ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR HARD POTENTIALS

ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR HARD POTENTIALS PART I : EXISTENCE, UNIQUENESS AND SMOOTHNESS L. DESVILLETTES AND C. VILLANI Abstract. We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity. Contents 1. Introduction 1 2. Preliminaries and main results 4 2.1. Notations 4 2.2. Main definitions 6 2.3. Main results 11 3. Appearance and propagation of moments 19 4. Ellipticity of the diffusion matrix 28 5. Approximated problems 31 5.1. The approximated nonlinear equation 32 5.2. Holder estimates for aij and bi 35 5.3. Uniqueness for a linear parabolic equation 38 6. Smoothing effects 41 7. Initial data with infinite entropy 50 8. Uniqueness by Gronwall's lemma 53 9. Uniqueness in a wider class 59 10. Maxwellian lower bound 62 References 65 1. Introduction The spatially homogeneous Landau equation (also called Fokker– Planck–Landau) is a common model in kinetic theory (Cf. [5, 24]). It reads 1

• landau equation


• rapidly than


• potentials


• maxwellian lower


• potentials has


• spatially homogeneous


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• nonnegative function