Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems

Coulombel, Jean-François; Guès, Olivier

doi:10.5802/aif.2581

Cited by 28 publications

(126 citation statements)

References 22 publications

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“…We justify the evolutionarity of the amplitude equation by making the link with the vanishing of the so-called Lopatinskii determinant. This part of our analysis is rather similar to an observation recently made in [15], see also [10]. Moreover, we prove that Hunter's stability condition holds true as soon as the amplitude equation is of evolutionary type, and exhibit a Hamiltonian structure under some technical assumptions.…”

Section: Introductionsupporting

confidence: 85%

“…Even though this result may not seem very surprising (it was more or less conjectured by Hunter, see [12, page 193]), it is in sharp contrast with the case of hyperbolic boundary value problems for which the uniform Lopatinskii condition fails in the hyperbolic region, in which there is a loss of derivatives. Indeed, the results in [10] point out an amplification of oscillations at the boundary in that situation, which means that the regime of weakly nonlinear geometric optics corresponds to initial oscillations with a much smaller amplitude than what we consider here.…”

Section: General Framework and Main Resultsmentioning

confidence: 67%

“…Furthermore, we assume for simplicity that 6 b βγj = 0 for all β, γ, j. Scale-invariance thus reduces to (10) (12) together with the second part in (13), for all α and β ∈ {1, . .…”

Section: Scale Invariancementioning

confidence: 99%

“…One is mostly algebraic and follows from (10) and (12) in Lemma 1. The former implies that for all k = 0, for all α, β, J αγ a γβ k −1+θα−θ β = J αγ a γβ , which can be written with matrices as…”

Section: Back To Scale Invariance and Homogeneity Propertiesmentioning

confidence: 99%

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On the Amplitude Equations for Weakly Nonlinear Surface Waves

Benzoni-Gavage

Coulombel

2012

Arch Rational Mech Anal

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Nonlocal generalizations of Burgers' equation were derived in earlier work by Hunter [Contemp. Math. 1989], and more recently by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [Diff. Int. Eq. 2009] under an appropriate stability condition originally pointed out by Hunter. The latter stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov [Adv. Math. 2011]. In this article, we show how the verification of Hunter's stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. When the original boundary value problem has a variational origin, we also show that the resulting amplitude equation has a Hamiltonian structure. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.

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Section: Introductionsupporting

confidence: 85%

Section: General Framework and Main Resultsmentioning

confidence: 67%

“…Furthermore, we assume for simplicity that 6 b βγj = 0 for all β, γ, j. Scale-invariance thus reduces to (10) (12) together with the second part in (13), for all α and β ∈ {1, . .…”

Section: Scale Invariancementioning

confidence: 99%

Section: Back To Scale Invariance and Homogeneity Propertiesmentioning

confidence: 99%

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On the Amplitude Equations for Weakly Nonlinear Surface Waves

Benzoni-Gavage

Coulombel

2012

Arch Rational Mech Anal

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“…[3,4] for the definitions. In [10], Coulombel and Guès show that the loss of regularity in (1.9), (1.10) in such a case is optimal. They also prove that the well posedness result with loss of regularity is independent of Lipschitzean zero order terms B but is not independent of bounded zero order terms.…”

Section: IImentioning

confidence: 99%

Regularity of Weakly Well Posed Hyperbolic Mixed Problems With Characteristic Boundary

Morando

Secchi

2011

J. Hyper. Differential Equations

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Abstract. We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be "weakly" well posed, in the sense that a unique L 2 -solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiȋ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.

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Regularity of weakly well posed non characteristic boundary value problems

Morando

Trebeschi

2012

J. Pseudo-Differ. Oper. Appl.

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We consider a boundary value problem for a system of linear partial differential equations with non regular coefficients.We assume the problem to be “weakly” well posed, in the sense that a unique L2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential regularity with respect to the data. This is the case of problems that do not satisfy the uniform Kreiss–Lopatinskii condition in the hyperbolic region of the frequency domain. Provided the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted Sobolev spaces

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Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems

Cited by 28 publications

References 22 publications

On the Amplitude Equations for Weakly Nonlinear Surface Waves

On the Amplitude Equations for Weakly Nonlinear Surface Waves

Regularity of Weakly Well Posed Hyperbolic Mixed Problems With Characteristic Boundary

Regularity of weakly well posed non characteristic boundary value problems

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