Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation

We consider the SQG equation with dissipation given by a fractional Laplacian of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <\frac{1}{2}$$\end{document}α<12. We introduce a notion of suitable weak solution, which exists for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}L2 initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) $$\end{document}12α1+αα(1-2α)+2.


Introduction
For α ∈ 0, 1 2 we consider the following fractional drift-diffusion equation where θ : R 2 × [0, ∞) → R is an active scalar, u : R 2 × [0, ∞) → R 2 is the velocity field and (− ) α corresponds to the Fourier multiplier with symbol |ξ | 2α . The system is usually complemented with the initial condition We will be particularly interested in the surface quasigeostrophic (SQG) equation where the velocity field u is determined from θ by the Riesz-transform R on R 2 . More precisely, we require There is a natural scaling invariance associated to the system: whenever (θ, u) solves (1), then so does the pair θ r (x, t) := r 2α−1 θ(r x, r 2α t) u r (x, t) = r 2α−1 u(r x, r 2α t). (4)

Main Result
Our main result shows that for every L 2 initial datum and every α ∈ 9 20 , 1 2 , there exists an almost everywhere smooth solution of the SQG equation and, more precisely, it provides a bound on the box-counting and Hausdorff dimension of the closed set of its singular points.  16 . For any α ∈ α 0 , 1 2 and any initial datum θ 0 ∈ L 2 (R 2 ) there is a Leray-Hopf weak solution (θ, u) of (1)-(3) (see Definition 3.1) and a relatively closed set Sing θ ⊂ R 2 × (0, ∞) such that Remark 1. 2 We will in fact prove a slightly stronger statement, namely that all suitable weak solutions θ of (1)-(3) on R 2 × (0, ∞) (see Definition 3.4) satisfy the estimate on the dimension of the 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 guarantee local smooth existence.

∞)] is compact with box-counting dimension at most
The regularity issue for the equation (1)-(3) is fully understood only in the subcritical and critical regime, namely for α ≥ 1 2 . The critical case (without boundaries) is now well-understood thanks to Kiselev, Nazarov and Volberg [21] and Caffarelli and Vasseur [3] (see also [11]) and one even has a description of the long time behaviour of the system [12,13]. On bounded domains, the critical case has been well-studied in a series of works initiated by [8]. In the supercritical range α < 1 2 , the global regularity of Leray-Hopf weak solutions to the SQG equation is an open problem related to the problem of global existence of classical solutions: in fact, it is well-known that Leray-Hopf weak solutions coincide with classical solutions as long as the latter exist. Constantin and Wu [9,10] obtained partial results by extending the program of [3] to the supercritical regime. In [3] the technique of De Giorgi for uniformly elliptic equations with measurable coefficients is adapted to prove the smoothness of Leray-Hopf weak solutions in three steps: the local boundedness of L 2 solutions, the Hölder continuity of L ∞ solutions, and the smoothness of Hölder solutions. While the L ∞ -bound for Leray-Hopf weak solutions still works in the supercritical case [9], only conditional regularity results are known regarding the second and third step of the scheme. For instance, Hölder solutions in C δ are smooth for δ > 1 − 2α, while for δ < 1 − 2α this is left open [10]. On the negative side, [1] established non-uniqueness of a class of (very) weak-solution for the system (1)-(3), even for subcritical dissipations. In this context Theorem 1.1 is, to our knowledge, the first a.e. smoothness / partial regularity result.
The integral quantities present in (5) are two non-equivalent localized versions of the dissipative part of the energy, i.e. the L 2 ((0, T ), W α,2 (R 2 ))-norm of θ , and are globally controlled through the latter. At this point, the careful reader will object that Theorem 1.3 cannot be used in a covering argument since the maximal function is not bounded on L 1 . This issue represents a mere technical difficulty though: it is resolved by introducing a suitable variant of the sharp 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 ∈ ( 1 4 , 1 2 )) whose smallness requirement features an L p -based excess quantity and can be met at some small scale by requiring (5). Theorem 5.3 on the other hand is obtained via an excess decay result and a linearization argument, in analogy with [24] for the classical Navier-Stokes equations and with [7] for the hyperdissipative Navier-Stokes equations. Nevertheless there are some novelties in our approach with respect to the corresponding results for Navier-Stokes: • Our ε-regularity result relies on the crucial observation (previously used in [3,10]) that the 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 terms of the average of the rescaled solution u r on B 1 as r → 0. The lack of control on the averages introduces a challenge to iterate the excess decay, since at each step we need to correct for this change of variable, in a similar spirit to [10]. • As a second ingredient, we introduce a new notion of suitable weak solution which enables us 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 solutions in an essential way and are not available for the Navier-Stokes equations. The freedom of choosing a suitable nonlinear power on the other hand is crucial in the context of the SQG equation: Indeed, the classical (local) energy controls naturally θ ∈ L ∞ ((0, T ), L 2 (R 2 )) ∩ L 2 ((0, T ), W α,2 (R 2 )) and hence, by interpolation, θ ∈ L 2(1+α) (R 2 × (0, T )). Yet, since 2(1 + α) < 3 for α < 1 2 , this is not enough to conclude a strong enough Caccioppoli-type inequality 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 [27] for Navier-Stokes, providing ε-regularity criterion at a fixed scale. On the other side, in order to give an estimate on the dimension of the singular set, the smallness (5) must be required in terms of differential quantities of θ , as it happens in the more refined result by Caffarelli, Kohn and Nirenberg for Navier-Stokes [4]. In the context of the SQG equation, the easier Corollary 6.1 below may be seen as the full analogue of Scheffer's result. Although Corollary 6.1 still establishes the compactness of the singular set, it does, in contrast to Navier-Stokes, not yield any estimate on the dimension of the singular set.
Moreover, by the decay of the L ∞ -norm of solutions (see Theorem 3.2 below), the εregularity criterion is verified for large times and we recover the eventual regularization of suitable weak solutions from L 2 -initial data for α ∈ ( 1 4 , 1 2 ) previously established for Leray-Hopf solutions in [30] for α close to 1 2 and in [14,20] for any α ∈ (0, 1 2 ).

A Conjecture on the Optimal Dimension Estimate
Theorem 1.1 leaves open the question 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 estimate of the dimension of the spacetime singular set is and In (7), P is the parabolic Hausdorff measure that is, for α < 1 2 , the Hausdorff measure resulting from restricting the class F of admissible covering sets to the spacetime We refer for instance to [7] for its construction for α > 1 2 . The cylindersQ r (x, t) are the natural choice for α < 1 2 because their diameter is less than 4r , at difference from the classical parabolic cylinders B r (x) × (t − r 2α , t] whose diameter is of the order of r 2α . 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 which makes the spacetime integral of f (θ ) onQ r dimensionless, i.e. scaling-invariant with respect to (4). The number appearing on the right-hand side of (7) corresponds then to dimension of the energy, whose dissipative part is the globally controlled quantity in the form of a spacetime integral which scales best. This would correspond to the result of Caffarelli, Kohn and Nirenberg [4] ) for the Navier-Stokes system (see [7,22,33] for fractional dissipations of order α ∈ [ 3 4 , 5 4 )) who proved that suitable weak solutions of the latter are smooth outside a closed set of dimension 1. In fact, for the Navier-Stokes system this bound on the dimension of the singular set is what the scaling of the equations and boundedness of the energy suggest.
Notice that the right-hand side of both (7) and (8) does not converge to 0 as α → 1 2 : this is due to the fact that the quantity that dictates the scaling-criticality of the equation, namely the L ∞ -norm of θ , is not of integral type and hence cannot be used in a covering argument of the type that we do in the proof of Theorem 1.1. In turn, this covering argument finds his pivotal quantity in the dissipative part of 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, since the tilting effect of the change of variables, which sets the space average of u to zero, forces us to work on balls in spacetime (rather than parabolic cylinders, see Sect. 6.4 and in particular Step 3 of the proof of Corollary 6.6). This effect of the change of variables constitutes a serious obstacle for any parabolic Hausdorff dimension estimate. However, our estimate is nonoptimal: to obtain the optimal estimate, one should replace 1+α α by 2 in the estimate of the dimension of the singular set in Theorem 1.1; however, the integrability exponent 1+α α represents the least possible exponent for which we are able to use a "nonlinear" localized energy inequality in an excess decay argument (cf. Lemma 3.8). An analogous difficulty appears for the ipodissipative Navier-Stokes equations for low fractional orders α < 3 4 where the Caccioppoli-type inequality as described before fails to be strong enough to control the cubic nonlinearity and indeed no estimate of the dimension of the singular set is known.

Structure of the Paper
The paper is structured as follows. After recalling some technical preliminaries in Sect. 2, we discuss in Sect 3 the global and local energy inequalities of the SQG equation and we define the notion of suitable weak solutions. The key compactness property of the latter is proven in Sect. 3.5 and leads to an excess decay result established in Sect. 4. The iteration of the excess decay on all scales is performed in Sect. 5 and requires to 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 Sect. 6. The proof of Theorem 1.1 is given in Sect. 7. In Sect. 8, we discuss the stability of the singular set with respect to variations of the fractional order of dissipation.

Notation
We use the following notation for space(time) averages of functions or vector fields f defined on We introduce the spacetime cylinder adapted to the parabolic scaling (4) of the equation In the upper half-space R 3 We will omit the center of the cylinders whenever (x, t) = (0, 0). Moreover, we use the following convention to describe spacetime Hölder spaces: For α, β ∈ (0, 1) and Q ⊂ R 2 × R we denote by C α,β (Q) the functions which are αand β-Hölder continuous in space and time respectively, namely such that the following semi-norm is finite Whenever α = β, we denote the above space just by C α (Q). Furthermore, we will also work with spatial Sobolev spaces of fractional order: For ⊆ R n , s ∈ (0, 1) and 1 ≤ p < ∞, we denote by Correspondingly, we define for f ∈ W s, p ( ) the Gagliardo semi-norm by In the special p = 2, we will sometimes denote W α,2 by H α and we recall that for = R n the Gagliardo semi-norm coïncides, up to a universal constant, with the seminorms (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 2 ) or X = W α,2 (R 2 )). Whenever we work on a parabolic cube Q r (x, t), we will use the short-hand notation

Riesz-Transform
We recall that the Riesz-transforms admit a singular integral representation. Indeed, for f : By Calderon-Zygmund they are bounded operators on L p for 1 < p < ∞ and from L ∞ to B M O.

Caffarelli-Silvestre Extension
We recall the following extension problem. We use the notation ∇, for differential operators defined on the upper half-space R n+1 + .
Theorem 2.1 (Caffarelli-Silvestre [2]) Let θ ∈ H α (R n ) with α ∈ (0, 1) and set b := 1 − 2α. Then there is a unique "extension" θ * of θ in the weighted space and the boundary condition Moreover, there exists a constant c n,α , depending only on n and α, with the following properties: (a) The fractional Laplacian (− ) α θ is given by the formula The following energy identity holdŝ (c) The following inequality holds for every extension η ∈ H 1 (R n+1

Poincaré Inequalities
Let α ∈ (0, 1), 1 ≤ p < n α and p * := pn n− pα . There exists a universal constant We will also need a weighted Poincaré inequality in the spirit of the classical work [17] for α = 1 (where on the other side much more general weights are admissible).
Let ω ∈ C ∞ c (R n ) be a radial, non-increasing weight such that ω ≡ 1 on B r /2 (x), ω ≡ 0 outside B r (x) and |∇ω| ≤ C r pointwise. We introduce the weighted average The following weighted Poincaré inequality is classical for α = 1 (see [23,Lemma 6.12]) and it is established for q = p in [16,Proposition 4]: Their proof extends to the other endpoint q = p * and hence to the range q ∈ [p, p * ] by interpolation.
In the case p = 2, we can rewrite the right-hand side of (10) and (11) in terms of the extension as follows.

(Sharp) Maximal Function
For a function f : as well as the sharp fractional maximal function (in space) In order to use a spacetime integral of θ # α,q in a covering argument, we need to know that it is globally controlled; to guarantee the latter, we are forced to choose q < ∞.

Lemma 2.4
Let α ∈ (0, 1), f ∈ W α,2 (R n ) and q ∈ ( √ 2, ∞). Then there exists a constant C = C(n, q) ≥ 1 (which is uniformly bounded for q bounded away from ∞) such that For α = 1, the equivalent of Lemma 2.4 is a simple consequence of the Poincaré inequality and the maximal function estimate. Indeed, by Poincaré we have almost everywhere the pointwise estimate for f ∈ W 1,1 loc and f ∈ W 1,2(1−1/q 2 ) loc respectively. Integrating in x and using the boundedness of the maximal function on L 2 and L 1+1/(q 2 −1) , 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) where D α,2 f is the n-dimensional version of (6), i.e. for z ∈ R n By taking the supremum over r > 0, we deduce from (17) that for almost every x and hence by the maximal function estimate on L 1+1/(q 2 −1)

Leray-Hopf Weak Solutions
We recall the notion of Leray-Hopf weak solutions. (2) in the sense of distributions, namely div u = 0 and (19) for any ϕ ∈ C ∞ c (R 2 × R).
Observe that from the weak formulation (19) it follows that for all (22) for s = 0 and almost every t ∈ (0, T ) and for almost every 0 < s < t < T (with c α given by Theorem 2.1). Indeed, the equality between the last term of the second line and the last term of (22) holds for every θ ∈ L 2 ((0, T ), W α,2 (R 2 )); in the smooth case this equality is a consequence of 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 > 0.
Theorem 3.2 ([10] Theorem 2.1) Let θ 0 ∈ L 2 (R 2 ) and let (θ, u) be a Leray-Hopf weak solution of (1)- (2). Then there exist a universal constant, independent on u, such that for any t > 0 In the particular case (3), where u = R ⊥ θ , we obtain as a consequence that for any Remark 3.3 In [10] 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 the assumption div u = 0 is used; the structure (3) is only used to deduce (24) from (23).

Suitable Weak Solutions
We are now ready to give our definition of suitable weak solution. Both this notion and the one of Leray-Hopf solution are given without requiring the coupling (3), since, in the proof of Theorem 1.3, we will need to work on a larger class of equations, 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 2 × (0, T ) is a suitable weak solution if the following two inequalities hold for almost every t ∈ (0, T ), all nonnegative test  (26) 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 3.5
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 every linear transformation η := (θ − M)/L. However, it can be proved (see for instance [7]) that the class of suitable weak solutions is stable under this transformation. Here on the other hand, since we use a "nonlinear" energy inequality (26), it is no longer obvious that the class of suitable weak solutions is stable under linear transformations; hence we require it already in the definition. The class of suitable weak solutions contains smooth solutions (see Sect. 3.3) and is non-empty (see Sect. 3.4) for any L 2 initial datum.

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 2 (R). We multiply (1) by f (θ )ϕ| y=0 and integrate in space to obtain for t ∈ [0, T ] By means of the divergence theorem, we compute for fixed time t where we integrated by parts in the third equality and used that the boundary terms vanish due to the hypothesis ∂ y ϕ(·, 0, ·) = 0. We obtain that for f ∈ C 2 (R) Observe that if f is moreover convex and nonnegative, both the left-and the righthand side of the above equality have a sign. In particular, we obtain (25) with an equality when choosing

Existence of Suitable Weak Solutions
For any α ∈ (0, 1 2 ) the existence of suitable weak solutions can be established from any initial datum θ 0 ∈ L 2 (R 2 ) by adding a vanishing viscosity term θ on the right-hand side and letting → 0. The key argument is a classical Aubin-Lions type compactness argument that we sketch in Appendix C.

Compactness
We establish the compactness of a sequence of suitable weak solutions with vanishing excess. Let (θ, u) be a solution of (1)-(2) on R 2 × (0, T ). For r > 0 and Q r (x, t) ⊆ R 2 × (0, T ), we define the excess as for p ∈ (3, ∞) and σ ∈ (0, 2α) 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, 0), we will denote the excess simply by E(θ, u; r ).

Remark 3.7 (Rescaling of the excess)
The excess behaves nicely under the natural rescaling (4). Indeed, for r > 0 we have and η solves We will need the following auxiliary Lemma. Lemma 3.9 (Tail estimate) Let α ∈ (0, 1 2 ] , σ ∈ (0, 2α) and 1 < p < ∞ . Then there exists a universal constant The first term is estimated using Young and the fact that For i ≥ 1, we estimate, using the fact that for We obtain the claim by raising the previous inequality to the power p.
We may assume q ∈ [2, (p − 1)(1 + α)). Since the excess uniformly bounds the L 3/2 loc -norm of η k , there exists by Banach-Alaoglu a limit η ∈ L 3/2 , up to extracting a subsequence. By the uniform boundedness established in Step 1, we may assume, up to extracting a further subsequence, that η k η weakly in L q (Q 13/16 ). We now claim that the latter convergence is in fact strong on the slightly smaller cube Q 3/4 . Indeed, fix ε > 0 and a family {φ δ } δ>0 of mollifiers in the space variable. For k, j ≥ 1 we estimate 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 Hölder and the uniform boundedness of where C does not depend on k ≥ 1 by Step 1. Since η k is uniformly bounded in L (1+α)( p−1) (Q 13/16 ), we have by interpolation for some ϑ ∈ (0, 1] and C ≥ 1 that We now fix δ small enough, independently of k, such that this contribution does not exceed ε 3 . As for the third term, we consider for fixed δ > 0 small, the family of curves As for the last term, we have that for 1+|x| 2+2α . We estimate the convolution on dyadic balls for fixed time. We set η k,i : We conclude by integrating in time that Summarizing, we have shown that uniformly in k ≥ 1. Hence the family of curves {t → η k * φ δ } k≥1 is an equicontinuous sequence with values in a bounded subset of W 1,∞ (B 3/4 ). By Arzela-Ascoli there exists a uniformly convergent subsequence (which we don't relabel), and in particular, there exists N = N (δ) > 0 such that for any k, j ≥ N we have Step 3: Conclusion.

By
Step 2 we can pass to the limit in the equation in Q 3/4 and deduce that η ∈

Decay of the Excess
In this section, we prove the self-improving property of the excess, namely that if the excess is small at any given Q r , there exists a small, fixed scale μ 0 ∈ (0, 1 2 ), independent of r , at which the excess decays between Q r and Q μ 0 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-local part of the velocity (see E V (v 2 k ; μ) in the proof of Proposition 4.1). More generally, one could prove this excess decay at scale μ 0 under the weaker assumption that the average of the rescaled velocity u r on B 1 is bounded uniformly in r for r ∈ (0, 1). However, since all L p -norms are supercritical with respect to the scaling (4) of the equation, we will not be able to guarantee such an assumption. In this section, we will also for the first time make use of the structure of the velocity field (3). Similar arguments should apply for velocity fields determined from θ by other singular integral operators.

Remark 4.2
If in (28) f = 0 we recover simply the SQG equation. We will need the freedom to subtract a function of time f from the velocity field u in order to satisfy the zero-average assumption (see Lemma 5.1).
We will need the following auxiliary Lemma.
Moreover for i, j = 1, 2 and x ∈ B 1/4 , we notice that the integral representation is no longer singular and we can compute by integration by parts where we used that the boundary terms at {|z| = 3 8 } and at infinity vanish. Observe that for 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 suitable weak solutions to (1)-(2) such that We set E k := E(θ k , u k ; 1) and M k := (θ k ) Q 1 . We will consider the rescaled and shifted sequence By construction, (η k ) Q 1 = 0 and E(η k , v k ; 1) = 1. In particular, we have for all μ ∈ (0, 1 2 ) that We will now take the limit k → ∞ and argue that (29) contradicts the excess decay dictated by the linear limit equation. Indeed, by Lemma 3.8, the sequence η k converges weakly to η in L 3/2 loc (R 2 ×[−1, 0]) and moreover, η k → η strongly in L p (Q 3/4 ). Hence we have for μ ∈ (0, 1 2 ) We also know from Lemma 3.8 that η ∈ L p (Q 3/4 ) solves the fractional heat equation In particular, we infer that for μ ∈ (0, 1 2 ) Let us now consider the non-local part of the excess. We split We estimate the second term by adding and subtracting [η k (t)] B 1 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 N L (η k ; 1) + E S (η k ; 1) ≤ 1 for all k ≥ 1, we obtain that We infer, using the strong converge of We conclude that Finally, let us consider the part of the excess which is related to the velocity v k . We observe that, using the structure of the velocity (28), for some cut-off χ between B 3/8 and B 1/2 . Correspondingly, we write By Calderon-Zygmund estimates, we infer that v 1 k → R ⊥ (ηχ ) =: v 1 strongly in L p (Q 3/4 (30). We conclude that We now come to the excess related to v 2 k . By construction, Correspondingly, we define w 1 k,ρ := R ⊥ (η k (1 − χ)χ ρ ) and w 2 k,ρ := R ⊥ M k E k χ ρ for some radially symmetric cut-off χ ρ between B ρ and B ρ+1 . By Calderon-Zygmund, we have for fixed time t that w 1 k, Let us consider first w 1 k,ρ . We apply Lemma 4.3 to w 1 k,ρ to deduce that for fixed time t Integrating in time, we infer that uniformly in ρ ≥ 1 We deduce that for μ ∈ (0, 1 4 ) we have We now come to the contribution of w 2 k,ρ . Observe that (− ) 1/2 w 2 k,ρ = 0 in B 1 for ρ ≥ 1 such that w 2 k,ρ is smooth in the inside of B 1 . Recall moreover, that we have the integral representation (in the principal value sense) so that, by spherical symmetry of χ ρ , we immediately infer w 2 k,ρ (0) = 0. Moreover, Thus for fixed k ≥ 1, we have so that excess associated to v k,2 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 ) > 0 , that for all μ ∈ (0, 1 4 ). We reach the desired contradiction for

Iteration of the Excess Decay
In this section, we prove the decay of the excess on all scales. We iteratively define shifted rescalings of (θ, u) verifying the zero average assumption of Proposition 4.1 as well as (28) and therefore allowing the decay of the excess when passing at scale μ 0 . From the decay of the excess on all scales, we deduce Hölder continuity by means of Campanato's Theorem. In contrast to Navier-Stokes, we need our estimates to be quantitative, since it is not known whether local smoothness for SQG follows from a mere L ∞ -bound; instead we need to prove spatial C δ -Hölder continuity of the velocity for a δ > 1 − 2α (see Lemma B.1). The main mechanism of the iteration is the invariance of the equation under the following change of variables realizing the zero average assumption on B 1/4 . The latter has been exploited previously in [3,10].

be a suitable weak solution to (1)-(3) on
where (θ 0 , u 0 ) is obtained from θ through the change of variables of Lemma 5.1. Then

Remark 5.4 (Role of the parameters)
The parameter p is crucial since it determines the dimension of the singular set (see proof of Theorem 1.1): the lower the power p, the better the dimension estimate. All the other parameters are of technical nature; yet, the range of admissible parameters is sufficiently large to allow us to conclude the desired estimate on the size of the singular set for all fractional orders for which the latter is meaningful (i.e. for α > α 0 , see Remark 6.7). We deliberately choose to leave all the parameters free to increase the readability of the paper; but one could also read the paper fixing the parameters as in the proof of Theorem 1.3. Let us now comment on the role of the single parameters in more detail: • p cannot go below the threshold 1+α α : This corresponds to spacetime integrability that guarantees the compactness of the ( p − 1)-energy inequality (see Lemma 3.8) which in turn is the crucial ingredient of the excess decay. The requirement p > 2α σ on the other hand is purely technical and harmless for σ close to 2α.
• σ captures the decay at infinity of the non-local part of both the fractional Laplacian and the velocity and should be thought arbitrarily close to 2α.
• γ describes the decay of the excess when passing to the smaller scale μ 0 . In order to apply the excess decay of Proposition 4.1 iteratively, we have to verify its smallness requirement along a sequence of by μ := μ 0 4 rescaled solutions which is possible only if the decay rate beats the supercritical scaling of the excess (see Remark 3.7), i.e. γ ≥ 1 − 2α.
• The exponent of the local Hölder continuity in space, δ , is obtained from γ , but considerably worsened. This stems form the fact that in order to use the decay of the excess on all scales to deduce Hölder continuity via Campanato's Theorem, we have to control the effect of the flow of Lemma 5.1. This loss in the Hölder continuity exponent is peculiar to SQG and is not observed in the similar results for Navier-Stokes.
Proof Let ε 1 > 0, C 1 ≥ 1 be the universal constants from Lemma 5.1. The proof relies on an iterative construction. We fix (x, t) ∈ Q 1 . We obtain the suitable weak solution (θ 0 , u 0 ) by applying Lemma 5.1 to (θ, u) at the point (x, t). This first change of variables does two things: It translates (x, t) to the origin (0, 0) and it produces a new suitable weak solution whose velocity u 0 has zero average on B 1/4 . Hereafter, the excess will always be centered in (0, 0).
Next, we want to deduce the Hölder 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 0 made on modified cylinders. 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 in Step 4 in terms of θ rather than θ 0 , and we apply Campanato's Theorem to deduce the Hölder continuity 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 constantε 2 ∈ (0, 1 2 ) (depending only on α, σ , γ and p) such that if ε 2 ∈ (0,ε 2 ] and if (θ, u) is a suitable weak solution to (1) on R 2 × (−2 2α , 0] with (36), then for every k ≥ 0 the excess of (θ k , u k ) (see (38)-(39)), decays at scale μ: where C 1 is the universal constant from Lemma 5.1. We proceed by induction on k ≥ 0. The case k = 0. Let ε 2 ∈ (0,ε 2 ] for someε 2 ∈ (0, 1 2 ) to be chosen later and assume that (36) holds. We only need to show (40). If The inductive step. By the inductive hypothesis, we can assume that We recall that (θ k , u k ) is obtained by applying the change of variables of Lemma 5.1 to (θ k−1,μ , u k−1,μ ) at the point (0, 0). Using the inductive assumption and that [u k−1 (s)] B 1/4 = 0 for s ∈ [−1, 0], we can verify the smallness assumption of Lemma 5.1. Indeed, Choosingε 2 even smaller, namely, By Lemma 5.1, Remark 3.7 and the inductive hypothesis, we deduce that showing the second inequality and, recalling the choice of ε 2 ∈ (0,ε 2 ), showing in particular that Since by construction [u k (s)] B 1/4 = 0 for s ∈ [−1, 0], we infer from Proposition 4.1 and the inductive assumption that Step 2: bound on the translation in the change of variables. We observe that θ k is just a shifted and rescaled (by μ k , according to the natural scaling (4)) version of θ 0 . Indeed, notice that by construction, one can verify inductively for k ≥ 1 where θ 0,μ k (y, s) := μ k(2α−1) θ 0 (μ k y, μ 2αk s) and We claim that the center of the cylinders don't move too much, namely for s ∈ [−1, 0] Indeed, for j ≥ 1 we estimate as long as x j (s) ∈ B 1/4 where we used that [u j−1 (μ 2α s)] B 1/4 = 0 uniformly in time. In particular, and hence for s ∈ [−1, 0] we have, using (41), Collecting terms, we have Step 3: Decay of a modified excess of θ 0 . We claim that for every r ∈ (0, μ 2 ) Observe that by the scaling of the excess μ 2α−1 E S (θ k ; μ) = E S (θ k,μ ; 1) and by (42) We introduce the set If s ∈ I k+1 we can ensure, by an appropriate choice of ε 2 , that r k (μ 2α s) ∈ B 3μ k+1 /4 . Indeed for s ∈ I k+1 by Step 2. It is thus enough to choose ε 2 (if necessary) even smaller, or more precisely, we set We now estimate, by adding and substracting (θ k,μ ) Q 1 and Hölder Combining the previous inequality with Step 1 and observing that μ (k+1)2α This gives (43) for r = μ k+1 for some k ≥ 1. For r ∈ (μ k+2 , μ k+1 ) instead, we observe that Step 4: Decay of a modified excess of θ . There exists a r 0 = r 0 ( u L p+1 (Q 3/2 ) ) > 0 such that for every r ∈ (0, r 0 ) and for every ( As long as x 0 (s) ∈ B 1/4 and |s| < 1 5 , we have the estimate In particular for 0 ≤ |s| p p+1 ≤ min 1 4 the estimate (44) holds. Let now Recalling that μ 2 ≤ 1 64 , we observe that for all r ∈ (0, r 0 ), (x, t) ∈ Q 1 and s ∈ (−r p p−1 , 0] (44) holds and we have Hence we can estimate by the triangular inequality and Hölder, by (45) and by Step 3 Step 5: By Campanato's Theorem, we deduce that θ is Hölder continuous in Q 1 . By a variant of Campanato's Theorem [18,Theorem 2.9.], we deduce from Step 4 that (37) holds. Indeed, observe that the sets B r (x) × (t − r p/( p− 1) , t] are nothing else but balls with respect to the metric d ((x, t), (y, s)) := max{|x − y|, |t − s| ( p−1)/ p } on spacetime where in time, as usual for parabolic equations, we only look at backwardin-time intervals. The proof of this version of Campanato's Theorem follows, for instance, line by line [28,Theorem 1] when replacing the parabolic metric by d.

"-Regularity Results and Proof of Theorem 1.3
In this section we prove some ε-regularity results, including Theorem 1.3 and its more precise version 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 [27] for the Navier-Stokes system. As in the case of Navier-Stokes, it implies that the singular set of suitable weak solutions is compact in spacetime (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 everywhere smoothness (or any estimate on the dimension of the singular set): The fact that the L ∞ -norm is a controlled quantity necessitates to rely on spacetime integrals of derivatives of θ to show local smoothness. In order to pass from Theorem 5.3 to an ε-regularity criterion involving only fractional space derivatives of θ ,which are globally controlled through the energy, we need to overcome the following difficulties: • The excess E S related to θ involves the spacetime average of θ. In particular, in order to use a standard Poincaré inequality (10) to pass to a differential quantity, we would need some fractional differentiability in time too. Using the parabolic structure of the equation, we will be able to circumvent this and to establish in Lemma 6.2 a Poincaré inequality 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 0 , so that we need some control on the tilting effect of the flow. We will see that at scale r , the flow shifts the center of the excess in space by at most r 2α−2/q u L ∞ L q (see (55)). As a consequence, at scale r , all quantities related to the excess of θ will no longer live on parabolic cylinders but rather on modified cylinders Q(x, t; r ) in spacetime, approximately of radius r 2α in time and r 2α−2/q in space. Morally q = ∞; however, since the Riesz-transform is bounded from L ∞ → B M O and not from L ∞ → L ∞ , we introduce the parameter q which should be thought to be arbitrarily large.
• We set the excess in L p for p > 1+α α in order to gain the compactness of the ( p − 1)-energy inequality. In order to exploit the L 2 W α,2 -control given by the energy via the nonlinear Poincaré inequality described in the first point, we are lacking some higher integrability in time. We bypass this issue by factoring out p − 2 powers of θ in L ∞ .

An Analogue of Scheffer's Theorem
We provide a first ε-regularity result featuring spacetime integrals of θ and u. Observe that in agreement with the previous discussion, smooth solutions of (1)-(3) do, in general, not verify the ε-regularity criterion (46) at any small scale.

Nonlinear Poincaré Inequality of Parabolic Type
We introduce the following scaling-invariant quantity which should be understood as a localized version of the dissipative part of the energy: t) y b |∇θ * | 2 (z, y, s) dz dy ds.
The following Lemma and its proof is inspired by [32], where a parabolic Poincaré inequality is obtained for the classical, linear heat equation, and by [26], where a nonlinear Poincaré inequality of similar nature is also crucially used in a ε-regularity result.
Step 1: By means of the weighted Poincaré inequality (11), we reduce the Lemma to an estimate on weighted space averages computed at two different times. To this aim, let ω ∈ C ∞ c (R 3 + ) be a weight such that ω| y=0 is a radial non-increasing function, 0 ≤ ω ≤ 1 and ω ≡ 1 on B 1 × [0, 1] and ω ≡ 0 outside B 2 × [0, 2). We estimate for fixed time where we used ω(·, 0) ≡ 1 on B 1 . Reusing this fact and Hölder, we bound the last term by 1 2 , so that we deduce by the weighted Poincaré inequality (11) The first term on the right-hand side can be expressed in terms of the extension by (14). Since the weight ω is independent of time, the second term can be estimated by Step 2: We use the equation to estimate the difference of two weighted space averages computed at different times. We use the weak formulation (22) y). We estimate the right-hand side of (22) from below and the left-hand side from above for s, t ∈ [−2 2α , 0]. As for the lower bound, we havê
Collecting terms, we have for almost every s, t ∈ [−2 2α , 0] that Combining this estimate with Step 1, we conclude.

The Non-local Part of Excess
We recall from the proof of Proposition 4.1 that the excess related to the velocity can be estimated in terms of θ. More precisely, we have the following: . Consider a velocity field of the form u(z, s) = R ⊥ θ(z, s) + f (s) for some f ∈ L 1 loc (R). There exists C = C( p) ≥ 1 such that After reducing the Lemma to r = 1 and (x, t) = (0, 0), the proof follows line-by-line the estimate of E V in the proof of Proposition 4.1. We now bound the quantity E N L in terms of a variant of the sharp maximal function introduced in Sect. 2.6.
then there exists a constant C = C( p) ≥ 1 such that Proof By translation and scaling invariance, we may assume (x, t) = (0, 0) and r = 1.
We estimate the argument of the supremum for fixed time s and radius R ≥ 1 4 by the triangular inequality and Hölder For z ∈ B 1/4 it holds B R ⊆ B 2R (z), so that by the triangular inequality and by averaging over z ∈ B 1/4 , we have We combine (49)-(50) and use Hölder to bring the power 1 + 1 q 2 −1 inside the integral. We obtain Observe that this is ensured through (48) and that the supremum can be estimated from above by 4 p such that the constant of the Lemma depends only on p.

Proof of the Theorem 1.3
The following Corollary of Theorem 5.3 gives a different version of the ε-regularity criterion in terms of θ rather than its composition with the flow θ 0 . Theorem 1.3 will be an immediate consequence. To this aim, we introduce the following modified balls and cylinders (backwards and centered in time respectively) which are enlarged in space in accordance with the "intrinsic effect" of the flow (see (55) where To shorten notation, we will often omit the dependence of K q on u and r , and whenever (x, t) = (0, 0) , we will omit to specify the center of the balls and cylinders. The following remark justifies that one should really think of B(x; r ) as a enlarged balls of radius r 2α−2/q .
Remark 6.7 (Justification of α 0 ) α 0 is the threshold until which both the smallness hypothesis of Corollary 6.6 is verified, at sufficiently small scale, for smooth solutions at any point (x, t) in spacetime and the dimension estimate of Theorem 1.1 is nontrivial. Indeed, for α > α 0 it holds 1 2α Before proceeding with the proof, let us show that Theorem 1.3 is an immediate consequence of Corollary 6.6.
Proof of Theorem 1.3 Let α , p and q as in the statement and assume that (5) holds. Observe that C(x, t; r ) ⊇ Q(x, t + r 2α /16; r ). By (18) and Hölder we deduce the pointwise estimate We infer that θ satisfies (52) at the radius r /4 and the point (x, t + r 2α /16). We deduce from Corollary 6.6 that θ is smooth in the interior of Q r /8 (x, t + r 2α /16) which contains the open ball B r /8 (x) × (t − r 2α /16, t + r 2α /16).
Step 1: We tune the free parameters σ and γ .

The Singular Set and Proof of Theorem 1.1
We recall the box-counting dimension of a (compact) set S ⊆ R 3 : For every δ ∈ (0, 1) we denote by N (δ) the minimal number of sets of diameter δ needed to cover S. We then define (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.  Step 2: Let α ∈ (α 0 , 1 2 ) (otherwise the dimension estimate is trivial by Remark 6.7). We show that for every q ≥ 8 we have Indeed, fix q ≥ 8 and define p q := 1+α α + 1 q . From Corollary 6.6, we know that if (x, s) ∈ S , then for every r ∈ (0, (t/2) where ε = ε(α) > 0 is universal and in particular independent of r . By Theorem 3.2, the threshold ε 3 depends on t > 0 and θ 0 L 2 only. Following Remark 6.5, we observe that with the notation of Remark 6.5 Observe that L q depends only on θ 0 L 2 and t > 0. For δ ∈ (0, δ 0 ) , we define the collection δ containing balls Observe that { δ } δ is a family of coverings of S consisting of Euclidean balls in spacetime. By the Vitali covering Lemma, there exists for every δ a countable, disjoint Therefore, setting η := 2 √ 2L q δ 2α−2/q and using the disjointness of {B i } i∈I , we can estimate the minimal number N (η) of sets of diameter η needed to cover S by We conclude that Step 3: Conclusion. By taking the limit q → ∞ in Step 2, we conclude that

Stability of the Singular Set
This section is devoted to the proof of Corollary 1.4. We observe that the ε-regularity criterion is "continuous" under strong L p -convergence. This convergence result together with the observation that smooth solutions satisfy the ε-regularity criterion of Theorem 5.3 will allow to deduce the required stability. where we denote by θ n,0 and u n,0 (and θ 0 and u 0 respectively) the change of variables of Lemma 5.1 as applied to θ n (and θ respectively).
By the regularity of θ and (59) Then η is smooth with respect to the space variable and η ∈ C 1−1/ p (Q 1/2 ) in spacetime. Moreover, there existsC > 1 such that The constantC can be chosen uniform in α as long as α is bounded away from 0.
By Banach-Anaoglu, for any fixed time T > 0, there exists θ ∈ L 2 (R 2 × [0, T ]) and a subsequence k → 0 such that θ k θ weakly in L 2 (R 2 × [0, T ]). We now claim that this convergence is in fact strong via an Aubin-Lions type argument in the same spirit as Step 2 of the proof of Lemma 3.8. Fix η > 0 and a family of mollifiers {φ δ } δ≥0 ⊆ C ∞ c (R 2 ) in space. For k, j ≥ 1 we estimate The first two contributions converge to 0 independently of k and j due to a bound of the form θ k − θ k * φ δ 2 L 2 (R 2 ×[0,T ]) ≤ Cδ 2α 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 3 . Having δ fixed, we claim that the family of curves {t → θ k (·, t)} k≥1 is equicontinuous and equibounded with values in W 1,∞ . Indeed, by the energy equality (65) with s = 0 and the Calderon-Zygmund estimate u k L 2 (R 2 ×[0,T ]) ≤ C θ k L 2 (R 2 ×[0,T ]) , we estimate By Ascoli-Arzela the sequence {θ k * φ δ } k≥1 converges uniformly on R 2 × [0, T ] and by uniqueness of limits, we infer that this limit must coincide with θ * φ δ . We can therefore choose N ≥ 1 big enough such that for any k, j ≥ N we have (θ j − θ k ) * φ δ L 2 (R 2 ×[0,T ]) ≤ η 3 . We conclude that for k, j ≥ N there holds θ k − θ j L 2 (R 2 ×[0,T ]) ≤ η. Since η was arbitrary, we conclude by uniqueness of limits that θ k → θ strongly in L 2 (R 2 × [0, T ]). By the uniform boundedness in L 2(1+α) (R 2 × [0, ∞)) and by (66) we also deduce that θ k → θ strongly in L r (R 2 × [0, T ]) for any 2 ≤ r < 2(1 + α) and strongly in L r (R 2 × [τ, T ]) for any τ > 0 and any 2 ≤ r < ∞. By Calderon-Zygmund, we infer that u → u := R ⊥ θ strongly in L 2 (R 2 × [0, T ]) (and L r respectively). Passing to 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 (− ) for any 0 ≤ s < t. For almost every t ∈ [0, T ] we can extract a further subsequence such that θ k (·, t) → θ(·, t) strongly in L 2 (R 2 ). 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 > 0 (and almost every 0 < s < t) by observing that up to changing θ on a set of measure 0, we may assume that θ is continuous with respect to the weak topology on L 2 (R 2 ). We are left to pass to the limit in the localized energy inequality for f (x) = (x−M) 2 . From the strong convergence established before, we infer that η k → η strongly in L r loc (R 2 × [τ, T ]) for 2 ≤ r < ∞. Up to extracting a further subsequence and a diagonal argument, we obtain that η k (t) → η(t) strongly in L r loc (R 2 ) for almost every t > 0 and any r ∈ N ≥2 . By interpolation, the former statement holds in fact for every 2 ≤ r < ∞. We deduce that for almost every t > 0 , for q = 2 and any q ≥ 4 |η| q ∂ t ϕ| y=0 + u|η| q · ∇ϕ| y=0 dx ds.