Optimal Gevrey regularity for supercritical quasi-geostrophic equations

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity with the same decay exponent as the linear part. This settles several open problems in \cite{Bis14, BMS15}.


Introduction
We consider the following two-dimensional dissipative surface quasi-geostrophic equation: where ν ≥ 0, 0 < γ ≤ 2, D = (−∆) corresponds to the Fourier multiplier |ξ| s , i.e.D s f (ξ) = |ξ| s f (ξ) whenever it is suitably defined under certain regularity assumptions on f .The scalar-valued unknown θ is the potential temperature, and u = D −1 ∇ ⊥ θ corresponds to the velocity field of a fluid which is incompressible.One can write u = (−R 2 θ, R 1 θ) where R j is the j th Riesz transform in 2D.The dissipative quasi-geostrophic equation (1.1) can be derived from general quasi-geostrophic equations in the special case of constant potential vorticity and buoyancy frequency [24].It models the evolution of the potential temperature θ of a geostrophic fluid with velocity u on the boundary of a rapidly rotating half space.As such it is often termed surface quasi-geostrophic equations in the literature.If θ is a smooth solution to (1.1), then it obeys the L p -maximum principle, namely Similar results hold when the domain R 2 is replaced by the periodic torus T 2 .Moreover, if θ 0 is smooth and in Ḣ− 1 2 (R 2 ), then one can show that , t > 0.
(1.3)More precisely, for the inviscid case ν = 0 one has conservation and for the dissipative case ν > 0 one has dissipation of the Ḣ− 1 2 -Hamiltonian.Indeed for ν = 0 by using the identity (below P <J is a smooth frequency projection to {|ξ| ≤ constant • 2 J }) one can prove the conservation of D − 1 2 θ 2 2 under the assumption θ ∈ L 3 t,x .The two fundamental conservation laws (1.2) and (1.3) play important roles in the wellposedness theory for both weak and strong solutions.In [27] Resnick proved the global existence of a weak solution for 0 < γ ≤ 2 in L ∞ t L 2 x for any initial data θ 0 ∈ L 2 x .In [23] Marchand proved the existence of a global weak solution in x (R 2 ), p ≥ 4 3 , when ν > 0 and 0 < γ ≤ 2. It should be pointed out that in Marchand's result, the non-dissipative case ν = 0 requires p > 4/3 since the embedding L 4 3 ֒→ Ḣ− 1  2 is not compact.On the other hand for the diffusive case one has extra L 2 t Ḣ γ 2 − 1 2 conservation by construction.In recent [10], non-uniqueness of stationary weak solutions were proved for ν ≥ 0 and γ < 3  2 .In somewhat positive direction, uniqueness of surface quasi-geostrophic patches for the non-dissipative case ν = 0 with moving boundary satisfying the arc-chord condition was obtained in [7].
Inspired by these preceding works, we develop in this paper an optimal local regularity theory for the supercritical quasi-geostrophic equation.Set ν = 1 in (1.1).If we completely drop the nonlinear term and keep only the linear dissipation term, then the linear solution is given by θ linear (t, x) = (e −tD γ θ 0 )(t, x). (1.6) Formally speaking, one has the identity e tD γ (θ linear (t, •)) = θ 0 for any t > 0. This shows that the best smoothing estimate one can hope for is where X is a working Banach space.The purpose of this work, rough speaking, is to show that for the nonlinear local solution to (1.1) (say taking ν = 1 for simplicity of notation), we have where ǫ 0 > 0 can be taken any small number, and X can be a Sobolev or Besov space.In this sense this is the best possible regularity estimate for this and similar problems.We now state in more detail the main results.To elucidate the main idea we first showcase the result on the prototypical L 2 -type critical H 2−γ space.The following offers a substantial improvement of Miura [22] and Dong [12].To keep the paper self-contained, we give a bare-hand harmonic-analysis-free proof.The framework we develop here can probably be applied to many other problems.
Our next result is devoted to the Besov case.In particular, we resolve the problem left open in [3], namely one can push to the optimal threshold α = γ.Moreover we cover the whole regime 1 ≤ p < ∞.
There exists T = T (γ, θ 0 , p, q) > 0 and a unique solution , where C > 0 is a constant depending on (γ, p, q).The techniques introduced in this paper may apply to many other similar models such as Burgers equations, generalized SQG models, and Chemotaxi equations (cf.recent very interesting works [4,14,8,28]).Also there are some promising evidences that a set of nontrivial multiplier estimates can be generalized from our work.All these will be explored elsewhere.The rest of this paper is organized as follows.In Section 2 we collect some preliminary materials along with the needed proofs.In Section 3 we give the nonlinear estimates for the H 2−γ case.In Section 4 we give the proof of Theorem 1.1.In Section 5 we give the proof of Theorem 1.2.

Notation and preliminaries
In this section we introduce some basic notation used in this paper and collect several useful lemmas.We define the sign function sgn(x) on R as: For any two quantities X and Y , we denote X Y if X ≤ CY for some constant C > 0. The dependence of the constant C on other parameters or constants are usually clear from the context and we will often suppress this dependence.We denote For any quantity X, we will denote by X+ the quantity X + ǫ for some sufficiently small ǫ > 0. The smallness of such ǫ is usually clear from the context.The notation X− is similarly defined.This notation is very convenient for various exponents in interpolation inequalities.For example instead of writing For any two quantities X and Y , we shall denote X ≪ Y if X ≤ cY for some sufficiently small constant c.The smallness of the constant c (and its dependence on other parameters) is usually clear from the context.The notation X ≫ Y is similarly defined.Note that our use of ≪ and ≫ here is different from the usual Vinogradov notation in number theory or asymptotic analysis.
We shall adopt the following notation for Fourier transform on R n : Similar notation will be adopted for the Fourier transform of tempered distributions.For real-valued Schwartz functions f : R n → R, g : R n → R, the usual Plancherel takes the form (note that ĝ(ξ) = ĝ(−ξ)) We shall denote for s > 0 the fractional Laplacian D s = (−∆) s/2 = |∇| s as the operator corresponding to the symbol |ξ| s .For any 0 ≤ r ∈ R, the Sobolev norm f Ḣr is defined as We will need to use the Littlewood-Paley (LP) frequency projection operators.To fix the notation, let Let φ(ξ) := φ 0 (ξ) − φ 0 (2ξ) which is supported in 1 2 ≤ |ξ| ≤ 7 6 .For any f ∈ S ′ (R n ), j ∈ Z, define Sometimes for simplicity we write f j = P j f , f ≤j = P ≤j f , and f [a,b] = a≤j≤b f j .Note that by using the support property of φ, we have P j P j ′ = 0 whenever |j − j ′ | > 1.For f ∈ S ′ with lim j→−∞ P ≤j f = 0, one has the identity and for general tempered distributions the convergence (for low frequencies) should be taken as modulo polynomials.
The Bony paraproduct for a pair of functions f, g ∈ S(R n ) take the form For s ∈ R, 1 ≤ p, q ≤ ∞, the Besov norm • B s p,q is given by The Besov space B s p,q is then simply Note that Schwartz functions are dense in B s p,q when 1 ≤ p, q < ∞.In the following lemma we give refined heat flow estimate and frequency localized Bernstein inequalities for the fractional Laplacian |∇| γ , 0 < γ < 2. Note that for γ > 2 and p = 2 there are counterexamples to the frequency Bernstein inequalities (cf.Li and Sire [20]).
Remark.For 1 < q < ∞ the first two inequalities also hold for γ = 2, one can see Proposition 2.5 and Proposition 2.7 below.On the other hand, the inequality (2.4) does not hold for γ = 2.One can construct a counterexample in dimension n = 1 as follows.Take g(x) = 1 4 (3 sin x − sin 3x) = (sin x) 3 which only has zeros of third order.Take h(x) with ĥ compactly supported in |ξ| ≪ 1 and h(x) > 0 for all x.Set which obviously has frequency localized to |ξ| ∼ 1 and have same zeros as g(x).Easy to check that Remark 2.2.For γ > 0 sufficiently small, one can give a direct proof for 1 ≤ q < ∞ as follows.WLOG consider g = P 1 f with g q = 1, and let One can then obtain Since g has Fourier support localized in {|ξ| ∼ 1}, one can obtain uniformly in 0 < s ≤ 1, T s g q n g q = 1.
Note that I(0) = 1.Thus for γ < γ 0 (n) sufficiently small one must have 1  2 ≤ I(γ) ≤ 3 2 .Remark.The inequality (2.5) was obtained by Wang-Zhang [25] by an elegant contradiction argument under the assumption that f ∈ C 0 (R n ) (i.e.vanishing at infinity) and f is frequency localized to a dyadic annulus.Here we only assume f ∈ L ∞ and is frequency localized.This will naturally include periodic functions and similar ones as special cases.Moreover we provide two different proofs.The second proof is self-contained and seems quite short.
Proof of Lemma 2.1.For the first inequality and (2.3), see [19] for a proof using an idea of perturbation of the Lévy semigroup.Since the constant c 2 > 0 depends only on (γ, n), the inequality (2.4) can be obtained from (2.3) by taking the limit q → 1. (Note that since f j = P j f ∈ L 1 and has compact Fourier support, f j can be extended to be an entire function on C n and its zeros must be isolated.) Finally for (2.5) we give two proofs.With no loss we can assume j = 1 and write f = P 1 f .By using translation we may also assume x 0 = 0.With no loss we assume The first proof is to use (2.2) which yields where c > 0 depends only on (γ, n).Then since f = P 1 f is smooth and One can then divide by t → 0 and obtain The second proof is more direct.We note that ψ(y)dy = 0 where ψ corresponds to the projection operator P 1 .Since 1 = (P 1 f )(0), we obtain In what follows we will give a different proof of (2.3) (and some stronger versions, see Proposition 2.5 and Proposition (2.7)) and some equivalent characterization.For the sake of understanding (and keeping track of constants) we provide some details.
Lemma 2.3.Let 0 < s < 1.Then for any g ∈ L 2 (R n ) with ĝ being compactly supported, we have where C 2s,n is a constant corresponding to the fractional Laplacian |∇| 2s having the asymptotics Similarly Proof.Note that where C 2s,n ∼ n s(1 − s).Now for each 0 < ǫ < 1, it is easy to check that (for the case 1 2 ≤ s < 1 one needs to make use of the regularised quantity g(x) − g(y) where Mg is the usual maximal function.By Lebesgue Dominated Convergence, we then obtain Now note that for each ǫ > 0, we have Therefore by using Fubini, symmetrising in x and y and Lebesgue Monotone Convergence, we obtain On the other hand, note that , where φ ∈ L 1 is a smooth function used in the kernel P ≤J .The desired equivalence then easily follows.
Lemma 2.4.Let 1 < q < ∞.Then for any a, b ∈ R, we have Proof.The first inequality is easy to check.To prove the second inequality, it suffices to show for any 0 < x < 1, ).The inequality is obvious for q = 2.If 2 < q < ∞, then we need to show If 1 < q < 2, then we need 2 ).It then suffices for us to show the inequality Thus the desired inequality follows.
Proposition 2.5.Let 1 < q < ∞ and 0 < γ ≤ 2. Then for any f ∈ L q (R n ) and any j ∈ Z, we have . (2.6) Consequently if P j f q = 1, then for any 0 < s ≤ 1, Also for any 0 < s ≤ 1, Remark.In [16], by using a strong nonlocal pointwise inequality (see also Córdoba-Córdoba [6]), Ju Proved an inequality of the form A close inspection of our proof below shows that the inequality (2.6) also works with P j f replaced by θ.Note that the present form works for any 1 < q < ∞.Furthermore in the regime q > 2, we have 4(q−1) > 2 q and hence the constant here is slightly sharper.
Remark.The inequality (2.7) was already obtained by Chamorro and P. Lemarié-Rieusset in [9].Remarkably modulo a q-dependent constant it is equivalent to the corresponding inequality for the more localized quantity (|∇| γ P j f )|P j f | q−2 P j f dx.The inequality (2.8) for q > 2 was obtained by Chen, Miao and Zhang [5] by using Danchin's inequality ∇(|P 1 f | q/2 ) 2 2 ∼ q,n P 1 f q q together with a fractional Chain rule in Besov spaces.The key idea in [5] is to show ∇P [N0,N1] (|P 1 f | q/2 ) 2 1 and in order to control the high frequency piece one needs the assumption q > 2 (so as to use |∇| 1+ǫ0 -derivative for ǫ 0 > 0 sufficiently small).Our approach here is different: namely we will not use Danchin's inequality and prove directly |∇| s0 (|P 1 f | q/2 ) 2 1 for some s 0 sufficiently small (depending on (q, n)).Together with some further interpolation argument we are able to settle the full range 1 < q < ∞.One should note that in terms of lower bound the inequality (2.8) is stronger than (2.7).
Proof.With no loss we can assume j = 1 and for simplicity write P 1 f as f .Assume first 0 < γ < 2. Then for some constant C γ,n ∼ n γ(2 − γ), we have (the rigorous justification of the computation below follows a smilar argument as in the proof of Lemma 2.3) where in the last two steps we have used Lemma 2.4 and Lemma 2.3 respectively.One may then carefully take the limit γ → 2 to get the result for γ = 2 (when estimating Next to show (2.7), we can use Remark 2.2 to obtain |∇| s g 2 ∼ q,n 1 for any 0 < s ≤ s 0 (n) and g = |f | q 2 −1 f .Since g 2 = 1 and ∇g 2 q,n 1, a simple interpolation argument then yields |∇| s g 2 ∼ q,n 1 uniformly for 0 < s ≤ 1.
Finally to show (2.8), we first use the simple fact that ∇(|g|) 2 ≤ ∇g 2 to get It then suffices for us to show |∇| s (|f | q 2 ) 2 q,n 1 for 0 < s ≤ s 0 (q, n) sufficiently small.To this end we consider the quantity To circumvent the problem of differentiating under the integral, one can further consider the regularized expression (later N → ∞) Thus by Hörmander we get T s P ≤N (|f |) q n,q f q = 1.For T (2) s one can use ∇f q 1 to get an upper bound which is uniform in 0 < s ≤ 1  2 .Therefore T sP ≤N (|f |) q q,n 1 for 0 < s ≤ 1 2 .One can then obtain for 0 < s ≤ s 0 (q, n) sufficiently small that 1  2 ≤ I(s) ≤ 3 2 .Finally view I(s) as where Q ≤N (ξ) = q(2 −N ξ), and q ∈ C ∞ c satisfies q(x) ≥ 0 for any x ∈ R n (such q can be easily constructed by taking q(x) = φ(x) 2 which corresponds to q = φ * φ).By using the integral representation of the operator |∇| s and a symmetrization argument (similar to what was done before), we can obtain and the desired result follows.
Proof.With no loss we assume j = 1 and write P 1 f simply as f .In view of the semigroup property of e −t|∇| γ it suffices to prove the inequality for 0 < t ≪ γ,q,n 1. Denote e −t|∇| γ f = K * f and observe that K is a positive kernel with K(z)dz = 1.Consider first 2 ≤ q < ∞.Clearly By Lemma 2.6 and Proposition 2.5, we then get 2 ≤ e −ct f q q .For the case 1 < q < 2, we observe Thus this case is also OK.
Proof.With no loss we consider dimension n = 1.The case n > 1 is similar except some minor changes in numerology.
(3) One can induct on α.The statement clearly holds for α = 0. Assume the statement holds for α ≤ m and any admissible F .Then for α = m + 1, we have , where F (x) = xF ′ (x) is again admissible.The result then follows from the inductive assumption, Leibniz and the estimates obtained in (1) and ( 2). ( , and in general for β ≥ 0, Note that for β ≥ 1 the summand corresponding to m = 0 is actually absent (this is allowed in our notation since we can take the term (OK) to be zero).Similarly one can check for any admissible F and t > 0, where F m are admissible functions.This then reduce matters to the estimate in (3).The result is obvious.
3. Nonlinear estimates: with f and ĝ being compactly supported, it holds that Proof.We first show (3.1).For simplicity of notation we shall write R ⊥ g as g.Note that in the final estimates the operator R ⊥ can be easily discarded since we are in the L 2 setting.On the Fourier side we express the LHS inside the absolute value as (up to a multiplicative constant) Observe that by a change of variable ξ → η − ξ (and dropping the tildes), we have We just need to bound N (g, f, f ).By frequency localization, we have Rewriting j N (g >j+9 , f j , f ) = j N (g j , f <j−9 , f ), we obtain where g ≪j corresponds to |η| ≪ 2 j , g ∼j means |η| ∼ 2 j , and g j means |η| 2 j .These notations are quite handy since only the relative sizes of the frequency variables η, ξ and ξ − η will play some role in the estimates.Note that we should have written g ≪j as g {l: 2 l ≪2 j } according to our convention of the notation ≪ but we ignore this slight inconsistency here for the simplicity of notation.
The estimates of (3.3) and (3.4) are much simpler.We omit the details.

Proof of theorem 1.1
To simplify numerology we conduct the proof for the case ǫ 0 = 1/2.Throughout this proof we shall denote s = 2 − γ.
Step 1.A priori estimate.Denote A = 1 2 D γ and f = e tA θ.It will be clear from Step 2 below that f is smooth and well-defined, and the following computations can be rigorously justified.Then f satisfies the equation Take J 0 > 0 which will be made sufficiently large later.Set N 0 = 2 J0 .Then Now for convenience of notation we denote D s e tA g 3 dx. Denote By Lemma 3.1 and noting that f l (t) 2 e N γ 0 t θ 0 2 , we get (see Remark 3.2) This implies for 0 where c 1 , c 2 , c 3 > 0 are constants depending on γ.
Thus as long as sup 0≤s≤t c 1 D s P >J0 f (s) 2 < 1 10 and t ≤ N −γ 0 , we obtain sup 0≤s≤t In particular, for any prescribed small constant ǫ 0 > 0, we can first choose J 0 sufficiently large such that Then by using (4.1) and choosing T 0 = T 0 (J 0 , θ 0 , ǫ 0 ) sufficiently small we can guarantee sup 0≤s≤T0 Step 2. Approximation system.For n = 1, 2, 3, • • • , define θ (n) as solutions to the system ) t=0 = P <n θ 0 .The solvability of the above regularized system is not an issue thanks to frequency cut-offs.It is easy to check that θ (n) has frequency supported in |ξ| 2 n and θ (n) 2 ≤ θ 0 2 .In particular for any s ≥ 0 we have where c > 0 is a constant.
For any integer J 0 to be fixed momentarily, it is not difficult to check that 1 2 Now fix J 0 sufficiently large such that By using the nonlinear estimate derived in Step 1 (easy to check that these estimates hold for θ (n) with slight changes of the constants c i if necessary), one can then find T 0 = T 0 (γ, θ 0 ) > 0 sufficiently small such that uniformly in n tending to infinity, we have By slightly shrinking T 0 further if necessary and repeating the argument for Ã = 4 3 A = 2 3 D γ , we have uniformly in n tending to infinity, Furthermore for any prescribed small constant ǫ 0 > 0, by using (4.3), we can choose J 0 and T 0 such that uniformly in n, Note that this implies sup 0≤t≤T0 The estimate (4.5) will be needed later.
Step 3. Strong contraction of θ n) .Then (below for simplicity of notation we write −R ⊥ as R) By using the divergence-free property, we have Clearly where we have used the uniform Sobolev estimates in Step 2. Note that It follows that 1 2 . By using the nonlinear estimates in Step 2 and (4.5), one can choose J 0 sufficiently large (and slightly shrink T 0 further if necessary) such that the term D s P >J0 θ (n) 2 becomes sufficiently small (to kill the implied constant pre-factors in the above inequality).This implies Thus for some constants c1 > 0, c2 > 0, we have The desired strong contraction of θ (n) → θ in C 0 t L 2 x follows easily.Step 4. Higher norms.By using the estimates in previous steps, we have for any 0 ≤ t ≤ T 0 , where the constant B 1 > 0 is independent of t.
It follows easily that for any 0 ≤ s ′ < s, x it suffices to consider the continuity at t = 0 (for t > 0 one can use the fact e x which controls frequencies |ξ| ≫ t −1/γ , and for the part |ξ| t −1/γ one uses C 0 t L 2 x ).Since we are in the Hilbert space setting with weak continuity in time, the strong continuity then follows from norm continuity at t = 0 which is essentially done in Step 1.

5.
Nonlinear estimates for Besov case: 0 < γ < 1 For σ = σ(ξ, η) we denote the bilinear operator Then for any ) and for some Ã1 > 0 sup Then for any Proof.For the first case see Theorem 3.7 in [3].The idea is to make a Fourier expansion in the η-variable: where A rough estimate on the number of derivatives required is n 0 = 2d + [d/2] + 1.Note that r > 1/2 and (by paying 2d derivatives) 2dr > d so that the resulting summation in k converges in l r -norm.For the second case, one can make a Fourier expansion in (ξ, η).

By using the estimates
where c > 0 is a small constant.Denote By using Lemma 5.1, Lemma 2.8 and some elementary computations, it is not difficult to check that for any We shall need to use these inequalities (sometimes without explicit mentioning) below.
Proof.The estimate for j ≤ J 0 + 6 is obvious.Observe that for j > J 0 + 6, Thus by Lemma 5.1, Estimate of (1).Note that for given integer ) can be included in the estimate of (3).

It suffices for us to estimate
) with (here to ensure |ξ| ≪ 2 j we need to take J 1 sufficiently large) . By an argument similar to that in Lemma 5.4, we get Estimate of (2).Clearly Estimate of (4).We first note that On the other hand by using that R ⊥ f is divergence-free, we have k≥j+10 where gj = g [j−2,j+2] .Clearly (2 js f j g <j−2 p ) l q j (j≥J0) j≥J0) → 0, uniformly in n as J 0 → ∞.A similar estimate also shows that the diagonal piece f j gj is OK.On the other hand (2 js f <j−2 g j p ) l q j (j≥J0) f ∞ (2 js P j θ 0 p ) l q j (j≥J0) → 0, uniformly in n as J 0 → ∞.
We now complete the proof of Theorem 1.2.This will be carried out in several steps below.
Step 1. Definition of approximating solutions.Define θ (0) ≡ 0. For n ≥ 0, define the iterates θ (n+1) as solutions to the following system where χ ∈ C ∞ c (R 2 ) satisfies 0 ≤ χ ≤ 1 for all x, χ(x) ≡ 1 for |x| ≤ 1, and χ(x) = 0 for |x| ≥ 2.Here we introduce the spatial cut-off χ so that θ (n+1) t=0 ∈ H k for all k ≥ 0 when we only assume θ 0 lies in L p type spaces.The scaling parameters λ n ≥ 1 are inductively chosen such that λ n > max{4λ n−1 , 2 n } and and by interpolation for 0 < s Also by Lemma 6.1, we have These estimates will be needed for the contraction estimate later.
Clearly we have the uniform boundedness of L p norm: This will often be used without explicit mentioning below.
Step 2 One can view f (n+1) as the unique limit of the sequence of solutions (f ) ∞ m=1 solving the regularized system = P ≤m χ(λ −1 n x)P ≤n+2 θ 0 .
By using the estimates in Section 2 (and the inductive assumption that f By using the fact that f It follows that for any T > 0 This together with interpolation implies f (n+1) ∈ C 0 t ([0, T ], W k,p ) for any T > 0, k ≥ 0. These estimates establish the (a priori) finiteness of the various Besov norms and associated time continuity needed in the following steps.
Then by using Lemma 2.1, we get for some constants C1 > 0, C2 > 0, Take an integer J 0 ≥ 10 which will be made sufficiently large later.By using the nonlinear estimates derived before (see Lemma 5.3-5.6),we then obtain f (6.9) The desired result follows.
It is clear that for any T > 0, η where the implied constant (in the notation "∼") depends only on (s 0 , p).By this simple observation, using Lemma 6.2, (6.2), (6.1), and choosing first J 1 sufficiently large and then T 1 sufficiently small, we obtain , where C θ0 is a constant depending on (θ 0 , s 0 , p, γ, q) and σ 0 > 0 depends on (s 0 , γ, p).This clearly yields the desired contraction in the Banach space C 0 t ([0, T 1 ], B s0 p,∞ ). .By the previous step and interpolation, we get f (n) converges strongly to the limit f also in C 0 t B s ′ p,1 ([0, T 1 ]) for any 0 < s ′ < 1 − γ + 2 p .We still have to show f ∈ C 0 t B 1−γ+ 2 p p,q ([0, T 1 ]).Since f (n) → f in C 0 t L p x we only need to consider the high frequency part.Denote s = 1 − γ + 2 p .By using the estimates in Step 3 and strong convergence in each dyadic frequency block, we have for any M ≥ 10, 1≤j≤M where A 1 > 0 is a constant independent of M .Thus (f j ) l q j L ∞ t L p x (j≥1, t∈[0,T1]) < ∞.Since P ≤M f ∈ C 0 t B s p,q for any M , and → 0, as M ′ > M → ∞, we obtain f ∈ C 0 t B s p,q .
Remark.An alternative argument to show time continuity is to use directly (6.2) to get time continuity at t = 0.
For t > 0 one can proceed similarly as the last part of Section 3 and show e ǫ0At f ∈ L ∞ t B 1−γ+ 2 p p,q for some ǫ 0 > 0 small and use it to "damp" the high frequencies.
Step .Recall θ n = e −tA f n (t, •).In view of strong convergence of f n to f , we have θ n → θ strongly in C 0 t B s ′ p,1 for any 0 < s ′ < 1 − γ + 2 p .Since for any 0 ≤ t 0 < t ≤ T 0 we have Thus θ is the desired local solution.One can regard (6.11) (together with some regularity assumptions) as a variant of the usual mild solution.Note that θ j = P j θ is smooth, and one can easy deduce from the integral formulation (6.11) the point-wise identity: ∂ t θ j = −D γ θ j − ∇ • P j (θR ⊥ θ).
From this one can proceed with the localized energy estimates and easily check the uniqueness of solution in . We omit the details.
Remark.Much better uniqueness results can be obtained by exploiting the specific form of R ⊥ θ in connection with the H −1/2 conservation law for non-dissipative SQG.Since this is not the focus of this work, we will not dwell on this issue here.