Abstract: We consider a special $2\times 2$ viscous hyperbolic system of conservation laws of the form $u_t+A(u)u_x=\ve u_{xx}$, where $A(u)=Df(u)$ is the Jacobian of some flux function $f$. For initial data with small total variation, we prove that the corresponding solutions satisfy a uniform BV bound, independent of $\ve$. Letting $\ve\to 0$, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system $u_t+f(u)_x=0$. Within the proof, we introduce two new Lyapunov functional which control the interaction of viscous waves of the same family. This provides an example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.
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