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Kwon, Bongsuk
Partial Differential Equations and their applications
Research Interests
  • Partial differential equations, hyperbolic conservation laws, stability of nonlinear waves

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Stability of planar shock fronts for multidimensional systems of relaxation equations

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Title
Stability of planar shock fronts for multidimensional systems of relaxation equations
Author
Kwon, Bongsuk
Keywords
Asymptotic behavior; Hyperbolic conservation laws; Hyperbolic relaxation system; Stability; Traveling waves
Issue Date
2011-10
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Citation
JOURNAL OF DIFFERENTIAL EQUATIONS, v.251, no.8, pp.2226 - 2261
Abstract
We investigate stability of multidimensional planar shock profiles of a general hyperbolic relaxation system whose equilibrium model is a system, under the necessary assumption of spectral stability and a standard set of structural conditions that are known to hold for many physical systems. Our main result, generalizing the work of Kwon and Zumbrun in the scalar relaxation case, is to establish the bounds on the Green's function for the linearized equation and obtain nonlinear L2 asymptotic behavior/sharp decay rate of perturbed weak shock profiles. To establish Green's function bounds, we use the semigroup approach in the low-frequency regime, and use the energy method for the high-frequency bounds, separately. For the system equilibrium case, the analysis of the linearized equation is complicated due to glancing phenomena. We treat this difficulty similarly as in the inviscid and viscous systems, under the constant multiplicity condition.
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DOI
10.1016/j.jde.2011.07.007
ISSN
0022-0396
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PHY_Journal Papers
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