## Viscous limits for piecewise smooth solutions to systems of conservation laws.(English)Zbl 0792.35115

Let $$u_ 1 + f(u)_ x$$ be a strictly hyperbolic system of conservation laws (one space dimension) and $$u^ \varepsilon_ t + f(u^ \varepsilon)_ x = eu^ \varepsilon_{xx}$$ the related viscous system. For scalar laws it is well known that weak entropic solutions $$u$$ of the first system may be recovered as strong limits for $$\varepsilon \to 0$$ of solutions $$u^ \varepsilon$$ of the second; this is also true for some special $$2 \times 2$$ systems. In this paper the authors prove that the same holds true for general $$n \times n$$ systems, in the case that the solution $$u$$ has only a finite number of noninteracting entropic shocks. More precisely they prove that $\sup_{0 \leq t \leq T} \int | u(x,t) - u^ \varepsilon(x,t)| dx \leq C_ \eta\varepsilon^ \eta$ for any $$\eta \in (0,1)$$; outside the shock curves the error is $$O(1)\varepsilon$$. The proof matches two different asymptotic expansions, one near and the other away from the shock curves; these expansions are justified through an energy estimate related to the stability theory for viscous shock profiles [T. Liu, Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 328 (1985; Zbl 0617.35058)].
Reviewer: A.Corli (Ferrara)

### MSC:

 35L65 Hyperbolic conservation laws 35L67 Shocks and singularities for hyperbolic equations 35C20 Asymptotic expansions of solutions to PDEs

Zbl 0617.35058
Full Text:

### References:

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