Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

Abstract

We are concerned with the vortex sheet solutions for the inviscid two-phase flow in two dimensions. In particular, the nonlinear stability and existence of compressible vortex sheet solutions under small perturbations are established by using a modification of the Nash-Moser iteration technique, where a priori estimates for the linearized equations have a loss of derivatives. Due to the jump of the normal derivatives of densities of liquid and gas, we obtain the normal estimates in the anisotropic Sobolev space, instead of the usual Sobolev space. New ideas and techniques are developed to close the energy estimates and derive the tame estimates for the two-phase flows.

Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results

“…For the 2D case, we make the conclusion about the local-in-time existence of vortex sheets under a "supersonic" stability condition. In the sense of a much lower regularity requirement for the initial data, our result for 2D vortex sheets essentially improves the recent result in [13] for vortex sheets in the liquid-gas two-phase flow.…”

“…For model (1), the free boundary value problem for vortex sheets is the problem for (13) in the domains Ω ± (t) with the boundary conditions (27) and initial data for U ± and ϕ. This free boundary problem totally coincides with that for vortex sheets for the nonisentropic Euler equations studied in [26,27].…”

“…The elementary symmetrization of conservations laws (1) giving the quasilinear symmetric hyperbolic system (13)/ (14), which looks like the symmetric system of the compressible Euler equations, implies a number of results almost for nothing. For example, we have the local-intime existence and uniqueness theorem [15,34] in Sobolev spaces H s for the Cauchy problem for system (13), with s ≥ 3.…”

“…For the perturbation of the "entropy" (we again denote it by S) we obtain the separate problem (cf. (13), (21))…”

The asymptotic limits of Riemann solutions for the isentropic drift‐flux model of compressible two‐phase flows

“…In a word, it can be proved rigorously in the sense of distributions that the limits of Riemann solutions of the isentropic drift‐flux model really converge to the corresponding ones of the weakly hyperbolic system under the same Riemann data . It is worthy to mention that the gas phase is usually much lighter than the liquid phase and thus is neglected in the mixture momentum equation such as in Huang et al, whose singular limit problem has also been considered recently. It is of interest to find that the delta shock wave has different propagation speed and strength for the limiting system when the gas phase is involved in the mixture momentum equation or not.…”