On the Cauchy Problem for the Cutoff Boltzmann Equation with Small Initial Data

Abstract

We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model $$-1\le \gamma < 0$$ with the small initial data in three dimensional space. Thus our result fixes the gap for the case $$\gamma =-1$$ in three dimensional space in the authors’ previous work (He and Jiang in J Stat Phys 168(2):470–481, 2017) where the estimate for the loss term was improperly used. The other gap in He and Jiang (2017) for the case $$\gamma =0$$ in two dimensional space is recently fixed by Chen et al. (Arch Ration Mech Anal 240:327–381, 2021). The initial data $$f_{0}$$ is non-negative and satisfies that $$\Vert \langle v \rangle ^{\ell _{\gamma }} f_{0}(x,v)\Vert _{L^{3}_{x,v}}\ll 1$$ and $$\Vert \langle v \rangle ^{\ell _{\gamma }} f_0\Vert _{L^{15/8}_{x,v}}<\infty $$ where $$\ell _{\gamma }=0$$ when $$\gamma =-1$$ and $$\ell _{\gamma }=(1+\gamma )^{+}$$ when $$-1<\gamma <0$$ . We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model $$-1<\gamma <0$$ when applying the Strichartz estimates.