Vanishing viscosity solutions of a 2 \times 2 triangular hyperbolic system with Dirichlet conditions on two boundaries Laura Spinolo 35L65hyperbolic systemsconservation lawsinitial boundary value problemsviscous approximations We consider the $2 \times 2$ parabolic systems \[ u^{\varepsilon}_t + A(u^{\varepsilon}) u^{\varepsilon}_x = \varepsilon u^{\varepsilon}_{xx} \] on a domain $(t,x) \in \left] 0, +\infty \right[ \times \left] 0,l \right[$ 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 $0$.\par We show that, if the initial and boundary data have sufficiently small total variation, then the solution $u^{\varepsilon}$ exists for all $t \geq 0$ and depends Lipschitz continuously in $L^1$ on the initial and boundary data.\par Moreover, as $\varepsilon \to 0^{+}$, the solutions $u^{\varepsilon}(t)$ converge in $L^1$ to a unique limit $u(t)$, which can be seen as the \emph{vanishing viscosity solution} of the quasilinear hyperbolic system \[ u_t + A(u)u_x = 0, \quad x \in \left] 0,l \right[. \] This solution $u(t)$ depends Lipschitz continuously in $L^1$ with respect to the initial and boundary data. We also characterize precisely in which sense the boundary data are assumed by the solution of the hyperbolic system. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.2843 10.1512/iumj.2007.56.2843 en Indiana Univ. Math. J. 56 (2007) 279 - 364 state-of-the-art mathematics http://iumj.org/access/