Vanishing viscosity solutions of a $2 \times 2$ triangular hyperbolic system with Dirichlet conditions on two boundaries

We consider the $2 \times 2$ parabolic systems \begin{equation*} u^{\epsilon}_t + A(u^{\epsilon}) u^{\epsilon}_x = \epsilon u^{\epsilon}_{xx} \end{equation*} on a domain $(t, x) \in ]0, + \infty[ \times ]0, l[$ with Dirichlet boundary conditions imposed at $x=0$ and at $x=l$. The matrix $A$ is assumed to be in triangular form and strictly hyperbolic, and the boundary is not characteristic, i.e. the eigenvalues of $A$ are different from 0. We show that, if the initial and boundary data have sufficiently small total variation, then the solution $u^{\epsilon}$ exists for all $t \geq 0$ and depends Lipschitz continuously in $L^1$ on the initial and boundary data. Moreover, as $\epsilon \to 0^+$, the solutions $u^{\epsilon}(t)$ converge in $L^1$ to a unique limit $u(t)$, which can be seen as the vanishing viscosity solution of the quasilinear hyperbolic system \begin{equation*} u_t + A(u)u_x = 0, \quad x \in ]0, l[. \end{equation*} This solution $u(t)$ depends Lipschitz continuously in $L^1$ w.r.t the initial and boundary data. We also characterize precisely in which sense the boundary data are assumed by the solution of the hyperbolic system. 2000 Mathematics Subject Classification: 35L65. Key words: Hyperbolic systems, conservation laws, initial boundary value problems, viscous approximations.