High frequency asymptotics for 2d viscous shocks
2000 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 49, no 4, 1623-1671 p.Article in journal (Refereed) Published
High frequency perturbations u of a weak planar two dimensional isentropic slightly viscous shock are shown to give a geometric optics expansion u = e(if/epsilon)(u(0) + epsilonu(1)), where the gradients of the phase functions delf solve eikonal equations related to Riemann problems for classical and over-compressive shocks. The main result is a rigorous expansion of the linearized equations describing in detail the perturbations and their refraction patterns caused by the viscous shock, when the wave length epsilon is large compared to the width of the shock. The expansion is based on the phases constructed from heteroclinic orbits related to the Riemann problems and on the principal term with L-2-norm //e(if)/(epsilon)u(0)//(L2) = O(1) obtained from the transport equations generated by the phases. The estimate of the remainder, //e(if/epsilon)epsilonu(1)//(L2) much less than 1, holds up to time 1 for sufficiently weak shocks and is proven by weighted energy estimates based on shock compressibility and relaxation techniques.
Place, publisher, year, edition, pages
2000. Vol. 49, no 4, 1623-1671 p.
nonlinear hyperbolic waves, conservation-laws, diffusion waves, stability, systems
Computer and Information Science
IdentifiersURN: urn:nbn:se:kth:diva-20768ISI: 000169744400015OAI: oai:DiVA.org:kth-20768DiVA: diva2:339465
QC 201005252010-08-102010-08-102012-02-27Bibliographically approved