Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

Abstract

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results largely extend the classical results of 12,13] and Lee-Liu-Yu [10] to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.2000 Mathematics Subject Classification. 35Q20; 82C40.

Explicit structure of the Fokker-Planck equation with potential

“…Later, the result was generalized to the Boltzmann equation with cutoff hard potentials [15]. Very recently, the authors of the current paper extend the pointwise result to more general potentials, the range −2 < γ < 1, and obtain an explicit relation between the decay rate and velocity weight assumption [16]. Let us point out some similarities and differences between the Fokker-Planck equation with potential and the Boltzmann equation with hard sphere or cutoff hard potentials.…”

Finite Speed of Propagation of the Relativistic Landau and Boltzmann Equations

“…Finally, we want to emphasis that this theorem reflects the difference in essence between the relativistic kinetic equations and the classical kinetic equations. For the classical kinetic equations (Boltzmann, Landau or Fokker-Planck), note that the speed of the transport part is v, which reflects the infinite speed of propagation of the solution (see for instance the Fokker-Planck equation with potentials [16], classical Boltzmann equation with hard sphere [19], for the linear problems regarding the classical Boltzmann equation with hard potentials and soft potentials [12,15,18], and for the Vlasov-Poisson-Boltzmann system [14]). Precisely, if imposing the initial data compactly supported in the x variable, then we only have spatial decay (exponentially, subexponentially or algebraically) of the solution to the classical kinetic equations.…”

Space-time behavior of the solution to the Boltzmann equation with soft potentials

“…Thus, it would be interesting to investigate the space-time behaviors of the solutions for different potentials. To this end, the pointwise approach has been initiated by [25,27,28] for the full nonlinear hard sphere case, and then generalized by [20,21,22] to hard and soft potential cases on the linear level.…”

“…Method of proof and plan of the paper. In order to study the spatially asymptotic behavior of the solution f to the full nonlinear Boltzmann equation (1.2), the following weight functions will be taken into account (which are motivated by the linear results [21,22]):…”

“…There, it is crucial that the estimates of linear problem can be obtained in the same weighted space as the initial data, which allows for the nonlinear iteration, then the authors achieve the estimate of nonlinear problem. However, for hard potential, as well as soft potential, this methodology does not work since one needs extra weights for maintaining the space-time structure even for the linear equation (see [20,21,22]). As a comparison, to obtain the spatially asymptotic behavior of the nonlinear equation, we circumvent the difficulty of nonlinear iteration due to mismatch of velocity weight on the linear level, and directly study the L ∞ ξ,β L ∞ x estimate of the solution f wi to the weighted full Boltzmann equation (1.12).…”