Traveling wave solutions of advection–diffusion equations with nonlinear diffusion

Monsaingeon, L. ; Novikov, A. ; Roquejoffre, J.-M.

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 705-735 / Harvested from Numdam

Résumé

Nous étudions lʼexistence dʼune classe particulière de solutions dʼondes pour une équation non linéaire parabolique dégénérée en présence dʼun écoulement cisaillé. Sous certaines hypothèses nous prouvons que ces solutions existent au moins pour des vitesses de propagation $ c\in ]{c}^{⁎},+\infty [$, où $ {c}^{⁎}>0$ est une vitesse critique calculée explicitement en fonction de lʼécoulement mais peut-être pas optimale. Nous prouvons également quʼune hypersurface de frontière libre sépare une zone $ u=0$ dʼune zone $ u>0$ et que, sous une hypothèse supplémentaire de non-dégénérescence, cette frontière peut être globalement paramétrée comme un graphe lipschitzien. Nous nous intéressons à des solutions qui, à lʼinfini dans la direction de propagation, sont planes et linéaires avec pente égale à la vitesse.

Abstract

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds $ c\in ]{c}_{⁎},+\infty [$, where $ {c}_{⁎}>0$ is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface separates a region where $ u=0$ and a region where $ u>0$, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region $ u>0$, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.

Détail

Publié le : 2013-01-01

MR 3082481 | Zbl 1288.35169

DOI : https://doi.org/10.1016/j.anihpc.2012.11.003

Bibliographie

[1] D.G. Aronson, P. Bénilan, Régularité des solutions de lʼéquation des milieux poreux dans $ {\mathbf{R}}^{N}$, C. R. Acad. Sci. Paris Sér. A–B 288 (1979), A103-A105 | MR 524760 | Zbl 0397.35034

[2] D.G. Aronson, L.A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), 351-366 | MR 712265 | Zbl 0556.76084

[3] P. Bénilan, M.G. Crandall, M. Pierre, Solutions of the porous medium equation in $ {\mathbf{R}}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), 51-87 | MR 726106 | Zbl 0552.35045

[4] H. Berestycki, L.A. Caffarelli, L. Nirenberg, Uniform estimates for regularization of free boundary problems, Analysis and Partial Differential Equations, Lect. Notes Pure Appl. Math. vol. 122, Dekker, New York (1990), 567-619 | Zbl 0702.35252

[5] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bull. Braz. Math. Soc. 22 (1991), 1-37 | MR 1159383 | Zbl 0784.35025

[6] L. Caffarelli, J.L. Vazquez, Viscosity solutions for the porous medium equation, Differential Equations, La Pietra, Florence, 1996, Proc. Sympos. Pure Math. vol. 65, Amer. Math. Soc., Providence, RI (1999), 13-26 | MR 1662747 | Zbl 0929.35072

[7] L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. vol. 43, Amer. Math. Soc., Providence, RI (1995) | MR 1351007 | Zbl 0834.35002

[8] L.A. Caffarelli, A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), 361-391 | MR 570687 | Zbl 0439.76085

[9] L.A. Caffarelli, J.L. Vázquez, N.I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), 373-401 | MR 891781 | Zbl 0644.35058

[10] L.A. Caffarelli, N.I. Wolanski, $ {C}^{1,\alpha }$ regularity of the free boundary for the N-dimensional porous media equation, Comm. Pure Appl. Math. 43 (1990), 885-902 | MR 1072396 | Zbl 0728.76103

[11] Y. Choquet-Bruhat, J. Leray, Sur le problème de Dirichlet, quasilinéaire, dʼordre 2, C. R. Acad. Sci. Paris Sér. A–B 274 (1972), A81-A85 | MR 291623

[12] P. Constantin, A. Kiselev, L. Ryzhik, Quenching of flames by fluid advection, Comm. Pure Appl. Math. 54 (2001), 1320-1342 | MR 1846800 | Zbl 1032.35087

[13] M.G. Crandall, H. Ishii, P.-L. Lions, Userʼs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67

[14] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer-Verlag, Berlin (2001) | MR 1814364 | Zbl 0691.35001

[15] I.C. Kim, H.K. Lei, Degenerate diffusion with a drift potential: a viscosity solutions approach, Discrete Contin. Dyn. Syst. 27 (2010), 767-786 | MR 2600689 | Zbl 1213.35178

[16] J.L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Math. Monogr., The Clarendon Press/Oxford University Press, Oxford (2007) | MR 2286292

[17] Y.B. ZelʼDovich, Y. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamics Phenomena, Academic Press, New York (1966)