Infinite speed of support propagation for the Derrida–Lebowitz–Speer–Spohn equation and quantum drift–diffusion models
- 139 Downloads
We show that weak solutions of the Derrida–Lebowitz–Speer–Spohn (DLSS) equation display infinite speed of support propagation. We apply our method to the case of the quantum drift–diffusion equation which augments the DLSS equation with a drift term and possibly a second-order diffusion term. The proof is accomplished using weighted entropy estimates, Hardy’s inequality and a family of singular weight functions to derive a differential inequality; the differential inequality shows exponential growth of the weighted entropy, with the growth constant blowing up very fast as the singularity of the weight becomes sharper. To the best of our knowledge, this is the first example of a nonnegativity-preserving higher-order parabolic equation displaying infinite speed of support propagation.
Mathematics Subject Classification (2010)35K25 35B05 35K55 35Q40 82D37
KeywordsDerrida–Lebowitz–Speer–Spohn equation DLSS equation Infinite speed of propagation Qualitative behaviour Higher-order parabolic equation
Unable to display preview. Download preview PDF.
- 7.Degond, P., Gallego, S., Mehats, F., Ringhofer, C.: Quantum Hydrodynamic models derived from the entropy principle. In: Ben Abdallah, N., Frosali, G.(eds) Quantum Transport—Modeling, Analysis, and Asymptotics, pp. 111–168, Springer, Berlin (2008)Google Scholar
- 9.Dolbeault, J., Gentil, I., Jüngel, A.: A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities. Preprint (2004)Google Scholar
- 10.Fischer, J.: A class of uniqueness for the Derrida–Lebowitz–Speer–Spohn equation and related quantum drift–diffusion models. Preprint (2012)Google Scholar
- 11.Fischer, J.: Optimal lower bounds on asymptotic support propagation rates for the thin-film equation. Preprint (2012)Google Scholar
- 12.Fischer, J.: Upper bounds on waiting times for the thin-film equation: the case of weak slippage. Preprint (2012)Google Scholar
- 16.Jüngel, A.; Matthes, D: A review on results for the Derrida-Lebowitz-Speer-Spohn equation. WSPC—Proceedings (2007)Google Scholar