Nonlinear variational surface waves

Austria, Lawrence; Hunter, John K.

doi:10.4310/cis.2013.v13.n1.a1

Cited by 3 publications

(5 citation statements)

References 24 publications

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“…When all the coefficients d αjβℓγm are equal to zero then b 2 ≡ 0 and b is positively homogeneous degree one. This is what happens when the energy W is quadratic in ∇u, as for instance in the simplified model for liquid crystals studied by Austria and Hunter [3,4], for which they find a kernel of the form The more complicated kernel associated with the full model for liquid crystals is also of degree one, see [4, pp. 101-103] for explicit formulas.…”

Section: Derivation Of Amplitude Equationsmentioning

confidence: 91%

“…Up to the addition of a Lagrange multiplier associated with the constraint |u| = 1 to this energy, (NLBVP) then corresponds to a model introduced by Saxton [21] and Alì and Hunter [1] for nematic liquid crystals. This specific boundary value problem and a simplified version of it were studied by Austria [4,3]. Otherwise, a most famous model that fits the abstract setting in (NLBVP) is given by the equations describing hyper-elastic materials with traction free boundary condition, on which there is abundant literature.…”

Section: Amplitude Equations In Abstract Variational Problems 21 Genmentioning

confidence: 99%

“…Remark 4. In the case when b 1 = 0, there is another way to go from the amplitude equation ( 2) to an equation looking like (3), which breaks the symmetry of the kernel but still yields a well-posedness result. Let us recall indeed from the proof of Lemma 2 that…”

Section: Well-posedness Theory Of Amplitude Equationsmentioning

confidence: 99%

“…Roughly speaking, amplitude equations associated with surface waves in variational problems are found to be locally wellposed. The abstract part in § 2 provides in particular a way of revisiting the case of elasticity that is much simpler than in [7] and also applies to more general energies, such as the Oseen-Frank energy for liquid crystals considered by Austria and Hunter [3,4]. The more specific part § 3 closes the loop about phase boundaries, which do not fit the abstract framework of § 2 and may nevertheless be viewed as a variational problem.…”

Section: Introductionmentioning

confidence: 99%

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Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems

Benzoni-Gavage

Coulombel

2017

Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics

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Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009].

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Section: Derivation Of Amplitude Equationsmentioning

confidence: 91%

Section: Amplitude Equations In Abstract Variational Problems 21 Genmentioning

confidence: 99%

Section: Well-posedness Theory Of Amplitude Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems

Benzoni-Gavage

Coulombel

2017

Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics

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“…The kernel (4.31) is the same as the one arising in a simplified model for weakly nonlinear Rayleigh wave modulation in elastodynamics, see [HIZ95], and in a related problem of magnetohydrodynamics, namely the plasma-vacuum interface problem [Sec15]. Further consideration on amplitude equations for weakly nonlinear surface waves may be found in [AHP02,AH13]. The kernel in (4.31) coincides of course with the one derived in [AH03] in the two-dimensional case.…”

Section: The Nonlocal Hamilton-jacobi Equation For the Leading Front ...mentioning

confidence: 99%

Weakly nonlinear surface waves in magnetohydrodynamics

Pierre,

Coulombel

2018

Preprint

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This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the incompressible magnetohydrodynamics equations. Current vortex sheets are piecewise smooth solutions to the incompressible magnetohydrodynamics equations that satisfy suitable jump conditions for the velocity and magnetic field on the (free) discontinuity surface. In this work, we complete an earlier work by Alì and Hunter [Quart. Appl. Math. 61(3), 451-474, 2003] and construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to 'surface waves' on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime that we consider here, their leading amplitude is governed by a nonlocal Hamilton-Jacobi type equation known as the 'HIZ equation' (standing for Hamilton-Il'insky-Zabolotskaya [J. Acoust. Soc. Am. 97(2), 891-897, 1995]) in the context of Rayleigh waves in elastodynamics.The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. Based on a suitable duality formula, we exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on a combination of mere algebra and arguments of combinatorial analysis. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the free boundary problem. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves [C. R. Math. Acad. Sci. Paris 349(23-24), 1239-1244, 2011 does not arise in the same way for the current vortex sheet problem.

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Weakly nonlinear surface waves in magnetohydrodynamics. I

Pierre

Coulombel

2020

Asymptotic Analysis

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This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.

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Nonlinear variational surface waves

Cited by 3 publications

References 24 publications

Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems

Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems

Weakly nonlinear surface waves in magnetohydrodynamics

Weakly nonlinear surface waves in magnetohydrodynamics. I

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