Nonlinear compressible vortex sheets in two space dimensions

Coulombel, Jean-François; Secchi, Paolo

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 1, pp. 85-139.

Nous construisons des nappes de tourbillon supersoniques pour les équations d'Euler compressibles isentropiques en deux dimensions d'espace. Il s'agit d'un problème non-linéaire hyperbolique à frontière libre présentant deux difficultés principales : la frontière libre est caractéristique et la condition dite de Lopatinskii n'est satisfaite que dans un sens faible, ce qui induit des estimations à perte. Néanmoins nous montrons l'existence de telles solutions régulières par morceaux des équations d'Euler en utilisant un schéma itératif de type Nash-Moser palliant les pertes de régularité. Notre analyse s'étend au cas de discontinuités non-caractéristiques et faiblement stables comme certaines ondes de choc pour les équations d'Euler ou les transitions de phase liquide- vapeur.

We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme. We also show how a similar analysis yields the existence of weakly stable shock waves in isentropic gas dynamics, and the existence of weakly stable liquid/vapor phase transitions.

Classification : 76N10, 35Q35, 35L50, 76E17
Mots clés : compressible Euler equations, vortex sheets, contact discontinuities, weak stability, loss of derivatives

[1] S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14 (1989), 173-230. | MR | Zbl

[2] S. Alinhac & S. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser, InterÉditions, 1991. | Zbl

[3] M. Artola & A. Majda, Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes, Phys. D 28 (1987), 253-281. | Zbl

[4] S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal. 31 (1998), 243-263. | MR | Zbl

[5] S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 150 (1999), 23-55. | MR | Zbl

[6] A. Blokhin & Y. Trakhinin, Stability of strong discontinuities in fluids and MHD, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, 2002, 545-652. | Zbl

[7] J. Chazarain & A. Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications 14, North-Holland Publishing Co., 1982. | Zbl

[8] J.-Y. Chemin, Dynamique des gaz à masse totale finie, Asymptotic Anal. 3 (1990), 215-220. | MR | Zbl

[9] J.-F. Coulombel, Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 401-443. | Numdam | MR | Zbl

[10] J.-F. Coulombel, Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl. 84 (2005), 786-818. | MR | Zbl

[11] J.-F. Coulombel & A. Morando, Stability of contact discontinuities for the nonisentropic Euler equations, Ann. Univ. Ferrara Sez. VII (N.S.) 50 (2004), 79-90. | Zbl

[12] J.-F. Coulombel & P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004), 941-1012. | Zbl

[13] J. A. Fejer & J. W. Miles, On the stability of a plane vortex sheet with respect to three-dimensional disturbances, J. Fluid Mech. 15 (1963), 335-336. | Zbl

[14] J. Francheteau & G. Métivier, Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Astérisque 268 (2000), 198. | Numdam | Zbl

[15] H. Freistühler, Some results on the stability of non-classical shock waves, J. Partial Differential Equations 11 (1998), 25-38. | Zbl

[16] O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15 (1990), 595-645. | MR | Zbl

[17] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222. | MR | Zbl

[18] E. Harabetian, A convergent series expansion for hyperbolic systems of conservation laws, Trans. Amer. Math. Soc. 294 (1986), 383-424. | MR | Zbl

[19] L. Hörmander, Implicit function theorems, Stanford university lecture notes, 1977.

[20] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 181-205. | MR | Zbl

[21] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. | MR | Zbl

[22] J.-L. Lions & E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, 1968. | Zbl

[23] A. Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), 93. | MR | Zbl

[24] A. Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), 95. | MR | Zbl

[25] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences 53, Springer, 1984. | MR | Zbl

[26] A. Majda & S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), 607-675. | Zbl

[27] G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986), 431-479. | MR | Zbl

[28] G. Métivier, Ondes soniques, J. Math. Pures Appl. 70 (1991), 197-268. | MR | Zbl

[29] G. Métivier, Stability of multidimensional shocks, in Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl. 47, Birkhäuser, 2001, 25-103. | MR | Zbl

[30] J. W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid Mech. 4 (1958), 538-552. | MR | Zbl

[31] A. Mokrane, Problèmes mixtes hyperboliques non-linéaires, Thèse, Université de Rennes I, 1987.

[32] A. Morando & P. Trebeschi, Stability of contact discontinuities for the nonisentropic Euler equations in two-space dimensions, preprint, 2007. | Zbl

[33] J. B. Rauch & F. J. I. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303-318. | Zbl

[34] S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), 49-75. | MR | Zbl

[35] P. Secchi, The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity, Differential Integral Equations 9 (1996), 671-700. | MR | Zbl

[36] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems, Arch. Rational Mech. Anal. 134 (1996), 155-197. | MR | Zbl

[37] D. Serre, Systems of conservation laws. 2, Cambridge University Press, 2000, Geometric structures, oscillations, and initial-boundary value problems. | MR | Zbl

[38] T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), 475-485. | MR | Zbl

[39] H. Beirão Da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981), 317-351. | Numdam | MR | Zbl