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November  2020, 40(11): 6275-6288. doi: 10.3934/dcds.2020279

On Morawetz estimates with time-dependent weights for the Klein-Gordon equation

Jungkwon Kim , Hyeongjin Lee , Ihyeok Seo , and  Jihyeon Seok

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

* Corresponding author: Ihyeok Seo

Received  November 2019 Revised  April 2020 Published  July 2020

Fund Project: This research was supported by NRF-2019R1F1A1061316

We obtain some new Morawetz estimates for the Klein-Gordon flow of the form
$ \begin{equation*} \big\| |\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^{2}_{x, t}(|(x, t)|^{-\alpha})} \lesssim \| f \|_{H^s} \end{equation*} $
where
$ \sigma, s\geq0 $
 and
$ \alpha>0 $
. The conventional approaches to Morawetz estimates with
$ |x|^{-\alpha} $
 are no longer available in the case of time-dependent weights
$ |(x, t)|^{-\alpha} $
. Here we instead apply the Littlewood-Paley theory with Muckenhoupt
$ A_2 $
 weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of
$ \sqrt{1-\Delta} $
 for low and high frequencies.
Keywords: Morawetz estimates, Klein-Gordon equation.
Mathematics Subject Classification: Primary: 35B45; Secondary: 35Q40.
Citation: Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279
References:
[1]

J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela, A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut, 21 (2008), 481-488.  doi: 10.5209/rev_REMA.2008.v21.n2.16405.

[2]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.

[3]

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.

[4]

L. Grafakos, Classical Fourier Analysis, 3$^{rd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[5]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279.  doi: 10.1007/BF01360915.

[6]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in applied mathematics, Adv. Math. Suppl. Stud., Academic Press, New York, 8 (1983), 93–128.

[7]

T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.  doi: 10.1142/S0129055X89000171.

[8]

Y. Koh and I. Seo, Global well-posedness for higher-order Schrödinger equations in weighted $L^2$ spaces, Comm. Partial Differential Equations, 40 (2015), 1815-1830.  doi: 10.1080/03605302.2015.1048551.

[9]

Y. Koh and I. Seo, On weighted $L^2$ estimates for solutions of the wave equation, Proc. Amer. Math. Soc., 144 (2016), 3047-3061.  doi: 10.1090/proc/12951.

[10]

Y. Koh and I. Seo, Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications, Discrete Contin. Dyn. Syst., 37 (2017), 4877-4906.  doi: 10.3934/dcds.2017210.

[11]

Y. Koh and I. Seo, Strichartz and smoothing estimates in weighted $L^2$ spaces and their applications, to appear in Indiana Univ. Math. J., arXiv: 1803.10430.

[12]

D. S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254.  doi: 10.2307/1998156.

[13]

H. Lee, I. Seo and J. Seok, Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential, Discrete Contin. Dyn. Syst., 40 (2020), 597-608.  doi: 10.3934/dcds.2020024.

[14]

W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770.  doi: 10.1090/S0002-9904-1963-11025-3.

[15]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. London Ser. A, 306 (1968), 291-296.  doi: 10.1098/rspa.1968.0151.

[16]

T. Ozawa and K. M. Rogers, Sharp Morawetz estimates, J. Anal. Math., 121 (2013), 163-175.  doi: 10.1007/s11854-013-0031-0.

[17]

M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. Lond. Math. Soc., 105 (2012), 393-423.  doi: 10.1112/plms/pds006.

[18]

I. Seo, A note on the Schrödinger smoothing effect, Math. Nachr., 292 (2019), 2481-2487.  doi: 10.1002/mana.201800502.

[19]

E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. doi: 10.1515/9781400883929.

show all references

References:
[1]

J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela, A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut, 21 (2008), 481-488.  doi: 10.5209/rev_REMA.2008.v21.n2.16405.

[2]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.

[3]

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.

[4]

L. Grafakos, Classical Fourier Analysis, 3$^{rd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[5]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279.  doi: 10.1007/BF01360915.

[6]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in applied mathematics, Adv. Math. Suppl. Stud., Academic Press, New York, 8 (1983), 93–128.

[7]

T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.  doi: 10.1142/S0129055X89000171.

[8]

Y. Koh and I. Seo, Global well-posedness for higher-order Schrödinger equations in weighted $L^2$ spaces, Comm. Partial Differential Equations, 40 (2015), 1815-1830.  doi: 10.1080/03605302.2015.1048551.

[9]

Y. Koh and I. Seo, On weighted $L^2$ estimates for solutions of the wave equation, Proc. Amer. Math. Soc., 144 (2016), 3047-3061.  doi: 10.1090/proc/12951.

[10]

Y. Koh and I. Seo, Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications, Discrete Contin. Dyn. Syst., 37 (2017), 4877-4906.  doi: 10.3934/dcds.2017210.

[11]

Y. Koh and I. Seo, Strichartz and smoothing estimates in weighted $L^2$ spaces and their applications, to appear in Indiana Univ. Math. J., arXiv: 1803.10430.

[12]

D. S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254.  doi: 10.2307/1998156.

[13]

H. Lee, I. Seo and J. Seok, Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential, Discrete Contin. Dyn. Syst., 40 (2020), 597-608.  doi: 10.3934/dcds.2020024.

[14]

W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770.  doi: 10.1090/S0002-9904-1963-11025-3.

[15]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. London Ser. A, 306 (1968), 291-296.  doi: 10.1098/rspa.1968.0151.

[16]

T. Ozawa and K. M. Rogers, Sharp Morawetz estimates, J. Anal. Math., 121 (2013), 163-175.  doi: 10.1007/s11854-013-0031-0.

[17]

M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. Lond. Math. Soc., 105 (2012), 393-423.  doi: 10.1112/plms/pds006.

[18]

I. Seo, A note on the Schrödinger smoothing effect, Math. Nachr., 292 (2019), 2481-2487.  doi: 10.1002/mana.201800502.

[19]

E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. doi: 10.1515/9781400883929.

Figure 1.  The range of $ (\alpha, \sigma) $ for (1) and (4) with $ s = 1/2 $
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