The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box [42]. Our methods consist of the generalization of the wellposedness theory for the Fokker-Planck equation [37, 38] and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations [27] and the Morrey estimates [56] to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem. 2010 Mathematics Subject Classification. Primary: 35Q84, 35Q20, 82C40, 35B45, 34C29, 35B65.
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.
In this paper, we study the local-in-time validity of the Hilbert expansion for the relativistic Landau equation. We justify that solutions of the relativistic Landau equation converge to small classical solutions of the limiting relativistic Euler equations as the Knudsen number shrinks to zero in a weighted Sobolev space. The key difficulty comes from the temporal and spatial derivatives of the local Maxwellian, which produce momentum growth terms and are uncontrollable by the standard L 2 -based energy and dissipation. We introduce novel time-dependent weight functions to generate additional dissipation terms to suppress the large momentum. The argument relies on a hierarchy of energy-dissipation structures with or without weights. As far as the authors are aware of, this is the first result of the Hilbert expansion for the Landau-type equation.
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