Skip to main content Accessibility help
×
Home

WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

  • Antoine Benoit (a1)

Abstract

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$ . In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

Copyright

References

Hide All
1.Benoit, A., Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip, Preprint.
2.Benoit, A., Problèmes aux limites, optique géométrique et singularités. PhD thesis, Université de Nantes (2015), https://hal.archives-ouvertes.fr/tel-01180449v1.
3.Benoit, A., Geometric optics expansions for hyperbolic corner problems, I: Self-interaction phenomenon, Anal. PDE 9(6) (2016), 13591418.
4.Kreiss, H.-O., Gustafsson, B. and Sundstrom, A., Stability theory of difference approximations for mixed initial boundary value problems. ii, Math. Comp. 26(119) (1972), 649686.
5.Coulombel, J.-F., Stability of finite difference schemes for hyperbolic initial boundary value problems II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10(1) (2011), 3798.
6.Hersh, R., Mixed problems in several variables, J. Math. Mech. 12 (1963), 317334.
7.Kreiss, H.-O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277298.
8.Lax, P. D., Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627646.
9.Lescarret, V., Wave transmission in dispersive media, Math. Models Meth. Appl. Sci. 17(4) (2007), 485535.
10.Métivier, G., The block structure condition for symmetric hyperbolic systems, Bull. Lond. Math. Soc. 32(6) (2000), 689702.
11.Métivier, G. and Zumbrun, K., Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations 211(1) (2005), 61134.
12.Osher, S., Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Am. Math. Soc. 176 (1973), 141164.
13.Rauch, Jeffrey, Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, vol. 133 (American Mathematical Society, Providence, RI, 2012).
14.Sarason, L. and Smoller, J. A., Geometrical optics and the corner problem, Arch. Rat. Mech. Anal. 56 (1974/75), 3469.
15.Trefethen, L. N., Stability of finite-difference models containing two boundaries or interfaces, Math. Comput. 45(172) (October 1985), 279300.
16.Williams, M., Nonlinear geometric optics for hyperbolic boundary problems, Comm. Partial Differ. Equ. 21(11–12) (1996), 18291895.
17.Williams, M., Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. Éc. Norm. Super. (4) 33(3) (2000), 383432.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

  • Antoine Benoit (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.