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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 $u_t+A(u)u_x=0$, $t\ge 0$, $x\in {\Bbb R}$, 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:
35L45First order hyperbolic systems, initial value problems
35L65Conservation laws
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B45A priori estimates for solutions of PDE
35K45Systems of second-order parabolic equations, initial value problems
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