Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the lifespan and the blowup mechanism of smooth solutions to a class of 2-D nonlinear wave equations with small initial data

Authors: Ding Bingbing, Ingo Witt and Yin Huicheng
Journal: Quart. Appl. Math. 73 (2015), 773-796
MSC (2010): Primary 35L65, 35J70, 35R35
DOI: https://doi.org/10.1090/qam/1410
Published electronically: September 11, 2015
MathSciNet review: 3432283
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Abstract: This paper is concerned with the lifespan of and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $ \partial _t^2u-\sum _{i=1}^2\partial _i(c_i^2(u)\partial _iu)$ $ =0$, where $ c_i(u)\in C^{\infty }(\mathbb{R}^n)$, $ c_i(0)\neq 0$, and $ (c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physical background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $ (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x))$ with $ u_0(x), u_1(x)\in C_0^{\infty }(\mathbb{R}^2)$, and $ \varepsilon >0$ is small, we will show that the classical solution $ u(t,x)$ stops to be smooth at some finite time $ T_{\varepsilon }$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $ \nabla _{t,x}u(t,x)$, while $ u(t,x)$ itself is continuous up to the blowup time  $ T_{\varepsilon }$.

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  • [1] R. K. Agarwal and D. W. Halt, A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows, Frontiers of Computational Fluid Dynamics 1994 (D. A. Caughey, M. M. Hafez, eds.), 1995.
  • [2] Giuseppe Alì and John K. Hunter, Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations, Comm. Pure Appl. Math. 60 (2007), no. 10, 1522-1557. MR 2342956 (2008m:35349), https://doi.org/10.1002/cpa.20199
  • [3] Serge Alinhac and Patrick Gérard, Pseudo-differential operators and the Nash-Moser theorem, Graduate Studies in Mathematics, vol. 82, American Mathematical Society, Providence, RI, 2007. Translated from the 1991 French original by Stephen S. Wilson. MR 2304160 (2007m:35001)
  • [4] Serge Alinhac, Blowup of small data solutions for a quasilinear wave equation in two space dimensions, Ann. of Math. (2) 149 (1999), no. 1, 97-127 (English, with English and French summaries). MR 1680539 (2000d:35147), https://doi.org/10.2307/121020
  • [5] Serge Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. II, Acta Math. 182 (1999), no. 1, 1-23. MR 1687180 (2000d:35148), https://doi.org/10.1007/BF02392822
  • [6] Serge Alinhac, An example of blowup at infinity for a quasilinear wave equation, Astérisque 284 (2003), 1-91 (English, with English and French summaries, Autour de l'analyse microlocale). MR 2003417 (2005a:35197)
  • [7] Alberto Bressan and Yuxi Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys. 266 (2006), no. 2, 471-497. MR 2238886 (2007i:35168), https://doi.org/10.1007/s00220-006-0047-8
  • [8] Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), no. 2, 267-282. MR 820070 (87c:35111), https://doi.org/10.1002/cpa.3160390205
  • [9] Ding Bingbing and Yin Huicheng, On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations 252 (2012), no. 5, 3608-3629. MR 2876666 (2012k:35330), https://doi.org/10.1016/j.jde.2011.11.018
  • [10] Robert T. Glassey, John K. Hunter, and Yuxi Zheng, Singularities of a variational wave equation, J. Differential Equations 129 (1996), no. 1, 49-78. MR 1400796 (97f:35139), https://doi.org/10.1006/jdeq.1996.0111
  • [11] H. Holden, K. H. Karlsen, and N. H. Risebro, A convergent finite-difference method for a nonlinear variational wave equation, IMA J. Numer. Anal. 29 (2009), no. 3, 539-572. MR 2520158 (2010k:65160), https://doi.org/10.1093/imanum/drn026
  • [12] Helge Holden and Xavier Raynaud, Global semigroup of conservative solutions of the nonlinear variational wave equation, Arch. Ration. Mech. Anal. 201 (2011), no. 3, 871-964. MR 2824468 (2012h:35219), https://doi.org/10.1007/s00205-011-0403-5
  • [13] Lars Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Pseudodifferential operators (Oberwolfach, 1986) Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 214-280. MR 0897781 (88j:35024), https://doi.org/10.1007/BFb0077745
  • [14] Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700 (98e:35103)
  • [15] John K. Hunter and Ralph Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), no. 6, 1498-1521. MR 1135995 (93a:76005), https://doi.org/10.1137/0151075
  • [16] John K. Hunter and Yu Xi Zheng, On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions, Arch. Rational Mech. Anal. 129 (1995), no. 4, 305-353. MR 1361013 (96m:35215), https://doi.org/10.1007/BF00379259
  • [17] F. John, Blow-up of radial solutions of $ u_{tt}=c^2(u_t)\Delta u$ in three space dimensions, Mat. Apl. Comput. 4 (1985), no. 1, 3-18 (English, with Portuguese summary). MR 808321 (87c:35114)
  • [18] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space $ {\bf R}^{n+1}$, Comm. Pure Appl. Math. 40 (1987), no. 1, 111-117. MR 865359 (88a:46035), https://doi.org/10.1002/cpa.3160400105
  • [19] Zhen Lei and Yuxi Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations 236 (2007), no. 1, 280-292. MR 2319927 (2008c:35181), https://doi.org/10.1016/j.jde.2007.01.024
  • [20] Fengbai Li and Wei Xiao, Interaction of four rarefaction waves in the bi-symmetric class of the pressure-gradient system, J. Differential Equations 252 (2012), no. 6, 3920-3952. MR 2875607 (2012m:35195), https://doi.org/10.1016/j.jde.2011.11.010
  • [21] Jun Li, Ingo Witt, and Huicheng Yin, On the blowup and lifespan of smooth solutions to a class of 2-D nonlinear wave equations with small initial data, Quart. Appl. Math. 73 (2015), no. 2, 219-251. MR 3357493, https://doi.org/10.1090/S0033-569X-2015-01374-2
  • [22] Hans Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math. 43 (1990), no. 4, 445-472. MR 1047332 (91i:35129), https://doi.org/10.1002/cpa.3160430403
  • [23] Hans Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math. 130 (2008), no. 1, 115-157. MR 2382144 (2009b:58062), https://doi.org/10.1353/ajm.2008.0009
  • [24] R. A. Saxton, Dynamic instability of the liquid crystal director, Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, ME, 1988) Contemp. Math., vol. 100, Amer. Math. Soc., Providence, RI, 1989, pp. 325-330. MR 1033527 (90k:35246), https://doi.org/10.1090/conm/100/1033527
  • [25] Kyungwoo Song and Yuxi Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1365-1380. MR 2505709 (2010f:35244), https://doi.org/10.3934/dcds.2009.24.1365
  • [26] Huicheng Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J. 175 (2004), 125-164. MR 2085314 (2005f:35203)
  • [27] Ping Zhang and Yuxi Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal. 166 (2003), no. 4, 303-319. MR 1961443 (2003m:35175), https://doi.org/10.1007/s00205-002-0232-7
  • [28] Ping Zhang and Yuxi Zheng, Conservative solutions to a system of variational wave equations of nematic liquid crystals, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 701-727. MR 2591971 (2012d:35214), https://doi.org/10.1007/s00205-009-0222-0
  • [29] Yuxi Zheng, Systems of conservation laws, Two-dimensional Riemann problems, Progress in Nonlinear Differential Equations and their Applications, 38, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1839813 (2002e:35155)
  • [30] Yuxi Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser. 22 (2006), no. 2, 177-210. MR 2215512 (2007b:35229), https://doi.org/10.1007/s10255-006-0296-5
  • [31] Yuxi Zheng and Zachary Robinson, The pressure gradient system, Methods Appl. Anal. 17 (2010), no. 3, 263-278. MR 2785874 (2012e:35156), https://doi.org/10.4310/MAA.2010.v17.n3.a2

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Additional Information

Ding Bingbing
Affiliation: Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: 13851929236@163.com

Ingo Witt
Affiliation: Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, D-37073 Göttingen, Germany
Email: iwitt@uni-math.gwdg.de

Yin Huicheng
Affiliation: Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: huicheng@nju.edu.cn

DOI: https://doi.org/10.1090/qam/1410
Keywords: Nonlinear wave equation, blowup, lifespan, blowup system, Klainerman fields, Nash-Moser-H\"ormander iteration
Received by editor(s): April 21, 2014
Received by editor(s) in revised form: July 19, 2014
Published electronically: September 11, 2015
Additional Notes: The first and third authors were supported by the NSFC (No. 10931007, No. 11025105), by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and by the DFG via the Sino-German project “Analysis of PDEs and Applications.” Some parts of this work were done when the first and third authors were visiting the Mathematical Institute of the University of Göttingen. The third author is the corresponding author.
The second author was supported by the DFG via the Sino-German project “Analysis of PDEs and Applications”.
Article copyright: © Copyright 2015 Brown University

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