Abstract:We consider the 2×2 parabolic systemson a domain (u^{ε}_{t}+A(u^{ε})u^{ε}_{x}=εu^{ε}_{xx}t,x) ∈ ]0,+∞[ × ]0,l[ with Dirichlet boundary conditions imposed atx=0 and atx=l. The matrixAis assumed to be in triangular form and strictly hyperbolic, and the boundary is not characteristic, i.e. the eigenvalues ofAare different from 0. We show that, if the initial and boundary data have sufficiently small total variation, then the solution u^{ε}exists for allt≥0 and depends Lipschitz continuously inL^{1}on the initial and boundary data. Moreover, as ε→0^{+}, the solutionsu^{ε}(t) converge inL^{1}to a unique limitu(t), which can be seen as thevanishing viscosity solutionof the quasilinear hyperbolic systemThis solutionu_{t}+A(u)u_{x}=0,x∈]0,l[.u(t) depends Lipschitz continuously inL^{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.

**Paper:**- Available from arXiv.
**Author(s):**- Laura V. Spinolo, <spinolo@sissa.it>
**Publishing information:****Comments:****Submitted by:**- <spinolo@sissa.it> August 9 2005.

[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | All Preprints | Preprint Server Homepage ]

Conservation Laws Preprint Server <conservation@math.ntnu.no> 2005-08-12 08:41:48 UTC