Differential and Integral Equations

On the stability of viscous shock fronts for certain conservation laws in two-dimensional space

Masataka Nishikawa

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Abstract

We prove nonlinear stability of planar shock front solutions for certain viscous scalar conservation laws in two space dimensions. Let us admit the planar shock front solution in x-direction, where the flux function is only in the x-direction and its convexity is not necessarily assumed. For either the nondegenerate or degenerate cases, if the initial disturbance is sufficiently small, then the solution approaches to the shifted planar shock front solution as t$ \rightarrow \infty$. Here, the shift function may have different asymptotic states in $y$-direction. The proofs are given by applying an elementary weighted energy method to the ``integrated equation" which is equivalent to the original one.

Article information

Source
Differential Integral Equations, Volume 10, Number 6 (1997), 1181-1195.

Dates
First available in Project Euclid: 1 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367438228

Mathematical Reviews number (MathSciNet)
MR1608065

Zentralblatt MATH identifier
0940.35132

Subjects
Primary: 35L67: Shocks and singularities [See also 58Kxx, 76L05]
Secondary: 76L05: Shock waves and blast waves [See also 35L67]

Citation

Nishikawa, Masataka. On the stability of viscous shock fronts for certain conservation laws in two-dimensional space. Differential Integral Equations 10 (1997), no. 6, 1181--1195. https://projecteuclid.org/euclid.die/1367438228


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