Large Amplitude Solutions in $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ to the Boltzmann Equation for Soft Potentials

In this paper we consider the Cauchy problem on angular cutoff Boltzmann equation near global Maxwillians for soft potentials either in whole space or torus. We establish existence of unique mild solutions $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ with polynomial velocity weights suitably large $p\leq \infty$, whenever initial perturbation weighted $L^p_vL^{\infty}_x$ norm can be arbitrarily but $L^1_xL^\infty_v$ and defect mass, energy, entropy are sufficiently small. The proof is based local time as well uniform a priori estimates via an interplay $L^{\infty}_{T}L^{\infty}_{x}L^1_v$.