Compactness properties and local existence of weak solutions to the landau equation

Abstract

© 2020 American Mathematical Society. We consider the Landau equation nearby the Maxwellian equilibrium. Based on the assumptions on the boundedness of mass, energy, and entropy in the sense of Silvestre [J. Diffential Equations 262 (2017), no. 3, 3034–3055], we enjoy the locally uniform ellipticity of the linearized Landau operator to derive local-in-time L∞x,v uniform bounds. Then we establish a compactness theorem for the sequence of solutions using the L∞x,v bounds and the standard velocity averaging lemma. Finally, we pass to the limit and prove the local existence of a weak solution to the Cauchy problem. The highlight of this work is in the low-regularity setting where we only assume that the initial condition f0 is bounded in L∞x,v, whose size determines the maximal time-interval of the existence of the weak solution