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

The modified Camassa-Holm equation in Lagrangian coordinates

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.

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. BealeT. 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. CamassaJ. 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. ChenY. LiuC. 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. ChertockJ.-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. FuG. GuiY. 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. GuiY. LiuP. 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. HimonasG. MisiolekG. 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. JiangL. 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. LiuP. J. OlverC. 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. BealeT. 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. CamassaJ. 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. ChenY. LiuC. 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. ChertockJ.-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. FuG. GuiY. 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. GuiY. LiuP. 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. HimonasG. MisiolekG. 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. JiangL. 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. LiuP. J. OlverC. 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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