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August  2018, 23(6): 2545-2592. doi: 10.3934/dcdsb.2018067

The modified Camassa-Holm equation in Lagrangian coordinates

Yu Gao 1,2,, and  Jian-Guo Liu 2,

1. 

Department of Mathematics of Harbin Institute of Technology, Harbin, 150001, China
2. 

Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA

* Corresponding author: Yu Gao

Received  June 2017 Revised  September 2017 Published  August 2018 Early access  February 2018

Fund Project: The second author is supported by KI-Net NSF RNMS (Grant No. 1107444) and NSF DMS (Grant No. 1514826).

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}≥ \frac{1}{||m_0||_{L^∞}||m_0||_{L^1}}.$ And there is a unique solution $X(ξ, t)$ to the Lagrange dynamics which is a strictly monotonic function of $ξ$ for any $t∈[0, T_{max})$: $X_ξ(·, t)>0$. As $t$ approaching $T_{max}$, we prove that the classical solution $m(·, t)$ in Eulerian coordinates has a unique limit $m(·, T_{max})$ in Radon measure space and there is a point $ξ_0$ such that $X_ξ(ξ_0, T_{max}) = 0$ which means $T_{max}$ is an onset time of collisions of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.

Keywords: Integrable system, Lagrange dynamics, finite time blow up, local classical solutions, global weak solutions.
Mathematics Subject Classification: Primary: 35G25, 35B44; Secondary: 35D30.
Citation: Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
References:
[1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-d euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[2]

L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414.  doi: 10.1007/s00220-014-1958-4.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, volume 20. Oxford University Press on Demand, 2000.

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.

[5]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.

[6]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[7]

R. Camassa, J. Huang and L. Lee, Integral and integrable algorithms for a nonlinear shallow-water wave equation, J. Comput. Phys., 216 (2006), 547-572.  doi: 10.1016/j.jcp.2005.12.013.

[8]

R. M. Chen, Y. Liu, C. Qu and S. Zhang, Oscillation-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251.  doi: 10.1016/j.aim.2014.12.003.

[9]

A. Chertock, J.-G. Liu and T. Pendleton, Convergence of a particle method and global weak solutions of a family of evolutionary PDEs, SIAM J. Numer. Anal., 50 (2012), 1-21.  doi: 10.1137/110831386.

[10]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, In Ann. Inst. Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757.

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.

[12]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[13]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Int. Eq., 14 (2001), 953-988. 

[14]

A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.  doi: 10.1007/BF00994638.

[15]

Y. Fu, G. Gui, Y. Liu and C. Qu, On the cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938.  doi: 10.1016/j.jde.2013.05.024.

[16]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.

[17]

Y. Gao and J.-G. Liu, Global convergence of a sticky particle method for the modified Camassa-Holm equation, SIAM J. Math. Anal., 49 (2017), 1267-1294.  doi: 10.1137/16M1102069.

[18]

G. Gui, Y. Liu, P. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.  doi: 10.1007/s00220-012-1566-0.

[19]

H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44 (2006), 1655-1680.  doi: 10.1137/040611975.

[20]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.  doi: 10.1080/03605300601088674.

[21]

A.A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522.  doi: 10.1007/s00220-006-0172-4.

[22]

Z. Jiang, L. Ni and Y. Zhou, Wave breaking of the Camassa-Holm equation, J. Nonlinear Sci., 22 (2012), 235-245.  doi: 10.1007/s00332-011-9115-0.

[23]

G. Leoni, A First Course in Sobolev Spaces, volume 105, American Mathematical Society Providence, RI, 2009.

[24]

Y. Liu, P. J. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Anal. Appl., 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.

[25]

H. P. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.  doi: 10.4310/AJM.1998.v2.n4.a10.

[26]

L. Molinet, On well-posedness results for Camassa-Holm equation on the line: A survey, J. Nonlinear Math. Phy., 11 (2013), 521-533.  doi: 10.2991/jnmp.2004.11.4.8.

[27]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021.  doi: 10.1016/j.jde.2011.01.030.

[28]

P. J. Olver and P. Rosenau, Tri-hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.

[29]

Z. Qiao, A new integrable equation with cuspons and w/m-shape-peaks solitons J. Math. Phys. , 47 (2006), 112701, 9 pp. doi: 10.1063/1.2365758.

[30]

C. Villani, Topics in Optimal Transportation, Number 58. American Mathematical Soc., 2003.

[31]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure. Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[32]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), 1815-1844.  doi: 10.1081/PDE-120016129.

[33]

Q. Zhang, Global wellposedness of cubic Camassa-Holm equations, Nonlinear Anal., 133 (2016), 61-73.  doi: 10.1016/j.na.2015.11.020.

[34]

Y. Zhou, Blow-up of solutions to the DGH equation, J. Funct. Anal., 250 (2007), 227-248.  doi: 10.1016/j.jfa.2007.04.019.

[35]

Y. Zhou and Z. Guo, Blow up and propagation speed of solutions to the DGH equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 657-670.  doi: 10.3934/dcdsb.2009.12.657.

show all references

References:
[1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-d euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[2]

L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414.  doi: 10.1007/s00220-014-1958-4.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, volume 20. Oxford University Press on Demand, 2000.

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.

[5]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.

[6]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[7]

R. Camassa, J. Huang and L. Lee, Integral and integrable algorithms for a nonlinear shallow-water wave equation, J. Comput. Phys., 216 (2006), 547-572.  doi: 10.1016/j.jcp.2005.12.013.

[8]

R. M. Chen, Y. Liu, C. Qu and S. Zhang, Oscillation-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251.  doi: 10.1016/j.aim.2014.12.003.

[9]

A. Chertock, J.-G. Liu and T. Pendleton, Convergence of a particle method and global weak solutions of a family of evolutionary PDEs, SIAM J. Numer. Anal., 50 (2012), 1-21.  doi: 10.1137/110831386.

[10]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, In Ann. Inst. Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757.

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.

[12]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[13]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Int. Eq., 14 (2001), 953-988. 

[14]

A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.  doi: 10.1007/BF00994638.

[15]

Y. Fu, G. Gui, Y. Liu and C. Qu, On the cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938.  doi: 10.1016/j.jde.2013.05.024.

[16]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.

[17]

Y. Gao and J.-G. Liu, Global convergence of a sticky particle method for the modified Camassa-Holm equation, SIAM J. Math. Anal., 49 (2017), 1267-1294.  doi: 10.1137/16M1102069.

[18]

G. Gui, Y. Liu, P. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.  doi: 10.1007/s00220-012-1566-0.

[19]

H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44 (2006), 1655-1680.  doi: 10.1137/040611975.

[20]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.  doi: 10.1080/03605300601088674.

[21]

A.A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522.  doi: 10.1007/s00220-006-0172-4.

[22]

Z. Jiang, L. Ni and Y. Zhou, Wave breaking of the Camassa-Holm equation, J. Nonlinear Sci., 22 (2012), 235-245.  doi: 10.1007/s00332-011-9115-0.

[23]

G. Leoni, A First Course in Sobolev Spaces, volume 105, American Mathematical Society Providence, RI, 2009.

[24]

Y. Liu, P. J. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Anal. Appl., 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.

[25]

H. P. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.  doi: 10.4310/AJM.1998.v2.n4.a10.

[26]

L. Molinet, On well-posedness results for Camassa-Holm equation on the line: A survey, J. Nonlinear Math. Phy., 11 (2013), 521-533.  doi: 10.2991/jnmp.2004.11.4.8.

[27]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021.  doi: 10.1016/j.jde.2011.01.030.

[28]

P. J. Olver and P. Rosenau, Tri-hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.

[29]

Z. Qiao, A new integrable equation with cuspons and w/m-shape-peaks solitons J. Math. Phys. , 47 (2006), 112701, 9 pp. doi: 10.1063/1.2365758.

[30]

C. Villani, Topics in Optimal Transportation, Number 58. American Mathematical Soc., 2003.

[31]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure. Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[32]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), 1815-1844.  doi: 10.1081/PDE-120016129.

[33]

Q. Zhang, Global wellposedness of cubic Camassa-Holm equations, Nonlinear Anal., 133 (2016), 61-73.  doi: 10.1016/j.na.2015.11.020.

[34]

Y. Zhou, Blow-up of solutions to the DGH equation, J. Funct. Anal., 250 (2007), 227-248.  doi: 10.1016/j.jfa.2007.04.019.

[35]

Y. Zhou and Z. Guo, Blow up and propagation speed of solutions to the DGH equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 657-670.  doi: 10.3934/dcdsb.2009.12.657.

Figure 1.  At $T_{max}$, $X_\xi(\cdot,T_{max})\geq0$ and $X_\xi(\xi,T_{max}) = 0$ for $\xi\in\{\xi_1,\xi_4\}\cup[\xi_{21},\xi_{22}]\cup[\xi_{31},\xi_{32}]$. $F_{T_{max}} = \{x_1,x_2,x_3,x_4\}$ and $\widehat{F}_{T_{max}} = \{x_2,x_3\}$
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