no. 9-10
Équations aux dérivées partielles
Large-amplitude internal fronts in two-fluid systems
[Fronts internes de grandes amplitudes pour des systèmes à deux fluides]
Chen, Robin Ming1 ; Walsh, Samuel2  ; Wheeler, Miles H.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
Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1073-1083.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.128
Classification : 35B32, 76B15, 35J60, 35J66
Chen, Robin Ming 1 ; Walsh, Samuel 2 ; Wheeler, Miles H. 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 
@article{CRMATH_2020__358_9-10_1073_0,
     author = {Chen, Robin Ming and Walsh, Samuel and Wheeler, Miles H.},
     title = {Large-amplitude internal fronts in two-fluid systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1073--1083},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {9-10},
     year = {2020},
     doi = {10.5802/crmath.128},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.128/}
}
TY  - JOUR
AU  - Chen, Robin Ming
AU  - Walsh, Samuel
AU  - Wheeler, Miles H.
TI  - Large-amplitude internal fronts in two-fluid systems
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 1073
EP  - 1083
VL  - 358
IS  - 9-10
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.128/
DO  - 10.5802/crmath.128
LA  - en
ID  - CRMATH_2020__358_9-10_1073_0
ER  - 
%0 Journal Article
%A Chen, Robin Ming
%A Walsh, Samuel
%A Wheeler, Miles H.
%T Large-amplitude internal fronts in two-fluid systems
%J Comptes Rendus. Mathématique
%D 2020
%P 1073-1083
%V 358
%N 9-10
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.128/
%R 10.5802/crmath.128
%G en
%F CRMATH_2020__358_9-10_1073_0
Chen, Robin Ming; Walsh, Samuel; Wheeler, Miles H. Large-amplitude internal fronts in two-fluid systems. Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1073-1083. doi : 10.5802/crmath.128. http://www.numdam.org/articles/10.5802/crmath.128/

[1] Akers, Benjamin F.; Ambrose, David M.; Wright, J. Douglas Gravity perturbed Crapper waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 470 (2014) no. 2161, 20130526 | DOI | MR | Zbl

[2] Ambrose, David M.; Strauss, Walter A.; Wright, J. Douglas Global bifurcation theory for periodic traveling interfacial gravity-capillary waves, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 4, pp. 1081-1101 | DOI | MR | Zbl

[3] Amick, Charles J. Bounds for water waves, Arch. Ration. Mech. Anal., Volume 99 (1987) no. 2, pp. 91-114 | DOI | MR | Zbl

[4] Amick, Charles J.; Turner, Robert E. L. Small internal waves in two-fluid systems, Arch. Ration. Mech. Anal., Volume 108 (1989) no. 2, pp. 111-139 | DOI | MR | Zbl

[5] Berestycki, Henri; Nirenberg, Louis Travelling fronts in cylinders, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 9 (1992) no. 5, pp. 497-572 | DOI | Numdam | MR | Zbl

[6] Buffoni, Boris; Toland, John Francis Analytic theory of global bifurcation. An introduction, Princeton Series in Applied Mathematics, Princeton University Press, 2003 | DOI | Zbl

[7] Chen, Robin Ming; Walsh, Samuel; Wheeler, Miles H. Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics (2019) (https://arxiv.org/abs/1907.04370)

[8] Chen, Robin Ming; Walsh, Samuel; Wheeler, Miles H. Global bifurcation for monotone fronts of elliptic equations (2020) (https://arxiv.org/abs/2005.00651)

[9] Chen, Robin Ming; Walsh, Samuel; Wheeler, Miles H. Global bifurcation of anti-plane shear fronts (2020) (Preprint,https://arxiv.org/abs/2008.09453)

[10] Choi, Wooyoung; Camassa, Roberto Long internal waves of finite amplitude, Phys. Rev. Lett., Volume 77 (1996) no. 9, pp. 1759-1762 | DOI

[11] Choi, Wooyoung; Camassa, Roberto Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., Volume 396 (1999), pp. 1-36 | DOI | MR | Zbl

[12] Constantin, Adrian; Strauss, Walter; Vărvărucă, Eugen Global bifurcation of steady gravity water waves with critical layers, Acta Math., Volume 217 (2016) no. 2, pp. 195-262 | DOI | MR | Zbl

[13] Córdoba, Diego; Enciso, Alberto; Grubic, Nastasia On the existence of stationary splash singularities for the Euler equations, Adv. Math., Volume 288 (2016), p. 922-941, | DOI | MR | Zbl

[14] Crapper, G. D. An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech., Volume 2 (1957), pp. 532-540 | DOI | MR | Zbl

[15] Dias, Frédéric; Vanden-Broeck, Jean-Marc On internal fronts, J. Fluid Mech., Volume 479 (2003), pp. 145-154 | DOI | MR | Zbl

[16] Dyachenko, Sergey A.; Hur, Vera Mikyoung Stokes waves with constant vorticity: folds, gaps and fluid bubbles, J. Fluid Mech., Volume 878 (2019), pp. 502-521 | DOI | MR | Zbl

[17] Dyachenko, Sergey A.; Hur, Vera Mikyoung Stokes waves with constant vorticity: I. Numerical computation, Stud. Appl. Math., Volume 142 (2019) no. 2, pp. 162-189 | DOI | MR | Zbl

[18] Helfrich, Karl R.; Melville, W. Kendall Long nonlinear internal waves, Annual review of fluid mechanics (Annual review of fluid mechanics), Volume 38, Annual Reviews, 2006, pp. 395-425 | DOI | MR | Zbl

[19] Hur, Vera Mikyoung; Vanden-Broeck, J.-M. A new application of Crapper’s exact solution to waves in constant vorticity flows, Eur. J. Mech. B Fluids, Volume 83 (2020), pp. 190-194 | DOI | MR

[20] Hur, Vera Mikyoung; Wheeler, Miles H. Exact free surfaces in constant vorticity flows, J. Fluid Mech., Volume 896 (2020), R1 | DOI | MR | Zbl

[21] von Kármán, Theodore The engineer grapples with non-linear problems, Bull. Am. Math. Soc., Volume 46 (1940), pp. 615-683 | DOI | MR | Zbl

[22] Kirchgässner, Klaus Wave-solutions of reversible systems and applications, J. Differ. Equations, Volume 45 (1982) no. 1, pp. 113-127 | DOI | MR | Zbl

[23] Laget, O.; Dias, Frédéric Numerical computation of capillary-gravity interfacial solitary waves, J. Fluid Mech., Volume 349 (1997), pp. 221-251 | DOI | MR | Zbl

[24] Lamb, Horace Hydrodynamics, Cambridge Mathematical Library, 6, Cambridge University Press, 1993 (with a foreword by Russel E. Caflisch) | MR | Zbl

[25] Makarenko, Nikolay I. Smooth bore in a two-layer fluid, Free boundary problems in continuum mechanics. International conference on free boundary problems in continuum mechanics, Novosibirsk, Russia, July 15-19, 1991 (ISNM. International Series of Numerical Mathematics), Volume 106, Birkhäuser (1992), pp. 195-204 | DOI | MR | Zbl

[26] Maklakov, Dmitri V. A note on the existence of pure gravity waves at the interface of two fluids, Physica D, Volume 401 (2020) no. 5, 132157 | DOI | MR

[27] Mielke, Alexander Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., Volume 10 (1988) no. 1, pp. 51-66 | DOI | MR | Zbl

[28] Mielke, Alexander Homoclinic and heteroclinic solutions in two-phase flow, Proceedings of the IUTAM/ISIMM Symposium on Structure and Dynamics of Nonlinear Waves in Fluids (Hannover, 1994) (Advanced Series in Nonlinear Dynamics), Volume 7, World Scientific (1995), pp. 353-362 | MR | Zbl

[29] Miyata, Manabu Long internal waves of large amplitude, Nonlinear water waves (International Union of Theoretical and Applied Mechanics), Springer, 1988, pp. 399-406 | DOI

[30] Pullin, Dale I.; Grimshaw, Roger Hamilton James Finite-amplitude solitary waves at the interface between two homogeneous fluids, Phys. Fluids, Volume 31 (1988) no. 12, pp. 3550-3559 | DOI | MR | Zbl

[31] Rabinowitz, Paul Henry Some global results for nonlinear eigenvalue problems, J. Funct. Anal., Volume 7 (1971), pp. 487-513 | DOI | MR | Zbl

[32] Rayleigh, Lord On the theory of long waves and bores, Proc. Roy. Soc. Lond., Ser. A, Volume 90 (1914) no. 619, pp. 324-328 | Zbl

[33] Sandstede, Björn; Scheel, Arnd Essential instability of pulses and bifurcations to modulated travelling waves, Proc. R. Soc. Edinb., Sect. A, Math., Volume 129 (1999) no. 6, pp. 1263-1290 | DOI | MR | Zbl

[34] Sandstede, Björn; Scheel, Arnd Essential instabilities of fronts: bifurcation, and bifurcation failure, Dyn. Syst., Volume 16 (2001) no. 1, pp. 1-28 | DOI | MR | Zbl

[35] Spielvogel, E. R. A variational principle for waves of infinite depth, Arch. Ration. Mech. Anal., Volume 39 (1970), pp. 189-205 | DOI | MR | Zbl

[36] Stokes, Georges Gabriel On the theory of oscillatory waves, Mathematical and Physical Papers, 1, Cambridge University Press, 1880, pp. 197-229 | DOI

[37] Sun, Shu-Ming Existence of large amplitude periodic waves in two-fluid flows of infinite depth, SIAM J. Math. Anal., Volume 32 (2001) no. 5, p. 1014-1031, | DOI | MR | Zbl

[38] Toland, John Francis Stokes waves, Topol. Methods Nonlinear Anal., Volume 7 (1996) no. 1, pp. 1-48 | DOI | MR | Zbl

[39] Turner, Robert E. L.; Vanden-Broeck, Jean-Marc Broadening of interfacial solitary waves, Phys. Fluids, Volume 31 (1988) no. 9, pp. 2486-2490 | DOI | Zbl

[40] Volpert, Aizik I.; Volpert, Vitaly A.; Volpert, Vladimir A. Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, 140, American Mathematical Society, 1994 (Translated from the Russian manuscript by James F. Heyda) | MR | Zbl

[41] Wheeler, Miles H. Solitary water waves of large amplitude generated by surface pressure, Arch. Ration. Mech. Anal., Volume 218 (2015) no. 2, pp. 1131-1187 | DOI | MR | Zbl

Cité par Sources :