Title: Well-posedness and scattering for the Boltzmann equations: Soft potential with cut-off,
Abstract: We prove the global existence of the unique mild solution for the Cauchy
problem of the cut-off Boltzmann equation for soft potential model $\gamma=2-N$
with initial data small in $L^N_{x,v}$ where $N=2,3$ is the dimension. The
proof relies on the existing inhomogeneous Strichartz estimates for the kinetic
equation by Ovcharov and convolution-like estimates for the gain term of the
Boltzmann collision operator by Alonso, Carneiro and Gamba. The global dynamics
of the solution is also characterized by showing that the small global solution
scatters with respect to the kinetic transport operator in $L^N_{x,v}$. Also
the connection between function spaces and cut-off soft potential model
$-N<\gamma<2-N$ is characterized in the local well-posedness result for the
Cauchy problem with large initial data.