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

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dc.creatorSpinolo, Laura V.
dc.date2005-08-08
dc.date2005-11-11
dc.date.accessioned2026-07-07T06:42:45Z
dc.date.available2026-07-07T06:42:45Z
dc.descriptionWe consider the $2 \times 2$ parabolic systems \begin{equation*} u^ε_t + A(u^ε) u^ε_x = εu^ε_{xx} \end{equation*} on a domain $(t, x) \in ]0, + \infty[ \times ]0, l[$ 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 $t \geq 0$ and depends Lipschitz continuously in $L^1$ on the initial and boundary data. Moreover, as $ε\to 0^+$, the solutions $u^ε(t)$ converge in $L^1$ to a unique limit $u(t)$, which can be seen as the vanishing viscosity solution of the quasilinear hyperbolic system \begin{equation*} u_t + A(u)u_x = 0, \quad x \in ]0, l[. \end{equation*} This solution $u(t)$ depends Lipschitz continuously in $L^1$ 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. 2000 Mathematics Subject Classification: 35L65. Key words: Hyperbolic systems, conservation laws, initial boundary value problems, viscous approximations.
dc.description56 pages, 3 figures, added references and two Remarks
dc.identifierhttps://arxiv.org/abs/math/0508142
dc.identifierhttp://arxiv.org/abs/math/0508142
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/102159
dc.subjectAnalysis of PDEs
dc.subject35L65
dc.titleVanishing viscosity solutions of a $2 \times 2$ triangular hyperbolic system with Dirichlet conditions on two boundaries
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