Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension

Trakhinin, Yuri; Wang, Tao

doi:10.1007/s00208-021-02180-z

Cited by 19 publications

(23 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, one can still expect to establish the tame estimates for the linearized equation. Based on this and Nash-Moser iteration, Trakhinin-Wang [66,67] recently proved the LWP for free-boundary compressible ideal MHD with or without surface tension. We also mention that Chen-Wang [5] and Trakhinin [63] proved the LWP for the current-vortex sheets, and Secchi-Trakhinin [56] proved the LWP for the full plasma-vacuum problem for compressible ideal MHD under the non-collinearity condition.…”

Section: Free-boundary Mhd Equations: Incompressible Casementioning

confidence: 96%

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Lindblad¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

We consider 3D free-boundary compressible ideal magnetohydrodynamic (MHD) system under the Rayleigh-Taylor sign condition. It describes the motion of a free-surface perfect conducting fluid in an electro-magnetic field. The local wellposedness was recently proved by Trakhinin and Wang [66] by using Nash-Moser iteration. In this paper, we prove the a priori estimates without loss of regularity for the free-boundary compressible MHD system in Lagrangian coordinates in anisotropic Sobolev space, with more regularity tangential to the boundary than in the normal direction. It is based on modified Alinhac good unknowns, which take into account the covariance under the change of coordinates to avoid the derivative loss; full utilization of the cancellation structures of MHD system, to turn normal derivatives into tangential ones; and delicate analysis in anisotropic Sobolev spaces. Our method is also completely applicable to compressible Euler equations and thus yields an alternative estimate for compressible Euler equations without the analysis of div-curl decomposition or the wave equation in Lindblad-Luo [42], that do not work for compressible MHD. To the best of our knowledge, we establish the first result on the energy estimates without loss of regularity for the free-boundary problem of compressible ideal MHD. * in the second equation of (1.17), we getIn the energy estimates, we need to commute ∇ A with ∂ I * and then integrate by parts. However, the commutator [∂ I * , A li ]∂ l f contains the following terms whose L 2 (Ω)-norms cannot be controlled in the anisotropic Sobolev space), which cannot be controlled even in the standard Sobolev spaces when i 0 = 0;where f = Q or v i and I ′ is a multi-index with I ′ = 1. To overcome such difficulty, we can use the ideas of the Alinhac good unknown method, i.e., we can rewrite ∂ I * (∇ A Q) and ∂ I * (∇ A • v) in terms of the sum of the covariant derivative part and the commutator part satisfying(1.25)

show abstract

Section: Free-boundary Mhd Equations: Incompressible Casementioning

confidence: 96%

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Lindblad¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We will assume without loss of generality that ϕ 0 L ∞ (R 2 ) ≤ 1/4, so that the change of variables is admissible on sufficiently short time interval [0, T ]. Here we introduce the cut-off function χ as in [20,[33][34][35][36] to avoid the assumption that the initial perturbations have compact support.…”

Section: Reformulated Problem In a Fixed Domainmentioning

confidence: 99%

“…In light of the nonlinear analysis in [8,12,[33][34][35][36], we neglect the last terms in (3.9) to consider the effective linear problem…”

Section: Linearizationmentioning

confidence: 99%

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

Trakhinin,

Wang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangent to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in threedimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash-Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.

show abstract

“…It is known that surface tension provides a stabilizing effect on the motion of free vacuum interfaces; see, for instance, Coutand-Shkoller [8] and Shatah-Zeng [22,23] for the incompressible Euler equations with surface tension, and Coutand et al [9] for the compressible isentropic case. Motived by these works, the authors [28] recently investigated the free-boundary ideal compressible MHD equations with surface tension and constructed the unique solution without assuming any Taylortype sign condition. The result obtained in [28] corresponds to the special case of vanishing vacuum magnetic field.…”

Section: Introductionmentioning

confidence: 99%

“…Motived by these works, the authors [28] recently investigated the free-boundary ideal compressible MHD equations with surface tension and constructed the unique solution without assuming any Taylortype sign condition. The result obtained in [28] corresponds to the special case of vanishing vacuum magnetic field. Therefore, it is natural to examine the stabilizing effect of surface tension on the evolution of moving interfaces in ideal compressible MHD for general vacuum magnetic field, that is, to study the local well-posedness for problem (1.1)-(1.3), (1.5), and (1.7) with s > 0.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness for Moving Interfaces with Surface Tension in Ideal Compressible MHD

Trakhinin¹,

Wang²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We study the local well-posedness for an interface with surface tension that separates a perfectly conducting inviscid fluid from a vacuum. The fluid flow is governed by the equations of three-dimensional ideal compressible magnetohydrodynamics (MHD), while the vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The fluid and vacuum magnetic fields are tangential to the interface. This renders a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. We introduce some suitable regularization to establish the solvability and tame estimates for the linearized problem. Combining the linear well-posedness result with a modified Nash-Moser iteration scheme, we prove the local existence and uniqueness of solutions of the nonlinear problem. The non-collinearity condition required by Secchi and Trakhinin [Nonlinearity 27(1): 105-169, 2014] for the case of zero surface tension becomes unnecessary in our result, which verifies the stabilizing effect of surface tension on the evolution of moving vacuum interfaces in ideal compressible MHD.

show abstract

Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension

Cited by 19 publications

References 37 publications

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

Well-posedness for Moving Interfaces with Surface Tension in Ideal Compressible MHD

Contact Info

Product

Resources

About