On the blowup and lifespan of smooth solutions to a class of 2-D nonlinear wave equations with small initial data
Authors:
Li Jun, Ingo Witt and Yin Huicheng
Journal:
Quart. Appl. Math. 73 (2015), 219-251
MSC (2010):
Primary 35L15, 35L65, 35L70.
DOI:
https://doi.org/10.1090/S0033-569X-2015-01374-2
Published electronically:
January 29, 2015
MathSciNet review:
3357493
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We are concerned with a class of two-dimensional nonlinear wave equations $\partial _t^2u-\text {div}(c^2(u)\nabla u)=0$ or $\partial _t^2u-c(u)\mathrm {div}(c(u)\nabla u)=0$ with small initial data $(u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x))$, where $c(u)$ is a smooth function, $c(0)\not =0$, $x\in \mathbb R^2$, $u_0(x), u_1(x)\in C_0^{\infty }(\mathbb R^2)$ depend only on $r=\sqrt {x_1^2+x_2^2}$, and $\varepsilon >0$ is sufficiently small. Such equations arise in a pressure-gradient model of fluid dynamics, as well as in a liquid crystal model or other variational wave equations. When $c’(0)\not = 0$ or $c’(0)=0$, $c”(0)\not = 0$, we establish blowup and determine the lifespan of smooth solutions.
References
- 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 (D. A. Caughey and M. M. Hafez), 1994.
- 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, DOI https://doi.org/10.2307/121020
- S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597–618. MR 1856402, DOI https://doi.org/10.1007/s002220100165
- 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
- Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), no. 2, 267–282. MR 820070, DOI https://doi.org/10.1002/cpa.3160390205
- 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, DOI https://doi.org/10.1016/j.jde.2011.11.018
- 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, DOI https://doi.org/10.1006/jdeq.1996.0111
- Paul Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations 18 (1993), no. 5-6, 895–916. MR 1218523, DOI https://doi.org/10.1080/03605309308820955
- 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 897781, DOI https://doi.org/10.1007/BFb0077745
- Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
- Akira Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J. 24 (1995), no. 3, 575–615. MR 1357032, DOI https://doi.org/10.14492/hokmj/1380892610
- Akira Hoshiga, The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space, Hokkaido Math. J. 37 (2008), no. 4, 669–688. MR 2474170, DOI https://doi.org/10.14492/hokmj/1249046363
- John K. Hunter and Ralph Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), no. 6, 1498–1521. MR 1135995, DOI https://doi.org/10.1137/0151075
- 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
- S. Klainerman, 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., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683
- 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, DOI https://doi.org/10.1002/cpa.3160400105
- S. Klainerman and Gustavo Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085, DOI https://doi.org/10.1002/cpa.3160360106
- Li Ta-tsien and Chen Yun-mei, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations 13 (1988), no. 4, 383–422. MR 920909, DOI https://doi.org/10.1080/03605308808820547
- 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, DOI https://doi.org/10.1002/cpa.3160430403
- Hans Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math. 45 (1992), no. 9, 1063–1096. MR 1177476, DOI https://doi.org/10.1002/cpa.3160450902
- Hans Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math. 130 (2008), no. 1, 115–157. MR 2382144, DOI https://doi.org/10.1353/ajm.2008.0009
- Jason Metcalfe and Christopher D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles, Discrete Contin. Dyn. Syst. 28 (2010), no. 4, 1589–1601. MR 2679723, DOI https://doi.org/10.3934/dcds.2010.28.1589
- 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, DOI https://doi.org/10.1090/conm/100/1033527
- 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, DOI https://doi.org/10.1017/S002776300000893X
- Huicheng Yin and Qingjiu Qiu, The blowup of solutions for 3-D axisymmetric compressible Euler equations, Nagoya Math. J. 154 (1999), 157–169. MR 1689178, DOI https://doi.org/10.1017/S0027763000025368
- Ping Zhang and Yuxi Zheng, On the second-order asymptotic equation of a variational wave equation, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 2, 483–509. MR 1899833, DOI https://doi.org/10.1017/S0308210500001748
- Ping Zhang and Yuxi Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math. 65 (2012), no. 5, 683–726. MR 2898888, DOI https://doi.org/10.1002/cpa.20380
- Yuxi Zheng, Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions, Comm. Partial Differential Equations 22 (1997), no. 11-12, 1849–1868. MR 1629498, DOI https://doi.org/10.1080/03605309708821323
References
- 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 (D. A. Caughey and M. M. Hafez), 1994.
- Serge Alinhac, Blowup of small data solutions for a quasilinear wave equation in two space dimensions, English, with English and French summaries, Ann. of Math. (2) 149 (1999), no. 1, 97–127. MR 1680539 (2000d:35147), DOI https://doi.org/10.2307/121020
- S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597–618. MR 1856402 (2002i:35127), DOI https://doi.org/10.1007/s002220100165
- Serge Alinhac, An example of blowup at infinity for a quasilinear wave equation, English, with English and French summaries, Autour de l’analyse microlocale, Astérisque 284 (2003), 1–91. MR 2003417 (2005a:35197)
- 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), DOI https://doi.org/10.1002/cpa.3160390205
- 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), DOI https://doi.org/10.1016/j.jde.2011.11.018
- 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), DOI https://doi.org/10.1006/jdeq.1996.0111
- Paul Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations 18 (1993), no. 5-6, 895–916. MR 1218523 (94g:35149), DOI https://doi.org/10.1080/03605309308820955
- L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Psuedodifferential Operators (Oberwolfach, 1986), Lecture Notes in Math. 1256, Springer, Berlin, 1987. MR 0897781 (88j:35024)
- L. Hörmander, Lectures on nonlinear hyperbolic equations, Mathematiques & Applications 26, Springer, Berlin, 1997. MR 1466700 (98e:35103)
- Akira Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J. 24 (1995), no. 3, 575–615. MR 1357032 (96m:35213)
- Akira Hoshiga, The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space, Hokkaido Math. J. 37 (2008), no. 4, 669–688. MR 2474170 (2010j:35332)
- John K. Hunter and Ralph Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), no. 6, 1498–1521. MR 1135995 (93a:76005), DOI https://doi.org/10.1137/0151075
- F. John, Blow-up of radial solutions of $u_{tt}=c^2(u_t)\Delta u$ in three space dimensions, English, with Portuguese summary, Mat. Apl. Comput. 4 (1985), no. 1, 3–18. MR 808321 (87c:35114)
- S. Klainerman, The null condition and global existence to nonlinear wave equations, mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683 (87h:35217)
- S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space $\textbf {R}^{n+1}$, Comm. Pure Appl. Math. 40 (1987), no. 1, 111–117. MR 865359 (88a:46035), DOI https://doi.org/10.1002/cpa.3160400105
- S. Klainerman and Gustavo Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085 (84a:35173), DOI https://doi.org/10.1002/cpa.3160360106
- Li Ta-tsien and Chen Yun-mei, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations 13 (1988), no. 4, 383–422. MR 920909 (89e:35102), DOI https://doi.org/10.1080/03605308808820547
- 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), DOI https://doi.org/10.1002/cpa.3160430403
- Hans Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math. 45 (1992), no. 9, 1063–1096. MR 1177476 (94a:35080), DOI https://doi.org/10.1002/cpa.3160450902
- Hans Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math. 130 (2008), no. 1, 115–157. MR 2382144 (2009b:58062), DOI https://doi.org/10.1353/ajm.2008.0009
- Jason Metcalfe and Christopher D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles, Discrete Contin. Dyn. Syst. 28 (2010), no. 4, 1589–1601. MR 2679723 (2011g:35257), DOI https://doi.org/10.3934/dcds.2010.28.1589
- 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), DOI https://doi.org/10.1090/conm/100/1033527
- 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)
- Huicheng Yin and Qingjiu Qiu, The blowup of solutions for 3-D axisymmetric compressible Euler equations, Nagoya Math. J. 154 (1999), 157–169. MR 1689178 (2000g:35170)
- Ping Zhang and Yuxi Zheng, On the second-order asymptotic equation of a variational wave equation, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 2, 483–509. MR 1899833 (2002m:35138), DOI https://doi.org/10.1017/S0308210500001748
- Ping Zhang and Yuxi Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math. 65 (2012), no. 5, 683–726. MR 2898888, DOI https://doi.org/10.1002/cpa.20380
- Yuxi Zheng, Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions, Comm. Partial Differential Equations 22 (1997), no. 11-12, 1849–1868. MR 1629498 (99c:35191), DOI https://doi.org/10.1080/03605309708821323
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35L15,
35L65,
35L70.
Retrieve articles in all journals
with MSC (2010):
35L15,
35L65,
35L70.
Additional Information
Li Jun
Affiliation:
Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email:
lijun@nju.edu.cn
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
Keywords:
Nonlinear wave equation,
blowup,
lifespan,
Klainerman-Sobolev inequality
Received by editor(s):
January 30, 2013
Published electronically:
January 29, 2015
Additional Notes:
The first and third authors were supported by the NSFC (No. 10931007, No. 11025105, No. 11001122), by the Doctoral Program Foundation of the Ministry of Education of China (No. 20090091110005), and by the DFG via the joint Sino-German project “Analysis of PDEs and application”. This work was done when the first and third authors were visiting the Mathematical Institute of the University of Göttingen. The second author was partially supported by the DFG via the Sino-German project “Analysis of PDEs and application”. The second author is the corresponding author.
Article copyright:
© Copyright 2015
Brown University