Vanishing Viscosity Solutions of a 2 × 2 Triangular Hyperbolic System with Dirichlet Conditions on Two Boundaries

We consider the 2 × 2 parabolic systems ${\mathrm{u}}_{\mathrm{t}}^{\mathrm{\varepsilon}}+\mathrm{A}\left({\mathrm{u}}^{\mathrm{\varepsilon}}\right){\mathrm{u}}_{\mathrm{x}}^{\mathrm{e}}=\mathrm{\varepsilon}{\mathrm{u}}_{\mathrm{x}\mathrm{x}}^{\mathrm{\varepsilon}}$ on a domain (t, x) ∈ ]0, + ∞[ × ]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 uε exists for all t ≥ 0 and depends Lipschitz continuously in L1 on the initial and boundary data. Moreover, as ε → 0+, the solutions uε(t) converge in L1 to a unique limit u(t), which can be seen as the vanishing viscosity solution of the quasilinear hyperbolic system ut + A(u)ux = 0, x ∈ ]0, l[. This solution u(t) depends Lipschitz continuously in L1 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.