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

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

Seongyeon Kim 1, , Yehyun Kwon 2, and  Ihyeok Seo 1,,

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.

Keywords: Strichartz estimates, regularity, elastic wave equation.
Mathematics Subject Classification: Primary: 35B45, 35B65; Secondary: 35L05.
Citation: Seongyeon Kim, Yehyun Kwon, Ihyeok Seo. Strichartz estimates and local regularity for the elastic wave equation with singular potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1897-1911. doi: 10.3934/dcds.2020344
References:
[1]

J. A. Barceló, L. Fanelli, A. Ruiz, M. 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.  Google Scholar

[2]

J. A. Barceló, M. Folch-Gabayet, S. Pérez-Esteva, A. 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.   Google Scholar

[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.  Google Scholar

[4]

N. Burq, F. Planchon, J. 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.  Google Scholar

[5]

N. Burq, F. Planchon, J. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

S. Kim, I. 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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

F. Planchon, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

show all references

References:
[1]

J. A. Barceló, L. Fanelli, A. Ruiz, M. 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.  Google Scholar

[2]

J. A. Barceló, M. Folch-Gabayet, S. Pérez-Esteva, A. 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.   Google Scholar

[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.  Google Scholar

[4]

N. Burq, F. Planchon, J. 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.  Google Scholar

[5]

N. Burq, F. Planchon, J. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

S. Kim, I. 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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

F. Planchon, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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