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Yuri Trakhinin - One of the best experts on this subject based on the ideXlab platform.

  • Structural stability of shock waves in 2D compressible elastodynamics
    Mathematische Annalen, 2019
    Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
    Abstract:

    We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material 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.

  • well posedness of the Linearized Problem for mhd contact discontinuities
    Journal of Differential Equations, 2015
    Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
    Abstract:

    Abstract We study the free boundary Problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh–Taylor sign condition [ ∂ p / ∂ N ] 0 on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the Linearized Problem for 2D planar MHD flows.

  • well posedness of the Linearized plasma vacuum interface Problem in ideal incompressible mhd
    Quarterly of Applied Mathematics, 2014
    Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
    Abstract:

    We consider the free boundary Problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the initial discontinuity, the well-posedness of the Linearized Problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard initial-boundary value Problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized Problem. For the regularized Problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .

  • well posedness of the Linearized plasma vacuum interface Problem
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Paolo Secchi, Yuri Trakhinin
    Abstract:

    We consider the free boundary Problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface we assume that the total pressure is continuous and that the magnetic field is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the Linearized Problem in conormal Sobolev spaces.

  • Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis
    Archive for Rational Mechanics and Analysis, 2005
    Co-Authors: Yuri Trakhinin
    Abstract:

    We study the initial-boundary value Problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the Linearized Problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W _2^1,^ σ . Despite the fact that the constant coefficients Linearized Problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients Problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics.

Dmitry Tkachev - One of the best experts on this subject based on the ideXlab platform.

Paolo Secchi - One of the best experts on this subject based on the ideXlab platform.

  • stability of the Linearized mhd maxwell free interface Problem
    Communications on Pure and Applied Analysis, 2014
    Co-Authors: Davide Catania, Marcello Dabbicco, Paolo Secchi
    Abstract:

    We consider the free boundary Problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the Problem, in particular in relation with the electric field in vacuum. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the Linearized Problem in the Sobolev space $H^1_{\tan}$ with conormal regularity. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method. An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the Problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum.

  • well posedness of the Linearized mhd maxwell free boundary Problem
    arXiv: Analysis of PDEs, 2013
    Co-Authors: Davide Catania, Marcello Dabbicco, Paolo Secchi
    Abstract:

    We consider the free boundary Problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The aim of this paper is the study of the stability of the Linearized Problem with variable coefficients for nonplanar plasma-vacuum interfaces. Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the Linearized Problem in the Sobolev space $H^1_{\tan}$ with conormal regularity. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method.

  • well posedness of the Linearized plasma vacuum interface Problem
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Paolo Secchi, Yuri Trakhinin
    Abstract:

    We consider the free boundary Problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface we assume that the total pressure is continuous and that the magnetic field is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the Linearized Problem in conormal Sobolev spaces.

Paola Trebeschi - One of the best experts on this subject based on the ideXlab platform.

  • Structural stability of shock waves in 2D compressible elastodynamics
    Mathematische Annalen, 2019
    Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
    Abstract:

    We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material 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.

  • well posedness of the Linearized Problem for mhd contact discontinuities
    Journal of Differential Equations, 2015
    Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
    Abstract:

    Abstract We study the free boundary Problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh–Taylor sign condition [ ∂ p / ∂ N ] 0 on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the Linearized Problem for 2D planar MHD flows.

  • well posedness of the Linearized plasma vacuum interface Problem in ideal incompressible mhd
    Quarterly of Applied Mathematics, 2014
    Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
    Abstract:

    We consider the free boundary Problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the initial discontinuity, the well-posedness of the Linearized Problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard initial-boundary value Problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized Problem. For the regularized Problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .

A M Blokhin - One of the best experts on this subject based on the ideXlab platform.