Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

Morando, Alessandro; Trebeschi, Paola; Wang, Tao

doi:10.1016/j.jde.2018.10.029

Cited by 18 publications

(23 citation statements)

References 26 publications

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“…This free boundary problem totally coincides with that for vortex sheets for the nonisentropic Euler equations studied in [26,27]. Note that in [26,27] the Euler equations are written in form (13) for the polytropic gas equation of state (for which ρP ρ = γP ). Recall that for the equation of state (9) we also have ρP ρ = γP .…”

Section: Vortex Sheetssupporting

confidence: 63%

“…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]. Note that in [26,27] the Euler equations are written in form (13) for the polytropic gas equation of state (for which ρP ρ = γP ).…”

Section: Vortex Sheetsmentioning

confidence: 90%

“…Even formally, by introducing the "entropy" function s = ln(S + 1), we can reduce (9) to the polytropic gas equation of state p = p(ρ, s) = Aρ γ e s , with A = γ − 1 (clearly, the last equation in (13) stays valid for s = ln(S + 1)). On the other hand, the fact that vortex sheets were formally considered in [26,27] for the polytropic gas equation of state is absolutely unessential because the results in [26,27] stay valid for a general equation of state (as for 2D vortex sheets for the isentropic Euler equations whose linear and structural stability was proved in [6,7] under the "supersonic" condition).…”

Section: Vortex Sheetsmentioning

confidence: 99%

“…That is, the linear and nonlinear (structural) stability results obtained in [26,27] for 2D vortex sheets can be directly carried over 2D vortex sheets for equations (1). Referring to [27], we conclude that a vortex sheet x 1 = ϕ(t, x 2 ) for the 2D case of (1) (with x = (x 1 , x 2 ) and u = (u 1 , u 2 )) is structurally stable under the "supersonic" condition…”

Section: Vortex Sheetsmentioning

confidence: 99%

“…The existence theorem in [27] was proved for a finite (not necessarily short) time but under the condition that the initial discontinuity is close to a rectilinear discontinuity corresponding to a constant background solution U ± = ( P ± , 0,û ± 2 , s ± ),φ = 0, provided that this solution satisfies the "supersonic" condition (28) (see [27] for the whole details: exact assumptions on the initial data, regularity of solutions, etc.). Clearly, instead of the assumption that the initial data are close to a piecewise constant solution in the existence theorem we could assume that the time of existence is sufficiently short (see, e.g., such a local-in-time existence theorem for current-vortex sheets [32]).…”

Section: Vortex Sheetsmentioning

confidence: 99%

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

Ruan

Trakhinin

2019

Physica D: Nonlinear Phenomena

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We consider two models of a compressible inviscid isentropic two-fluid flow. The first one describes the liquid-gas two-phase flow. The second one can describe the mixture of two fluids of different densities or the mixture of fluid and particles. Introducing an entropy-like function, we reduce the equations of both models to a symmetric form which looks like the compressible Euler equations written in the nonconservative form in terms of the pressure, the velocity and the entropy. Basing on existing results for the Euler equations, this gives a number of instant results for both models. In particular, we conclude that all compressive shock waves in these models exist locally in time. 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 a recent result for vortex sheets in the liquid-gas two-phase flow.

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Section: Vortex Sheetssupporting

confidence: 63%

Section: Vortex Sheetsmentioning

confidence: 90%

Section: Vortex Sheetsmentioning

confidence: 99%

Section: Vortex Sheetsmentioning

confidence: 99%

Section: Vortex Sheetsmentioning

confidence: 99%

See 3 more Smart Citations

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

Ruan

Trakhinin

2019

Physica D: Nonlinear Phenomena

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

Wang,

Xin

2023

Comm Pure Appl Math

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We establish the local existence and uniqueness of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two‐phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well‐posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably‐chosen nonlinear approximate problems for the two‐phase compressible viscous non‐resistive MHD.

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On the evolution equation of compressible vortex sheets

Morando

Secchi

Trebeschi

2020

Mathematische Nachrichten

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We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo-differential equation of order two. In agreement with the classical stability analysis, if the Mach number satisfies < √ 2, the symbol is elliptic and the problem is ill-posed. On the contrary, if > √ 2 then the problem is weakly stable, and we are able to derive a wave-type a priori energy estimate for the solution, with no loss of regularity with respect to the data. Then we prove the well-posedness of the problem, by showing the existence of the solution in weighted Sobolev spaces. K E Y W O R D S compressible Euler equations, contact discontinuities, linear stability, loss of derivatives, vortex sheet, weak stability M S C ( 2 0 1 0 ) 35L50, 35Q35, 76E17, 76N10 √ 2 and are violently unstable when < √ 2, while planar vortex sheets are always violently unstable in three space dimensions. This kind of instabilities is the analogue of the Kelvin-Helmholtz instability for incompressible fluids.Artola and Majda [1] studied certain instabilities of two-dimensional supersonic vortex sheets by analyzing the interaction with highly oscillatory waves through geometric optics. A rigorous mathematical theory on nonlinear stability and local-in-time existence of two-dimensional supersonic vortex sheets was first established by 9] based on their linear stability results in [7] and a Nash-Moser iteration scheme.Characteristic discontinuities, especially vortex sheets, arise in a broad range of physical problems in fluid mechanics, oceanography, aerodynamics, plasma physics, astrophysics, and elastodynamics. The linear results in [7] have been generalized to cover the two-dimensional nonisentropic flows [15], the three-dimensional compressible steady flows [25,27], and the two-dimensional two-phase flows [18]. Recently, the methodology in [7] has been developed to deal with several constant Mathematische Nachrichten. 2020;293:945-969.

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Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

Cited by 18 publications

References 26 publications

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

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

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

On the evolution equation of compressible vortex sheets

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