Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates

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, $$u_t^{\epsilon }+A(u^{\epsilon })u_x^{\epsilon }=\epsilon (B(u^{\epsilon })u_x^{\epsilon }{)}_x\mathrm{.}$$ We prove global in time uniform $BV$ bound for solution to this parabolic system when $\epsilon >0$ provided that the initial data is small in $BV$ and the matrix $A(u)$ and $B(u)$ commutate. Moreover, in the case where the system is conservative, we show that the sequence $(u^{\epsilon }{)}_{\epsilon >0}$ admits a limit $u$, which is the unique global weak solution to the limiting strictly hyperbolic system. We provide a concrete application of this result in the study of the visco-dispersive limit of the Navier-Stokes-Korteweg system.