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

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.