Abstract
We prove nonlinear stability in L 1 of planar shock front solutions to a viscous conservation law in two spatial dimensions and obtain an expression for the asymptotic form of small perturbations. The leading-order behavior is shown rigorously to be governed by an effective diffusion coefficient depending on forces transverse to the shock front. The proof is based on a spectral analysis of the linearized problem.
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Goodman, J., Miller, J.R. Long-Time Behavior of Scalar Viscous Shock Fronts in Two Dimensions. Journal of Dynamics and Differential Equations 11, 255–277 (1999). https://doi.org/10.1023/A:1021977329306
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DOI: https://doi.org/10.1023/A:1021977329306