Uniform asymptotic stability for convection-reaction-diffusion equations in the inviscid limit towards Riemann shocks

DOI10.4171/AIHPC/90arXiv2201.13436OpenAlexW4315589423MaRDI QIDQ6389757

Publication date: 31 January 2022

Abstract: The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter

e p s i l o n

. The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale

e p s i l o n

-dependent topology reduces to the classical

W^{1, i n f t y}

-topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine-Hugoniot conditions and the non-uniform shift arising merely from phasing out the non-decaying

0

-mode, as in the classical stability analysis for fronts of reaction-diffusion equations.

Full work available at URL: https://doi.org/10.4171/aihpc/90

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Shocks and singularities for hyperbolic equations (35L67) Stability in context of PDEs (35B35) Singular perturbations in context of PDEs (35B25) Stability problems for infinite-dimensional dissipative dynamical systems (37L15) Initial value problems for second-order parabolic equations (35K15) Semilinear parabolic equations (35K58)