De Giorgi-Nash-Moser and H{ö}rmander theories: new interplay

Abstract

We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows Hölder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of Hörmander shows, under general "bracket" conditions, the regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent extensions of the De Giorgi-Nash-Moser theory to hypoelliptic equations of Kolmogorov (kinetic) type with rough coefficients. These equations combine a first-order skewsymmetric operator with a second-order elliptic operator involving derivatives in only certain variables, and with rough coefficients. We then discuss applications to the Boltzmann and Landau equations in kinetic theory and present a program of research with some open questions.

Global mild solutions of the Landau and non-cutoff Boltzmann equations

“…Notice that for the Boltzmann equation without cutoff, the regularity issue has also been studied in many recent works, for instance, Silvestre [72], Imbert-Silvestre [56], Imbert-Mouhot-Silvestre [55], and Chen-Hu-Li-Zhan [20]. See also a recent survey by Mouhot [66] and references therein.…”

Variation on a Theme by Kiselev and Nazarov: Hölder Estimates for Nonlocal Transport-Diffusion, Along a Non-Divergence-Free Bmo Field

“…Another related study, albeit slightly more distant from (1), is that of the kinetic Fokker-Planck equation. On that mater, we refer the reader to Mouhot [19] and references therein, and, for example, to the regularity result by Golse and Vasseur [13].…”

Variation on a theme by Kiselev and Nazarov: H{ö}lder estimates for non-local transport-diffusion, along a non-divergence-free BMO field

“…Another related study, albeit slightly more distant from (1), is that of the kinetic Fokker-Planck equation. On that mater, we refer the reader to C. Mouhot [17] and the references therein, and, e.g., to the regularity result by F. Golse, A. Vasseur [12].…”

“…where ψ denotes the solution of the Cauchy problem (17). The constant A, which is implicit in the definition of A p r , has to be chosen large enough.…”

“…The proof of proposition 18 is inspired by [14] and [9], though the fractional derivative requires some additional care. Thanks to (19), the zero-average property of atoms is obviously propagated by (17). Let x(s) be the solution to the following ODE, which tracks the average flow on a ball of size r. It is obviously well defined for v ∈ L 1 loc (R + × R d ):…”