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Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin
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 |
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.
References:
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Remarks on the breakdown of smooth solutions for the 3-d euler equations, Comm. Math. Phys., 94 (1984), 61-66.
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L. Brandolese,
Local-in-space criteria for blowup in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414.
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A. Bressan,
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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.
|
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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. |

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