Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations

Abstract

In this paper, we are concerned with the existence of weak solutions of the time-dependent Thomas-Fermi equations. We derive approximate solutions by the fractional step Lax-Friedrichs scheme and establish uniform boundedness of approximate solutions. Based on the uniform energy-type estimates, we establish that the entropy dissipation measures of the weak solution of the one-dimensional time-dependent Thomas-Fermi equations for weak entropy-entropy flux pairs, generated by compactly supported C0 & INFIN; test functions, are confined in a compact set in Hloc-1. We prove that the Young measure must be a Dirac measure by the Tartar-Murat commutator relation. The convergence of approximate solutions is established by using the compensated compactness method.