Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation

We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=\mu +{\mu }^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}{L}_{x,v}^r}+\mathrm{\parallel }f{\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}{L}_{x,v}^2}$ remains bounded for some sufficiently large $r>0$. As a byproduct, we establish $L_{t,x,v}^2$ stability for initial data $f_0\in L_{x,v}^r\cap L_{x,v}^2$. Our approach employs dilated dyadic decompositions in phase space $(v,\xi ,\eta )$ to capture hypoellipticity and to reduce the fractional derivative structure $(-{\mathrm{\Delta }}_v{)}^s$ of the Boltzmann collision operator to zeroth order. The difficulties posed by the large solution are overcome through the negative-order hypoelliptic estimate that gains integrability in $(t,x)$.