Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients
- 28 Downloads
The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work (Chen et al. in Adv Math 311:18–60, 2017b) 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 Coulombel (SIAM J Math Anal 34(1):142–172, 2002; Ann Inst H Poincaré Anal Non Linéaire 21(4):401–443, 2004) and Coulombel and Secchi (Indiana Univ Math J 53(4):941–1012, 2004). Motivated by our method introduced in the constant-coefficient case (Adv Math 311:18–60, 2017b), 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.
Mathematics Subject Classification35Q31 35Q35 74F10 76E17 76N99
R. M. Chen’s research was supported in part by the Simons Foundation under Grant 354996 and the NSF Grant DMS-1613375. J. Hu’s research was supported in part by the NSF Foundation under Grant DMS-1312800. D. Wang was supported in part by the NSF Grants DMS-1312800 and DMS-1613213. The authors would like to thank the anonymous referees for valuable comments and suggestions.
- 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]
- 5.Chen, R.M., Hu, J., Wang, D.: Nonlinear stability and existence of compressible vortex sheets in two-dimensional elastodynamics (2018) (preprint)Google Scholar
- 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)Google Scholar
- 10.Francheteau, J., Métivier, G.: Existence de chocs faibles pour des systemes quasi-linéaires hyperboliques multidimensionnels. Astérisque no. 268 (2000)Google Scholar
- 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)Google Scholar
- 13.Joseph, D.: Applied Mathematical Sciences, 84. Fluid dynamics of viscoelastic liquids. Springer, New York (1990)Google Scholar
- 14.Meyer, Y.: Remarques sur un théorème de J.-M. Bony. Rend. Circ. Mat. Palermo (2) 1, 1–20 (1981)Google Scholar