The global existence of generalized solutions to the time-dependent Thomas–Fermi equations. (June 2022) - Wang, Shu; Ren, Yabo
Abstract: In this paper, we are concerned with the global existence of generalized solutions of one-dimensional time-dependent Thomas–Fermi equations of quantum theory of atoms. We will use the vanishing artificial viscosity method. First, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E ε, σ and the a priori L ∞ estimate, 0 < 2 σ ≤ n ε, σ ≤ M ( t ), u ε, σ ≤ M ( t ), where M ( t ) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions ( n ε, σ, u ε, σ ) . Second, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫ − ∞ x n ε, σ ( y, t ) − 2 σ d y, deduces the uniform L ∞ estimate, 0 < 2 σ ≤ n ε, σ ≤ M, u ε, σ ≤ M, independent of the time t and ε, σ . Finally, the compensated compactness theory is applied to prove the almost everywhere convergence of ( n ε, σ, u ε, σ ) as ε and σ tend to zero. The limit ( n ( x, t ), u ( x, t ) ) is a global generalized solution.