TY - JOUR
T1 - A multidimensional nonlinear sixth-order quantum diffusion equation
AU - Bukal, Mario
AU - Jüngel, Ansgar
AU - Matthes, Daniel
N1 - Funding Information:
✩ The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P20214, P22108, and I395; the Austria–Croatia Project HR 01/2010; the Austria–France Project FR 07/2010; and the Austria–Spain Project ES 08/2010 of the Austrian Exchange Service (ÖAD). * Corresponding author. E-mail addresses: [email protected] (M. Bukal), [email protected] (A. Jüngel), [email protected] (D. Matthes).
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Entropy-dissipation estimate
KW - Gradient flow
KW - Higher-order diffusion equations
KW - Quantum diffusion model
UR - http://www.scopus.com/inward/record.url?scp=84875692983&partnerID=8YFLogxK
U2 - 10.1016/j.anihpc.2012.08.003
DO - 10.1016/j.anihpc.2012.08.003
M3 - Article
AN - SCOPUS:84875692983
SN - 0294-1449
VL - 30
SP - 337
EP - 365
JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
IS - 2
ER -