The Vlasov-Poisson-Landau System with the Specular-Reflection Boundary Condition

Abstract

We consider the Vlasov-Poisson-Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular reflection boundary condition. We provide a new improved L 2 → L ∞ framework: L 2 energy estimate combines only with S p L estimate for the ultra-parabolic equation.

Kinetic Fokker-Planck and Landau Equations with Specular Reflection Boundary Condition

“…Such an equation is useful for proving the global in time well-posedness of (1.4) for sufficiently small initial data (see, for example, [4], [10], [15]). Furthermore, we can rewrite (1.4) in the divergence form…”

“…See the details in [4], [9], [15]. To establish the existence of the finite energy strong solution to the problem (1.5) (see Definition 1.3), we work with the equation…”

“…For the related works, we refer the reader to [1], [12], [13], [14]. The boundary-value problem for the Landau equation is considered in the articles [2], [4], [10]. Our paper serves as a foundation of the linear theory for the nonlinear Landau equation used in the works [4], [10].…”

“…The boundary-value problem for the Landau equation is considered in the articles [2], [4], [10]. Our paper serves as a foundation of the linear theory for the nonlinear Landau equation used in the works [4], [10]. The article is organized as follows.…”

“…4 and the energy identity (3.10) in Lemma 3.6, we haveu ∈ L ∞ ((0, T ), L 2 (Ω × R 3 )). (B.1)By this and the Cauchy-Schwartz inequality,ˆΩ×R 3 u(t, x, v)(φ(0, x, v) − φ(t, x, v)) dxdv ≤ u L∞((0,T ),L2(Ω×R 3 )) φ(0, •) − φ(t, •) L2(Ω×R 3 ) .…”