Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Abstract

Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We model the fluids by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density such that the sound speed is positive. We prove that, for a short time, there exists a unique solution of the equations with the same structure.The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity which becomes fairly standard after rotating the velocity according to the interface normal. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al. in the setting of a compressible liquid-vacuum interface.Although already considered by Coutand et al. [10] and Lindblad [17], we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.

Existence of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics

“…Coulombel and Secchi [7,8] proved the nonlinear stability of 2D supersonic vortex sheets for the compressible isentropic Euler equations, and the nonisentropic case was proved by Morando and Trebeschi [19] and Morando et al [20]. It is known that the surface tension has the stabilizing effect on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, see Cheng et al [4] and Shatah and Zeng [21,22] for the local well-posedness of the twophase incompressible Euler equations with surface tension and Stevens [23] for the compressible case.…”

“…But if (𝜂 0 , 𝑝 0 , 𝑣 0 , 𝑏 0 , 𝜌 0 ) satisfy the (𝑚 − 1)-th order compatibility conditions: ⟦ 𝐕 𝓁 ∇𝜂 0 ,𝜌 0 ( ∇𝓁 𝑈 0 , 𝜕 𝓁 3 𝑈 0 ) ⟧ = 0, 𝓁 = 0, … , 𝑚 − 1, (A. 23) then (𝑝 𝛿 0 , 𝑣 𝛿 0 , 𝑏 𝛿 0 ) → (𝑝 0 , 𝑣 0 , 𝑏 0 ) in 𝐻 𝑚 (Ω ± ) as 𝛿 → 0. Indeed, we claim that 𝑈 𝛿,(𝑗) 0 → 𝑈 0 in 𝐻 𝑚 (Ω ± ) as 𝛿 → 0 for 𝑗 = 0, … , 𝑚 − 1.…”

Global well-posedness of the free-interface incompressible Euler equations with damping

“…The presence of surface tension is necessary for the stability of vortex sheets in the two-phase incompressible Euler equations; without surface tension, we have the well-known Kelvin-Helmholtz instability, see Caflisch and Orellana [5] and Ebin [15]. For the short-time structural stability of vortex sheets in the two-phase compressible Euler equations, we refer to Coulombel and Secchi [11,12] and Stevens [33].…”

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

“…Surface tension has been proved to suppress the instability of vortex sheets in three dimensions by Ambrose-Masmoudi [3] for incompressible irrotational flows, by Cheng et al [11] and Shatah-Zeng [29] for incompressible rotational flows, and by Stevens [30] for compressible flows. Numerical and experimental studies of free-interface MHD flows with surface tension have been provided in Samulyak et al [25] and the references therein.…”