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

TL;DR

The paper develops a novel entropy-solution formula for one-dimensional scalar hyperbolic conservation laws with fluxes that may degenerate in convexity and initial data in $L^\infty$, extending the Lax–Oleinik framework to nonuniformly convex flux. The approach yields a comprehensive global dynamic description of entropy solutions, including six initial-wave criteria, backward characteristic triangles, shock-generation points with precise regularity, four invariants and divides with exact locations and speeds, and sharp asymptotics in both $L^\infty$ and $L^p_{\rm loc}$, along with generalized $N$-waves. It also characterizes lifespans and termination of characteristics, the formation and development of shocks for five continuous shock-generation-point types, and directional limits at discontinuities, with extensions to more general scalar laws $U(u)_t+F(u)_x=0$. Overall, the results provide a unified, technically robust framework for understanding entropy solutions under convexity degeneracy and offer precise criteria, invariants, and asymptotics that enhance both theory and potential applications.

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.

Figures (24)

• Figure 2.1: $u^\pm(x,t)\in\mathcal{U}(x,t)\subset[u^+(x,t),u^-(x,t)]$.
• Figure 2.8: The shock curve $x(t)$ for the case that $t_p^+\neq t_p^-$. The left/right figure illustrates the case of shocks formatted by the compression of local characteristics emitting from points on the left/right of $x_0$.
• Figure 2.9: ${\rm Con}(x,t,u)={\rm Con}(x',t',\hat{u})$ as in \ref{['con1']}.
• Figure 3.1: $a=\overline{{\rm D}}_- \Phi(x_0)\geq\underline{{\rm D}}_+ \Phi(x_0)=b$.
• Figure 3.2: The positional relations of $\Phi(\cdot\,;x_0,c)$ and $F(\cdot\,;t,c)$ for $t_p^\pm\in (0,\infty)$.

Theorems & Definitions (122)

• Definition 2.1: Weak Solutions
• Definition 2.2: Entropy Solutions
• Definition 2.3
• Theorem 2.4: New formula for entropy solutions
• Remark 2.1
• Corollary 2.5
• Remark 2.2
• Corollary 2.6: Refined semigroup properties of entropy solutions determined by the new solution formula
• Remark 2.3
• Lemma 2.7