Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

Abstract

The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyse the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an apriori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained. Also, the non-negativity of the solution is rigorously shown.

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

“…In [18], the authors conjectured that it remained still open to determine whether the regularity indices 1 + 1 2s is sharp or not. On the other hand, Duan-Liu-Xu [12] and Morimoto-Sakamoto [22] studied the Cauchy problem for the Boltzmann equation with the initial datum belonging to critical Besov space. Motivated by those works, we intend to study the inhomogeneous non-cutoff Kac equation in critical Besov space and then improve the Gelfand-Shilov regularizing properties and Gevrey regularizing properties.…”

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

“…In the part of global-in-time existence, we establish those trilinear estimates to the nonlinear Landau collision operator, which play a key role achieving the global solution. For that end, our proof heavily depends on spectral properties of Landau operator with Maxwellian molecules (see Section 2 for details) in contrast with [19,29]. Finally, the standard continuity argument enables us to obtain Theorem 1.1.…”

“…Very recently, Duan, Liu and the last author of this paper [19] first introduced the Chemin-Lerner type spaces involving the microscopic velocity and established the global existence of strong solutions near Maxwellian for the cut-off Boltzmann equation. Subsequently, Morimoto and Sakamoto [29] extended their result to the non-cutoff Boltzmann equation by using the triple norm that was introduced by Alexandre-Morimoto-Ukai-Xu-Yang [2,5]. However, there are few results concerning the global existence for the Landau equation in spatially critical Besov spaces.…”

“…The Bony's para-product decomposition have been widely used in the study of fluid dynamics, see for example [10,11,13,14,15] and references therein, however, there are few results available that one applies the Besov space theory to the global existence of kinetic equations. The recent works [19,29] are devoted to the Boltzmann equation. We would like to mention that Theorem 1.1 should be our first effort, and the research of Landau equation in cases of hard potentials and soft potentials, which is left in near future.…”

Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

“…Following this result, [16] considers the problem under the same conditions in the above space replacing B s 2,1 by B s 2,r with 1 ≤ r ≤ 2 and s > 3/2. Also, it is proved in [13] that the Cauchy problem is well-posed in the same space for the Boltzmann equation without angular cutoff. It should be also noted that the use of the Besov space in this paper is strongly motivated by [8], therefore, we here provide another aspect of applications of the Besov space to the problem.…”