BOUNDED SOLUTIONS OF THE BOLTZMANN EQUATION IN THE WHOLE SPACE

Radjesvarane Alexandre (1) , Yoshinori Morimoto (2) , Seiji Ukai (3) , Chao-Jiang Xu (4) , Tong Yang (5)

1 IRENAV - Institut de Recherche de l'Ecole Navale
2 Graduate School of Human and Environmental Studies
3 Mr. - retaite
4 LMRS - Laboratoire de Mathématiques Raphaël Salem
5 Pr. - Department of mathematics

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infonity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cuto and non-cuto collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial innity), and solutions in the whole space having a limit equilibrium state at the spatial innity are included in our category.

Mots clés: Boltzmann equation; local existence; locally uniform Sobolev space

Domaines: Equations aux dérivées partielles [math.AP]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. BOUNDED SOLUTIONS OF THE BOLTZMANN EQUATION IN THE WHOLE SPACE. Kinetic and Related Models , 2011, 4 (1), pp.17-40. ⟨10.3934/krm.2011.4.17⟩. ⟨hal-01116726⟩