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Vanishing viscosity solutions of nonlinear hyperbolic systems. (English) Zbl 1082.35095
The authors study the Cauchy problem for a generally nonconservative strictly hyperbolic \(n\times n\)-system \(u_t+A(u)u_x=0\), \(t\geq 0\), \(x\in {\mathbb R}\), assuming that the initial data have small total variation. They show that the solutions of the viscous approximations \(u_t+A(u)u_x=\varepsilon u_{xx}\) are defined globally in time and satisfy uniform \(BV\) estimates, independent of \(\varepsilon\). Moreover, these viscous approximations depend continuously on the initial data in the \(L^1\) metric with a Lipschitz constant independent of \(t,\varepsilon\), and converge as \(\varepsilon\to 0\) to a unique limit, depending Lipschitz continuously on the initial data. The limit vector can be regarded as the unique viscosity solution of the hyperbolic Cauchy problem. In the conservative case when \(A(u)=df(u)\), every viscosity solution is a weak entropy solution of the system of conservation laws \(u_t+f(u)_x=0\). Under the additional assumption that the characteristic fields are either genuinely nonlinear or linearly degenerate, the viscosity solutions coincide with the unique limits of Glimm or front-tracking approximations.

35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B45 A priori estimates in context of PDEs
35K45 Initial value problems for second-order parabolic systems
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