UNIFORM Lp-STABILITY ESTIMATE FOR THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION NEAR VACUUM

We present a new uniform Lp-stability theory for the spatially inhomogeneous Boltzmann equation near vacuum via the nonlinear functional approach proposed by the first author. Our stability analysis is based on new nonlinear functionals which are equivalent to the pth power of Lp-distance. The L1-nonlinear functionals play the key role of "modulators" which make the accumulative functional be non-increasing in time t along classical solutions.

PROPAGATION OF SINGULARITIES IN THE SOLUTIONS TO THE BOLTZMANN EQUATION NEAR EQUILIBRIUM

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of Boudin–Desvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

THE BOLTZMANN EQUATION AND THE GROUP TRANSFORMATIONS

This paper is devoted to the investigation of group properties of the nonlinear Boltzmann equation. The complete Lie group of invariant transformations for the spatially inhomogeneous Boltzmann equation is constructed. The generalization to the Lie-Backlund groups is given for the spatially homogeneous case. It is shown that there are only two non-trivial group transformations for the Boltzmann equation in the wide class of Lie and Lie-Backlund transformations. Some consequences of these symmetry properties are discussed. The special role of Galileo group and the analogy between the spatially homogeneous Boltzmann equation and the full equation are also investigated.