October  2018, 38(10): 5205-5220. doi: 10.3934/dcds.2018230

Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics

1. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

2. 

Department of Mathematics, Southwestern University of Finance and economics, Sichuan 611130, China

3. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

4. 

College of Mathematics and statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Zeng Rong and Shouming Zhou

Received  February 2018 Published  July 2018

Fund Project: The first author (Yang) is supported by Chongqing Normal University (Grant No. YKC18038). The third author (Zhou) is partly supported by the Science and Technology Research Program of Chongqing Municipal Education Commission, Natural Science Foundation of Chongqing. The fourth author (Mu) is supported by NSFC (Grant No. 11571062 and 11771062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and the Fundamental Research Funds for the Central Universities(Grant No. 106112016CDJXZ238826)

It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution $u = u(t,x)$, an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities $u$ and $v = 2 \arctan u_x$ along each characteristic, it is obtained that the Cauchy problem with general initial data $u_0∈ H^1(\mathbb{R})$ has a unique global conservative solution.

Citation: Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
References:
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A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

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

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A. BressanG. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discrete. Contin. Dyn. Syst., 35 (2015), 25-42.

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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.

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R. CamassaD. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

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G. M. CocliteH. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.

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A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

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A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

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A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.

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A. Costantin and J. Escher, Global existence of solutions and breaking waves for a shallow water equation, Ann. Sc. Norm. Super. Pisa CL. Sci., 26 (1998), 303-328.

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A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.

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A. Constantin and W. Strauss, Stablity of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.

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C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8 (2011), 159-168. doi: 10.1142/S0219891611002366.

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J. Eckhardt, The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data, Arch. Ration. Mech. Anal., 224 (2017), 21-52. doi: 10.1007/s00205-016-1066-z.

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Q. FengF. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on times scales, J. Math. Anal. Appl., 382 (2011), 772-784. doi: 10.1016/j.jmaa.2011.04.077.

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A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.

[21]

X. HaoL. LiuY. Wu and Q. Sun, Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions, Nonlinear Anal., 73 (2010), 1653-1662. doi: 10.1016/j.na.2010.04.074.

[22]

X. HeA. Qian and W. Zou, Existence and concentration of positive solutions for quasi-linear Schrodinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168. doi: 10.1088/0951-7715/26/12/3137.

[23]

F. Li, Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shell system, J. Differential Equations, 249 (2010), 1241-1257. doi: 10.1016/j.jde.2010.05.005.

[24]

O. Lopes, Stability of peakons for the generalized Camassa-Holm equation, Electron. J. Differential Equations, 2003 (2003), 1-12.

[25]

X. MaP. Wang and W. Wei, Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains, J. Funct. Anal., 274 (2018), 252-277. doi: 10.1016/j.jfa.2017.10.002.

[26]

T. Qian and M. Tang, Peakons and periodic cusp waves in a generalized Camassa-Holm equation, Chaos Solitons Fractals, 12 (2001), 1347-1460. doi: 10.1016/S0960-0779(00)00117-X.

[27]

J. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26 (2005), 1149-1162. doi: 10.1016/j.chaos.2005.02.021.

[28]

L. Tian and X. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19 (2004), 621-637. doi: 10.1016/S0960-0779(03)00192-9.

[29]

P. Wang, The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height, Pacific J. Math., 267 (2014), 489-509. doi: 10.2140/pjm.2014.267.489.

[30]

P. Wang and L. Zhao, Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds, Nonlinear Anal., 130 (2016), 1-17. doi: 10.1016/j.na.2015.09.021.

[31]

P. Wang and D. Zhang, Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature, J. Differential Equations, 262 (2017), 5534-5564. doi: 10.1016/j.jde.2017.02.010.

[32]

P. Wang and D. Zhang, Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature, J. Differential Equations, 262 (2017), 5534-5564. doi: 10.1016/j.jde.2017.02.010.

[33]

Z. Yin, On the blow-up scenario for the generalized Camassa-Holm equation, Comm. Partial Differential Equations, 29 (2004), 867-877. doi: 10.1081/PDE-120037334.

[34]

Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Analysis, 66 (2007), 460-471. doi: 10.1016/j.na.2005.11.040.

[35]

Z. Yin, On the Cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys., 10 (2003), 10-15. doi: 10.2991/jnmp.2003.10.1.2.

[36]

S. ZhouZ. QiaoZhijunC. Mu and L. Wei, Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude, J. Differential Equations, 263 (2017), 910-933. doi: 10.1016/j.jde.2017.03.002.

[37]

S. Zhou and C. Mu, Global conservative solutions and dissipative solutions of the generalized Camassa-Holm equation, Discrete. Contin. Dyn. Syst., 4 (2013), 1713-1739. doi: 10.3934/dcds.2013.33.1713.

show all references

References:
[1]

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

[2]

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

[3]

A. BressanG. Chen and Q. Zhang, Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discrete. Contin. Dyn. Syst., 35 (2015), 25-42.

[4]

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.

[5]

R. CamassaD. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[6]

G. M. CocliteH. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.

[7]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[9]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[10]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.

[11]

A. Costantin and J. Escher, Global existence of solutions and breaking waves for a shallow water equation, Ann. Sc. Norm. Super. Pisa CL. Sci., 26 (1998), 303-328.

[12]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.

[13]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[14]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[15]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x.

[16]

A. Constantin and W. Strauss, Stablity of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.

[17]

C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8 (2011), 159-168. doi: 10.1142/S0219891611002366.

[18]

J. Eckhardt, The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data, Arch. Ration. Mech. Anal., 224 (2017), 21-52. doi: 10.1007/s00205-016-1066-z.

[19]

Q. FengF. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on times scales, J. Math. Anal. Appl., 382 (2011), 772-784. doi: 10.1016/j.jmaa.2011.04.077.

[20]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.

[21]

X. HaoL. LiuY. Wu and Q. Sun, Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions, Nonlinear Anal., 73 (2010), 1653-1662. doi: 10.1016/j.na.2010.04.074.

[22]

X. HeA. Qian and W. Zou, Existence and concentration of positive solutions for quasi-linear Schrodinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168. doi: 10.1088/0951-7715/26/12/3137.

[23]

F. Li, Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shell system, J. Differential Equations, 249 (2010), 1241-1257. doi: 10.1016/j.jde.2010.05.005.

[24]

O. Lopes, Stability of peakons for the generalized Camassa-Holm equation, Electron. J. Differential Equations, 2003 (2003), 1-12.

[25]

X. MaP. Wang and W. Wei, Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains, J. Funct. Anal., 274 (2018), 252-277. doi: 10.1016/j.jfa.2017.10.002.

[26]

T. Qian and M. Tang, Peakons and periodic cusp waves in a generalized Camassa-Holm equation, Chaos Solitons Fractals, 12 (2001), 1347-1460. doi: 10.1016/S0960-0779(00)00117-X.

[27]

J. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26 (2005), 1149-1162. doi: 10.1016/j.chaos.2005.02.021.

[28]

L. Tian and X. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19 (2004), 621-637. doi: 10.1016/S0960-0779(03)00192-9.

[29]

P. Wang, The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height, Pacific J. Math., 267 (2014), 489-509. doi: 10.2140/pjm.2014.267.489.

[30]

P. Wang and L. Zhao, Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds, Nonlinear Anal., 130 (2016), 1-17. doi: 10.1016/j.na.2015.09.021.

[31]

P. Wang and D. Zhang, Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature, J. Differential Equations, 262 (2017), 5534-5564. doi: 10.1016/j.jde.2017.02.010.

[32]

P. Wang and D. Zhang, Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature, J. Differential Equations, 262 (2017), 5534-5564. doi: 10.1016/j.jde.2017.02.010.

[33]

Z. Yin, On the blow-up scenario for the generalized Camassa-Holm equation, Comm. Partial Differential Equations, 29 (2004), 867-877. doi: 10.1081/PDE-120037334.

[34]

Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Analysis, 66 (2007), 460-471. doi: 10.1016/j.na.2005.11.040.

[35]

Z. Yin, On the Cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys., 10 (2003), 10-15. doi: 10.2991/jnmp.2003.10.1.2.

[36]

S. ZhouZ. QiaoZhijunC. Mu and L. Wei, Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude, J. Differential Equations, 263 (2017), 910-933. doi: 10.1016/j.jde.2017.03.002.

[37]

S. Zhou and C. Mu, Global conservative solutions and dissipative solutions of the generalized Camassa-Holm equation, Discrete. Contin. Dyn. Syst., 4 (2013), 1713-1739. doi: 10.3934/dcds.2013.33.1713.

Figure 1.  The Lipschitz continuous text function $\varphi^\epsilon$ introduced at (32)
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