A Revisiting of the $$L^2$$ L 2 -Stability Theory of the Boltzmann Equation Near Global Maxwellians

Abstract

We study the L 2 -stability theory of the Boltzmann equation near a global Maxwellian. When an initial datum is a perturbation of a global Maxwellian, we show that the L 2 -distance between two classical solutions can be controlled by the initial data in a Lipschitz manner, which illustrates the Lipschitz continuity of the solution operator for the Boltzmann equation in L 2 -topology. Our local-in-time L 2 -stability results cover cutoff very soft potentials as well as non-cutoff hard and soft potentials. These cases were not treated in the previous work (Ha et al. in Arch Ration Mech Anal 197:657-688, 2010). Thus, our results together with the results in Ha et al. (2010) complete the L 2 -stability theory for the Boltzmann equation near a global Maxwellian. For this L 2 -stability estimate, we use the coercivity estimate of the linearized collision operator, the smallness of perturbation in a mixed Lebesgue norm, and Strichartz-type estimates of perturbation. We also show that for all classical solutions available in the literature, the Lipschitz constant can be chosen as independent of time to obtain the uniform L 2 -stability of the Boltzmann equation.

Large-time behavior of the two-species relativistic Landau–Maxwell system inRx3

“…For the estimates of the nonlinear terms, we use micro-macro decomposition (2.2) and the Cauchy inequality with . For the nonlinear term N 3 , we cite the following estimate from [16]: …”

“…[12][13][14][15][16][17][18]. We will discuss the L 2 -stability of the relativistic Landau-Maxwell system in this paper.…”