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.