American Institute of Mathematical Sciences

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September  2012, 11(5): 1723-1752. doi: 10.3934/cpaa.2012.11.1723

Generalized and weighted Strichartz estimates

Jin-Cheng Jiang 1, , Chengbo Wang 2, and  Xin Yu 2,

1. 

Institute of Mathematics, Academic Sinica, Taipei, Taiwan 10617, Taiwan
2. 

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United States, United States

Received  January 2011 Revised  November 2011 Published  March 2012

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension $2$ and $3$.
Keywords: Generalized Strichartz estimates, weighted Strichartz estimates, Strauss conjecture, semilinear wave equations., angular regularity.
Mathematics Subject Classification: 35L05, 35L70, 35J1.
Citation: Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723
References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,", Springer-Velag Berlin Heidelberg, (1976).   Google Scholar

[2]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture notes in Mathematics, (2003).   Google Scholar

[3]

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

[4]

D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation,, Nonlinear Anal., 65 (2006), 697.  doi: 10.1016/j.na.2005.09.040.  Google Scholar

[5]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications,, Forum Math., 23 (2011), 181.  doi: 10.1515/FORM.2011.009.  Google Scholar

[6]

D. Fang and C. Wang, Almost global existence for some semilinear wave equations with almost critical regularity,, \arXiv{1007.0733}., ().   Google Scholar

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J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1 (1984), 309.   Google Scholar

[8]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[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.  doi: 10.1353/ajm.1997.0038.  Google Scholar

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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.   Google Scholar

[11]

K. Hidano and Y. Kurokawa, Local existence of minimal-regularity radial solutions to semi-linear wave equations,, preprint., ().   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9947-09-05053-3.  Google Scholar

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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.   Google Scholar

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K. Hidano and K. Yokoyama, A remark on the almost global existence theorems of Keel, Smith and Sogge,, Funkcial. Ekvac., 48 (2005), 1.  doi: 10.1619/fesi.48.1.  Google Scholar

[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.  doi: 10.1016/j.matpur.2010.10.001.  Google Scholar

[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.  doi: 10.1007/BF02868477.  Google Scholar

[18]

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

[19]

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

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[21]

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

[22]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations,, Amer. J. Math., 118 (1996), 1047.  doi: 10.1353/ajm.1996.0042.  Google Scholar

[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.  doi: 10.1016/j.jfa.2004.07.005.  Google Scholar

[24]

J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles,, Houston J. Math., 30 (2004), 259.   Google Scholar

[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.  doi: 10.1137/050627149.  Google Scholar

[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.  doi: 10.1215/S0012-7094-98-09117-7.  Google Scholar

[27]

C. D. Sogge, "Fourier Integrals in Classical Analysis,", Cambridge Tracts in Mathematics, (1993).   Google Scholar

[28]

C. D. Sogge, "Lectures on Nonlinear Wave Equations," Second edition,, International Press, (2008).   Google Scholar

[29]

C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds,, J. Anal. Math., 112 (2010), 1.  doi: 10.1007/s11854-010-0023-2.  Google Scholar

[30]

E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.   Google Scholar

[31]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971).   Google Scholar

[32]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation,, With an appendix by Igor Rodnianski, 4 (2005), 187.  doi: 10.1155/IMRN.2005.187.  Google Scholar

[33]

R. S. Strichartz, Multipliers for spherical harmonic expansions,, Trans. Amer. Math. Soc., 167 (1972), 115.  doi: 10.2307/1996130.  Google Scholar

[34]

R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation,, Duke Math. J., 44 (1977), 705.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[35]

T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation,, Comm. Partial Differential Equations, 25 (2000), 1471.  doi: 10.1080/03605300008821556.  Google Scholar

[36]

T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[37]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation,, Trans. Amer. Math. Soc., 353 (2001), 795.  doi: 10.1090/S0002-9947-00-02750-1.  Google Scholar

[38]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978).   Google Scholar

[39]

G. N. Watson, A treatise on the theory of Bessel functions,, Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, (1944).   Google Scholar

[40]

X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture,, Differential Integral Equations, 24 (2011), 443.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,", Springer-Velag Berlin Heidelberg, (1976).   Google Scholar

[2]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture notes in Mathematics, (2003).   Google Scholar

[3]

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

[4]

D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation,, Nonlinear Anal., 65 (2006), 697.  doi: 10.1016/j.na.2005.09.040.  Google Scholar

[5]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications,, Forum Math., 23 (2011), 181.  doi: 10.1515/FORM.2011.009.  Google Scholar

[6]

D. Fang and C. Wang, Almost global existence for some semilinear wave equations with almost critical regularity,, \arXiv{1007.0733}., ().   Google Scholar

[7]

J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1 (1984), 309.   Google Scholar

[8]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[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.  doi: 10.1353/ajm.1997.0038.  Google Scholar

[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.   Google Scholar

[11]

K. Hidano and Y. Kurokawa, Local existence of minimal-regularity radial solutions to semi-linear wave equations,, preprint., ().   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9947-09-05053-3.  Google Scholar

[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.   Google Scholar

[15]

K. Hidano and K. Yokoyama, A remark on the almost global existence theorems of Keel, Smith and Sogge,, Funkcial. Ekvac., 48 (2005), 1.  doi: 10.1619/fesi.48.1.  Google Scholar

[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.  doi: 10.1016/j.matpur.2010.10.001.  Google Scholar

[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.  doi: 10.1007/BF02868477.  Google Scholar

[18]

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

[19]

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

[20]

H. Lindblad, Blow up for solutions of $\square u = |u|^p$ with small initial data,, Comm. Partial Differential Equations, 15 (1990), 757.  doi: 10.1080/03605309908820708.  Google Scholar

[21]

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

[22]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations,, Amer. J. Math., 118 (1996), 1047.  doi: 10.1353/ajm.1996.0042.  Google Scholar

[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.  doi: 10.1016/j.jfa.2004.07.005.  Google Scholar

[24]

J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles,, Houston J. Math., 30 (2004), 259.   Google Scholar

[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.  doi: 10.1137/050627149.  Google Scholar

[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.  doi: 10.1215/S0012-7094-98-09117-7.  Google Scholar

[27]

C. D. Sogge, "Fourier Integrals in Classical Analysis,", Cambridge Tracts in Mathematics, (1993).   Google Scholar

[28]

C. D. Sogge, "Lectures on Nonlinear Wave Equations," Second edition,, International Press, (2008).   Google Scholar

[29]

C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds,, J. Anal. Math., 112 (2010), 1.  doi: 10.1007/s11854-010-0023-2.  Google Scholar

[30]

E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.   Google Scholar

[31]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971).   Google Scholar

[32]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation,, With an appendix by Igor Rodnianski, 4 (2005), 187.  doi: 10.1155/IMRN.2005.187.  Google Scholar

[33]

R. S. Strichartz, Multipliers for spherical harmonic expansions,, Trans. Amer. Math. Soc., 167 (1972), 115.  doi: 10.2307/1996130.  Google Scholar

[34]

R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation,, Duke Math. J., 44 (1977), 705.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[35]

T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation,, Comm. Partial Differential Equations, 25 (2000), 1471.  doi: 10.1080/03605300008821556.  Google Scholar

[36]

T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[37]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation,, Trans. Amer. Math. Soc., 353 (2001), 795.  doi: 10.1090/S0002-9947-00-02750-1.  Google Scholar

[38]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978).   Google Scholar

[39]

G. N. Watson, A treatise on the theory of Bessel functions,, Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, (1944).   Google Scholar

[40]

X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture,, Differential Integral Equations, 24 (2011), 443.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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