Abstract
In this paper, we first establish a kind of weighted space-time
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: JP15K04955
Award Identifier / Grant number: JP18K03365
Funding statement: The first author was supported in part by JSPS KAKENHI Grants JP15K04955 and JP18K03365. The second author was supported by the Shanghai Sailing Program (no. 17YF1400700) and Fundamental Research Funds for the Central Universities (no. 17D110913).
Acknowledgements
The authors would like to express their sincere gratitude to the referee for his careful review and helpful comments.
References
[1] R. Agemi, Global existence of nonlinear elastic waves, Invent. Math. 142 (2000), no. 2, 225–250. 10.1007/s002220000084Search in Google Scholar
[2] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd ed., Texts Appl. Math. 4, Springer, New York, 1993. 10.1007/978-1-4612-0883-9Search in Google Scholar
[3] Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math. 124 (2014), 1–38. 10.1007/s11854-014-0025-6Search in Google Scholar
[4] K. Hidano, Small solutions to semi-linear wave equations with radial data of critical regularity, Rev. Mat. Iberoam. 25 (2009), no. 2, 693–708. 10.4171/RMI/579Search in Google Scholar
[5] K. Hidano, Regularity and lifespan of small solutions to systems of quasi-linear wave equations with multiple speeds. I: Almost global existence, RIMS Kôkyûroku Bessatsu B65 (2017), 37–61. Search in Google Scholar
[6] K. Hidano, J. Jiang, S. Lee and C. Wang, Weighted fractional chain rule and nonlinear wave equations with minimal regularity, preprint (2018), https://arxiv.org/abs/1605.06748v3. 10.4171/rmi/1130Search in Google Scholar
[7] 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), no. 3–4, 267–306. 10.57262/ade/1355703087Search in Google Scholar
[8] K. Hidano, C. Wang and K. Yokoyama, Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations, Math. Ann. 366 (2016), no. 1–2, 667–694. 10.1007/s00208-015-1346-1Search in Google Scholar
[9]
K. Hidano and K. Yokoyama,
Space-time
[10]
T. Hoshiro,
On weighted
[11] T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal. 63 (1976), no. 3, 273–294. 10.1007/BF00251584Search in Google Scholar
[12] J.-C. Jiang, C. Wang and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal. 11 (2012), no. 5, 1723–1752. 10.3934/cpaa.2012.11.1723Search in Google Scholar
[13] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 112, Chapman & Hall/CRC, Boca Raton, 2000. 10.1201/9781482285789Search in Google Scholar
[14] F. John, Formation of singularities in elastic waves, Trends and Applications of Pure Mathematics to Mechanics (Palaiseau 1983), Lecture Notes in Phys. 195, Springer, Berlin (1984), 194–210. 10.1007/3-540-12916-2_58Search in Google Scholar
[15] F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 (1988), no. 5, 615–666. 10.1002/cpa.3160410507Search in Google Scholar
[16] M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, J. Anal. Math. 87 (2002), 265–279. 10.1007/BF02868477Search in Google Scholar
[17] S. Klainerman, On the work and legacy of Fritz John, 1934–1991, Comm. Pure Appl. Math. 51 (1998), 991–1017. 10.1002/(SICI)1097-0312(199809/10)51:9/10<991::AID-CPA3>3.0.CO;2-TSearch in Google Scholar
[18] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268. 10.1002/cpa.3160460902Search in Google Scholar
[19] S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 (1996), 307–321. 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-HSearch in Google Scholar
[20] H. Kubo, Lower bounds for the lifespan of solutions to nonlinear wave equations in elasticity, Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math. 301, Birkhäuser, Basel (2012), 187–212. 10.1007/978-3-0348-0454-7_10Search in Google Scholar
[21] H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math. 118 (1996), no. 1, 1–16. 10.1353/ajm.1996.0002Search in Google Scholar
[22] H. Lindblad, Counterexamples to local existence for quasilinear wave equations, Math. Res. Lett. 5 (1998), no. 5, 605–622. 10.4310/MRL.1998.v5.n5.a5Search in Google Scholar
[23] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357–426. 10.1006/jfan.1995.1075Search in Google Scholar
[24] M. Y. Liu and C. B. Wang, Global existence for some 4-D quasilinear wave equations with low regularity, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 4, 629–640. 10.1007/s10114-017-7138-7Search in Google Scholar
[25] 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), no. 1, 1–20. 10.1016/j.jfa.2004.07.005Search in Google Scholar
[26] J. Metcalfe, Elastic waves in exterior domains. I. Almost global existence, Int. Math. Res. Not. IMRN 2006 (2006), Article ID 69826. 10.1155/IMRN/2006/69826Search in Google Scholar
[27] 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), no. 1, 188–209. 10.1137/050627149Search in Google Scholar
[28] J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 (2007), no. 3, 521–549. 10.1007/s00209-006-0083-2Search in Google Scholar
[29] J. Metcalfe and B. Thomases, Elastic waves in exterior domains. II. Global existence with a null structure, Int. Math. Res. Not. IMRN 2007 (2007), no. 10, Article ID rnm034. 10.1093/imrn/rnm034Search in Google Scholar
[30] E. Y. Ovcharov, Radial Strichartz estimates with application to the 2-D Dirac–Klein–Gordon system, Comm. Partial Differential Equations 37 (2012), no. 10, 1754–1788. 10.1080/03605302.2011.632047Search in Google Scholar
[31] G. Ponce and T. C. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 (1993), no. 1–2, 169–177. 10.1080/03605309308820925Search in Google Scholar
[32] T. C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math. 123 (1996), no. 2, 323–342. 10.1007/s002220050030Search in Google Scholar
[33] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2) 151 (2000), no. 2, 849–874. 10.2307/121050Search in Google Scholar
[34] T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33 (2001), no. 2, 477–488. 10.1137/S0036141000378966Search in Google Scholar
[35] H. F. Smith and D. Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2) 162 (2005), no. 1, 291–366. 10.4007/annals.2005.162.291Search in Google Scholar
[36] C. D. Sogge, Lectures on Non-Linear Wave Equations, 2nd ed., International Press, Boston, 2008. Search in Google Scholar
[37] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, And Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. 10.1515/9781400883929Search in Google Scholar
[38] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. IMRN 2005 (2005), no. 4, 187–231. 10.1155/IMRN.2005.187Search in Google Scholar
[39] Q. Wang, A geometric approach for sharp local well-posedness of quasilinear wave equations, Ann. PDE 3 (2017), no. 1, Article ID 12. 10.1007/s40818-016-0013-5Search in Google Scholar
[40]
D. Zha,
Space-time
[41] Y. Zhou and Z. Lei, Global low regularity solutions of quasi-linear wave equations, Adv. Differential Equations 13 (2008), no. 1–2, 55–104. 10.57262/ade/1355867360Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston