Global well-posedness of slightly supercritical active scalar equations

Abstract

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions

On the Regularity Issues of a Class of Drift-Diffusion Equations with Nonlocal Diffusion

“…As already observed by several authors in the literature, the SQG equation (in the inviscid or supercritical case) may be the simplest physical PDE model that the issue of global regularity still remains open. The results in this paper improve and extend some noticeable results of SQG equation in [25,19,17] to the drift-diffusion equation (1.1) with a general velocity field given by (1.2). But since we only use the representation formula (1.2) (and the divergence-free condition of u in some cases) and do not use the exclusive properties of the Riesz transform (cf.…”

“….2), partially generalizing the result of[19] on the slightly supercritical SQG and Burgers equations.Theorem 1.1. Assume that θ 0 ∈ H s (R d ), s > d 2 + 1,and either Case (I) or Case (II) is considered with α ∈]0, 1], σ ∈ [0, 1[ and some constant c 0 = c α,σ > 0.…”

“…≤ R, the supercritical SQG equation has a unique global solution as long as α depending on R sufficiently close to 1. For the SQG equation with general diffusion operator L, Dabkowski et al in [19] considered the slightly supercritical case, where the operator L defined by (1.3) and (1.8) satisfies (1.22) below, and they obtained the global well-posedness of smooth solution by applying the method of nonlocal maximum principle. They also showed the global result for the multiplier operator L = m(D) under some suitable assumptions on m(ζ) = m(|ζ|).…”

“…For the Burgers equation with fractional diffusion, the subcritical and critical cases can be treated as the corresponding cases of SQG equation to obtain the global results; while for the supercritical case, Kiselev et al in [27] proved that the shock singularity similar to the inviscid case occurs in the supercritical case (see also [1]). For the Burgers equation with a general L defined by (1.3) and (1.8), the authors in [19] proved that under (1.22) below and other mild conditions on m, the equation is globally well-posed for smooth data; whereas under lim ν→0+ 1 ν m(r −1 )dr < ∞, finite time blowup will happen for some smooth data. The Córdoba-Córdoba-Fontelos (CCF) equation corresponds to the equation (1.1) with d = 1, and u = Hθ, (1.17) and H is the usual 1D Hilbert transform.…”

“…For Theorem 1.1, the local well-posedness result is stated and proved in the appendix section 7, then we introduce a MOC ω(ξ) defined by (4.1) and prove that the evolution of the considered equation (1.1)-(1.2) obeys this MOC, which implies the needed Hölder regularity of the solution. Compared with [19], which adapted the same method for the slightly supercritical SQG equation, the MOC (4.1) has a much simpler form, and we use a different way to estimate the contribution (3.16) so that we can avoid the difficulty encountered in considering the general u defined by (1.2) (since the treating in [19] uses the special structure of u = R ⊥ θ and do not extend). Proposition 1.3 concerns the uniform-in-ǫ improvement of the eventual Hölder regularity from the L ∞ -weak solution for the solution of ǫ-regularized equation 1.24, which indeed plays a core role in the proof of Theorems 1.4, 1.5 and 1.7.…”