Communications on Pure and Applied Mathematics · 2003 · 40 citations · 18 references
Natural SciencesMathematicsPhysicsDifferential EquationsFluid DynamicsFluid MechanicsInviscid FlowNavier-stokes EquationsPartial Differential EquationsShock WavesViscous FlowWeak FormulationConjugationConservation LawsHyperbolic Partial Differential EquationsParabolic Partial Differential EquationsPseudodifferential OperatorsShock DynamicsShock Wave StructureShocks In Inviscid Flow
Abstract In this paper we prove the existence of curved, multidimensional viscous shocks and also justify the small‐viscosity limit. Starting with a curved, multidimensional (inviscid) shock solution to a system of hyperbolic conservation laws, we show that the shock can be obtained as a small‐viscosity limit of solutions to an associated parabolic problem (viscous shocks). The two main hypotheses are a natural Evans function assumption on the viscous profile, together with a restriction on how much the shock can deviate from flatness. The main tools are a conjugation lemma that removes x N / ϵ dependence from the linearization of the parabolic problem about the viscous profile, new degenerate Kreiss‐type symmetrizers used to prove an L 2 estimate for the linearized problem, and a finite‐regularity calculus of semiclassical and mixed type (classical‐semiclassical) pseudodifferential operators. © 2003 Wiley Periodicals, Inc.
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