GEOMETRIC OPTICS WITH CRITICAL VANISHING VISCOSITY FOR ONE DIMENSIONAL SEMILINEAR INITIAL VALUE PROBLEM

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GEOMETRIC OPTICS WITH CRITICAL VANISHING VISCOSITY FOR ONE-DIMENSIONAL SEMILINEAR INITIAL VALUE PROBLEM STEPHANE JUNCA Abstract. We study propagations of high frequency oscillations for one dimensional semi-linear hyperbolic system with small parabolic perturbation. We obtain a new de- generate parabolic system for profile, and valid an asymptotic development in a spirit of Joly, Metivier and Rauch. key words: geometric optics, small viscosity, profile, phase, non stationary phase, max- imum principle, energy estimates, interpolation, weakly coupled parabolic systems. 1. Introduction Joly, Metivier and Rauch in [13] give a rigorous and deep description of solutions to one dimensional nonlinear hyperbolic equations with smooth and highly oscillatory initial data. On the other hand, numerous physical problems involve parabolic partial differential equations with small viscosity, for instance Navier-Stokes system with large Reynolds number. Then it is of interest to put the Joly, Metivier, Rauch framework with a small diffusion. The first step to study such multiphase expansions is the semilinear case. So we wish to investigate propagations of high frequency oscillations for following nonlinear parabolic system with highly oscillatory initial data u?0 and small positive viscosity ?: ?? ∂2u? ∂x2 + ∂u? ∂t +A(t, x) ∂u? ∂x = F (t, x, u?)(1.1) u?(0, x) = u?0(x).

initial phase

parabolic system

system

?? ?

nonlinear hyperbolic

real linear

almost everywhere

∂tu ?

there exists

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Publié par

pefav

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GEOMETRIC OPTICS WITH CRITICAL VANISHING VISCOSITY FOR ONE-DIMENSIONAL SEMILINEAR INITIAL VALUE PROBLEM ´ STEPHANE JUNCA

Abstract. We study propagations of high frequency oscillations for one dimensional semi-linear hyperbolic system with small parabolic perturbation. We obtain a new de-generate parabolic system for proﬁle, and valid an asymptotic development in a spirit of Joly,Me´tivierandRauch. key words: geometric optics, small viscosity, proﬁle, phase, non stationary phase, max-imum principle, energy estimates, interpolation, weakly coupled parabolic systems.

1. Introduction Joly,M´etivierandRauchin[13]givearigorousanddeepdescriptionofsolutionsto one dimensional nonlinear hyperbolic equations with smooth and highly oscillatory initial data. On the other hand, numerous physical problems involve parabolic partial diﬀerential equations with small viscosity, for instance Navier-Stokes system with large Reynolds number.ThenitisofinteresttoputtheJoly,Me´tivier,Rauchframeworkwithasmall diﬀusion. The ﬁrst step to study such multiphase expansions is the semilinear case. So we wish to investigate propagations of high frequency oscillations for following nonlinear parabolic system with highly oscillatory initial data u ε 0 and small positive viscosity ν : (1.1) − ν∂∂ 2 xu 2 ε + ∂∂ut ε + A ( t, x ) ∂∂ux ε = F ( t, x, u ε ) (1.2) u ε (0 , x ) = u ε 0 ( x ) . The matrix A ( t, x ) is a smooth N × N real matrix with N distinct real eigenvalues: λ 1 ( t, x ) < λ 2 ( t, x ) < ∙ ∙ ∙ < λ N ( t, x ) , in such a way that ∂ t + A ( t, x ) ∂ x is a strictly hyperbolic operator . In this paper, we will focus our attention on the special case of a rapidly oscillating data with a given vectorial phase ϕ 0 ( x ), frequency 1 /ε and a viscosity equals to the square of the wavelength: (1.3) u ε (0 , x ) = u ε 0 ( x ) = U 0  x, ϕ 0 ε ( x )  , (1.4) ν = ε 2 . So, with such oscillating data, the viscous coeﬃcient has critical size (see next section), such that we expect to have ε 2 ∂ 2 x u ε of order one. And we hope to see interactions of high oscillations with the viscous term. Indeed, it is the aim of this paper to justify a geometric optics expansion: u ε ( t, x ) = U  t, x, ϕ ( tε,x )  + o (1) , 1

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