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

Seongyeon Kim; Yehyun Kwon; Ihyeok Seo

doi:10.3934/dcds.2020344

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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 and 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.
[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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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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