SMALL DATA SOLUTIONS OF 2-D QUASILINEAR WAVE EQUATIONS UNDER NULL CONDITIONS

doi:10.1016/S0252-9602(17)30121-2

Abstract

Abstract:

For the 2-D quasilinear wave equation satisfying null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation (∂_t²-△_x)u+ satisfying null conditions with small initial data and the coefficients depending simultaneously on u and ∂u. Through construction of an approximate solution, combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction.

Key words: global existence, null condition, weighted energy estimate, continuous induction

Yingbo LIU, Ingo WITT. SMALL DATA SOLUTIONS OF 2-D QUASILINEAR WAVE EQUATIONS UNDER NULL CONDITIONS[J].Acta mathematica scientia,Series B, 2018, 38(1): 125-150.

Trendmd

References

[1] Alinhac S. The null condition for quasilinear wave equations in two space dimensions I. Invent Math, 2001, 145(3):597-618
[2] Hoshiga A. The initial value problems for quasi-linear wave equations in two space dimensions with small data. Adv Math Sci Appl, 1995, 5(1):67-89
[3] Katayama S. Global existence for systems of nonlinear wave equations in two space dimensions Ⅱ. Publ RIMS, 1995, 31:645-665
[4] Katayama S. Lifespan of solutions for two space dimensional wave equations with cubic nonlinearity. Comm Paritial Differ Equ, 2001, 26:205-235
[5] Li T T, Zhou Y. Life-span of classical solutions to nonlinear wave equations in two-space-dimensions Ⅱ. J Partial Differ Equ, 1993, 6:17-38
[6] Alinhac S. Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions Ⅱ. Acta Math, 1999, 182(1):1-23
[7] Alinhac S. The null condition for quasilinear wave equations in two space dimensions Ⅱ. Amer J Math, 2001, 123(6):1071-1101
[8] Godin P. Lifespan of solutions of semilinear wave equations in two space dimensions. Comm Partial Differ Equ, 1993, 18(5/6):895-916
[9] Hörmander L. The lifespan of classical solutions of nonlinear hyperbolic equations. Pseudodifferential operators//Lecture Notes in Math, 1256. Berlin:Springer, 1987:214-280
[10] Hörmander L. Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26. Heidelberg:Springer-Verlag, 1997
[11] Lei Z. Global well-posedness of incompressible elastodynamics in two dimensions. Comm Pure Appl Math, 2016, 69(11):2072-2106
[12] Lei Z, Sideris T C, Zhou Y. Almost global existence for 2-D incompressible isotropic elastodynamics. Trans Amer Math Soc, 2015, 367(11):8175-8197
[13] Li T T, Zhou Y. Life-span of classical solutions to nonlinear wave equations in two space dimensions. J Math Pures Appl, 1994, (9)73(3):223-249
[14] Li T T, Zhou Y. Nonlinear stability for two-space-dimensional wave equations with higher-order perturbations. Nonlinear World, 1994, 1(1):35-58
[15] Klainerman S. Remarks on the global Sobolev inequalities in the Minkowski space Rⁿ⁺¹. Comm Pure Appl Math, 1987, 40:111-117
[16] Klainerman S, Ponce G. Global, small amplitude solutions to nonlinear evolution equations. Comm Pure Appl Math, 1983, 36(1):133-141
[17] Lindblad H. On the lifespan of solutions of nonlinear wave equations with small initial data. Comm Pure Appl Math, 1990, 43(4):445-472
[18] Christodoulou D. Global solutions of nonlinear hyperbolic equations for small initial data. Comm Pure Appl Math, 1986, 39(2):267-282
[19] John F. Blow-up of radial solutions of u_tt=c²(ut)△u in three space dimensions. Mat Apl Comput, 1985, 4(1):3-18
[20] John F, Klainerman S. Almost global existence to nonlinear wave equations in three space dimensions. Comm Pure Appl Math, 1984, 37(4):443-455
[21] Klainerman S. The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics, Part 1(Santa Fe, N M, 1984)//Lectures in Appl Math 23. Providence, RI:Amer Math Soc, 1986:293-326
[22] Yin H C. The blowup mechanism of small data solution for the quasilinear wave equations in three space dimensions. Acta Math Sin Engl Ser, 2001, 18(1):35-76
[23] Alinhac S. An example of blowup at infinity for quasilinear wave equations. Asterisque, 2003, 284:1-91
[24] Lindblad H. Global solutions of nonlinear wave equations. Comm Pure Appl Math, 1992, 45(9):1063-1096
[25] Lindblad H. Global solutions of quasilinear wave equations. Amer J Math, 2008, 130(1):115-157
[26] Ding B B, Witt I, Yin H C. Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations. Commun Pure Appl Anal, 2017, 16(3):719-744
[27] Ding B B, Witt I, Yin H C. Blowup of classical solutions for a class of 3-D quasilinear wave equations with small initial data. Differ Integral Equ, 2015, 28(9/10):941-970
[28] Ding B B, Liu Y B, Yin H C. The small data solutions of general 3D quasilinear wave equations I. SIAM J Math Anal, 2015, 47(6):4192-4228
[29] Ding B B, Witt I, Yin H C. The small data solutions of general 3D quasilinear wave equations Ⅱ. J Differ Equ, 2016, 261(2):1429-1471
[30] Alinhac S. Temps de vie des solutions des équations d'Euler compressibles axisymétriques en dimension deux. Invent Math, 1993, 111:627-670
[31] Ding B B, Witt I, Yin H C. The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases. J Differ Equ, 2015, 258(2):445-482