Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

TL;DR

This paper develops a new analytical approach to validate weakly nonlinear geometric optics for hyperbolic conservation laws, demonstrating $O( ext{epsilon}^2)$ accuracy for entropy solutions with small BV initial data, including non-compact support cases.

Contribution

The authors introduce a novel method combining auxiliary approximations, wave front-tracking, and semigroup estimates to rigorously validate geometric optics expansions for hyperbolic systems.

Findings

$L^1$-estimate between entropy solution and geometric optics expansion is $O( ext{epsilon}^2)$

Validation holds for both compact and non-compact support initial data

Approach applies to systems with linearly degenerate characteristic fields

Abstract

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the $L^1-$estimate between the entropy solution and the geometric optics expansion function is bounded by $O(\varepsilon^2)$, {\it independent of} the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.