April  2021, 41(4): 1897-1911. doi: 10.3934/dcds.2020344

Strichartz estimates and local regularity for the elastic wave equation with singular potentials

1. 

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

2. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

* Corresponding author: Ihyeok Seo

Received  July 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

Fund Project: The second author is supported by a KIAS Individual Grant (MG073701) at Korea Institute for Advanced Study and NRF-2020R1F1A1A01073520. The thrid author is supported by NRF-2019R1F1A1061316

We obtain weighted $ L^2 $ estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lamé operator $ -\Delta^\ast $. This improves upon the previous result in [1] which relies on a stronger condition to guarantee the self-adjointness of $ -\Delta^\ast $. Furthermore, by establishing local energy estimates for the elastic wave equation we also prove that the solution has local regularity.

Citation: Seongyeon Kim, Yehyun Kwon, Ihyeok Seo. Strichartz estimates and local regularity for the elastic wave equation with singular potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1897-1911. doi: 10.3934/dcds.2020344
References:
[1]

J. A. BarcelóL. FanelliA. RuizM. C. Vilela and N. Visciglia, Resolvent and Strichartz estimates for elastic wave equations, Appl. Math. Lett., 49 (2015), 33-41.  doi: 10.1016/j.aml.2015.04.013.

[2]

J. A. BarcelóM. Folch-GabayetS. Pérez-EstevaA. Ruiz and M. C. Vilela, Limiting absorption principles for the Navier equation in elasticity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 817-842. 

[3]

M. Beals and W. Strauss, $L^p$ estimates for the wave equation with a potential, Comm. Partial Differential Equations, 18 (1993), 1365-1397.  doi: 10.1080/03605309308820977.

[4]

N. BurqF. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.  doi: 10.1016/S0022-1236(03)00238-6.

[5]

N. BurqF. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1682.  doi: 10.1512/iumj.2004.53.2541.

[6]

F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman: "The uncertainty principle", Proc. Amer. Math. Soc., 108 (1990), 407-409.  doi: 10.2307/2048289.

[7]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.

[8]

R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79 (1980), 249-254.  doi: 10.2307/2043245.

[9]

L. Cossetti, Bounds on eigenvalues of perturbed Lamé operators with complex potentials, Preprint, preprint, arXiv: 1904.08445.

[10]

S. Cuccagna, On the wave equation with a potential, Comm. Partial Differential Equations, 25 (1999), 1549-1565.  doi: 10.1080/03605300008821559.

[11]

P. D'Ancona, On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations, Commun. Pure Appl. Anal., 19 (2020), 609-640.  doi: 10.3934/cpaa.2020029.

[12]

V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Comm. Partial Differential Equations, 28 (2003), 1325-1369.  doi: 10.1081/PDE-120024371.

[13]

M. Goldberg, L. Vega and N. Visciglia, Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not., (2006), Art. ID 13927, 16 pp. doi: 10.1155/IMRN/2006/13927.

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[15]

S. KimI. Seo and J. Seok, Note on Strichartz inequalities for the wave equation with potential, Math. Inequal. Appl., 23 (2020), 377-382.  doi: 10.7153/mia-2020-23-29.

[16]

S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268.  doi: 10.1002/cpa.3160460902.

[17]

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon, 1970.

[18]

H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.  doi: 10.1006/jfan.1995.1075.

[19]

D. Maharani, J. Widjaja and M. Wono Setya Budhi, Boundedness of Mikhlin Operator in Morrey Space, J. Phys.: Conf. Ser., 1180 (2019), 012002. doi: 10.1088/1742-6596/1180/1/012002.

[20]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, 1983, reprinted by Dover Publications, N.Y., 1994.

[21]

S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc., 136 (2008), 1237-1249.  doi: 10.1090/S0002-9939-07-08934-4.

[22]

F. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, $L^p$ estimates for the wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst., 9 (2003), 427-442.  doi: 10.3934/dcds.2003.9.427.

[23]

A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J., 76 (1994), 913-940.  doi: 10.1215/S0012-7094-94-07636-9.

[24]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.

[25] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. 
[26]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[27]

K. Yajima, The $W^{k, p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47 (1995), 551-581.  doi: 10.2969/jmsj/04730551.

show all references

References:
[1]

J. A. BarcelóL. FanelliA. RuizM. C. Vilela and N. Visciglia, Resolvent and Strichartz estimates for elastic wave equations, Appl. Math. Lett., 49 (2015), 33-41.  doi: 10.1016/j.aml.2015.04.013.

[2]

J. A. BarcelóM. Folch-GabayetS. Pérez-EstevaA. Ruiz and M. C. Vilela, Limiting absorption principles for the Navier equation in elasticity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 817-842. 

[3]

M. Beals and W. Strauss, $L^p$ estimates for the wave equation with a potential, Comm. Partial Differential Equations, 18 (1993), 1365-1397.  doi: 10.1080/03605309308820977.

[4]

N. BurqF. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.  doi: 10.1016/S0022-1236(03)00238-6.

[5]

N. BurqF. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1682.  doi: 10.1512/iumj.2004.53.2541.

[6]

F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman: "The uncertainty principle", Proc. Amer. Math. Soc., 108 (1990), 407-409.  doi: 10.2307/2048289.

[7]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.

[8]

R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79 (1980), 249-254.  doi: 10.2307/2043245.

[9]

L. Cossetti, Bounds on eigenvalues of perturbed Lamé operators with complex potentials, Preprint, preprint, arXiv: 1904.08445.

[10]

S. Cuccagna, On the wave equation with a potential, Comm. Partial Differential Equations, 25 (1999), 1549-1565.  doi: 10.1080/03605300008821559.

[11]

P. D'Ancona, On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations, Commun. Pure Appl. Anal., 19 (2020), 609-640.  doi: 10.3934/cpaa.2020029.

[12]

V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Comm. Partial Differential Equations, 28 (2003), 1325-1369.  doi: 10.1081/PDE-120024371.

[13]

M. Goldberg, L. Vega and N. Visciglia, Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not., (2006), Art. ID 13927, 16 pp. doi: 10.1155/IMRN/2006/13927.

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[15]

S. KimI. Seo and J. Seok, Note on Strichartz inequalities for the wave equation with potential, Math. Inequal. Appl., 23 (2020), 377-382.  doi: 10.7153/mia-2020-23-29.

[16]

S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268.  doi: 10.1002/cpa.3160460902.

[17]

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon, 1970.

[18]

H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.  doi: 10.1006/jfan.1995.1075.

[19]

D. Maharani, J. Widjaja and M. Wono Setya Budhi, Boundedness of Mikhlin Operator in Morrey Space, J. Phys.: Conf. Ser., 1180 (2019), 012002. doi: 10.1088/1742-6596/1180/1/012002.

[20]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, 1983, reprinted by Dover Publications, N.Y., 1994.

[21]

S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc., 136 (2008), 1237-1249.  doi: 10.1090/S0002-9939-07-08934-4.

[22]

F. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, $L^p$ estimates for the wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst., 9 (2003), 427-442.  doi: 10.3934/dcds.2003.9.427.

[23]

A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J., 76 (1994), 913-940.  doi: 10.1215/S0012-7094-94-07636-9.

[24]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.

[25] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. 
[26]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[27]

K. Yajima, The $W^{k, p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47 (1995), 551-581.  doi: 10.2969/jmsj/04730551.

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