# American Institute of Mathematical Sciences

July  2019, 18(4): 1735-1767. doi: 10.3934/cpaa.2019082

## Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition

 1 Department of Mathematics, Faculty of Education, Mie University, 1577 Kurima-machiya-cho Tsu, Mie Prefecture 514-8507, Japan 2 Department of Applied Mathematics, Donghua University, Shanghai 201620, China

* Corresponding author

Received  May 2018 Revised  November 2018 Published  January 2019

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.

Citation: Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082
##### References:
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Math. Sci. (RIMS), Kyoto, (2017), 37–61. Google Scholar [6] 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. Google Scholar [7] K. Hidano and K. Yokoyama, Global existence for a system of quasi-linear wave equations in 3D satisfying the weak null condition, International Mathematics Research Notices. IMRN, to appear., doi: 10.1093/imrn/rny024. Google Scholar [8] L. Hórmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin), 26. Springer-Verlag, Berlin, 1997. Google Scholar [9] F. John, Nonlinear Wave Equations, Formation of Singularities, Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. University Lecture Series, 2. American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002. Google Scholar [10] F. Pusateri and J. Shatah, Space-time resonances and the null condition for first-order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540. doi: 10.1002/cpa.21461. Google Scholar [11] M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc., 17 (2004), 109-153. doi: 10.1090/S0894-0347-03-00443-0. Google Scholar [12] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ${\mathbb R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117. doi: 10.1002/cpa.3160400105. Google Scholar [13] S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math., 49 (1996), 307-321. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H. Google Scholar [14] H. Lindblad and I. Rodnianski, The weak null condition for Einstein's equations, C. R. Math. Acad. Sci. Paris, 336 (2003), 901-906. doi: 10.1016/S1631-073X(03)00231-0. Google Scholar [15] J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition, Japan. J. Math. (N.S.), 31 (2005), 391-472. doi: 10.4099/math1924.31.391. Google Scholar [16] J. Metcalfe and C. D. Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math., 159 (2005), 75-117. doi: 10.1007/s00222-004-0383-2. Google Scholar [17] 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. Google Scholar [18] R. Racke, Lectures on Nonlinear Evolution Equations. Initial Value Problems, Aspects of Mathematics, E19. Friedr. Vieweg & Sohn, Braunschweig, 1992. doi: 10.1007/978-3-663-10629-6. Google Scholar [19] Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z., 191 (1986), 165-199. doi: 10.1007/BF01164023. Google Scholar [20] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849–874. doi: 10.2307/121050. Google Scholar [21] 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), 477-488. doi: 10.1137/S0036141000378966. Google Scholar [22] C. D. Sogge, Global existence for nonlinear wave equations with multiple speeds, in Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001. eds. W. Beckner, A. Nagel, A. Seeger and H.F. Smith), 353–366, Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/conm/320. Google Scholar [23] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation. With an appendix by Igor Rodnianski, Int. Math. Res. Not., 2005, 187–231. doi: 10.1155/IMRN.2005.187. Google Scholar [24] K. Yokoyama, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, 52 (2000), 609-632. doi: 10.2969/jmsj/05230609. Google Scholar [25] D. Zha, Some remarks on quasilinear wave equations with null condition in 3-D, Math. Methods Appl. Sci., 39 (2016), 4484-4495. doi: 10.1002/mma.3876. Google Scholar

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##### References:
 [1] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145 (2001), 597-618. doi: 10.1007/s002220100165. Google Scholar [2] S. Alinhac, Geometric Analysis of Hyperbolic Differential Equations: An Introduction, London Mathematical Society Lecture Note Series, 374. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139107198. Google Scholar [3] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. Google Scholar [4] J. Ginibre and G. Velo, Conformal invariance and time decay for nonlinear wave equations. I, Ann. Inst. H. Poincaré Phys. Théor., 47 (1987), 221-261. 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, in Harmonic Analysis and Nonlinear Partial Differential Equations (eds. H. Kubo and H. Takaoka), RIMS Kôkyûroku Bessatsu B65, Res. Inst. Math. Sci. (RIMS), Kyoto, (2017), 37–61. Google Scholar [6] 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. Google Scholar [7] K. Hidano and K. Yokoyama, Global existence for a system of quasi-linear wave equations in 3D satisfying the weak null condition, International Mathematics Research Notices. IMRN, to appear., doi: 10.1093/imrn/rny024. Google Scholar [8] L. Hórmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin), 26. Springer-Verlag, Berlin, 1997. Google Scholar [9] F. John, Nonlinear Wave Equations, Formation of Singularities, Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. University Lecture Series, 2. American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002. Google Scholar [10] F. Pusateri and J. Shatah, Space-time resonances and the null condition for first-order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540. doi: 10.1002/cpa.21461. Google Scholar [11] M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc., 17 (2004), 109-153. doi: 10.1090/S0894-0347-03-00443-0. Google Scholar [12] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ${\mathbb R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117. doi: 10.1002/cpa.3160400105. Google Scholar [13] S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math., 49 (1996), 307-321. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H. Google Scholar [14] H. Lindblad and I. Rodnianski, The weak null condition for Einstein's equations, C. R. Math. Acad. Sci. Paris, 336 (2003), 901-906. doi: 10.1016/S1631-073X(03)00231-0. Google Scholar [15] J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition, Japan. J. Math. (N.S.), 31 (2005), 391-472. doi: 10.4099/math1924.31.391. Google Scholar [16] J. Metcalfe and C. D. Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math., 159 (2005), 75-117. doi: 10.1007/s00222-004-0383-2. Google Scholar [17] 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. Google Scholar [18] R. Racke, Lectures on Nonlinear Evolution Equations. Initial Value Problems, Aspects of Mathematics, E19. Friedr. Vieweg & Sohn, Braunschweig, 1992. doi: 10.1007/978-3-663-10629-6. Google Scholar [19] Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z., 191 (1986), 165-199. doi: 10.1007/BF01164023. Google Scholar [20] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849–874. doi: 10.2307/121050. Google Scholar [21] 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), 477-488. doi: 10.1137/S0036141000378966. Google Scholar [22] C. D. Sogge, Global existence for nonlinear wave equations with multiple speeds, in Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001. eds. W. Beckner, A. Nagel, A. Seeger and H.F. Smith), 353–366, Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/conm/320. Google Scholar [23] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation. With an appendix by Igor Rodnianski, Int. Math. Res. Not., 2005, 187–231. doi: 10.1155/IMRN.2005.187. Google Scholar [24] K. Yokoyama, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, 52 (2000), 609-632. doi: 10.2969/jmsj/05230609. Google Scholar [25] D. Zha, Some remarks on quasilinear wave equations with null condition in 3-D, Math. Methods Appl. Sci., 39 (2016), 4484-4495. doi: 10.1002/mma.3876. Google Scholar
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