Preprint 2005-032

Vanishing viscosity solutions of a 2×2 triangular hyperbolic system with Dirichlet conditions on two boundaries

Abstract: We consider the 2×2 parabolic systems

u^ε_t+A(u^ε)u^ε_x=εu^ε_xx

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 u^ε exists for all t≥0 and depends Lipschitz continuously in L¹ on the initial and boundary data. Moreover, as ε→0⁺, the solutions u^ε(t) converge in L¹ to a unique limit u(t), which can be seen as the vanishing viscosity solution of the quasilinear hyperbolic system

u_t+A(u)u_x=0,  x∈]0,l[.

This solution u(t) depends Lipschitz continuously in L¹ 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.

Paper:

Available from arXiv.

Author(s):

Laura V. Spinolo, <spinolo@sissa.it>

Publishing information:

Comments:

Submitted by:

<spinolo@sissa.it> August 9 2005.