L p solutions to the Cauchy problem of the BGK equation

Abstract

The BGK model of the Boltzmann equation plays an important role in the kinetic theory of rarefied gases. Some existence and uniqueness results of solutions to its Cauchy problem were established for large initial data under various circumstances [see, for example, Perthame, B., “Global existence to the BGK model of Boltzmann equation,” J. Differ. Equations 82, 191–205 (1989)]. In this paper, by establishing some weighted Lp estimates of the hydrodynamical quantities of a gas, we prove the existence theorem of the Lp solutions to the Cauchy problem and establish the propagation properties of the Lp moments for this kind of solutions.

Global weak solutions to the cometary flow equation with a self-generated electric field

“…Especially we will show that methods suggested in Refs. [11,32,8,14,43] are applicable in the present situation.…”

Classical solutions for the ellipsoidal BGK model with fixed collision frequency

“…The uniqueness problem was studied in a more stringent functional space in [23,25]. L p extension of [24,25] were made in [38]. The classical solutions near the global equilibrium was studied in [4,14,36].…”

On the Cauchy problem for Wigner–Poisson–BGK equation in the Wiener algebra

“…The system – covers and is motivated by the following relevant well‐known systems: Boltzmann–BGK equation: A lot of attention has been paid in the last years to the Equation . For instance, the existence and uniqueness of the solution of the Equation were proved in . Wigner–Poisson equation: There have been many mathematical studies of the system , which models the charge transport in a semiconductor device under the Coulomb potential. Such as, it has been studied in the whole space (see and the references therein), in a bounded spatial domain with periodic , or absorbing , or time‐dependent inflow , boundary conditions, and on a discrete lattice . Relaxation‐time Wigner–Poisson: It has been studied on X 1 = L 2 ([0,1] × R v ;(1 + v 2 ) d x d v )by Arnold , where f e q is chosen as a steady state of the Wigner–Poisson system …”

“…In other words, we do not succeed in repeating the analogous strategies for this paper: The natural choices of the functional setting for the study of the WP and relaxation‐time WP problem are, respectively, the Hilbert space and L 2 ([0,1] × R v ,(1+| v | 2 ) d x d v ). However, by , the collision operator Q [ f ]needs a higher order weight in L 2 space, for example, , which leads to demanding exploit forth the regularity of the function φ . It is clear that the function φ given by does not satisfy the need of the regularity. The choice of the | v | 2 weight was also already seen to be convenient to control the L 1 ‐norm of the collision operator Q [ f ]on .…”

“…It is clear that the function φ given by does not satisfy the need of the regularity. The choice of the | v | 2 weight was also already seen to be convenient to control the L 1 ‐norm of the collision operator Q [ f ]on . That is, the Hilbert space is also a natural choice of the functional setting for the study of the BGK or VP‐BGK problem, for example, . However, the nonlinear operator is not well defined in such space.…”