We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.
The global @math estimate we prove in Theorem is similar to an estimate in @cite_16 for the Boltzmann equation. The techniques in the proof are completely different. The propagation of Gaussian bounds that we give in Theorem is reminiscent of the result in @cite_10 . That result is for the space-homogeneous Boltzmann equation with cut-off, which is in some sense the opposite of the Landau equation in terms of the angular singularity in the cross section.
@cite_16 @cite_10
230142061 2018379915
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Holder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in ( L^ ) which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded. For the spatially homogeneous Boltzmann equation with cutoff hard potentials, it is shown that solutions remain bounded from above uniformly in time by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.