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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
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