On Kolmogorov entropy compactness estimates for scalar conservation laws without uniform convexity

TL;DR

This paper extends Kolmogorov entropy compactness estimates for scalar conservation laws to cases with non-uniform convexity or a single inflection point, broadening understanding of solution regularity and complexity.

Contribution

It provides new quantitative compactness estimates for scalar conservation laws with less restrictive convexity conditions on the flux function.

Findings

Extended entropy estimates to non-uniform convex flux functions.

Quantified the solution set complexity for broader flux conditions.

Revealed irreversibility features in more general scalar conservation laws.

Abstract

In the case of scalar conservation laws $$ u_{t} + f(u)_{x}~=~0,\qquad t\geq 0, x\in\mathbb{R}, $$ with uniformly strictly convex flux $f$, quantitative compactness estimates - in terms of Kolmogorov entropy in ${\bf L}^{1}_{loc}$ - were established in~\cite{DLG,AON1} for sets of entropy weak solutions evaluated at a fixed time $t>0$, whose initial data have a uniformly bounded support and vary in a bounded subset of ${\bf L}^\infty$. These estimates reflect the irreversibility features of entropy weak discontinuous solutions of these nonlinear equations. We provide here an extension of such estimates to the case of scalar conservation laws with a smooth flux function $f$ that either is strictly (but not necessarily uniformly) convex or has a single inflection point with a polynomial degeneracy.