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On the existence, uniqueness, and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux in such a way that preserves the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes.

In the regime 1 < β < 2, in [CCC + 12], the Cauchy problem for the inviscid case (κ = 0) of (1.1) was shown to be locally well-posed in H 4 . This result was sharpened in [HKZ15], where local well-posedness was established in H β+1+ǫ , for any ǫ > 0. For blow-up of a closely related non-local transport equation, we refer the reader to [Don14]. Local well-posedness in the borderline space H β+1 remains an outstanding open problem for these models, especially in light of recent ill-posedness results for the Euler equation (cf. [BL15,EM20]) and complementary results on the impossibility of uniform continuity of the solution operator (cf. [BL19,HM10,Inc15,MY16] for Euler) and (cf. [Inc18]) for the inviscid SQG). Positive results are, however, available for rather mild regularizations up to a threshold. Such a threshold was identified in [CW12] for the inviscid generalized SQG equations in the regime β ∈ [0, 1) and established to be sharp in [Kwo20] for the particular case of the 2D incompressible Euler equation. In an upcoming paper by the authors (cf. [JKM20]), alternative mechanisms for recovering well-posedness are studied in the spirit of [CW12] for the full range β ∈ (1, 2], with the β = 2 endpoint modified accordingly. In contrast, continuity of the solution operator in borderline spaces has been shown to hold for the Navier-Stokes equations, (cf. most recently [FGH19]). To the best of our knowledge, analogous results for similar hydrodynamic models, particularly for (1.1) in the regime of parameters treated here, are not known and remain an interesting unresolved issue.
In this paper, we address the problem of existence, uniqueness of solutions, and the smoothing effect for the corresponding Cauchy problem of (1.1) when κ ∈ (0, 1) and β ∈ (1, 2], with β = 2 modified as described in (1.2), particularly for arbitrarily large initial data belonging to the borderline Sobolev spaces, H β+1−κ . These spaces are identified by imposing norm invariance under the scaling symmetry of the equations. This scaling is defined by θ λ (t, x) = λ κ−β θ(λ κ t, λx). (1.3) In specific, if θ is a solution of (1.1) with initial data θ 0 , then θ λ is also a solution of (1.1) with initial data (θ 0 ) λ . The homogeneous Sobolev norm, · Ḣβ+1−κ , of a solution remains invariant under (1.3). Consequently, H β+1−κ is referred to as a scaling-critical space for (1.1). Although the modification of the β = 2 endpoint breaks this scaling symmetry, due to the slightly supercritical nature of the velocity, we nevertheless consider by analogy the space H 3−κ+ǫ as the borderline space corresponding to this case. The existence and uniqueness of solutions for large initial data in critical Sobolev spaces for the critical and supercritical SQG equation, i.e., β = 1 case, was established by Miura in [Miu06], while the instantaneous smoothing effect was later established in [Don17,Bis14], using different approaches. For the subcritical SQG equation, using a mild solution approach, analyticity was established in [DL08] in the critical Lebesgue spaces and in [Bis12] in Besov spaces. It was observed in [Miu06] that the main difficulty when working in the critical space setting for the supercritically dissipative SQG equation is in obtaining a suitable continuity estimate for the bilinear term. Indeed, the classical Fujita-Kato mild solution approach cannot be carried out in this setting as the low degree of dissipation cannot, alone, accommodate the loss of derivatives from the nonlinearity. One can nevertheless establish such a continuity estimate by exploiting cancellation through the underlying commutator structure in the equation. Insofar as existence is concerned, one must therefore identify a suitable approximation procedure that respects this commutator structure. In the case of the supercritically dissipative gSQG equation, the difficulty in obtaining the desired continuity estimate is compounded by the increasingly singular velocities that characterize the family. Indeed, a direct adaptation of the analysis in [Miu06] to the gSQG family breaks down in the range κ ≤ β − 1 without a more delicate treatment of the nonlinearity. ∃ & ! H 3−κ+ǫ Gevrey smoothing GWP [MX11] Gevrey smoothing When β ∈ (1, 2), the criticality regimes are more nuanced; one identifies an additional "subcritical" region, κ > β − 1, "critical" line, κ = β − 1, and "supercritical" regime κ < β − 1. The supercritical regime represents the case where the equation becomes "fully" quasilinear, in the particular sense that the coefficients of the nonlinearity depend on derivatives of the solution of an order that exceeds that of the linear term. Indeed, we see that in these additional critical and supercritical regimes, κ necessarily restricts to κ ∈ (0, 1). Although we observe that one can take advantage of the additional commutator structure identified by Chae, Constantin, Córdoba, Gancedo, and Wu in [CCC + 12] to successfully carry out the approaches in [Miu06] and [Bis14], a direct adaptation of the analysis there is limited to the subcritical regime κ > β − 1. To overcome this limitation, one must make use of the more subtle commutator structure identified by Hu, Kukavica, and Ziane in [HKZ15]. In the critical space setting, however, the approximation procedure proposed by Miura in [Miu06] cannot accommodate these additional commutators. In this paper, we propose a new approximation scheme in which one modifies the flux in such a way that ultimately preserves the underlying commutator structure (see Section 5). Through this approximation, we are then able to obtain the desired continuity properties for the critical and supercritical regimes represented by κ ≤ β − 1. In either regime, we must carry out a more delicate analysis at the level of the Littlewood-Paley operators to accommodate the commutator estimates within the critical space setting, as well as extend these estimates appropriately to the Gevrey classes (see Section 4). Analogous results for the β = 2 endpoint case are also established. As we remarked earlier, since the equation corresponding to (1.2) does not possess a scaling symmetry, we instead work arbitrarily close to what would formally be the scaling critical space, that is, in H 3−κ+ǫ . With the appropriate commutator estimates and approximation scheme in hand, we prove our main results, Theorem 3.1 and Theorem 3.2 in Section 6 and Section 7, respectively. As an immediate consequence of establishing Gevrey regularity, we obtain asymptotic decay of all derivatives with respect to the critical Sobolev topology (cf. Corollary 3.1).
Previous works on the existence theory for various regimes in the parameter space (κ, β) ∈ (0, 2] × (0, 2] of (1.1) have been carried out in [CIW08,MX12] for the diagonal case κ = β, i.e., the so-called "modified SQG" equation, and in [MX11] for the regime 1 < β < 2, β < κ < 2, where local well-posedness was studied, and in [MX12], for the regime 1 < β < 2, 2β − 2 < κ < β, where global well-posedness was studied. A closely related generalization of (1.1) was also considered in [LX19], where global existence of weak solutions, global regularity for a slightly supercritical regularization, and eventual regularity of solutions were established. We point out that the smoothing effect in this paper is proved for an equation that belongs outside of the general class of systems treated in [OT01,BB15] as well as the general semilinear parabolic equation that includes the incompressible Navier-Stokes equations as a special case, treated in [CGZ20]. Indeed, this paper establishes a Gevrey regularity smoothing effect for a quasilinear parabolic equation of the form ∂ t θ + γΛ κ θ = b(Λ α θ, ∇θ), where α, κ ∈ (0, 1), particularly, for one in which α > κ. Some context for these results (displayed in larger font) in the (κ, β)-plane is given in Figure 1.

Mathematical Preliminaries
Denote by S the space of Schwartz class functions on R 2 and by S ′ (R 2 ) the space of tempered distributions. For p ∈ [1, ∞], we let L p and L p loc denote the spaces of Lebesgue integrable and locally Lebesgue integrable functions of order p on R 2 , respectively. The norm on L p is defined as We recall that L p is a Banach space with this norm and that L p loc ⊂ S ′ , for all p ∈ [1, ∞]. In the particular case p = 2, L 2 can be endowed with the following inner product: so that L 2 becomes a Hilbert space. Given f ∈ S ′ , letf denote the Fourier transform of f ; we will also use the notation, F , to denote the Fourier transform. We recall that F is an isometry on L 2 and in particular that We define the fractional laplacian operator, Λ = (−∆) 1/2 , and its powers, Λ σ , σ ∈ R, by For σ ∈ R, the homogeneous and the inhomogeneous Sobolev spaces on R 2 are defined aṡ The spaces (2.1), (2.2) can be endowed with inner products given by With this in hand, H σ is a Hilbert space for all σ ∈ R, whereas, in dimension-two,Ḣ σ is a Hilbert space if and only if σ < 1. The inhomogeneous spaces are nested H σ ′ ⊂ H σ , whenever σ ′ > σ, and moreover the embedding is compact over any compact set K ⊂ R 2 . On the other hand, the homogeneous spaces are, in general, only related by the following interpolation inequality: For for some constant depending only on σ, σ 1 , σ 2 . A related interpolation inequality that will be also be useful is stated as follows: for σ ∈ R and σ 1 > −1 > −σ 2 , there exists a constant, C, depending on σ, σ 1 , σ 2 such that 2.1. Littlewood-Paley Decomposition. We will make use of the characterization of Sobolev spaces in terms of the Littlewood-Paley decomposition. We review this decomposition now and refer the reader [BCD11,Che95] for additional details. First, let us define We denote by Q ′ the topological dual space of Q. Note that Q ′ can be identified with the space of tempered distributions modulo polynomials, that is, as where the vector space of polynomials on R 2 is denoted by P. Let us denote by B(r), the open ball of radius r centered at the origin and by A(r 1 , r 2 ) , the open annulus with inner and outer radii r 1 and r 2 , and centered at the origin. There exist two non-negative radial functions χ, φ ∈ S with supp χ ⊂ B(1) and supp φ ⊂ A(2 −1 , 2) such that for χ j (ξ) := χ(2 −j ξ) and φ j (ξ) := φ(2 −j ξ), one has j∈Z φ j (ξ) = 1, whenever ξ ∈ R 2 \{0}, χ+ j≥0 φ j ≡ 1, and one has the following almost-orthogonality conditions: We will make use the shorthand so that, in particular, A j = A j−1,j+1 . With this notation, observe that (2.5) We denote the (homogeneous) Littlewood-Paley dyadic blocks by △ j and S j , which are both defined in terms of its Fourier transform by The following localization properties of {△ j } j∈Z and {S j } j∈Z are a direct consequence of (2.5): and, in particular that On the other hand, when restricted to Q ′ one has and, in particular that In light of the Littlewood-Paley decomposition, one can see that the homogeneous Sobolev spaces,Ḣ σ , can be identified with the homogeneous Besov spaces,Ḃ σ 2,2 , whose norm is given by In particular, we have for some constant C, independent of σ. The relation between the Littlewood-Paley blocks and the fractional laplacian is conveniently captured by the following Bernstein-type inequalities, which we will make copious use of throughout the paper.
Lemma 2.1 (Bernstein inequalities). Let σ ∈ R and 1 ≤ p ≤ q ≤ ∞. Then where c ′ , c are constants that depends on p, q and σ.

Gevrey classes.
We introduce the Gevrey classes in this section. These spaces identify a scale of subspaces between the analytic class of functions, C ω , and the class of smooth functions, C ∞ . We will consider a Littlewood-Paley characterization of the Gevrey classes inspired by the spectral characterization of the Gevrey norm introduced by Foias and Temam in their seminal paper [FT89]. We remark that the Littlewood-Paley characterization was also adopted in [Bis14]; for an extension to L p -based Besov spaces, see [BMS15]. Let α ∈ (0, 1] and λ > 0. Then we define the Gevrey operator, G λ α , of order α and radius λ by F (G λ α f )(ξ) = e λ|ξ| αf (ξ). We will also make use of the notation (2.6) We define theḢ σ -based Gevrey norm by Then the homogeneous (α, λ, σ)-Gevrey classes are defined aṡ Finiteness of f Ġλ α,σ , for some σ ∈ R and α, λ > 0, automatically yields estimates on higher-order derivatives. Indeed, one has for all multi-indices, β ∈ N 2 0 , where N 0 := N ∪ {0}. Thus, with the Sobolev embedding theorem, (2.8) implies uniform bounds on all orders of derivatives. In the particular case α = 1, if f ∈ L 2 satisfies (2.8) for all β ∈ N 2 0 , for some λ > 0, then f is real analytic at each x ∈ R 2 with analyticity radius λ. On the other hand if (2.8) is satisfied with α < 1, we say that f belongs to a subanalytic Gevrey class, which is a subclass of smooth functions. For additional properties of Gevrey classes and applications to a wide-class of equations, the reader is referred to [LO97,OT00,OT01,BB15,MZ17].
The main scenario of interest in this paper is when f is a time-dependent function, globally defined in time. In this case, if f satisfies f (t) ∈Ġ λ(t) α,σ , for all t > 0, for some monotonically increasing function λ = λ(t), then (2.8) yields temporal decay of all higher-order derivatives of f ; this is one of the main motivations for using the Gevrey norm.
We will distinguish between the (α, λ)-Gevrey operators, G λ α , and the related fractional heat propagator, In particular, given θ 0 ∈ L 2 , one has that θ(t; Observe that the (κ, λ)-Gevrey operators can be rewritten as fractional heat propagators appropriately rescaled: One may thus alternatively view the (α, λ, σ)-Gevrey classes as the space of functions for which that is, for which the inverse of the heat propagator belongs to theḢ σ . It can be viewed as a parabolic analog of the so-called X s,b -spaces in the dispersive PDE literature, which are defined in terms of inverses of dispersive operators such as the Airy kernel or Schrödinger propagator (cf. [Bou93a,Bou93b,Tao06]). Lastly, let us recall the following interpolation-type inequality for Gevrey operators that was originally proved in [OT00], but stated here in a slightly more generalized form.
Proof. Let R > 0, to be chosen later, and fix ρ ∈ (0, 1]. By Plancherel's theorem, we have We estimate I as Remark 2.1. For the rest of the paper, we adopt the convention that C denotes a positive constant whose magnitude may change from line-to-line. Dependencies on other parameters will typically be suppressed in performing estimates, but may be specified in statements of lemmas, propositions, or theorems when they are relevant or for clarity.

Statements of Main Results
Our main result for (1.1) when β ∈ (1, 2) in the regime of supercritical dissipation is stated in the following theorem.
Theorem 3.1. Let β ∈ (1, 2), κ ∈ (0, 1), and σ c := 1 + β − κ. For each θ 0 ∈ H σc , there exists T > 0 and a unique solution, θ, of (1.1) such that Moreover, for any 0 < α < κ and δ > 0 sufficiently small, there exists an increasing function λ : We remark that in the assertion of local existence above, the standard critical space phenomenon where T depends on θ 0 in a manner beyond exclusively through its critical norm θ 0 Ḣ1+β−κ is observed. On the other hand, in the small data, global existence setting, we observe that (3.1) along with the Sobolev embedding theorem implies temporal decay of all higher-order derivatives of the corresponding solution in the critical norm. As we will see in the proof of Theorem 3.1, λ can be chosen as for ε > 0 chosen sufficiently small. With this in mind, we have the following immediate corollary.
We lastly observe that (3.1) is nearly optimal in the sense that θ(t) ∈Ġ λ(t) α,σc+δ , for all α ∈ (0, κ), where the optimal result is represented by the endpoint α = κ. This is consistent with the results obtained in [Bis14] for the supercritical SQG equation in the critical Sobolev spaces. The analysis in [Bis14] was subsequently extended to the L p -based critical Besov spaces in [BMS15]. In light of these results, it would be interesting to extend Theorem 3.1 to the Besov space setting as well.
Our second main result establishes the analogous statement for the modified β = 2 endpoint case defined by (1.2).

Commutator estimates
In (1.1), the expression for θ(x, t) in terms of u(x, t) is given by a singular integral; the strength of the singularity of the operator is quantified by the parameter β ∈ (1, 2). The parameter, β, belonging to this range precludes one from obtaining a suitable continuity-type estimate on the bilinear term. To overcome this difficulty, we exploit observations made in [CCC + 12] and [HKZ15] in which additional commutators are identified that allow one to allocate derivatives more effectively. We will require the following lemma, the proof of which is provided in Appendix A. It is essentially a classical product estimate, but we provide it in a frequency-restricted dualized form, as this is the natural form in which it appears in the apriori analysis below. It will be frequently deployed in proving the required commutator estimates.
Suppose that suppĥ ⊂ A j , for some j ∈ Z. Then for each σ ∈ (−1, 1) and ǫ ∈ (0, 2) such that σ > ǫ − 1, there exists a constant C > 0, depending only on σ, ǫ, and a sequence We observe that Lemma 4.1 immediately imply the following corollary, which will be useful to have in-hand for proving the commutator estimates below.
Suppose that suppĥ ⊂ A j , for some j ∈ Z. Then for each σ ∈ [0, 1) and ǫ ∈ (0, 1), there exists a constant C > 0, depending only on σ, ǫ, and a sequence {c j } ∈ ℓ 2 (Z) with {c j } ℓ 2 ≤ 1 such that Commutator estimates in Sobolev classes. We will require two commutator estimates, stated below in Lemma 4.2 and Lemma 4.3, for the trilinear interactions that appear naturally in the energy arguments carried out in Section 5, Section 6, Section 7 below. Lemma 4.2 essentially arises as an intermediate step in establishing a classical version of the commutator estimate proved by H. Miura in [Miu06]. We include a proof in Appendix A. The trilinear form of Lemma 4.2 is crucial as some of the commutators we appeal to can ultimately only be formed between a triad interaction of functions. On the other hand, due to the expression of the velocity in terms of fractional laplacians and partial derivatives, we will require a variation of the commutator estimate appearing in [Miu06] to accommodate these types of operators; this is established in Lemma 4.3.
In what follows, we will denote the commutator of two operators, S and T , by We adopt the convention that [ where f is a scalar function, and I denotes the identity operator.
Remark 4.1. Note that the upper bound Cc j 2 (ρ1−ρ2−1)j f Ḣ1−ρ 1 h L 2 g Ḣ1+ρ 2 can also be established above by directly applying Proposition 2 in [Miu06]. Thus, the bound we provide allows for additional flexibility in the allocation of derivatives.
Lemma 4.3. Let β ∈ (1, 2), ρ 1 ∈ (0, 2), ρ 2 ∈ (−1, 1) be such that ρ 2 > ρ 1 − 1. Let f, g ∈ L 2 . Then there exists a constant C > 0, depending only on β, ρ 1 , ρ 2 , such that Proof. Let h ∈ L 2 . It will be convenient to define the following functional: Indeed, by the Plancherel theorem, observe that For convenience, we will suppress the dependence of A on ξ, η. Observe that where the fact β ∈ (1, 2) is invoked to obtain the last inequality. Let ϕ := ξ−η |η| and ϑ := η |η| . We observe that for fixed ξ and η, we have We have the following inequality: giving us, since β < 2, If |ϕ| ≤ 1, we have for 1 < β < 2 If |ϕ| > 1, then by the mean value theorem, since 1 < β < 2, Using the above inequalities in (4.1), we get Application of Lemma 4.1 with σ = ρ 2 , ǫ = ρ 1 gives us where c j is independent of h and j c 2 j ≤ 1. Since L 2 =Ḃ 0 2,2 , this completes the proof. 4.2. Commutator estimates in Gevrey classes. Next we prove a commutator estimate for operators which can be expressed as a product of Fourier multiplier operators given by In what follows, it will be convenient to introduce the operator, E λ α , given by We will also let D denote Proof. As we will see below, the proof will make use of the fact that the symbol of D is dominated by that of Λ, and so in order to avoid redundancy in the argument, it will suffice to treat the case D = ∂ ℓ .
First, let us define Then, using Plancherel's theorem, we see that By (4.5), it is therefore equivalent to obtain bounds for L λ,α,σ+ρ j,ℓ . For convenience, we will now suppress the indices on m.

Dissipative perturbation of a linear conservation law with modified flux
The proof of our first main result, Theorem 3.1, will rely on an approximating sequence that is determined by a linear scalar conservation law that is dissipatively perturbed by the appropriate power of the fractional laplacian. To be specific, given q sufficiently smooth, we will consider the following initial value problem: Note that one formally has div F −θ (θ) = −(∇ ⊥ Λ β−2 θ)· ∇θ = u· ∇θ. Hence, one recovers equation (1.1) in the case q = −θ. The purpose of this particular modification to the flux is to accommodate additional commutators in the study of (1.1) that the "standard" approximating sequence of linear transport equations cannot handle. We observe that when β < 1 + κ, that is, σ c < 2, where σ c = β + 1 − κ denotes the critical Sobolev exponent, no modification is required and one may simply use the standard approximating sequence by a linear transport equation, as indicated by (5.2). However, a modification is crucial for treating the regime corresponding to β ≥ 1 + κ, i.e., σ c ≥ 2, of the more singular velocities in (1.1). In this regard, the proposed equation (5.1) faithfully respects the more nuanced commutator structure of the generalized SQG equations required to treat the more singular regime of β ≥ 1 + κ. We will first establish existence and uniqueness of solutions to (5.1) Theorem 5.1. Let β ∈ (1, 2), κ ∈ (0, 1), and σ c = 1 + β − κ. Given T > 0, suppose q ∈ L 2 (0, T ;Ḣ σc+κ/2 ). Then for each θ 0 ∈ H σc , (5.1) has a unique solution θ ∈ C([0, T ]; H σc ) ∩ L 2 (0, T ;Ḣ σc+ κ 2 ) satisfying sup 0≤t≤T θ(t) H σc ≤ θ 0 H σc exp C q 2 L 2 TḢ σc +κ/2 . Theorem 5.1 can be proved by an artificial viscosity argument. A sketch of the proof is provided in Appendix B. The reader is referred to [Kum] for additional details. With this in hand, we will only develop apriori estimates for solutions to (5.1). We ultimately find it expedient to perform these estimates in Gevrey classes and simply specialize them to the case where the exponential rate, λ, in the Gevrey norm is identically zero to obtain estimates in the corresponding Sobolev spaces.

Apriori estimates. Given
Then ∇· v = 0. We will begin by establishing L 2 estimates. Then we will proceed to establishing the claimed Sobolev space estimates. Lastly, we provide estimates in the Gevrey norm topologyĠ λ α,σc+δ .

L 2 estimates.
Since v is divergence-free, integrating by parts, we have v· ∇h, h = 0, for any h sufficiently smooth. Combined with the skew self-adjointness of the operator ∇Λ β−2 , we deduce By Lemma 4.3 with ρ 1 = ρ 2 = κ/2, and Young's inequality, we have Taking the inner product in L 2 of (5.1) with θ and using (5.5) yields (5.6) 5.1.2. Preparatory estimates. It will be economical at this point to introduce the Gevrey operator and derive the estimates with this operator included since the commutator estimates that we apply will reduce accordingly to the Sobolev setting upon setting the rate, λ, in the Gevrey norm to be identically zero. Since we will have to make different choices for various parameters in each setting, we will then specialize to the Sobolev setting first, and then return to the Gevrey setting again afterwards. For the remainder of Section 5, let us assume that Observe that λ ≡ 0 is allowed. To help contain expressions, we will make use of the notations Upon applying the operator G λ(t) α Λ σc+δ j to (5.1) and invoking (5.7) with ϕ = Λ σc+δ j θ, one has Then by (5.3), taking the inner product in L 2 of (5.8) with G We will now treat the trilinear terms; it will be divided into two cases: β < 1 + κ and β ≥ 1 + κ.
Case: β ≥ 1 + κ. Observe that in this case σ c ≥ 2. We decompose the term involving the flux in (5.9) as Observe that by integrating by parts, we derive J 2 , J 4 = 0, so that it suffices to treat J 1 , J 3 and J 5 .

Sobolev space estimates.
We will now specialize the estimates from the previous section to the Sobolev setting by simply taking λ(t) ≡ 0. Upon particular choices of the parameters δ, ν, ν ′ , we will derive estimates in L ∞ TḢ σc ∩ L 2 TḢ σc+ κ 2 . These estimates will ultimately be leveraged to establish existence and uniqueness. To this end, let us choose the parameter δ = δ(κ, β) to be defined by Observe that 0 < δ < κ/2.
With λ ≡ 0 and these choices for δ, ν, ν ′ , upon returning to (5.19) and (5.20), then applying the Bernstein inequalities, we derive Upon dividing both sides by Λ σc+δ j θ L 2 , then integrating in time, we obtain Observe that With this in hand, we take the ℓ 2 -norm in j, followed by the L κ/δ -norm in time to obtain We treat T 1 (T ) by applying Minkowski's inequality and the Lebesgue dominated convergence theorem to deduce We treat T 2 (T ) by applying the Hardy-Littlewood-Sobolev's inequality followed by the Cauchy-Schwarz inequality, we obtain Referring back to (5.19) and (5.20) with this choice, we obtain We divide both sides by 2 (δ− κ 2 )j Λ σc+δ j θ L 2 , use Bernstein's inequality, then integrate in time to obtain Using (5.22), and taking the ℓ 2 -norm in j, followed by the L 2 -norm in time, we have Applying the Hardy-Littlewood-Sobolev inequality followed by Holder's inequality, we obtain Upon returning to (5.27) and applying (5.28) and (5.29), we obtain TḢ σc -estimates. Finally, we obtain an estimate of θ L ∞ TḢ σc . For this, we must return to (5.19) and (5.20) and, instead of (5.21), we make the choice δ = 0. We then choose Then (5.19) and (5.20) become where we used Young's inequality in the last step. We subtract the last term on the right, sum in j and integrate in time to obtain . Upon combining this with (5.6), we deduce Summary of Sobolev space estimates. Collecting the estimates (5.26), (5.30), (5.31), we arrive at (5.32) 5.1.4. Gevrey class estimates. In this section, we will obtain an apriori estimate for (5.1) in Gevrey classes. First, for a given measurable function Φ : (0, T ] −→Ḣ σc+δ , we define Φ(·) XT := ess sup whereĠ λ α,σ denotes the homogeneous (α, λ, σ)-Gevrey class defined in (2.7). Let us also choose λ(t) to be For convenience, we will often drop the dependence on t. Suppose that α satisfies 0 < α < κ.
Let us assume Upon returning to (5.19) and (5.20), we choose These choices yield Now observe that from Lemma 2.2, applied with ρ = 1, s 1 = α/2, s 2 = κ/2, f = Λ σc+δ θ j , it follows that α κ εγ α/κ t α/κ−1 Λ σc+δ+α/2 θ j 2 where C depends on α and κ. We will fix ε small enough such that On the other hand, recall that E λ α is defined by (4.4). Using the elementary estimate we first observe that for any function f ∈Ġ Upon using the estimate in (5.34), (5.35) dividing both sides by Λ σc+δ θ j L 2 and applying Bernstein's inequality, we obtain for some {c j } ∈ ℓ 2 (Z) such that {c j } ℓ 2 ≤ 1. By Gronwall's inequality and Bernstein's inequality, we obtain Similarly, we estimate the last two terms in (5.36) to obtain We multiply both sides by (γt) δ/κ , take the ℓ 2 (Z)-norm on, apply the Minkowski inequality, and take the supremum over 0 < t ≤ T , to obtain Applying the Lebesgue dominated convergence theorem, we obtain lim T →0 Upon possibly taking ε smaller so that Cε ≤ 1/2, we obtain the following apriori estimate for θ(·) XT : (5.38) 6. Existence, uniqueness, and smoothing for 1 < β < 2: Proof of Theorem 3.1 We will now carry out the proof of Theorem 3.1. We do so by introducing a sequence of approximating equations that will satisfy the dissipative perturbation of the conservation law (5.1) at each level of the approximation. In particular, we consider the following approximation scheme.
As a result, we can express J ′ 1 in terms of a commutator.
By (5.37), for arbitrary initial datum θ 0 , T can be chosen sufficiently small such that T ≤ T 0 and This condition also holds if θ 0 Ḣσc is small enough and T = ∞. By induction, and (6.18), we have the uniform-in-n bound on θ n (· ) XT given by It is now straightforward to check that the solution θ will also satisfy the above bounds.
7. Existence, uniqueness, and smoothing for the endpoint case β = 2: Proof of Theorem 3.2 In this section, we prove our second main result, Theorem 3.2. This is carried out by proving apriori estimates for the norms θ H σ and θ Ġλ α,σ . The existence of a solution then can be carried out by using a standard artificial viscosity approximation; we refer the reader to [Kum] for additional details. 7.1. Existence. To simplify the treatment, we only consider the case when σ ∈ (3−κ, 3) and α ∈ (σ−3+κ, κ). Let δ 1 be chosen such that Denote by δ 2 = 2(σ − 3 + κ − δ 1 ).
An application of the Gronwall inequality and Holder's inequality then establishes uniqueness.
Appendix A.
We will treat the cases L 1 , L 2 and L 3 separately.
Using similar methods as above, it is easy to see that ∂ t θ ǫ L ∞ T H σc−2 is bounded uniformly in ǫ. Application of Aubin-Lions theorem (cf. [CF88]) guarantees the existence of a weak limit θ in L ∞ T H σc . It is then straightforward to show that θ is a weak solution of (5.1). This completes the proof.