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Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation
Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry
Univ. Littoral Côte d'Opale, UR2597, LMPA, Laboratoire de Mathématiques Pures et Appliquées, F-62100, France |
In this article we investigate the possible losses of regularity of the solution for hyperbolic boundary value problems defined in the strip $ \mathbb{R}^{d-1}\times \left[0,1 \right] $.
This question has already been widely studied in the half-space geometry in which a full characterization is almost completed (see [
Crudely speaking the question addressed here is "can several boundaries make the situation becomes worse?".
Here we focus our attention to one special case of loss (namely the elliptic degeneracy of [
References:
[1] |
A. Benoit,
Geometric optics expansions for linear hyperbolic boundary value problems and optimality of energy estimates for surface waves, Differ. Integral Equ., 27 (2014), 531-562.
|
[2] |
A. Benoit, WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: time depending loss of derivatives, preprint, https://hal.archives-ouvertes.fr/hal-02391809. Google Scholar |
[3] |
A. Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip, to appear in Indiana U. Math. J..
doi: 10.1512/iumj.2007.56.2851. |
[4] |
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 (2002), 1073-1104.
doi: 10.1017/S030821050000202X. |
[5] |
S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations, Oxford University Press, 2007.
![]() |
[6] |
J. F. Coulombel,
Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl., 84 (2005), 786-818.
doi: 10.1016/j.matpur.2004.10.005. |
[7] |
J. F. Coulombel, Stabilité Multidimensionnelle D'interfaces Dynamiques. Application Aux Transitions De Phase Liquide-vapeur, Ph. D thesis, ENS Lyon, 2002. Google Scholar |
[8] |
J. F. Coulombel and O. Guès,
Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems, Ann. Inst. Fourier (Grenoble), 60 (2010), 2183-2233.
|
[9] |
J. Chazarain and A. Piriou, Introduction À La Théorie Des équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981. |
[10] |
J. F. Hersh,
Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334.
|
[11] |
H. O. Kreiss,
Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23 (1970), 277-298.
doi: 10.1002/cpa.3160230304. |
[12] |
V. Lescarret,
Wave transmission in dispersive media, Math. Models Methods Appl. Sci., 17 (2007), 485-535.
doi: 10.1142/S0218202507002005. |
[13] |
A. Marcou,
Rigorous weakly nonlinear geometric optics for surface waves, Asymptot. Anal., 69 (2010), 125-174.
|
[14] |
G. Métivier,
The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702.
doi: 10.1112/S0024609300007517. |
[15] |
L. Sarason,
On hyperbolic mixed problems, Arch. Rational Mech. Anal., 18 (1965), 310-334.
doi: 10.1007/BF00251670. |
[16] |
M. Sablé-Tougeron,
Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension $2$, Arch. Rational Mech. Anal., 101 (1988), 261-292.
doi: 10.1007/BF00253123. |
[17] |
M. Williams,
Nonlinear geometric optics for hyperbolic boundary problems, Commun. Partial Differ. Equ., 21 (1996), 1829-1895.
doi: 10.1080/03605309608821247. |
[18] |
M. Williams,
Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup., 33 (2000), 383-432.
doi: 10.1016/S0012-9593(00)00116-6. |
show all references
References:
[1] |
A. Benoit,
Geometric optics expansions for linear hyperbolic boundary value problems and optimality of energy estimates for surface waves, Differ. Integral Equ., 27 (2014), 531-562.
|
[2] |
A. Benoit, WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: time depending loss of derivatives, preprint, https://hal.archives-ouvertes.fr/hal-02391809. Google Scholar |
[3] |
A. Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip, to appear in Indiana U. Math. J..
doi: 10.1512/iumj.2007.56.2851. |
[4] |
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 (2002), 1073-1104.
doi: 10.1017/S030821050000202X. |
[5] |
S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations, Oxford University Press, 2007.
![]() |
[6] |
J. F. Coulombel,
Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl., 84 (2005), 786-818.
doi: 10.1016/j.matpur.2004.10.005. |
[7] |
J. F. Coulombel, Stabilité Multidimensionnelle D'interfaces Dynamiques. Application Aux Transitions De Phase Liquide-vapeur, Ph. D thesis, ENS Lyon, 2002. Google Scholar |
[8] |
J. F. Coulombel and O. Guès,
Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems, Ann. Inst. Fourier (Grenoble), 60 (2010), 2183-2233.
|
[9] |
J. Chazarain and A. Piriou, Introduction À La Théorie Des équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981. |
[10] |
J. F. Hersh,
Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334.
|
[11] |
H. O. Kreiss,
Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23 (1970), 277-298.
doi: 10.1002/cpa.3160230304. |
[12] |
V. Lescarret,
Wave transmission in dispersive media, Math. Models Methods Appl. Sci., 17 (2007), 485-535.
doi: 10.1142/S0218202507002005. |
[13] |
A. Marcou,
Rigorous weakly nonlinear geometric optics for surface waves, Asymptot. Anal., 69 (2010), 125-174.
|
[14] |
G. Métivier,
The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702.
doi: 10.1112/S0024609300007517. |
[15] |
L. Sarason,
On hyperbolic mixed problems, Arch. Rational Mech. Anal., 18 (1965), 310-334.
doi: 10.1007/BF00251670. |
[16] |
M. Sablé-Tougeron,
Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension $2$, Arch. Rational Mech. Anal., 101 (1988), 261-292.
doi: 10.1007/BF00253123. |
[17] |
M. Williams,
Nonlinear geometric optics for hyperbolic boundary problems, Commun. Partial Differ. Equ., 21 (1996), 1829-1895.
doi: 10.1080/03605309608821247. |
[18] |
M. Williams,
Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup., 33 (2000), 383-432.
doi: 10.1016/S0012-9593(00)00116-6. |
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