2) R × (cid:2) n , ∞ (cid:1)(cid:3)(cid:1) ≤ α (cid:18) αα (1 − α ) + 2 (cid:19) . (cid:3) Stability of the singular set
This section is devoted to the proof of Corollary 1.4. We observe that the ε -regularity cri-terion is “continuous” under strong L p -convergence. This convergence result together with theobservation that smooth solutions satisfy the ε -regularity criterion of Theorem 5.3 will allow todeduce the required stability. Lemma 8.1.
Let α n ∈ ( , ) be such that α n → α ∈ ( , ] and consider a sequence of suitableweak solutions θ n to (1) – (3) with α = α n on R × [ − , such that θ n → θ strongly in L p ( R × [ − , and assume that θ is a classical solution to (1) – (3) on R × [ − , . Then, there existsa universal constant C > such that uniformly for any ( x, t ) ∈ Q lim n →∞ E ( θ n, , u n, ; 14 ) ≤ CE ( θ , u ; 14 ) where we denote by θ n, and u n, (and θ and u respectively) the change of variables of Lemma5.1 as applied to θ n (and θ respectively).Proof. We fix ( x, t ) ∈ Q and apply the change of variables of Lemma 5.1 to θ n and θ respectively.We denote the corresponding flow by x n, and x . Moreover, we estimate for s ∈ [ − , | x n, ( s ) − x ( s ) | ≤ | s | − p Z s − Z B ( x + x n, ( σ )) | u n ( y, σ + t ) − u ( y, σ + t ) | p dy dσ p + Z s − Z B ( x ) | u ( y + x n, ( σ ) , σ + t ) − u ( y + x ( σ ) , σ + t ) | dy dσ ≤ p | s | − p (cid:18)Z tt − Z | u n − u | p dy dσ (cid:19) p + sup σ ∈ [ t − ,t ] [ u ( σ )] Lip( R ) Z s | x n, ( σ ) − x ( σ ) | dσ . By Calderon-Zygmund, the strong convergence of θ n implies that u n → u strongly L p . Calling C = sup σ ∈ [ t − ,t ] [ u ( σ )] Lip( R ) we deduce by Gr¨onwall’s inequality that uniformly in s ∈ [ − , n →∞ | x n, ( s ) − x ( s ) | ≤ lim n →∞ p (1 + Ce C ) (cid:18)Z tt − Z | u n − u | p dy dσ (cid:19) p = 0 . (59)We now claim that θ n, → θ strongly in L p ( R × [ − (1 / α , . Fix ε > . We split as before Z − ( ) α Z | θ n, − θ | p ( y, s ) dy ds ≤ Z − ( ) α Z | θ n ( x n, ( s ) + y, s + t ) − θ ( x n, ( s ) + y, s + t ) | p dy ds + Z − ( ) α Z | θ ( x n, ( s ) + y, s + t ) − θ ( x ( s ) + y, s + t ) | p dy ds . Using the strong convergence of θ n in L p ( R × [ − , ε for n big enough. Using (59) and absolute continuity, there exists R ≥ n big enough Z − ( ) α Z | y |≥ R | θ ( x n, ( s ) + y, s + t ) − θ ( x ( s ) + y, s + t ) | p dy ds ≤ ε . By the regularity of θ and (59), we estimate the remaining contribution of the integral overthe set {| y | ≤ R } by [ θ ] p Lip( R × [ t − (1 / α ) ,t ]) k u n − u k pL p ( R × [ t − ,t ]) R p , which by the strong L p -convergence does not exceed ε for n big enough. The strong convergence of θ n, → θ in L p ( R × [ − (1 / α , u n, → u strongly in L p ( R × [ − (1 / α , . As animmediate consequence we obtainlim n →∞ E S ( θ n, ; 14 ) + lim n →∞ E V ( u n, ; 14 ) = E S ( θ ; 14 ) + E V ( u ; 14 ) . and lim n →∞ E NL ( θ n, ; 14 ) . lim n →∞ ( E NL ( θ ; 14 ) + − Z − (1 / α sup R ≥ R ) σp − Z B R | θ n, − θ | p ! p + − Z − (1 / α | [ θ ] B − [ θ n, ] B | p ! p ) = E NL ( θ ; 14 ) . (cid:3) Proof of Corollary 1.4.
We argue by contradiction and we let p := α − and σ := α + as inthe proof of Corollary 6.1. We may assume that there is a sequence of orders α n ∈ ( , ) andinitial data ¯ θ n, ∈ H such that • lim n →∞ α n = , • k ¯ θ n, k H ( R ) ≤ R for all n ≥ , • the local smooth solutions θ n to (1)–(3) with α = α n and initial data ¯ θ n, , which existon an interval [0 , T ], with T = C k ¯ θ n, k − H ≥ CR − bounded from below uniformly in n (see [12]), blow-up in finite time.Since θ n ∈ L ∞ ([0 , T ] , H ) implies, for instance, that ∇ θ n ∈ L ([0 , T ] , L /α ) , θ n satisifes theweak-strong uniqueness criterion of [14] and hence θ n coincides on [0 , T ] with the uniquesuitable weak solution to (1)–(3) with α = α n and initial data ¯ θ n, . We have the uniform boundsup n ≥ k θ n k L ∞ ( R × [ T / , ∞ )) + sup n ≥ k θ n k L ∞ ([ T / , ∞ ) ,L ( R )) ≤ C (60)given by (20) and Theorem 3.2 . Up to subsequence, we have that ¯ θ n, → ¯ θ strongly in L ( R ) . Fix
T > . By (60), θ n is uniformly bounded in L ( R × [ T / , T ]) and hence θ n ⇀ θ weakly in L . The strong convergence in L p ( R × [ T / , T ]) is established as in the proof of Lemma 3.8, The authors of [14] state their result in the form of an asymptotic stability result with respect to perturbationsof the initial datum and the right-hand side; however, in absence of any perturbation their energy yield thecorresponding weak-strong uniqueness statement, as previously observed for instance in [24]. Step 2, using an Aubin-Lions type argument. The strong convergence for any
T > θ is a Leray–Hopf solution to (1)–(3) with α = and initial datum ¯ θ . This Leray–Hopf solutionis smooth by [4]. The blow-up of the strong solutions means that there exists ( x n , t n ) ∈ Sing θ n for n ≥ . By Theorem 1.1 (and more precisely, noticing that Step 1 in its proof can be madeuniform in n ), there exists M > n ≥ x n , t n ) ∈ B M × [ T , M ] . Up to subsequence, we may assume that ( x n , t n ) → (¯ x, ¯ t ) ∈ B M × [ T , M ] . Rescaling thesequence of solutions by one single factor (related to T ) and translating them, we may assumethat T = − x, ¯ t ) = (0 , ∈ Q . By a further n -dependent temporal translation (by2 t n → t n <
0. By the continuity of translations in L p , we still have inthis way that θ n → θ strongly in L p ( R × [ − , r ∈ (0 ,
1] suchthat for any ( x, t ) ∈ Q E ( θ r, , u r, ; 14 ) < ε , (61)where we denote by θ r, and u r, the change of variables of Lemma 5.1 applied to ( θ r , u r ) (seeRemark 5.2) and by ε the constant from Theorem 5.3. This now gives rise to a contradic-tion: Indeed, by Lemma 8.1, uniformly for every ( x, t ) ∈ Q , we have the lower semicontinuitylim n →∞ E (( θ n ) r, , ( u n ) r, ; ) ≤ CE ( θ r, , u r, ; ) such that for every n big enough, the smallnessrequirement of Theorem 5.3 holds and we deduce that θ n,r is smooth in Q / , namely θ n issmooth in Q r/ . This contradicts the fact that the singular points ( x n , t n ) ∈ Q r/ for n bigenough. We are thus left to prove (61). Indeed, E S ( θ r, ; 14 ) = r α − − Z r α tr α t − ( r ) α − Z B r ( rx + rx ( r α ( s − t )) | θ ( y, s ) − r − α ( θ r, ) Q | p dy ds ! p ≤ r α − [ θ ] Lip( R × [ − (2r) α , , where we used again, as in the proof of Lemma 8.1, the fact that the flow x is Lipschitz for aregular solution θ . Similarly, E V ( u r, ; ) ≤ r α [ u ] L ∞ ([ − (2 r ) α , R )) . Finally for the non-localpart of the excess, we rewrite E NL ( θ r, ; 14 )= r α − − Z − ( r ) α sup R ≥ r (cid:16) r R (cid:17) σp − Z B R ( x ) | θ ( y + x ( r α s ) , s + r α t ) − r − α [ θ r, ] B | dy ! p ds p . For fixed time s ∈ [ − ( r/ α , , we estimate the supremum splitting it on the two sets { ≥ R ≥ r } and { R ≥ } . We get that E NL ( θ ,r ; 14 ) ≤ Cr σp ([ θ ] Lip( R × [ − (2 r ) α , + k θ k L ∞ ( R × [ − (2 r ) α , ) . (cid:3) Appendix A. Local spacetime regularity of the fractional heat equation
Lemma A.1.
Let α ∈ (0 , and p ≥ . Consider η ∈ L / loc ( R × [ − , with ( η ) Q = 0 and E S ( η ; 1) + E NL ( η ; 1) < + ∞ which solves ∂ t η + ( − ∆) α η = 0 in Q / . Then η is smooth withrespect to the space variable and η ∈ C − /p ( Q / ) in spacetime. Moreover, there exists ¯ C > such that k η k L ∞ ([ − (1 / α , ,C ( B )) + k η k C − p ( Q ) ≤ ¯ C ( E S ( η ; 1) + E NL ( η ; 1)) . The constant ¯ C can be chosen uniform in α as long as α is bounded away from .Proof. Using the linearity of the equation, we can assume by a standard regularization argumentthat η ∈ C ∞ ( R × ( − , ηϕ | y =0 where ϕ is a smooth cut-offbetween Q ∗ / and Q ∗ / with ∂ y ϕ ( · , , · ) = 0 and obtain, arguing as in Lemma 3.6, Z η ( x, t ) ϕ ( x, , t ) dx + 2 c α Z t Z y b |∇ η ∗ | ϕ dx dy ds = Z t Z η ( x, s ) ∂ t ϕ ( x, , s ) dx ds + c α Z t Z y b ( η ∗ ) ∆ b ϕ dx dy ds . Taking the supremum over t ∈ [ − (11 / α ,
0] and recalling the support of ϕ , we obtain byLemma 3.9 thatsup t ∈ [ − (11 / α , Z B η ( x, t ) dx ≤ C Z Q η dx ds + Z Q ∗ y b | η ∗ | dx dy ds ≤ C (cid:0) E S ( η ; 1) + E NL ( η ; 1) (cid:1) . (62)Let now p ( x, t ) be the fractional heat kernel on R × (0 , ∞ ) with explicit form e −| ξ | α t in Fourierspace. By scaling invariance, p ( x, t ) = t − /α p ( t − / (2 α ) x,
1) and p is C ∞ and bounded for t > . Let now P ∗ ( x, y ) := [ p ( · , ∗ ( x, y ) be the Caffarelli-Silvestre extension to R of x p ( x,
1) andobserve that from scaling, p ∗ ( x, y, t ) = t − /α P ∗ ( t − / (2 α ) x, t − / (2 α ) y ) . Fix a cut-off ϕ between B ∗ / and B ∗ / which is radially symmetric in x and y . We define ˜ p ( x, y, t ) := p ∗ ( x, y, t ) ϕ ( x, y ) . We proceed as in [30, Proposition 4.1] to obtain for ( x, t ) ∈ B / η ( x, t ) = Z B η ( z, − (3 / α ) p ( x − z, t + (3 / α ) ϕ ( x − z, dz + Z t − (3 / α Z B ∗ y b η ∗ ( x, y, τ )∆ b ˜ p ( x − z, y, t − τ ) dz dy dτ . We argue as in [30, Proposition 4.1] that y b ∆ b ˜ p ( x, y, t ) = div( y b ∇ ˜ p ( x, y, t )) is a smooth functionin x and y supported in B ∗ / \ B / ∗ which remains bounded as t → . We conclude that forany multi-index β with | β | ≥ k ∂ βx η k L ∞ ( Q ) . k η k L ∞ L ( Q ) + Z Q ∗ y b ( η ∗ ) dx dy dt . E S ( η ; 1) + E NL ( η ; 1) . (63) To get the spacetime regularity, we observe that for x ∈ B / we can write (for fixed time) | ( − ∆) α η ( x, t ) | = Z | y |≤ | η ( x, t ) − η ( y, t ) || x − y | n +2 α dy + Z | y | > | η ( x, t ) − η ( y, t ) || x − y | n +2 α dy . [ η ( t )] Lip(B ) + X i ≥ ( i − α − σ ) Z B i +1 \ B i | η ( x, t ) − η ( y, t ) | ( i − n + σ ) dy + Z B \ B | η ( x, t ) − η ( y, t ) | dy . [ η ( t )] Lip(B ) + sup R ≥ R σ − Z B R | η ( x, t ) − [ η ( t )] | dx + Z B | η ( x, t ) | dx . Using (63) and the equation, we conclude k ∂ t η k L p L ∞ ( Q ) . E S ( η ; 1) + E NL ( η ; 1) , which proves that η ∈ C − p ([ − (1 / α , , L ∞ ( B )) recalling that W ,p ( R ) ֒ → C − p ( R ) . (cid:3) Appendix B. C δ -H¨older continuous solutions are classical for δ > − α In [10] it is proved that solutions of (1)–(3) with u ∈ L ∞ ([0 , T ] , C δ ( R )), δ > − α , aresmooth. The following Lemma provides a localized version of this result. Lemma B.1.
Let θ : R × ( − , → R be a bounded solution of (1) – (3) . If θ ∈ L ∞ (( − , , C δ ( B )) for some δ > − α , then θ ∈ C ,δ − (1 − α ) ( B / × ( − / , in spacetime and in particular, isa classical solution.Proof. Step 1: We show that θ ∈ L ∞ ([ − / , , C ,δ − (1 − α ) ( B / )) . This follows from showing that u ∈ L ∞ (( − , , C δ ( B )) and the general result on fractionaladvection-diffusion equations [30, Theorem 1.1]. Let us write u = u + u , where u = R ⊥ ( θχ )and u = R ⊥ ( θ (1 − χ )) for χ a smooth cut-off in space between B / and B . We estimate bySchauder estimates [28, Proposition 2.8] k u k L ∞ (( − , ,C δ ) ≤ C k θχ k L ∞ (( − , ,C δ ) ≤ C k θ k L ∞ (( − , ,C δ ( B )) . Regarding u , we observe that [ u ( t )] C k is bounded, uniformly in t ∈ ( − , k ≥ k = 1: For x ∈ B and fixed time, by integration by parts (theboundary term at infinity vanishes using the uniform boundedness of θ ) ∂ j R i u ( x ) = c Z x i − z i | x − z | ∂ j ((1 − χ ( z )) θ ( z ) dz = − c Z | z |≥ ∂ z j (cid:18) x i − z i | x − z | (cid:19) (1 − χ ( z )) θ ( z ) dz , so that, using that for x ∈ B and z ∈ B c / we have | x − z | ≥ , we have | ∂ j R i u ( x ) | ≤ C k θ k L ∞ Z | x − z |≥ | x − z | dz < + ∞ . We have obtained that k u k L ∞ (( − , ,C δ ( B )) ≤ C ( k θ k L ∞ (( − , ,C δ ( B )) + k θ k L ∞ ( R × ( − , ) . Step 2: We show that θ ∈ C ,δ − (1 − α ) (( − / , , L ∞ ( B / )) . Observe that θ solves a heat equation with right-hand side u ·∇ θ ∈ L ∞ (( − / , , C δ − (1 − α ) ( B / )),so that ∂ t θ ∈ L ∞ (( − / , , C δ − (1 − α ) ( B / )) . In particular, θ ∈ Lip(( − / , , C δ − (1 − α ) ( B / )) . Repeating the argument, we obtain that θ ∈ C ,δ − (1 − α ) ( B / × ( − / , . Higher regularity then follows from energy estimates. (cid:3)
Appendix C. Existence of suitable weak solutions
Proof of Theorem 3.6.
Fix θ ∈ L ( R ). For ǫ >
0, we consider the system with added vanishingviscosity term ( ∂ t θ + u · ∇ θ + ( − ∆) α θ = ǫ ∆ θu = ∇ ⊥ ( − ∆) − θ , (64)complemented with the initial datum θ ( · ,
0) = θ . For any ǫ >
0, the system (64) admits aglobal smooth solution θ ǫ : R × (0 , ∞ ) → R . Moreover, for any t > θ ǫ ( · , t ) ∈ L ( R ) and forany 0 ≤ s < t , we have the energy equality Z θ ǫ ( x, t ) dx + 2 Z ts Z h | ( − ∆) α θ ǫ | ( x, τ ) + ǫ |∇ θ ǫ | ( x, τ ) i dx dτ = Z θ ǫ ( x, s ) dx . (65)Theorem 3.2 also applies to (64), so that θ ǫ is in L ∞ for t > ǫ bound k θ ǫ ( t ) k L ∞ ≤ Ct − α k θ k L . (66)Finally, with the obvious modifications of the computation in Section 3.3, we have for anynonnegative test function ϕ ∈ C ∞ c ( R ), locally constant in y in a neighbourhood of y = 0, anynonnegative and convex f ∈ C ( R ) and any t > Z R ϕ ( x, , t ) f ( θ ǫ )( x, t ) dx + c α Z t Z R y b |∇ θ ∗ ǫ | f ′′ ( θ ∗ ǫ ) ϕ dx dy ds ≤ Z t Z R [ f ( θ ǫ )( ∂ t ϕ | y =0 + ǫ ∆ ϕ | y =0 ) + u ǫ f ( θ ǫ ) · ∇ ϕ | y =0 ] dx ds + c α Z t Z R y b f ( θ ∗ ǫ )∆ b ϕ dx dy ds =: C ( ǫ ) + D ( ǫ ) . We want to pass to the limit ǫ →
0. From (65) with s = 0 and the Sobolev embedding of˙ W α, ( R ) ֒ → L − α ( R ), we infer by interpolation that the family { θ ǫ } ǫ> is uniformly boundedin L ∞ ([0 , ∞ ) , L ) ∩ L ([0 , ∞ ) , ˙ W α, ) ֒ → L α ) ( R × [0 , ∞ )) . By Banach-Anaoglu, for any fixed time
T >
0, there exists θ ∈ L ( R × [0 , T ]) and a subsequence ǫ k → θ ǫ k ⇀ θ weakly in L ( R × [0 , T ]). We now claim that this convergence isin fact strong via an Aubin-Lions type argument in the same spirit as Step 2 of the proof ofLemma 3.8. Fix η > { φ δ } δ ≥ ⊆ C ∞ c ( R ) in space. For k, j ≥ k θ ǫ j − θ ǫ k k L ( R × [0 ,T ]) ≤ k θ ǫ j − θ ǫ j ∗ φ δ k L + k θ ǫ k − θ ǫ k ∗ φ δ k L + k ( θ ǫ j − θ ǫ k ) ∗ φ δ k L . The first two contributions converge to 0 independently of k and j due to a bound of theform k θ ǫ k − θ ǫ k ∗ φ δ k L ( R × [0 ,T ]) ≤ Cδ α obtained as in (27) with η k replaced by θ ε k . We now choose δ small enough such that this contribution does not exceed η . Having δ fixed, we claimthat the family of curves { t θ ǫ k ( · , t ) } k ≥ is equicontinuous and equibounded with values in W , ∞ . Indeed, by the energy equality (65) with s = 0 and the Calderon-Zygmund estimate k u ǫ k k L ( R × [0 ,T ]) ≤ C k θ ǫ k k L ( R × [0 ,T ]) , we estimate k ∂ t θ ǫ k ∗ φ δ k L ([0 ,T ] ,W , ∞ ) ≤ k div( u ǫ k θ ǫ k ) ∗ φ δ k L W , ∞ + k θ ǫ k ∗ ( − ∆) α φ δ k L W , ∞ + ǫ k k θ ǫ k ∗ ∆ φ δ k L W , ∞ ≤ (cid:0) k u ǫ k k L ( R × [0 ,T ]) k θ k L + 2 k θ ǫ k k L ( R × [0 ,T ]) (cid:1) k φ δ k W , ∞ ≤ C ( δ ) . By Ascoli-Arzela the sequence { θ k ∗ φ δ } k ≥ converges uniformly on R × [0 , T ] and by uniquenessof limits, we infer that this limit must coincide with θ ∗ φ δ . We can therefore choose N ≥ k, j ≥ N we have k ( θ ǫ j − θ ǫ k ) ∗ φ δ k L ( R × [0 ,T ]) ≤ η . We concludethat for k, j ≥ N there holds k θ ǫ k − θ ǫ j k L ( R × [0 ,T ]) ≤ η . Since η was arbitrary, we conclude byuniqueness of limits that θ ǫ k → θ strongly in L ( R × [0 , T ]) . By the uniform boundedness in L α ) ( R × [0 , ∞ )) and by (66) we also deduce that θ k → θ strongly in L r ( R × [0 , T ]) for any2 ≤ r < α ) and strongly in L r ( R × [ τ, T ]) for any τ > ≤ r < ∞ . By Calderon-Zygmund, we infer that u ǫ → u := R ⊥ θ strongly in L ( R × [0 , T ]) (and L r respectively). Passingto the limit k → ∞ in the equation (64), we infer that θ is a distributional solution to (1)–(3).We are left to pass to the limit in the global and local energy (in-)equality. Consider first (65).By Banach Anaoglu and uniqueness of limit ( − ∆) α θ ǫ k ⇀ ( − ∆) α θ weakly in L ( R × [0 , T ]) andby weak lower semicontinuity Z ts Z | ( − ∆) α θ | ( x, τ ) dx dτ ≤ lim inf k →∞ Z ts Z | ( − ∆) α θ ǫ k | ( x, τ ) dx dτ for any 0 ≤ s < t . For almost every t ∈ [0 , T ] we can extract a further subsequence such that θ ǫ k ( · , t ) → θ ( · , t ) strongly in L ( R ). By passing to the limit in (65), we thereby obtain (20) and(21) for almost every 0 < s < t . We obtain it for every t > < s < t ) byobserving that up to changing θ on a set of measure 0, we may assume that θ is continuous withrespect to the weak topology on L ( R ) . We are left to pass to the limit in the localized energyinequality for f ( x ) = ( x − M ) L and f ( x ) = | x − ML | q for q > L > M ∈ R . Let us denote η k := ( θ ǫ k − M ) /L , η := ( θ − M ) /L and let us fix τ, R > ϕ ⊆ B R (0) × [0 , R ] × [ τ, T ] . From the strong convergence established before, we infer that η k → η strongly in L rloc ( R × [ τ, T ]) for 2 ≤ r < ∞ . Up to extracting a further subsequence anda diagonal argument, we obtain that η k ( t ) → η ( t ) strongly in L rloc ( R ) for almost every t > r ∈ N ≥ . By interpolation, the former statement holds in fact for every 2 ≤ r < ∞ . We deduce that for almost every t > , for q = 2 and any q ≥ k →∞ Z R ϕ ( x, , t ) f ( θ ǫ k ) ( x, t ) dx = lim k →∞ Z R ϕ ( x, , t ) | η k | q ( x, t ) dx = Z R ϕ ( x, , t ) | η | q ( x, t ) dx , lim k →∞ C ( ǫ k ) = Z t Z R [ | η | q ∂ t ϕ | y =0 + u | η | q · ∇ ϕ | y =0 ] dx ds . We recall that from the Poisson formula θ ∗ ǫ k ( x, y, t ) = ( P ( · , y ) ∗ θ ǫ k ( · , t ))( x ) and Young’s convo-lution inequality k θ ∗ ǫ k k L q ( R × [0 ,R ] × [ τ,T ] , y b ) ≤ C ( R ) k P (1 , · ) k L k θ ǫ k k L q ( R × [ τ,T ]) = C ( R, α ) k θ ǫ k k L q ( R × [ τ,T ]) , where we used that k P ( · , y ) k L = k P ( · , k L for y > . We deduce that θ ∗ ǫ k → θ ∗ strongly in L q ( R × [0 , R ] × [ τ, T ] , y b ) . By linearity, η ∗ k = θ ∗ ǫk − ML so that η ∗ k → η ∗ strongly in L q ( B R × [0 , R ] × [ τ, T ] , y b ) . Hence lim k →∞ D ( ǫ k ) = Z t Z R y b | η ∗ | q ∆ b ϕ dx dy ds . Moreover, we also deduce ∇ η ∗ k ⇀ ∇ η ∗ and ∇| η ∗ k | q ⇀ ∇| η ∗ | q weakly in L ( B R (0) × [0 , R ] × [ τ, T ] , y b ) and we infer by weak lower semicontinuity and the positivity of ϕ that Z t Z R y b |∇ η ∗ | ϕ dx dy ds ≤ lim inf k →∞ Z t Z R y b |∇ η ∗ k | ϕ dx dy ds Z t Z R y b |∇| η ∗ | q | ϕ dx dy ds ≤ lim inf k →∞ Z t Z R y b |∇| η ∗ k | q | ϕ dx dy ds for any t > . Passing to the limit k → ∞ , we obtain (25) and (26) for almost every t > . (cid:3) Acknowledgements . The authors thank the anonymous referee for pointing out a mistakein a previous version of the manuscript, which led to a substantial improvement in the paper.The authors have been supported by the SNF Grant 182565 “Regularity issues for the Navier-Stokes equations and for other variational problems”. MC has been supported by the NSF underGrant No. DMS-1638352. Both authors acknowledge gratefully the hospitality of the Institutefor Advanced Studies, where part of this work was done.
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