Viscous approximation of triangular system in 1-d with nonlinear viscosity

TL;DR

This work addresses the vanishing viscosity limit for a $2\times2$ triangular hyperbolic system with nonlinear viscosity that commutes with the convective part. It extends the BV-estimates and traveling-wave decomposition framework of Bianchini–Bressan to nonlinear, commuting viscosity matrices, proving global existence of a smooth viscous solution with uniform total variation and establishing convergence to the hyperbolic limit as $\varepsilon\to0$ under small $BV$ data. The approach hinges on a gradient decomposition into viscous traveling waves, new variables to handle cross-terms, parabolic regularization, and intricate interaction-energy estimates (including length functionals and transversal interactions) to control nonlinear effects. The results contribute to the theoretical understanding of viscous regularization and selection of physically relevant solutions in non-Temple-class systems, with implications for applications in chromatography and porous-media flows.

Abstract

We study the vanishing viscosity limit for $2\times2$ triangular system of hyperbolic conservation laws when the viscosity coefficients are non linear. In this article, we assume that the viscosity matrix $B(u)$ is commutating with the convective part $A(u)$. We show the existence of global smooth solution to the parabolic equation satisfying uniform total variation bound in $\varepsilon$ provided that the initial data is small in $BV$. This extends the previous result of Bianchini and Bressan [Commun. Pure Appl. Anal. (2002)] which was considering the case $B(u)=I$.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Proposition 2.3
• Remark 3.1
• Remark 3.2