IUMJ

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

Authors: Laura V. Spinolo

Issue: Volume 56 (2007), Issue 1, 279-364

Abstract:

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.