COMPACTNESS PROPERTIES AND LOCAL EXISTENCE OF WEAK SOLUTIONS TO THE LANDAU EQUATION

Abstract

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(infinity) uniform bounds. Then we establish a compactness theorem for the sequence of solutions using the L-x,v(infinity) 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 f(0) is bounded in L-x,v(infinity) whose size determines the maximal time-interval of the existence of the weak solution.