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Compactness in kinetic transport equations and hypoellipticity. (English) Zbl 1231.42023

The authors establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. It is shown that the relative compactness in all variables of a bounded family of nonnegative functions \(f_\lambda(x,v)\in L^1\) satisfying some appropriate transport relation \[ v\cdot \nabla_x f_\lambda =(1-\Delta_x)^{\beta/2}(1-\Delta_v)^{\alpha/2} g_\lambda \] may be inferred solely from additional integrability and compactness with respect to \(v\).

MSC:

42B37 Harmonic analysis and PDEs
35H10 Hypoelliptic equations
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