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A symmetric Random Walk defined by the time-one map of a geodesic flow
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 |
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 [
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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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. |
[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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