Shock waves in conservation laws with physical viscosity

We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to...

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Hlavní autoři:	Liu, Tai-Ping, Zeng, Yanni
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Providence, Rhode Island American Mathematical Society 2015
Vydání:	1
Edice:	Memoirs of the American Mathematical Society
Témata:	Conservation laws (Mathematics) Green''s functions Mathematics Shock waves Shock waves > Mathematics Conservation laws Green’s function nonlinear stability wave interactions compressible Navier-Stokes equations magneto-hydrodynamics physical viscosity large time behavior pointwise estimates quasilinear hyperbolic-parabolic systems shock waves
ISBN:	9781470410162, 1470410168
ISSN:	0065-9266, 1947-6221
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Abstract	We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small, but independent. Our assumptions on the viscosity matrix are general so that our results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. Our analysis depends on accurate construction of an approximate Green’s function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that we can close the nonlinear term through the Duhamel’s principle.
AbstractList	We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small, but independent. Our assumptions on the viscosity matrix are general so that our results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. Our analysis depends on accurate construction of an approximate Green’s function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that we can close the nonlinear term through the Duhamel’s principle. The authors study the perturbation of a shock wave in conservation laws with physical viscosity. They obtain the detailed pointwise estimates of the solutions. In particular, they show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small but independent. The authors' assumptions on the viscosity matrix are general so that their results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. The authors' analysis depends on accurate construction of an approximate Green's function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that the author can close the nonlinear term through Duhamel's principle.
Author	Liu, Tai-Ping Zeng, Yanni
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Keywords	Conservation laws Green’s function nonlinear stability wave interactions compressible Navier-Stokes equations magneto-hydrodynamics physical viscosity large time behavior pointwise estimates quasilinear hyperbolic-parabolic systems shock waves
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Notes	Includes bibliographical references (p. 167-168) Volume 234, number 1105 (fifth of 5 numbers), March 2015
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Snippet	We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In... The authors study the perturbation of a shock wave in conservation laws with physical viscosity. They obtain the detailed pointwise estimates of the solutions....
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SubjectTerms	Conservation laws (Mathematics) Green''s functions Mathematics Shock waves Shock waves -- Mathematics
TableOfContents	Introduction -- Preliminaries -- Green’s functions for Systems with Constant Coefficients -- Green’s Function for Systems Linearized Along Shock Profiles -- Estimates on Green’s Function -- Estimates on Crossing of Initial Layer -- Estimates on Truncation Error -- Energy Type Estimates -- Wave Interaction -- Stability Analysis -- Application to Magnetohydrodynamics Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminaries -- Chapter 3. Green's functions for Systems with Constant Coefficients -- Chapter 4. Green's Function for Systems Linearized Along Shock Profiles -- Chapter 5. Estimates on Green's Function -- Chapter 6. Estimates on Crossing of Initial Layer -- Chapter 7. Estimates on Truncation Error -- Chapter 8. Energy Type Estimates -- Chapter 9. Wave Interaction -- Chapter 10. Stability Analysis -- Chapter 11. Application to Magnetohydrodynamics -- References -- Back Cover
Title	Shock waves in conservation laws with physical viscosity
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Volume	234
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