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

Author:

Wang Shu1,Ren Yabo1ORCID

Affiliation:

1. Faculty of Science, Beijing University of Technology, Beijing, China

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 [Formula: see text] test functions, are confined in a compact set in [Formula: see text]. 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.

Publisher

AIP Publishing

Subject

Mathematical Physics,Statistical and Nonlinear Physics

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