The global existence of generalized solutions to the time-dependent Thomas–Fermi equations

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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 Eε,σ and the a priori L∞ estimate, 0<2σ≤nε,σ≤M(t), uε,σ≤M(t), where M(t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (nε,σ,uε,σ). Second, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫−∞xnε,σ(y,t)−2σdy, deduces the uniform L∞ estimate, 0<2σ≤nε,σ≤M, uε,σ≤M, independent of the time t and ε,σ. Finally, the compensated compactness theory is applied to prove the almost everywhere convergence of (nε,σ,uε,σ) as ε and σ tend to zero. The limit (n(x,t),u(x,t)) is a global generalized solution.