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) no. 4, pp. 705-735.

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.

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.

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