| Popis: |
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 $\epsilon$. The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale $\epsilon$-dependent topology reduces to the classical $W^{1,\infty}$-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. |
