New Formula for Entropy Solutions for Scalar Hyperbolic Conservation Laws with Flux Functions of Convexity Degeneracy and Global Dynamic Patterns of Solutions

TL;DR

This paper introduces a novel formula for entropy solutions of scalar hyperbolic conservation laws with degenerate convex flux functions, revealing new structural, invariant, and asymptotic properties of solutions.

Contribution

It generalizes the Lax-Oleinik formula and uncovers new solution structures, invariants, and asymptotic behaviors for entropy solutions with degenerate convex flux functions.

Findings

New criteria for all six types of initial waves.

Identification of four new invariants of entropy solutions.

Exact determination of global solution structures.

Abstract

We are concerned with a new solution formula and its applications to the analysis of properties of entropy solutions of the Cauchy problem for one-dimensional scalar hyperbolic conservation laws, wherein the flux functions exhibit convexity degeneracy and the initial data are in $L^\infty$. We first introduce/validate the novel formula for entropy solutions for the Cauchy problem, which generalizes the Lax-Oleinik formula. Then, by employing this formula, we obtain a series of fine properties of entropy solutions and discover several new structures and phenomena, which include: (i) Series of results on the fine structures of entropy solutions, especially including the new criteria for all six types of initial waves for the Cauchy problem, the new structures of entropy solutions inside the backward characteristic triangle, and the new features of the formation and development of shocks such as all five types of continuous shock generation points, along with their criteria and the optimal regularities of the corresponding resulting shocks; (ii) Series of results on the global structures of entropy solutions, including the four new invariants of entropy solutions, the new criteria for the locations and speeds of divides, and the exact determination of the global structures of entropy solutions; (iii) Series of new results on the asymptotic behaviors of entropy solutions, including the asymptotic profiles and decay rates of entropy solutions for initial data in $L^\infty$, respectively in the $L^\infty$--norm and the $L^p_{{\rm loc}}$--norm. Through these results above, we obtain the global dynamic patterns of entropy solutions for scalar hyperbolic conservation laws with the flux functions satisfying (1.3) and general initial data in $L^\infty$. Moreover, the new solution formula is also extended to more general scalar hyperbolic conservation laws.