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Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients. (English) Zbl 1440.35254

Summary: The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work [R. M. Chen et al., Adv. Math. 311, 18–60 (2017; Zbl 1373.35227)] on the linear stability with constant coefficients, the problem has a free boundary which is characteristic, and also the Kreiss-Lopatinskii condition is not uniformly satisfied. In addition, the roots of the Lopatinskii determinant of the para-linearized system may coincide with the poles of the system. Such a new collapsing phenomenon causes serious difficulties when applying the bicharacteristic extension method of J.-F. Coulombel [SIAM J. Math. Anal. 34, No. 1, 142–172 (2002; Zbl 1029.35171); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, No. 4, 401–443 (2004; Zbl 1072.35120)] and J.-F. Coulombel and P. Secchi [Indiana Univ. Math. J. 53, No. 4, 941–1012 (2004; Zbl 1068.35100)]. Motivated by our method introduced in the constant-coefficient case [Zbl 1373.35227], we perform an upper triangularization to the para-linearized system to separate the outgoing mode into a closed form where the outgoing mode only appears at the leading order. This procedure results in a gain of regularity for the outgoing mode, which allows us to overcome the loss of regularity of the characteristic components at the poles and hence to close all the energy estimates. We find that, analogous to the constant-coefficient case, elasticity generates notable stabilization effects, and there are additional stable subsonic regions compared with the isentropic Euler flows. Moreover, since our method does not rely on the construction of the characterisic curves, it can also be applied to other fluid models such as the non-isentropic Euler equations and the MHD equations.

MSC:

35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76E17 Interfacial stability and instability in hydrodynamic stability
76N15 Gas dynamics (general theory)
76G25 General aerodynamics and subsonic flows
76B47 Vortex flows for incompressible inviscid fluids
35B35 Stability in context of PDEs
35R35 Free boundary problems for PDEs
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References:

[1] Alinhac, S., Existence d’ondes de rarefaction pour des systemes quasi-linéaires hyperboliques multidimensionnels, Commun. Partial Differ. Equ., 14, 2, 173-230 (1989) · Zbl 0692.35063 · doi:10.1080/03605308908820595
[2] Bony, JM, 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 · doi:10.24033/asens.1404
[3] Chen, G.-Q., Secchi, P., Wang, T.: Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime (2017a). arXiv:1707.02672 [math.AP] · Zbl 1420.35206
[4] Chen, RM; Hu, J.; Wang, D., Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math., 311, 18-60 (2017) · Zbl 1373.35227 · doi:10.1016/j.aim.2017.02.014
[5] Chen, R.M., Hu, J., Wang, D.: Nonlinear stability and existence of compressible vortex sheets in two-dimensional elastodynamics (2018) (preprint)
[6] Coulombel, J-F, Weak stability of nonuniformly stable multidimensional shocks, SIAM J. Math. Anal., 34, 1, 142-172 (2002) · Zbl 1029.35171 · doi:10.1137/S0036141001392803
[7] Coulombel, J-F, Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 4, 401-443 (2004) · Zbl 1072.35120 · doi:10.1016/j.anihpc.2003.04.001
[8] Coulombel, J-F; Secchi, P., The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53, 4, 941-1012 (2004) · Zbl 1068.35100 · doi:10.1512/iumj.2004.53.2526
[9] Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, 4th edn. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 325. Springer, Berlin (2016) · Zbl 1364.35003
[10] Francheteau, J., Métivier, G.: Existence de chocs faibles pour des systemes quasi-linéaires hyperboliques multidimensionnels. Astérisque no. 268 (2000) · Zbl 0996.35001
[11] Gurtin, M.E.: An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, vol. 158. Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New York (1981) · Zbl 0559.73001
[12] Hu, X.; Wang, D., Global existence for the multi-dimensional compressible viscoelastic flows, J. Differ. Equ., 250, 2, 1200-1231 (2011) · Zbl 1208.35111 · doi:10.1016/j.jde.2010.10.017
[13] Joseph, D., Applied Mathematical Sciences, 84 (1990), New York: Springer, New York · Zbl 0698.76002
[14] Meyer, Y.: Remarques sur un théorème de J.-M. Bony. Rend. Circ. Mat. Palermo (2) 1, 1-20 (1981) · Zbl 0473.35021
[15] Morando, A.; Trebeschi, P., Two-dimensional vortex sheets for the nonisentropic Euler equations: linear stability, J. Hyperbolic Differ. Equ., 5, 3, 487-518 (2008) · Zbl 1170.35018 · doi:10.1142/S021989160800157X
[16] Ruan, L.; Wang, D.; Weng, S.; Zhu, C., Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: linear stability, Commun. Math. Sci., 14, 735-776 (2016) · Zbl 1334.35253 · doi:10.4310/CMS.2016.v14.n3.a7
[17] Serre, D., Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0936.35001
[18] Trakhinin, Y., Existence of compressible current-vortex sheets: variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177, 331-366 (2005) · Zbl 1094.76066 · doi:10.1007/s00205-005-0364-7
[19] Trakhinin, Y., On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci., 28, 8, 917-945 (2005) · Zbl 1112.76087 · doi:10.1002/mma.600
[20] Wang, Y-G; Yu, F., Stability of contact discontinuities in three-dimensional compressible steady flows, J. Differ. Equ., 255, 1278-1356 (2013) · Zbl 1288.35064 · doi:10.1016/j.jde.2013.05.014
[21] Wang, Y-G; Yuan, H., Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66, 341-388 (2015) · Zbl 1321.35149 · doi:10.1007/s00033-014-0404-y
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