Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

Bingbing Ding; Ingo Witt; Huicheng Yin

doi:10.3934/cpaa.2017035

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May 2017, 16(3): 719-744. doi: 10.3934/cpaa.2017035

Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

Bingbing Ding ^1, , Ingo Witt ^2, and Huicheng Yin ^1,,

1.	School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical, Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, 210023, China
2.	Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, Göttingen, D-37073, Germany

^* Corresponding author

Received January 2016 Revised March 2016 Published February 2016

Fund Project: The first author and the third author were supported by the NSFC (No. 11571177) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The second author was supported by the DFG via the Sino-German project "Analysis of PDEs and application.".

For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in ^[1,2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

Keywords: Weak null condition, lifespan, blowup, blowup system, Nash-Moser-Hörmander iteration.

Mathematics Subject Classification: 35L05, 35L72.

Citation: Bingbing Ding, Ingo Witt, Huicheng Yin. Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations. Communications on Pure and Applied Analysis, 2017, 16 (3) : 719-744. doi: 10.3934/cpaa.2017035

References:

[1]	S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), 97-127. doi: 10.2307/121020.
[2]	S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23. doi: 10.1007/BF02392822.
[3]	S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. Ⅱ, Amer. J. Math., 123 (2001), 1071-1101.
[4]	S. Alinhac, An example of blowup at infinity for quasilinear wave equations, Asterisque, 284 (2003), 1-91.
[5]	D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205.
[6]	D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014.
[7]	Binbing Ding, Yingbo Liu and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅰ, SIAM Journal on Mathematical Analysis, 47 (2015), 4192-4228. doi: 10.1137/151004793.
[8]	Binbing Ding, Ingo Witt and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅱ, J. Differential Equations, 261 (2016), 1429-1471. doi: 10.1016/j.jde.2016.04.002.
[9]	Binbing Ding, Ingo Witt and Huicheng Yin, Blowup of classical solutions for 2-D quasilinear wave equations with small initial data, Quart. Appl. Math., 73 (2015), 773-796. doi: 10.1090/qam/1410.
[10]	Binbing Ding, Ingo Witt and Huicheng Yin, On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations, 252 (2012), 3608-3629. doi: 10.1016/j.jde.2011.11.018.
[11]	P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955.
[12]	Fei Guo and phenomena Wave-breaking, decay properties and limit behaviour of solutions of the Degasperis-Procesi equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 805-824. doi: 10.1017/S0308210511000321.
[13]	Fei Guo and Weiwei Peng, Blowup solutions for the generalized two-component CamassaHolm system on the circle, Nonlinear Anal., 105 (2014), 120-133. doi: 10.1016/j.na.2014.03.021.
[14]	L. Hömander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, MittagLeffler report No. 5,1985.
[15]	L. Hömander, Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26, Springer Verlag, Heidelberg, 1997.
[16]	A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J., 24 (1995), 575-615. doi: 10.14492/hokmj/1380892610.
[17]	F. John, Blow-up of radial solutions of u_tt = c²(u_t)∆u in three space dimensions, Mat. Apl. Comput., 4 (1985), 3-18.
[18]	F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403.
[19]	M. Keel, H. 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.
[20]	S. Klainerman, The Null Condition and Global Existence to Nonlinear Wave Equations, Lectures in Appl. Math. , 23, Amer. Math. Soc. , Providence, RI, 1986.
[21]	S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ℝⁿ⁺¹, Comm. Pure Appl. Math., 40 (1987), 111-117. doi: 10.1002/cpa.3160400105.
[22]	Yutian Lei, Singularity analysis of Ginzburg-Landau energy related to p-wave superconductivity, Z. Angew. Math. Phys., 64 (2013), 1249-1266. doi: 10.1007/s00033-012-0284-y.
[23]	Ta-tsien Li and Yun-mei Chen, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations, 13 (1988), 383-422. doi: 10.1080/03605308808820547.
[24]	H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. doi: 10.1002/cpa.3160430403.
[25]	H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. doi: 10.1002/cpa.3160450902.
[26]	H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157. doi: 10.1353/ajm.2008.0009.
[27]	H. Lindblad, M. Nakamura and C. D. Sogge, Remarks on global solutions for nonlinear wave equations under the standard null conditions, J. Differential Equations, 254 (2013), 1396-1436. doi: 10.1016/j.jde.2012.10.022.
[28]	J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, preprint, arXiv: 1407.6320.
[29]	Sijue Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220. doi: 10.1007/s00222-010-0288-1.
[30]	Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8.

show all references

References:

[1]	S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), 97-127. doi: 10.2307/121020.
[2]	S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23. doi: 10.1007/BF02392822.
[3]	S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. Ⅱ, Amer. J. Math., 123 (2001), 1071-1101.
[4]	S. Alinhac, An example of blowup at infinity for quasilinear wave equations, Asterisque, 284 (2003), 1-91.
[5]	D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205.
[6]	D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014.
[7]	Binbing Ding, Yingbo Liu and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅰ, SIAM Journal on Mathematical Analysis, 47 (2015), 4192-4228. doi: 10.1137/151004793.
[8]	Binbing Ding, Ingo Witt and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅱ, J. Differential Equations, 261 (2016), 1429-1471. doi: 10.1016/j.jde.2016.04.002.
[9]	Binbing Ding, Ingo Witt and Huicheng Yin, Blowup of classical solutions for 2-D quasilinear wave equations with small initial data, Quart. Appl. Math., 73 (2015), 773-796. doi: 10.1090/qam/1410.
[10]	Binbing Ding, Ingo Witt and Huicheng Yin, On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations, 252 (2012), 3608-3629. doi: 10.1016/j.jde.2011.11.018.
[11]	P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955.
[12]	Fei Guo and phenomena Wave-breaking, decay properties and limit behaviour of solutions of the Degasperis-Procesi equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 805-824. doi: 10.1017/S0308210511000321.
[13]	Fei Guo and Weiwei Peng, Blowup solutions for the generalized two-component CamassaHolm system on the circle, Nonlinear Anal., 105 (2014), 120-133. doi: 10.1016/j.na.2014.03.021.
[14]	L. Hömander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, MittagLeffler report No. 5,1985.
[15]	L. Hömander, Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26, Springer Verlag, Heidelberg, 1997.
[16]	A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J., 24 (1995), 575-615. doi: 10.14492/hokmj/1380892610.
[17]	F. John, Blow-up of radial solutions of u_tt = c²(u_t)∆u in three space dimensions, Mat. Apl. Comput., 4 (1985), 3-18.
[18]	F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403.
[19]	M. Keel, H. 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.
[20]	S. Klainerman, The Null Condition and Global Existence to Nonlinear Wave Equations, Lectures in Appl. Math. , 23, Amer. Math. Soc. , Providence, RI, 1986.
[21]	S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ℝⁿ⁺¹, Comm. Pure Appl. Math., 40 (1987), 111-117. doi: 10.1002/cpa.3160400105.
[22]	Yutian Lei, Singularity analysis of Ginzburg-Landau energy related to p-wave superconductivity, Z. Angew. Math. Phys., 64 (2013), 1249-1266. doi: 10.1007/s00033-012-0284-y.
[23]	Ta-tsien Li and Yun-mei Chen, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations, 13 (1988), 383-422. doi: 10.1080/03605308808820547.
[24]	H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. doi: 10.1002/cpa.3160430403.
[25]	H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. doi: 10.1002/cpa.3160450902.
[26]	H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157. doi: 10.1353/ajm.2008.0009.
[27]	H. Lindblad, M. Nakamura and C. D. Sogge, Remarks on global solutions for nonlinear wave equations under the standard null conditions, J. Differential Equations, 254 (2013), 1396-1436. doi: 10.1016/j.jde.2012.10.022.
[28]	J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, preprint, arXiv: 1407.6320.
[29]	Sijue Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220. doi: 10.1007/s00222-010-0288-1.
[30]	Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8.

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Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

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