16 research outputs found
Global Newtonian limit for the Relativistic Boltzmann Equation near Vacuum
We study the Cauchy Problem for the relativistic Boltzmann equation with near
Vacuum initial data. Unique global in time "mild" solutions are obtained
uniformly in the speed of light parameter . We furthermore prove that
solutions to the relativistic Boltzmann equation converge to solutions of the
Newtonian Boltzmann equation in the limit as on arbitrary time
intervals , with convergence rate for any . This may be the first proof of unique global in time validity of the
Newtonian limit for a Kinetic equation.Comment: 35 page
Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System
In the study of solutions to the relativistic Boltzmann equation, their
regularity with respect to the momentum variables has been an outstanding
question, even local in time, due to the initially unexpected growth in the
post-collisional momentum variables which was discovered in 1991 by Glassey &
Strauss \cite{MR1105532}. We establish momentum regularity within energy spaces
via a new splitting technique and interplay between the Glassey-Strauss frame
and the center of mass frame of the relativistic collision operator. In a
periodic box, these new momentum regularity estimates lead to a proof of global
existence of classical solutions to the two-species relativistic
Vlasov-Boltzmann-Maxwell system for charged particles near Maxwellian with hard
ball interaction.Comment: 23 pages; made revisions which were suggested by the referee; to
appear in Comm. Math. Phy
Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
In this paper it is shown that unique solutions to the relativistic Boltzmann
equation exist for all time and decay with any polynomial rate towards their
steady state relativistic Maxwellian provided that the initial data starts out
sufficiently close in . If the initial data are continuous then
so is the corresponding solution. We work in the case of a spatially periodic
box. Conditions on the collision kernel are generic in the sense of
(Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves
the open question of global existence for the soft potentials.Comment: 64 page
Hilbert Expansion from the Boltzmann equation to relativistic Fluids
We study the local-in-time hydrodynamic limit of the relativistic Boltzmann
equation using a Hilbert expansion. More specifically, we prove the existence
of local solutions to the relativistic Boltzmann equation that are nearby the
local relativistic Maxwellian constructed from a class of solutions to the
relativistic Euler equations that includes a large subclass of near-constant,
non-vacuum fluid states. In particular, for small Knudsen number, these
solutions to the relativistic Boltzmann equation have dynamics that are
effectively captured by corresponding solutions to the relativistic Euler
equations.Comment: 50 page
Global Solution to the Relativistic Enskog Equation With Near-Vacuum Data
We give two hypotheses of the relativistic collision kernal and show the
existence and uniqueness of the global mild solution to the relativistic Enskog
equation with the initial data near the vacuum for a hard sphere gas.Comment: 6 page