Comptes Rendus
Partial Differential Equation
Large-amplitude internal fronts in two-fluid systems
Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1073-1083.

In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate “double stagnation” point.

Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.

Dans cette note, nous présentons des résultats d’existence d’ondes de Mascaret de grandes amplitudes. Cela correspond à des ondes progressives pour l’équation d’Euler incompressible à deux phases en deux dimensions d’espace. Les fluides sont délimités au-dessus et au-dessous par des parois horizontales et sont soumis à leurs gravités. Nous obtenons des courbes continues de solutions à ce système qui bifurquent de la solution triviale où l’interface est plate. A la limite, l’interface interne se renverse, entre en contact avec la paroi supérieure, ou développe un point de « double stagnation »  très dégénéré.

Notre construction est rendue possible grâce à une nouvelle méthode abstraite pour la continuation globale des solutions de type front monotone aux équations elliptiques, posées sur des cylindres infinis. Cette théorie est assez robuste et, en particulier, peut traiter des équations entièrement non linéaires ainsi que des problèmes quasi-linéaires avec des conditions aux limites de transmission.

Published online:
DOI: 10.5802/crmath.128
Classification: 35B32, 76B15, 35J60, 35J66

Robin Ming Chen 1; Samuel Walsh 2; Miles H. Wheeler 3

1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
2 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
3 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Robin Ming Chen; Samuel Walsh; Miles H. Wheeler. Large-amplitude internal fronts in two-fluid systems. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1073-1083. doi : 10.5802/crmath.128.

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