Concentrations in kinetic transport equations and hypoellipticity

Abstract

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family $f_\lambda(x,v)\in L^p$ satisfying some appropriate transport relation $$v\cdot\nabla_x f_\lambda = (1-\Delta_x)^\frac{\beta}{2}(1-\Delta_v)^\frac{\alpha}{2}g_\lambda$$ may be inferred solely from its compactness in $v$. This method is introduced as an alternative to the lack of known suitable averaging lemmas in $L^1$ when the right-hand side of the transport equation has very low regularity, due to an external force field for instance. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field.