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Weakly stable multidimensional shocks. (English) Zbl 1072.35120
The author considers the system of \(N\) conservation laws in \(\mathbb R \times \mathbb R ^d\) (time-space) \[ \sum\limits_{j=0}^{d}\partial_jf_j(u)=0, \] where \(x_0\equiv t\) is the time variable, \(x\equiv (x_1,x_2,\dots,x_d)\) is the space variable and \(\partial_j\equiv \partial /\partial x_j\). The fluxes \(f_0,f_1,\dots,f_d\) are \(C^{\infty }\) functions defined on an open set \(U\subset \mathbb R ^N\) with values in \(\mathbb R ^N\).
The author studies the linear stability of multidimensional shock waves for the system under consideration in the case where Majda’s uniform stability condition is violated. An energy estimate is obtained. It is shown that the solutions to the linearized problem have singularities localized along bicharacteristic curves originating from the boundary. The application to isentropic gas dynamics is discussed.

MSC:
35L67 Shocks and singularities for hyperbolic equations
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
76L05 Shock waves and blast waves in fluid mechanics
35L35 Initial-boundary value problems for higher-order hyperbolic equations
76N15 Gas dynamics (general theory)
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References:
[1] Adams, R.A, Sobolev spaces, (1975), Academic Press · Zbl 0186.19101
[2] Alinhac, S, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. partial differential equations, 14, 2, 173-230, (1989) · Zbl 0692.35063
[3] Alinhac, S, Blowup for nonlinear hyperbolic equations, (1995), Birkhäuser Boston · Zbl 0820.35001
[4] Arnol’d, V.I, Ordinary differential equations, (1992), Springer-Verlag · Zbl 0659.58012
[5] Benzoni-Gavage, S, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear anal., 31, 1-2, 243-263, (1998) · Zbl 0928.76015
[6] Benzoni-Gavage, S, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. rational mech. anal., 150, 1, 23-55, (1999) · Zbl 0980.76023
[7] Benzoni-Gavage, S; Rousset, F; Serre, D; Zumbrun, K, Generic types and transitions in hyperbolic initial-boundary value problems, Proc. roy. soc. Edinburgh sect. A, 132, 1073-1104, (2002) · Zbl 1029.35165
[8] S. Benzoni-Gavage, D. Serre, First order systems of hyperbolic partial differential equations with applications, in preparation · Zbl 1113.35001
[9] Blokhin, A.M; Trakhinin, Y.L, Stability of fast parallel MHD shock waves in polytropic gas, Eur. J. mech. B fluids, 18, 2, 197-211, (1999) · Zbl 0940.76019
[10] Bony, J.M, Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. sci. école norm. sup. (4), 14, 2, 209-246, (1981) · Zbl 0495.35024
[11] Chazarain, J; Piriou, A, Introduction to the theory of linear partial differential equations, (1982), North-Holland · Zbl 0487.35002
[12] Coulombel, J.F, Weak stability of nonuniformly stable multidimensional shocks, SIAM J. math. anal., 34, 1, 142-172, (2002) · Zbl 1029.35171
[13] Dafermos, C.M, Hyperbolic conservation laws in continuum physics, (2000), Springer-Verlag · Zbl 0940.35002
[14] Francheteau, J; Métivier, G, Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Astérisque, vol. 268, (2000) · Zbl 0996.35001
[15] Freistühler, H, Some results on the stability of non-classical shock waves, J. partial differential equations, 111, 1, 25-38, (1998) · Zbl 0903.35006
[16] Freistühler, H; Fries, C; Rohde, C, Existence, bifurcation, and stability of profiles for classical and non-classical shock waves, (), 287-309, 814 · Zbl 1002.35087
[17] Hersh, R, Mixed problems in several variables, J. math. mech., 12, 317-334, (1963) · Zbl 0149.06602
[18] Hirsch, M.W, Differential topology, (1994), Springer-Verlag · Zbl 0121.18004
[19] Hörmander, L, Lectures on nonlinear hyperbolic differential equations, (1997), Springer-Verlag · Zbl 0881.35001
[20] Kato, T, Perturbation theory for linear operators, (1976), Springer-Verlag
[21] Kreiss, H.O, Initial boundary value problems for hyperbolic systems, Comm. pure appl. math., 23, 277-298, (1970) · Zbl 0193.06902
[22] Lax, P.D, Hyperbolic systems of conservation laws. II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803
[23] Majda, A, The existence of multi-dimensional shock fronts, Mem. amer. math. soc., 43, 281, (1983) · Zbl 0517.76068
[24] Majda, A, The stability of multi-dimensional shock fronts, Mem. amer. math. soc., 41, 275, (1983) · Zbl 0506.76075
[25] Majda, A, Compressible fluid flow and systems of conservation laws in several space variables, (1984), Springer-Verlag · Zbl 0537.76001
[26] Métivier, G, The block structure condition for symmetric hyperbolic systems, Bull. London math. soc., 32, 6, 689-702, (2000) · Zbl 1073.35525
[27] Métivier, G, Stability of multidimensional shocks, (), 25-103 · Zbl 1017.35075
[28] Meyer, Y, Remarques sur un théorème de J.M. bony, Suppl. rend. circ. mat. Palermo ser. II, 1, 1-20, (1981) · Zbl 0473.35021
[29] A. Mokrane, Problèmes mixtes hyperboliques non-linéaires, Ph.D. Thesis, Université de Rennes I, 1987
[30] Ohkubo, T; Shirota, T, On structures of certain L2-well-posed mixed problems for hyperbolic systems of first order, Hokkaido math. J., 4, 82-158, (1975) · Zbl 0304.35067
[31] Ralston, J.V, Note on a paper of kreiss, Comm. pure appl. math., 24, 6, 759-762, (1971)
[32] Sablé-Tougeron, M, Existence pour un problème de l’élastodynamique Neumann non linéaire en dimension 2, Arch. rational mech. anal., 101, 3, 261-292, (1988) · Zbl 0652.73019
[33] Serre, D, Systems of conservation laws. 1, (1999), Cambridge University Press
[34] Serre, D, Systems of conservation laws. 2, (2000), Cambridge University Press
[35] Serre, D, La transition vers l’instabilité pour LES ondes de choc multi-dimensionnelles, Trans. amer. math. soc., 353, 12, 5071-5093, (2001) · Zbl 1078.35521
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