Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates
Boris Haspot, Animesh Jana
TL;DR
The paper analyzes the vanishing viscosity limit for a 1-D strictly hyperbolic system with a nonlinear, non-singular viscosity matrix $B(u)$ that commutes with the flux Jacobian $A(u)$. It proves global-in-time uniform BV bounds for solutions to the parabolic regularization $u_t+A(u)u_x=\varepsilon(B(u)u_x)_x$ starting from small BV data, by developing a two-step strategy: a parabolic smoothing phase up to a short time $\hat t$, followed by a global BV bound built from intricate transversal interaction estimates. A key innovation is the gradient decomposition of $u_x$ into a carefully crafted traveling-wave basis $\tilde r_i(u,v_i,\sigma_i)$ and the introduction of effective fluxes $z_i$ and $\hat z_i$, which allow control of cross-family and higher-order terms through a hierarchy of remainder estimates and Lyapunov-type functionals. The paper also derives closed equations for the effective fluxes and establishes a BV bound that is uniform in $\varepsilon$, enabling convergence results to a Liu-admissible weak solution in the conservative case and validating the vanishing viscosity limit. Together, these results extend prior BV-vanishing-viscosity theories to non-constant viscosity matrices under commutativity, providing a framework for selecting physically relevant solutions in non-conservative hyperbolic systems.
Abstract
We consider the following parabolic approximation for hyperbolic system of conservation laws in 1-D with non-singular viscosity matrix $B(u)$ and $A(u)$ strictly hyperbolic, \[u_t+A(u)u_x = \varepsilon(B(u)u_x)_x.\] We prove global in time uniform $BV$ bound for solution to this parabolic system when $\varepsilon>0$ provided that the initial data is small in $BV$ and the matrix $A(u)$ and $B(u)$ commutate.