8 research outputs found
Stability of Rarefaction Waves for the Non-cutoff Vlasov-Poisson-Boltzmann System with Physical Boundary
In this paper, we are concerned with the Vlasov-Poisson-Boltzmann (VPB)
system in three-dimensional spatial space without angular cutoff in a
rectangular duct with or without physical boundary conditions. Near a local
Maxwellian with macroscopic quantities given by rarefaction wave solution of
one-dimensional compressible Euler equations, we establish the time-asymptotic
stability of planar rarefaction wave solutions for the Cauchy problem to VPB
system with periodic or specular-reflection boundary condition. In particular,
we successfully introduce physical boundaries, namely, specular-reflection
boundary, to the models describing wave patterns of kinetic equations.
Moreover, we treat the non-cutoff collision kernel instead of the cutoff one.
As a simplified model, we also consider the stability and large time behavior
of the rarefaction wave solution for the Boltzmann equation.Comment: All comments are welcom
Spectral gap formation to kinetic equations with soft potentials in bounded domain
It has been unknown in kinetic theory whether the linearized Boltzmann or
Landau equation with soft potentials admits a spectral gap in the spatially
inhomogeneous setting. Most of existing works indicate a negative answer
because the spectrum of two linearized self-adjoint collision operators is
accumulated to the origin in case of soft interactions. In the paper we rather
prove it in an affirmative way when the space domain is bounded with an inflow
boundary condition. The key strategy is to introduce a new Hilbert space with
an exponential weight function that involves the inner product of space and
velocity variables and also has the strictly positive upper and lower bounds.
The action of the transport operator on such space-velocity dependent weight
function induces an extra non-degenerate relaxation dissipation in large
velocity that can be employed to compensate the degenerate spectral gap and
hence give the exponential decay for solutions in contrast with the
sub-exponential decay in either the spatially homogeneous case or the case of
torus domain. The result reveals a new insight of hypocoercivity for kinetic
equations with soft potentials in the specified situation.Comment: 43 pages. The paper has been rewritten to treat the problem in
bounded domain with the inflow boundary condition. All comments are welcom