Transverse bifurcation of viscous slow MHD shocks

Barker, Blake; Monteiro, Rafael; Zumbrun, Kevin

doi:10.48550/arxiv.1901.09153

Cited by 1 publication

(5 citation statements)

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“…Returning to elastodynamics, we note that according to our knowledge there were no studies of shock waves for system (1). In this connection, we can only mention characteristic discontinuities, namely, vortex sheets for system (1) whose linear and structural stability was recently studied by Chen, Hu and Wang [11,12,24].…”

Section: Lopatinski Conditions)mentioning

confidence: 99%

“…uniformly stable fast shock waves found in [37] are structurally stable. Regarding slow MHD shock waves, some results about their stability can be found in [28,16,10,18,17,1].…”

Section: Lopatinski Conditions)mentioning

confidence: 99%

“…Taking into account the divergence constraints (2), we easily symmetrize the system of conservation laws (1) by rewriting it as…”

Section: Introductionmentioning

confidence: 99%

“…The goal of this paper is the study of structural stability of shock waves for the system of hyperbolic conservation laws (1). By structural stability we mean the local-in-time existence and uniqueness in Sobolev spaces of shock front solutions.…”

Section: Introductionmentioning

confidence: 99%

“…By structural stability we mean the local-in-time existence and uniqueness in Sobolev spaces of shock front solutions. If in the first three scalar equations of system (1) we set formally F = 0, then we get the system of isentropic gas dynamics (for the 2D case). The structural stability of shock waves in isentropic gas dynamics was proved by Majda [32] provided that the uniform stability condition found by him in [31] is satisfied at each point of the initial shock front.…”

Section: Introductionmentioning

confidence: 99%

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Structural stability of shock waves in 2D compressible elastodynamics

2019

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We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic fluid in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role.In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss-Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

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Section: Lopatinski Conditions)mentioning

confidence: 99%

“…uniformly stable fast shock waves found in [37] are structurally stable. Regarding slow MHD shock waves, some results about their stability can be found in [28,16,10,18,17,1].…”

Section: Lopatinski Conditions)mentioning

confidence: 99%

“…Taking into account the divergence constraints (2), we easily symmetrize the system of conservation laws (1) by rewriting it as…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%