Well-posedness of Cauchy problem for Landau equation in critical Besov space

Abstract

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space L 2 v (B 3/2 2,1 ), then the Cauchy problem of Landau equation admits a global solution belongs to L ∞ t L 2 v (B 3/2 2,1 ). The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.2010 Mathematics Subject Classification. Primary: 35H20, 35E15; Secondary:76P05, 82C40.

Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials

“…The unique existence of weak solutions in the L 2 ∩ L ∞ setting was initiated by Alonso-Morimoto-Sun-Yang [9], and we mention the recent Silvestre-Snelson's work [57]. The well-posedness for Landau equations can be found in [15,17,18,31,35] and references therein Finally we mention two techniques used frequently when investigate the regularity property of kinetic equations, one referring to De Giorgi-Nash-Moser theory with the help of the averaging lemma and another to Hörmander's hypoelliptic theory. Interested readers may refer to [9,14,33,37,[42][43][44][45][46]56] for the De Giorgi type argument, and to [2, 13, 19, 22, 24, 30, 38-40, 47, 48] for the application of hypoelliptic techniques to kinetic equations.…”

On the uniqueness of weak solutions for Cauchy problem to Landau equation

“…但是在这一类函数空间中的 解的存在性, 特别是整体存在性, 仍然是一个具有挑战性的问题. 最近, 在临界正则性空间中, Cao 等 [18] 在 Maxwellian 分子情形下建立了 Landau 方程强解的整体存在性.…”

Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

“…This article concerns the existence of a solution to the Cauchy problem for the Landau equation, as well as the smoothing properties of that solution, which is a topic studied in many previous works, as stated above. Among them we refer [35,17,18,16] concerning the existence result, [32,28] about higher regularity such as (ultra-)analyticity, in the spatially homogeneous case, while in the spatially inhomogeneous case, [5] concerning renormalized solution with defect measure, [22,33,14,13,12,19] in a close to equilibrium setting, and recent regularity results by [11,20,23] under boundedness conditions on the mass, energy, entropy densities. Also we want to mention the related works on the non cut-off Boltzmann equation, e.g., the papers by [1,30,29,21,2,3,4,26,31,6].…”