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December  2020, 19(12): 5475-5486. doi: 10.3934/cpaa.2020248

Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry

Antoine Benoit

Univ. Littoral Côte d'Opale, UR2597, LMPA, Laboratoire de Mathématiques Pures et Appliquées, F-62100, France

Received  March 2020 Revised  June 2020 Published  December 2020 Early access  October 2020

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 [16,7,6]). In this setting it is known that several behaviours are possible, for example, a loss of a derivative on the boundary only or a loss of a derivative on the boundary combined with one or a half loss in the interior.

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 [16]) and we show that (in terms of losses of regularity) the situation is exactly the same as the one described in the half-space, meaning that the instability does not meet the geometry. This result has to be compared with the one of [2] in which the geometry has a real impact on the behaviour of the solution.

Keywords: Hyperbolic boundary value problem, strip geometry, loss of derivative.
Mathematics Subject Classification: Primary: 35L50, 35B30; Secondary: 78A05.
Citation: Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations, Oxford University Press, 2007.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[9]

J. Chazarain and A. Piriou, Introduction À La Théorie Des équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981.  Google Scholar

[10]

J. F. Hersh, Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334.   Google Scholar

[11]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23 (1970), 277-298.  doi: 10.1002/cpa.3160230304.  Google Scholar

[12]

V. Lescarret, Wave transmission in dispersive media, Math. Models Methods Appl. Sci., 17 (2007), 485-535.  doi: 10.1142/S0218202507002005.  Google Scholar

[13]

A. Marcou, Rigorous weakly nonlinear geometric optics for surface waves, Asymptot. Anal., 69 (2010), 125-174.   Google Scholar

[14]

G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702.  doi: 10.1112/S0024609300007517.  Google Scholar

[15]

L. Sarason, On hyperbolic mixed problems, Arch. Rational Mech. Anal., 18 (1965), 310-334.  doi: 10.1007/BF00251670.  Google Scholar

[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.  Google Scholar

[17]

M. Williams, Nonlinear geometric optics for hyperbolic boundary problems, Commun. Partial Differ. Equ., 21 (1996), 1829-1895.  doi: 10.1080/03605309608821247.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations, Oxford University Press, 2007.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[9]

J. Chazarain and A. Piriou, Introduction À La Théorie Des équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981.  Google Scholar

[10]

J. F. Hersh, Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334.   Google Scholar

[11]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23 (1970), 277-298.  doi: 10.1002/cpa.3160230304.  Google Scholar

[12]

V. Lescarret, Wave transmission in dispersive media, Math. Models Methods Appl. Sci., 17 (2007), 485-535.  doi: 10.1142/S0218202507002005.  Google Scholar

[13]

A. Marcou, Rigorous weakly nonlinear geometric optics for surface waves, Asymptot. Anal., 69 (2010), 125-174.   Google Scholar

[14]

G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702.  doi: 10.1112/S0024609300007517.  Google Scholar

[15]

L. Sarason, On hyperbolic mixed problems, Arch. Rational Mech. Anal., 18 (1965), 310-334.  doi: 10.1007/BF00251670.  Google Scholar

[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.  Google Scholar

[17]

M. Williams, Nonlinear geometric optics for hyperbolic boundary problems, Commun. Partial Differ. Equ., 21 (1996), 1829-1895.  doi: 10.1080/03605309608821247.  Google Scholar

[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.  Google Scholar

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