A multidimensional nonlinear sixth-order quantum diffusion equation
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.

DOI : 10.1016/j.anihpc.2012.08.003
Classification : 35B45, 35G25, 35K55
Mots-clés : Higher-order diffusion equations, Quantum diffusion model, Entropy-dissipation estimate, Gradient flow 
@article{AIHPC_2013__30_2_337_0,
     author = {Bukal, Mario and J\"ungel, Ansgar and Matthes, Daniel},
     title = {A multidimensional nonlinear sixth-order quantum diffusion equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {337--365},
     publisher = {Elsevier},
     volume = {30},
     number = {2},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.08.003},
     mrnumber = {3035980},
     zbl = {1288.35283},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.003/}
}
TY  - JOUR
AU  - Bukal, Mario
AU  - Jüngel, Ansgar
AU  - Matthes, Daniel
TI  - A multidimensional nonlinear sixth-order quantum diffusion equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 337
EP  - 365
VL  - 30
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.003/
DO  - 10.1016/j.anihpc.2012.08.003
LA  - en
ID  - AIHPC_2013__30_2_337_0
ER  - 
%0 Journal Article
%A Bukal, Mario
%A Jüngel, Ansgar
%A Matthes, Daniel
%T A multidimensional nonlinear sixth-order quantum diffusion equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 337-365
%V 30
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.003/
%R 10.1016/j.anihpc.2012.08.003
%G en
%F AIHPC_2013__30_2_337_0
Bukal, Mario; Jüngel, Ansgar; Matthes, Daniel. A multidimensional nonlinear sixth-order quantum diffusion equation. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 337-365. doi : 10.1016/j.anihpc.2012.08.003. http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.003/

[1] J. Becker, G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter 17 (2005), 291-307

[2] P. Bleher, J. Lebowitz, E. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Comm. Pure Appl. Math. 47 (1994), 923-942 | MR | Zbl

[3] M. Bukal, A. Jüngel, D. Matthes, Entropies for radially symmetric higher-order nonlinear diffusion equations, Commun. Math. Sci. 9 (2011), 353-382 | MR | Zbl

[4] L. Chen, M. Dreher, Quantum semiconductor models, M. Demuth, B.-W. Schulze, I. Witt (ed.), Partial Differential Equations and Spectral Theory, Oper. Theory Adv. Appl. vol. 211 (2011), 1-72

[5] P. Degond, F. Méhats, C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys. 118 (2005), 625-665 | MR | Zbl

[6] B. Derrida, J. Lebowitz, E. Speer, H. Spohn, Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett. 67 (1991), 165-168 | MR | Zbl

[7] J. Evans, V. Galaktionov, J.R. King, Unstable sixth-order thin film equation: I. Blow-up similarity solutions, Nonlinearity 20 (2007), 1799-1841 | MR | Zbl

[8] J.C. Flitton, J.R. King, Moving-boundary and fixed-domain problems for a sixth-order thin-film equation, European J. Appl. Math. 15 (2004), 713-754 | MR | Zbl

[9] U. Gianazza, G. Savaré, G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal. 194 (2009), 133-220 | MR | Zbl

[10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. vol. 840, Springer (1981) | MR | Zbl

[11] R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal. 29 (1998), 1-17 | MR | Zbl

[12] A. Jüngel, D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity 19 (2006), 633-659 | MR | Zbl

[13] A. Jüngel, D. Matthes, The Derrida–Lebowitz–Speer–Spohn equation: Existence, nonuniqueness, and decay rates of the solutions, SIAM J. Math. Anal. 39 (2008), 1996-2015 | MR | Zbl

[14] A. Jüngel, J.-P. Milišić, A sixth-order nonlinear parabolic equation for quantum systems, SIAM J. Math. Anal. 41 (2009), 1472-1490 | MR | Zbl

[15] J.R. King, The isolation oxidation of silicon: The reaction-controlled case, SIAM J. Appl. Math. 49 (1989), 1064-1080 | MR | Zbl

[16] M.D. Korzec, P.L. Evans, A. Münch, B. Wagner, Stationary solutions of driven fourth- and sixth-order Cahn–Hilliard-type equations, SIAM J. Appl. Math. 69 (2008), 348-374 | MR | Zbl

[17] P.-L. Lions, C. Villani, Régularité optimale de racines carrées, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1537-1541 | MR

[18] D. Matthes, R. Mccann, G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations 34 (2009), 1352-1397 | MR | Zbl

[19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983) | MR | Zbl

[20] A. Potter, An elementary version of the Leray–Schauder theorem, J. Lond. Math. Soc. 5 (1972), 414-416 | MR | Zbl

[21] J. Simon, Compact sets in the space L p (0,T;B), Ann. Mat. Pura Appl. 146 (1987), 65-96 | MR | Zbl

[22] C. Villani, Mass transportation, transportation inequalities and dissipative equations, M.C. Carvalho, J. Rodrigues (ed.), Recent Advances in the Theory and Applications of Mass Transport, Contemp. Math. vol. 353, Amer. Math. Soc., Providence (2004), 95-109

[23] E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. IIB, Springer, New York (1990) | MR

Cité par Sources :