Necessary and sufficient conditions for $a$-contraction
Cooper Faile
TL;DR
The paper addresses when small shocks of general hyperbolic systems with a single convex entropy are local attractors under $a$-contraction with shifts. It develops a precise limit computation for the Hessian of the maximal-dissipation functional $D_{max}$ as the shock size $s\to0$, yielding a negative-definiteness condition on a subspace spanned by eigenvectors of the flux Jacobian. This leads to necessary and sufficient conditions, depending on the sign of a matrix involving $\nabla^2\eta$, $f'$, and $f''$, that determine whether small interior shocks can be stabilized via $a$-contraction; these conditions are shown to hold for certain intermediate-family systems and are tested in an MHD application. The results extend the prior theory, which covered extremal shocks and some rich systems, by establishing local-attractor results for interior families in a generic setting, and they demonstrate both possibilities and limits of $a$-contraction in physically relevant models.
Abstract
In this paper we investigate the theory of $a$-contraction with shifts with the intention of extending it to intermediate families. The theory of $a$-contraction with shifts is used to prove orbital $L^2$ stability to shock solutions of conservation laws. In this setting there are strong results for scalar laws and the extremal families of $n\times n$ systems of conservation laws. The only known results showing contraction of interior families are for the contact family the full Euler system and the case of rich systems due to Serre and Vasseur '16. This investigation culminates in finding necessary and sufficient conditions for which small shocks of general systems are local attractors with respect to the $a$-contraction theory.