Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws
We are concerned with entropy solutions of hyperbolic systems of conservation laws in several space variables. The Euler equations of gas dynamics and magnetohydrodynamics (MHD) are prototypes of hyperbolic conservation laws. In general, there are two types of discontinuities in the entropy solutions: shock waves and characteristic discontinuities, in which characteristic discontinuities can be either vortex sheets or entropy waves. In gas dynamics and MHD, across a vortex sheet, the tangential velocity field has a jump while the normal velocity is continuous; across an entropy wave, the entropy has a jump while the velocity field is continuous. A vortex sheet or entropy wave front is a part of the unknowns, which is a free boundary. Compressible vortex sheets and entropy waves, along with shock and rarefaction waves, occur ubiquitously in nature and are fundamental waves in the entropy solutions to multidimensional hyperbolic conservation laws. The local stability of shock and rarefaction waves has been relatively better understood. In this paper we discuss the stability issues for vortex sheets/entropy waves and present some recent developments and further open problems in this direction. First we discuss vortex sheets and entropy waves for the Euler equations in gas dynamics and some recent developments for a rigorous mathematical theory on their nonlinear stability/instability. Then we review our recent study and present a supplement to the proof on the nonlinear stability of compressible vortex sheets under the magnetic effect in three-dimensional MHD. The compressible vortex sheets in three dimensions are unstable in the regime of pure gas dynamics. Our main concern is whether such vortex sheets can be nonlinearly stabilized under the magnetic fields. To achieve this, we first set up the current-vortex sheet problem as a free boundary problem; then we establish high-order energy estimates of the solutions to the linearized problem, which shows that the current-vortex sheets are linearly stable when the jump of the tangential velocity is dominated by the jump of the non-paralleled tangential magnetic fields; and finally we develop a suitable iteration scheme of the Nash–Moser–Hörmander type to obtain the existence and nonlinear stability of compressible current-vortex sheets, locally in time. Some further open problems and several related remarks are also presented.
KeywordsEuler Equation Rarefaction Wave Free Boundary Problem Entropy Solution Vortex Sheet
The research of Gui-Qiang Chen was supported in part by the National Science Foundation under Grants DMS-0935967, DMS-0807551, the UK EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal Society—Wolfson Research Merit Award (UK). The research of Ya-Guang Wang was supported in part by the National Science Foundation of China under Grants 10971134 and 11031001. The second author would like to express his gratitude to the Department of Mathematics of Northwestern University (USA) for the hospitality, where this work was initiated when he visited there during the Spring Quarter 2005. Both authors would like to thank Y. Trakhinin for the helpful discussion on this problem.
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