The large amplitude solution of the Boltzmann equation with soft potential

Abstract

In this paper, we deal with (angular cut-off) Boltzmann equation with soft potential ($-3<\gamma <0$). In particular, we construct a unique global solution in $L_{x,v}^{\infty }$ which converges to global equilibrium asymptotically provided that initial data has a large amplitude but with sufficiently small relative entropy. Because frequency multiplier is not uniformly positive anymore, unlike hard potential case, time-involved velocity weight will be used to derive sub-exponential decay of the solution. Motivated by recent development of $L^2\mbox{-}L^\infty$ approach also, we introduce some modified estimates of quadratic nonlinear terms. Linearized collision kernel will be treated in a subtle manner to control singularity of soft potential kernel.

34 pages

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