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Yuri Trakhinin - One of the best experts on this subject based on the ideXlab platform.
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Structural stability of shock waves and current-Vortex Sheets in shallow water magnetohydrodynamics
Zeitschrift für angewandte Mathematik und Physik, 2020Co-Authors: Yuri TrakhininAbstract:We study the structural stability of shock waves and current-Vortex Sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [25] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-Vortex Sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-Vortex Sheets.
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structural stability of shock waves and current Vortex Sheets in shallow water magnetohydrodynamics
arXiv: Analysis of PDEs, 2019Co-Authors: Yuri TrakhininAbstract:We study the structural stability of shock waves and current-Vortex Sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [Morando A., Trakhinin Y., Trebeschi P. Math. Ann. (2019), this https URL] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-Vortex Sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-Vortex Sheets.
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symmetrizations of rmhd equations and stability of relativistic current Vortex Sheets
Classical and Quantum Gravity, 2013Co-Authors: Heinrich Freistuhler, Yuri TrakhininAbstract:We consider the equations of relativistic magnetohydrodynamics (RMHD) in the case of special relativity. Starting by computations in the fluid's rest frame and then applying Lorentz transformations, we derive a covariant symmetric formulation of RMHD in terms of the primitive (physical) variables. This symmetric system is important for the study of various initial boundary value problems. We also find a so-called secondary symmetrization whose direct consequence is the extension of the sufficient stability condition obtained earlier for non-relativistic planar current–Vortex Sheets to the relativistic case. As in non-relativistic settings, this implies the local-in-time existence of corresponding smooth nonplanar current–Vortex Sheets.
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symmetrizations of rmhd equations and stability of relativistic current Vortex Sheets
arXiv: Analysis of PDEs, 2012Co-Authors: Heinrich Freistuhler, Yuri TrakhininAbstract:We consider the equations of relativistic magnetohydrodynamics (RMHD) in the case of special relativity. For the fluid rest frame a nonconservative reformulation of the RMHD equations gives a symmetric system for the vector of primitive (physical) variables. By applying the Lorentz transformation to this system we find a concrete form of symmetric matrices in the LAB-frame. The resulting symmetric system in terms of primitive variables is important for the study of various initial boundary value problems for the RMHD equations. We also find a so-called secondary symmetrization whose direct consequence is the extension of the sufficient stability condition obtained earlier for non-relativistic planar current-Vortex Sheets to the relativistic case. As in non-relativistic settings, this implies the local-in-time existence of corresponding smooth nonplanar current-Vortex Sheets.
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The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics
Archive for Rational Mechanics and Analysis, 2009Co-Authors: Yuri TrakhininAbstract:We prove the local-in-time existence of solutions with a surface of current-Vortex sheet (tangential discontinuity) of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity. This paper is a natural completion of our previous analysis ( Trakhinin in Arch Ration Mech Anal 177:331–366, 2005) where a sufficient condition for the weak stability of planar current-Vortex Sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities. The original nonlinear problem is a free boundary hyperbolic problem. Since the free boundary is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces $${H^m_*}$$ . The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates. Therefore, we prove our existence theorem by a suitable Nash–Moser-type iteration scheme.
Gui-qiang Chen - One of the best experts on this subject based on the ideXlab platform.
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nonlinear stability of relativistic Vortex Sheets in three dimensional minkowski spacetime
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Paolo Secchi, Gui-qiang Chen, Tao WangAbstract:We are concerned with the nonlinear stability of Vortex Sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic Vortex Sheets is obtained by analyzing the roots of the Lopatinskiĭ determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing the error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic Vortex Sheets under small initial perturbations by a Nash–Moser iteration scheme.
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Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
Archive for Rational Mechanics and Analysis, 2018Co-Authors: Gui-qiang Chen, Paolo Secchi, Tao WangAbstract:We are concerned with the nonlinear stability of Vortex Sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic Vortex Sheets is obtained by analyzing the roots of the Lopatinski\ui determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic Vortex Sheets under small initial perturbations by a Nash--Moser iteration scheme.
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l 1 stability of Vortex Sheets and entropy waves in steady compressible supersonic euler flows over lipschitz walls
arXiv: Analysis of PDEs, 2012Co-Authors: Gui-qiang Chen, Vaibhav KukrejaAbstract:We establish the well-posedness of compressible Vortex Sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls under a $BV$ boundary perturbation. In particular, when the total variation of the incoming flow perturbation around the background strong Vortex sheet/entropy wave is small, we prove that the two-dimensional steady supersonic Euler flows containing a strong Vortex sheet/entropy wave past a Lipschitz wall are $L^{1}$--stable. Both the Lipschitz wall (whose boundary slope function has small total variation) and incoming flow perturb the background strong Vortex sheet/entropy wave. The weak waves are reflected after nonlinear waves interact with the strong Vortex sheet/entropy wave and the wall boundary. Using the wave-front tracking method, the existence of solutions in $BV$ over Lipschitz walls is first shown, when the incoming flow perturbation of the background strong Vortex sheet/entropy wave has small total variation. Then we establish the $L^{1}$--contraction of the solutions with respect to the incoming flows. To achieve this, a Lyapunov functional, equivalent to the $L^{1}$--distance between two solutions containing strong Vortex Sheets/entropy waves, is carefully constructed to include the nonlinear waves generated both by the wall boundary and from the incoming flow. This functional is then shown to decrease in the flow direction, leading to the $L^{1}$--stability, as well as the uniqueness, of the solutions. Furthermore, the uniqueness of solutions extends to a larger class of viscosity solutions.
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Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics
Archive for Rational Mechanics and Analysis, 2007Co-Authors: Gui-qiang Chen, Ya-guang WangAbstract:Compressible Vortex Sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible Vortex Sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible Vortex Sheets predicts stability when the Mach number \(M > \sqrt{2}\) and instability when \(M \sqrt{2}\) . For the Euler equations in three dimensions, every compressible Vortex sheet is violently unstable and this instability is the analogue of the Kelvin–Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible Vortex Sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-Vortex Sheets in MHD does not meet the uniform Kreiss–Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-Vortex Sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash–Moser–Hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-Vortex Sheets, locally in time, in three-dimensional MHD.
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Stability of compressible Vortex Sheets in steady supersonic euler flows over lipschitz walls
SIAM Journal on Mathematical Analysis, 2007Co-Authors: Gui-qiang Chen, Yongqian Zhang, Dianwen ZhuAbstract:We are concerned with the stability of compressible Vortex Sheets in two‐dimensional steady supersonic Euler flows over Lipschitz walls under a $BV$ boundary perturbation, since steady supersonic Euler flows are important in many physical situations. It is proved that steady compressible Vortex Sheets in supersonic flow are stable in structure globally, even under the $BV$ perturbation of the Lipschitz walls. In order to achieve this, we develop a modified Glimm difference scheme and identify a Glimm‐type functional to obtain the required $BV$ estimates by incorporating the Lipschitz boundary and the strong Vortex Sheets naturally and by tracing the interaction not only between the boundary and weak waves but also between the strong Vortex Sheets and weak waves. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution and the corresponding approximate strong Vortex Sheets to a strong compressible Vortex sheet of the entropy solution. The asym...
Tao Wang - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear stability and existence of compressible Vortex Sheets in 2D elastodynamics
Journal of Differential Equations, 2020Co-Authors: Robin Ming Chen, Tao Wang, Dehua Wang, Difan YuanAbstract:Abstract The nonlinear stability and local existence of compressible Vortex Sheets for the two-dimensional isentropic elastic fluid are established in the usual Sobolev spaces. The problem has a characteristic free boundary, and the Kreiss–Lopatinskiĭ condition is satisfied only in a weak form. This paper completes the previous works [6] , [7] of the first three authors where the weakly linear stability of the rectilinear Vortex Sheets is proved by means of an upper triangularization technique. Our proof is based on certain higher-order energy estimates and an appropriate modification of the Nash–Moser iteration. In particular, the estimate for the normal derivatives of the characteristic variables can be recovered from that for the linearized divergences and vorticities.
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nonlinear stability of relativistic Vortex Sheets in three dimensional minkowski spacetime
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Paolo Secchi, Gui-qiang Chen, Tao WangAbstract:We are concerned with the nonlinear stability of Vortex Sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic Vortex Sheets is obtained by analyzing the roots of the Lopatinskiĭ determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing the error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic Vortex Sheets under small initial perturbations by a Nash–Moser iteration scheme.
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Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
Archive for Rational Mechanics and Analysis, 2018Co-Authors: Gui-qiang Chen, Paolo Secchi, Tao WangAbstract:We are concerned with the nonlinear stability of Vortex Sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic Vortex Sheets is obtained by analyzing the roots of the Lopatinski\ui determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic Vortex Sheets under small initial perturbations by a Nash--Moser iteration scheme.
Paolo Secchi - One of the best experts on this subject based on the ideXlab platform.
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anisotropic regularity of linearized compressible Vortex Sheets
Journal of Hyperbolic Differential Equations, 2020Co-Authors: Paolo SecchiAbstract:We consider supersonic Vortex Sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a p...
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Anisotropic regularity of linearized compressible Vortex Sheets
Journal of Hyperbolic Differential Equations, 2020Co-Authors: Paolo SecchiAbstract:We consider supersonic Vortex Sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a pseudo-differential equation that describes the time evolution of the discontinuity front of the Vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if [Formula: see text], and the well-posedness holds in standard weighted Sobolev spaces. Our aim in this paper is to improve this result, by showing the existence in functional spaces with additional weighted anisotropic regularity in the frequency space.
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Anisotropic regularity of linearized compressible Vortex Sheets
arXiv: Analysis of PDEs, 2020Co-Authors: Paolo SecchiAbstract: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, in [10] the authors have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the Vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[v\cdot\tau]|>2\sqrt{2}\,c$, and the well-posedness was obtained in standard weighted Sobolev spaces. The aim of the present paper is to improve the result of [10], by showing the existence of the solution in function spaces with some additional weighted anisotropic regularity in the frequency space.
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nonlinear stability of relativistic Vortex Sheets in three dimensional minkowski spacetime
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Paolo Secchi, Gui-qiang Chen, Tao WangAbstract:We are concerned with the nonlinear stability of Vortex Sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic Vortex Sheets is obtained by analyzing the roots of the Lopatinskiĭ determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing the error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic Vortex Sheets under small initial perturbations by a Nash–Moser iteration scheme.
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Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
Archive for Rational Mechanics and Analysis, 2018Co-Authors: Gui-qiang Chen, Paolo Secchi, Tao WangAbstract:We are concerned with the nonlinear stability of Vortex Sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic Vortex Sheets is obtained by analyzing the roots of the Lopatinski\ui determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic Vortex Sheets under small initial perturbations by a Nash--Moser iteration scheme.
Heinrich Freistuhler - One of the best experts on this subject based on the ideXlab platform.
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symmetrizations of rmhd equations and stability of relativistic current Vortex Sheets
Classical and Quantum Gravity, 2013Co-Authors: Heinrich Freistuhler, Yuri TrakhininAbstract:We consider the equations of relativistic magnetohydrodynamics (RMHD) in the case of special relativity. Starting by computations in the fluid's rest frame and then applying Lorentz transformations, we derive a covariant symmetric formulation of RMHD in terms of the primitive (physical) variables. This symmetric system is important for the study of various initial boundary value problems. We also find a so-called secondary symmetrization whose direct consequence is the extension of the sufficient stability condition obtained earlier for non-relativistic planar current–Vortex Sheets to the relativistic case. As in non-relativistic settings, this implies the local-in-time existence of corresponding smooth nonplanar current–Vortex Sheets.
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symmetrizations of rmhd equations and stability of relativistic current Vortex Sheets
arXiv: Analysis of PDEs, 2012Co-Authors: Heinrich Freistuhler, Yuri TrakhininAbstract:We consider the equations of relativistic magnetohydrodynamics (RMHD) in the case of special relativity. For the fluid rest frame a nonconservative reformulation of the RMHD equations gives a symmetric system for the vector of primitive (physical) variables. By applying the Lorentz transformation to this system we find a concrete form of symmetric matrices in the LAB-frame. The resulting symmetric system in terms of primitive variables is important for the study of various initial boundary value problems for the RMHD equations. We also find a so-called secondary symmetrization whose direct consequence is the extension of the sufficient stability condition obtained earlier for non-relativistic planar current-Vortex Sheets to the relativistic case. As in non-relativistic settings, this implies the local-in-time existence of corresponding smooth nonplanar current-Vortex Sheets.
