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Di Perna, Ronald J.; Majda, Andrew

The validity of nonlinear geometric optics for weak solutions of conservation laws. (English) Zbl 0582.35081

Commun. Math. Phys. 98, No. 3, 313-347 (1985).
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

Cited in 26 Documents

MSC:

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

Keywords:

nonlinear geometric optics; weak solutions; nonlinear hyperbolic system; conservation laws; equation of Burgers’ type; initial data; compact support; bounded variation; periodic initial data
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\textit{R. J. Di Perna} and \textit{A. Majda}, Commun. Math. Phys. 98, No. 3, 313--347 (1985; Zbl 0582.35081)
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References:

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