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Generalized and weighted Strichartz estimates
1. | Institute of Mathematics, Academic Sinica, Taipei, Taiwan 10617, Taiwan |
2. | Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United States, United States |
References:
[1] |
J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction," Springer-Velag Berlin Heidelberg, 1976. |
[2] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture notes in Mathematics, 10. Amer. Math. Soc., 2003. |
[3] |
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[4] |
D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation, Nonlinear Anal., 65 (2006), 697-706.
doi: 10.1016/j.na.2005.09.040. |
[5] |
D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.
doi: 10.1515/FORM.2011.009. |
[6] |
D. Fang and C. Wang, Almost global existence for some semilinear wave equations with almost critical regularity,, \arXiv{1007.0733}., ().
|
[7] |
J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309-323. |
[8] |
J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.
doi: 10.1006/jfan.1995.1119. |
[9] |
V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.
doi: 10.1353/ajm.1997.0038. |
[10] |
K. Hidano, Morawetz-Strichartz estimates for spherically symmetric solutions to wave equations and applications to semi-linear Cauchy problems, Differential Integral Equations, 20 (2007), 735-754. |
[11] |
K. Hidano and Y. Kurokawa, Local existence of minimal-regularity radial solutions to semi-linear wave equations,, preprint., ().
|
[12] |
K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations, Illinois J. Math., 52 (2008), 365-388. |
[13] |
K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809.
doi: 10.1090/S0002-9947-09-05053-3. |
[14] |
K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well-posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations, 17 (2012), 267-306. |
[15] |
K. Hidano and K. Yokoyama, A remark on the almost global existence theorems of Keel, Smith and Sogge, Funkcial. Ekvac., 48 (2005), 1-34.
doi: 10.1619/fesi.48.1. |
[16] |
J. Kato and T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications, J. Math. Pures Appl., 95 (2011), 48-71.
doi: 10.1016/j.matpur.2010.10.001. |
[17] |
M. Keel, H. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, Dedicated to the memory of Thomas H. Wolff. J. Anal. Math., 87 (2002), 265-279.
doi: 10.1007/BF02868477. |
[18] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[19] |
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. |
[20] |
H. Lindblad, Blow up for solutions of $\square u = |u|^p$ with small initial data, Comm. Partial Differential Equations, 15 (1990), 757-821.
doi: 10.1080/03605309908820708. |
[21] |
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. |
[22] |
H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.
doi: 10.1353/ajm.1996.0042. |
[23] |
S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal., 219 (2005), 1-20.
doi: 10.1016/j.jfa.2004.07.005. |
[24] |
J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles, Houston J. Math., 30 (2004), 259-281. |
[25] |
J. Metcalfe and C. D. Sogge, Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal., 38 (2006), 188-209.
doi: 10.1137/050627149. |
[26] |
S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equation, Duke Math. J., 19 (1998), 393-408.
doi: 10.1215/S0012-7094-98-09117-7. |
[27] |
C. D. Sogge, "Fourier Integrals in Classical Analysis," Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993. |
[28] |
C. D. Sogge, "Lectures on Nonlinear Wave Equations," Second edition, International Press, Boston, MA, 2008. |
[29] |
C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.
doi: 10.1007/s11854-010-0023-2. |
[30] |
E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[31] |
E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. |
[32] |
J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, With an appendix by Igor Rodnianski, Int. Math. Res. Not., 4 (2005) 187-231.
doi: 10.1155/IMRN.2005.187. |
[33] |
R. S. Strichartz, Multipliers for spherical harmonic expansions, Trans. Amer. Math. Soc., 167 (1972), 115-124.
doi: 10.2307/1996130. |
[34] |
R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[35] |
T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations, 25 (2000), 1471-1485.
doi: 10.1080/03605300008821556. |
[36] |
T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106. American Mathematical Society, Providence, RI, 2006. |
[37] |
D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.
doi: 10.1090/S0002-9947-00-02750-1. |
[38] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978. |
[39] |
G. N. Watson, A treatise on the theory of Bessel functions, Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge 1995. |
[40] |
X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential Integral Equations, 24 (2011), 443-468. |
[41] |
Y. Zhou, Blow up of classical solutions to $\square u = |u|^{1+\alpha$ in three space dimensions, J. Partial Differential Equations, 5 (1992), 21-32. |
[42] |
Y. Zhou, Life span of classical solutions to $\square u=|u|^p p$ in two space dimensions, Chinese Ann. Math. Ser. B, 14 (1993), 225-236. |
show all references
References:
[1] |
J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction," Springer-Velag Berlin Heidelberg, 1976. |
[2] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture notes in Mathematics, 10. Amer. Math. Soc., 2003. |
[3] |
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[4] |
D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation, Nonlinear Anal., 65 (2006), 697-706.
doi: 10.1016/j.na.2005.09.040. |
[5] |
D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.
doi: 10.1515/FORM.2011.009. |
[6] |
D. Fang and C. Wang, Almost global existence for some semilinear wave equations with almost critical regularity,, \arXiv{1007.0733}., ().
|
[7] |
J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309-323. |
[8] |
J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.
doi: 10.1006/jfan.1995.1119. |
[9] |
V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.
doi: 10.1353/ajm.1997.0038. |
[10] |
K. Hidano, Morawetz-Strichartz estimates for spherically symmetric solutions to wave equations and applications to semi-linear Cauchy problems, Differential Integral Equations, 20 (2007), 735-754. |
[11] |
K. Hidano and Y. Kurokawa, Local existence of minimal-regularity radial solutions to semi-linear wave equations,, preprint., ().
|
[12] |
K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations, Illinois J. Math., 52 (2008), 365-388. |
[13] |
K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809.
doi: 10.1090/S0002-9947-09-05053-3. |
[14] |
K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well-posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations, 17 (2012), 267-306. |
[15] |
K. Hidano and K. Yokoyama, A remark on the almost global existence theorems of Keel, Smith and Sogge, Funkcial. Ekvac., 48 (2005), 1-34.
doi: 10.1619/fesi.48.1. |
[16] |
J. Kato and T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications, J. Math. Pures Appl., 95 (2011), 48-71.
doi: 10.1016/j.matpur.2010.10.001. |
[17] |
M. Keel, H. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, Dedicated to the memory of Thomas H. Wolff. J. Anal. Math., 87 (2002), 265-279.
doi: 10.1007/BF02868477. |
[18] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[19] |
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. |
[20] |
H. Lindblad, Blow up for solutions of $\square u = |u|^p$ with small initial data, Comm. Partial Differential Equations, 15 (1990), 757-821.
doi: 10.1080/03605309908820708. |
[21] |
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. |
[22] |
H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.
doi: 10.1353/ajm.1996.0042. |
[23] |
S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal., 219 (2005), 1-20.
doi: 10.1016/j.jfa.2004.07.005. |
[24] |
J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles, Houston J. Math., 30 (2004), 259-281. |
[25] |
J. Metcalfe and C. D. Sogge, Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal., 38 (2006), 188-209.
doi: 10.1137/050627149. |
[26] |
S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equation, Duke Math. J., 19 (1998), 393-408.
doi: 10.1215/S0012-7094-98-09117-7. |
[27] |
C. D. Sogge, "Fourier Integrals in Classical Analysis," Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993. |
[28] |
C. D. Sogge, "Lectures on Nonlinear Wave Equations," Second edition, International Press, Boston, MA, 2008. |
[29] |
C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.
doi: 10.1007/s11854-010-0023-2. |
[30] |
E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[31] |
E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. |
[32] |
J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, With an appendix by Igor Rodnianski, Int. Math. Res. Not., 4 (2005) 187-231.
doi: 10.1155/IMRN.2005.187. |
[33] |
R. S. Strichartz, Multipliers for spherical harmonic expansions, Trans. Amer. Math. Soc., 167 (1972), 115-124.
doi: 10.2307/1996130. |
[34] |
R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[35] |
T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations, 25 (2000), 1471-1485.
doi: 10.1080/03605300008821556. |
[36] |
T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106. American Mathematical Society, Providence, RI, 2006. |
[37] |
D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.
doi: 10.1090/S0002-9947-00-02750-1. |
[38] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978. |
[39] |
G. N. Watson, A treatise on the theory of Bessel functions, Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge 1995. |
[40] |
X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential Integral Equations, 24 (2011), 443-468. |
[41] |
Y. Zhou, Blow up of classical solutions to $\square u = |u|^{1+\alpha$ in three space dimensions, J. Partial Differential Equations, 5 (1992), 21-32. |
[42] |
Y. Zhou, Life span of classical solutions to $\square u=|u|^p p$ in two space dimensions, Chinese Ann. Math. Ser. B, 14 (1993), 225-236. |
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