The Mach stem equation and amplification in strongly nonlinear geometric optics

Coulombel, Jean Francois; Williams, Mark L.

doi:10.1353/ajm.2017.0026

Cited by 10 publications

(21 citation statements)

References 34 publications

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“…The question is to know whether or not the boundary condition (5.11c) determines on its own the trace U osc |x d =0,ψ d =0 . As already explained in [CGW11] and [CW17], the answer depends on the existence of a resonance between two incoming frequencies that generates an outgoing frequency. Such a resonance pattern is excluded by Assumption 7.…”

Section: Ansatz and Main Resultsmentioning

confidence: 89%

Weakly nonlinear multiphase geometric optics for hyperbolic quasilinear boundary value problems: Construction of a leading profile

Kilque¹

2021

Preprint

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We investigate in this paper the existence of the leading profile of a WKB expansion for quasilinear initial boundary value problems with a highly oscillating forcing boundary term. The framework is weakly nonlinear, as the boundary term is of order O(ε) where the frequencies are of order O(1/ε). We consider here multiple phases on the boundary, generating a countable infinite number of phases inside the domain, and we therefore use an almost periodic functional framework. The major difficulties of this work are the lack of symmetry in the leading profile equation and the occurrence of infinitely many resonances (opposite to the simple phase case studied earlier) The leading profile is constructed as the solution of a quasilinear problem, which is solved using a priori estimates without loss of derivatives. The assumptions of this work are illustrated with the example of isentropic Euler equations in space dimension two. in res 6.3. Reducing the system 6.4. A priori estimate on the linearized system for the oscillating resonant part 6.5. A priori estimate for the linearized Burgers equations 6.6. Construction of a solution 6.7. Conclusion and perspectives Appendix A. Additional proofs A.1. Proof of Proposition 2.21 A.2. Proof of Lemma 2.22 A.3. Proof of Proposition 6.16 References

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Section: Ansatz and Main Resultsmentioning

confidence: 89%

Weakly nonlinear multiphase geometric optics for hyperbolic quasilinear boundary value problems: Construction of a leading profile

Kilque¹

2021

Preprint

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“…We do not see how to prove an analogue of Theorem 2.12 in this case, since the application of Proposition 4.6 requires that all D(ǫ, k, k − r) be "small". Another reason for our focus on the D( φ in ǫ ) case is its greater relevance to the Mach stem problem; this is explained in [CW17].…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

Hyperbolic boundary problems with large oscillatory coefficients: multiple amplification

Williams¹

2019

Preprint

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We study weakly stable hyperbolic boundary problems with highly oscillatory coefficients that are large, O(1), compared to the small wavelength ǫ of oscillations. Such problems arise, for example, in the study of classical questions concerning the stability of Mach stems and compressible vortex sheets. For such applications one seeks to prove energy estimates that are in an appropriate sense "uniform" with respect to the small wavelength ǫ, but the large oscillatory coefficients are a formidable obstacle to obtaining such estimates. In this paper we analyze a simplified form of the linearized problems that are relevant to the above stability questions, and obtain results that are both positive and negative. On the one hand we identify favorable structural conditions under which it is possible to prove uniform estimates, and then do so by a new approach. We also construct examples showing that large oscillatory coefficients can give rise to an instantaneous multiple amplification of the amplitude of solutions relative to data; for example, boundary data of a given amplitude O(1) can immediately give rise to a solution of amplitude O( 1 ǫ K ), where K > 1. 1 We use the examples of multiple amplification to confirm the optimality of our uniform estimates when the favorable structural conditions hold. When those conditions do not hold, we explain how multiple amplification of infinite order may rule out useful estimates. Contents

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“…Such constructions of geometric optics expansions in this context have a long story starting (in the author's knowledge) by the formal constructions given by [AM87] and [MR83] (see also [Maj88]) to explain the appearance of singularities in (non linear) fluid dynamics such as Mach stems formation. We also refer to [CW17] (and the many references therein) for the construction of such rigorous geometric optics expansions.…”

Section: Introductionmentioning

confidence: 99%

WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

Benoit

2021

J. Hyper. Differential Equations

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We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

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The Mach stem equation and amplification in strongly nonlinear geometric optics

Cited by 10 publications

References 34 publications

Weakly nonlinear multiphase geometric optics for hyperbolic quasilinear boundary value problems: Construction of a leading profile

Weakly nonlinear multiphase geometric optics for hyperbolic quasilinear boundary value problems: Construction of a leading profile

Hyperbolic boundary problems with large oscillatory coefficients: multiple amplification

WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

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