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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.

##### MSC:
 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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