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Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. (English) Zbl 0935.35101

In this long paper the author studies the solution \(u^0\) of a strictly hyperbolic conservation law \(u^0_t + f(u^0)_x = 0\) with \(u^0\in \mathbb{R}^n\), \(x\in \mathbb{R}\) and its viscous approximation \(u^\varepsilon_t+ f(u^\varepsilon)_x = \varepsilon u^\varepsilon_{xx}\). The initial condition is the same for both equations, and it is assumed that it contains exactly one shock discontinuity of sufficiently weak shock strength \(\delta\). Assuming also that for \(t\leq T\) the solution \(u^0\) develops no other discontinuities, the main result is a pointwise estimate for the difference \(|u^0(x,t) - u^\varepsilon(x,t)|\) for \(\varepsilon\) small and some time range depending on \(\delta\) and \(\varepsilon\).
The estimates are derived by studying the features that distinguish the viscous conservation law from the hyperbolic one:
(i) Smooth shock profiles appear instead of discontinuities;
(ii) nonlinear diffusion waves are present;
(iii) interaction between the shock layer and the diffusion waves generates higher-order coupling waves.
Although the detailed analysis requires a lot of notation and lengthy calculation, the author tries to motivate the approach first before he jumps into the details.
Reviewer: J.Härterich

MSC:

35L65 Hyperbolic conservation laws
35B25 Singular perturbations in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
35B45 A priori estimates in context of PDEs
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