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The validity of nonlinear geometric optics for weak solutions of conservation laws. (English) Zbl 0582.35081
It is shown that the solution u(x,t) to the genuinely nonlinear hyperbolic system of n conservation laws $u_ t+f(u)_ x=0,\quad u(x,0)=u_ 0+\epsilon v(x)$ in one space dimension satisfies $u(x,t)=u_ 0+\epsilon \sum^{n}_{j=1}\sigma^ j(\phi^ j,\tau)r_ j(u_ 0)+O(\epsilon^ 2)\quad for\quad \epsilon \to 0,$ where $$r_ j$$ are right eigenvectors of the matrix $$f'(u_ 0)$$, and the scalar functions $$\sigma^ j$$ are solutions of an equation of Burgers’ type. It is shown that for initial data v with compact support and bounded variation the terms $$O(\epsilon^ 2)$$ is uniformly small for all $$t>0$$ in the $$L_ 1$$-norm. Similar estimates are given for periodic initial data v.
Reviewer: H.-D.Alber

##### MSC:
 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems 35A35 Theoretical approximation in context of PDEs 78A05 Geometric optics
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##### References:
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