Mathematical Theories Of The Stability Of Multi-dimensional Contact Discontinuities

In this doctoral thesis, we study the stability of multi-dimensional contact discontinu-ities in two types of quasi-linear conservation laws and discuss the stability mechanisms.Mainly, we investigate the linear stability of current-vortex sheets in compressible magne-tohydrodynamics (MHD) equations, and the nonlinear structural stability of contact discon-tinuities in compressible steady ideal hydrodynamic equations. Moreover, we also analyzethe asymptotic behaviour of supersonic vortex sheet with high frequency oscillatory waveperturbations.In the introduction, we briefly review some physical backgrounds and related math-ematical theories and recall some known works of stability of multi-dimensional contactdiscontinuities. Besides, we state the main results and the structure of this thesis.In Chapter2, we first consider the linear stability of current-vortex sheets for two di-mensional compressible isentropic unsteady MHD equations. For this free boundary prob-lem with characteristic boundary, we have obtained the necessary and sufficient conditionsfor the stability of planar current-vortex sheets by computing the Kreiss-Lopatinskii condi-tions and obtain the a priori estimates of solutions to the corresponding linearized problem.We find that the linearized boundary value problem is not uniformly stable but weakly stablein the sense of Kreiss-Lopatinskii condition, so there is a loss of regularity in the a prioriestimates with respect to the source terms both in the interior domain and on the boundary.Besides, this stability condition implies the stabilization effects of magnetic fields on vor-tex sheets. Compared with the stability criterion of vortex sheets for two dimensional gasdynamics, due to the magnetic field the critical Mach number in the stability condition forcurrent-vortex sheets is smaller than that for the vortex sheets in gas dynamics, which allowssome unstable vortex sheets in gas dynamics to become stable in the MHD systems. Next, from Chapter3to Chapter5, we analyze the nonlinear structural stability of con-tact discontinuities in three dimensional compressible isentropic steady Euler equations. InChapter3, we study the linear stability of planar contact discontinuities at first. By develop-ing Kreiss, Coulombel and Secchi’s arguments for characteristic boundary value problems ofhyperbolic equations, we obtain a necessary and sufficient condition for the linear stability ofsupersonic planar contact discontinuities and a priori estimates of solutions to the linearizedproblem. The weak stability of this contact discontinuity also results in a loss of regularitywith respect to the source terms in the interior domain and on the boundary in the a prior-i estimates of solutions to the linearized problem. We discuss contact discontinuities withtangential velocity fields on two sides of the discontinuous front parallel or non-parallel.Moreover, the stability conditions illustrate that not only tangential velocity fields on twosides of the discontinuous front should be supersonic in the direction of the coming flow, butalso their orthogonal projection on some space-like hyperplane must be supersonic in orderto guarantee the stability of contact discontinuities. In Chapter4, we consider the linear sta-bility of a perturbation of a planar contact discontinuity. Based on the results of Chapter3and using the calculus of paradifferential operators, we get the a priori estimates of solutionsto the variable coefficient linearized problem. Further, taking advantage of the control ofnon-characteristic components of unknowns by the equations and the problems satisfied byvorticities of velocity fields, we get energy estimates of high order derivatives of solutions tothe linearized problem. In Chapter5, the nonlinear structural stability of contact discontinu-ities in three dimensional compressible isentropic steady Euler flows is obtained. As there isa loss of regularity in the estimates of solutions to the linearized problem derived in Chapter4, we adapt the Nash-Moser-Ho¨rmander iteration scheme to study the nonlinear stability ofsupersonic contact discontinuities in three dimensional compressible isentropic steady Eulerequations.Finally, in Chapter6, we employ the method of nonlinear geometric optics to investigatethe stability of supersonic contact discontinuities in three dimensional compressible isen-tropic steady Euler equations under the perturbation of high frequency oscillatory waves.For the structurally stable supersonic planar contact discontinuity given in Chapter5, weperturb this contact discontinuity with a high frequency oscillatory wave of incidence withamplitude O(ε2). We find that there are three specific angles of incidence resulting in an am-plification of amplitudes of reflected waves and refracted waves to O(ε). Thus, this implies the instability of interactions of supersonic vortex sheet in three dimensional compressibleisentropic steady Euler equations with some high frequency oscillatory waves. This resultcoincides with the weak stability of this supersonic vortex sheet obtained in Chapter3.