A multidimensional nonlinear sixth-order quantum diffusion equation

Bukal, Mario; Jüngel, Ansgar; Matthes, Daniel

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 337-365.

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.

Classification : 35B45, 35G25, 35K55
Mots clés : Higher-order diffusion equations, Quantum diffusion model, Entropy-dissipation estimate, Gradient flow

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