May  2014, 34(5): 1995-2011. doi: 10.3934/dcds.2014.34.1995

Convergence analysis of the vortex blob method for the $b$-equation

1. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China

2. 

Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, United States

Received  February 2013 Revised  July 2013 Published  October 2013

In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak solution to the b-equation when the initial data is a nonnegative Radon measure with compact support.
Citation: Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995
References:
[1]

J. T. Beale and A. Majda, Vortex methods II: Higher order accuracy in two and three dimensions,, Math. Comput., 39 (1982), 29.  doi: 10.2307/2007618.  Google Scholar

[2]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem,, Oxford Lecture Ser. Math. Appl. 20, (2000).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

D. Chae and J.-G. Liu, Blow-up,Zero Alpha limit and the Liouville type theorem for the Euler-Poincaré equations,, Comm. Math. Phys., 314 (2012), 671.  doi: 10.1007/s00220-012-1534-8.  Google Scholar

[7]

A. Chertock, P. Du Toit and J. Marsden, Integration of the EPDiff equation by particle methods,, ESAIM Math. Model. Numer. Anal., 46 (2012), 515.  doi: 10.1051/m2an/2011054.  Google Scholar

[8]

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

[9]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60.  doi: 10.1016/j.jfa.2005.07.008.  Google Scholar

[10]

G. M. Coclite, K. H. Karlsen and N. H. Risebro, A convregent finite defference scheme for the Camassa-Holm equation with general $H^1$ initial data,, SIAM J. Numer. Anal., 46 (2008), 1554.  doi: 10.1137/060673242.  Google Scholar

[11]

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

[12]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[13]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[14]

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

[15]

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

[16]

G. H. Cottet and P. D. Koumoutsakos, Vortex Methods: Theory and Practice,, Cambridge University Press, (2000).  doi: 10.1017/CBO9780511526442.  Google Scholar

[17]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta. Mech., 127 (1998), 193.  doi: 10.1007/BF01170373.  Google Scholar

[18]

A. Degasperis and M. Procesi, Asymptotic Integrability,, Symmetry and perturbation theory (Rome, (1998), 23.   Google Scholar

[19]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, (Russian) Teoret. Mat. Fiz., 133 (2002), 170.  doi: 10.1023/A:1021186408422.  Google Scholar

[20]

K. E. Dika and L. Molinet, Stability of multipeakons,, Ann. I. H. Poincaré-AN., 26 (2009), 1517.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[21]

Y. Duan and J.-G. Liu, Error Estimate of the Particle Method For The B-Equation,, Submitted to SIAM J. Numer. Anal., ().   Google Scholar

[22]

J. Goodman, T. Y. Hou and J. Lowengrub, Convergence of the Point Vortex Method for the 2-D Euler Equations,, Comm. Pure Appl. Math., 43 (1990), 415.  doi: 10.1002/cpa.3160430305.  Google Scholar

[23]

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

[24]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323.  doi: 10.1137/S1111111102410943.  Google Scholar

[25]

D. D. Holm, J. T. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy,, Neuroimage, 23 (2004).  doi: 10.1016/j.neuroimage.2004.07.017.  Google Scholar

[26]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry,, From finite to infinite dimensions. With solutions to selected exercises by David C. P. Ellis. Oxford Texts in Applied and Engineering Mathematics, (2009).   Google Scholar

[27]

Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62 (2009), 125.   Google Scholar

[28]

J.-G. Liu and Z. Xin, Convergence of vortex methods for weak solutions to 2-D Euler equations with vortex sheets data,, Comm. Pure Appl. Math., 48 (1995), 611.  doi: 10.1002/cpa.3160480603.  Google Scholar

[29]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inv. Problems, 19 (2003), 1241.  doi: 10.1088/0266-5611/19/6/001.  Google Scholar

[30]

T. Matsuo and Y. Miyatake, Conservative finite difference schemes for Degasperis-Procesi equation,, J. Comput. Appl. Math, 236 (2012), 3728.  doi: 10.1016/j.cam.2011.09.004.  Google Scholar

[31]

G. D. Rocca, M. C. Lombrado, M. Sammartino and V. Sciacca, Singularity tracking for Camassa-Holm and Prandtl's equations,, Appl. Numer. Math, 56 (2006), 1108.  doi: 10.1016/j.apnum.2005.09.009.  Google Scholar

[32]

Z. Xin and P. Zhang, On the weak solutions to a shallow water wave equation,, Comm. Pure Appl. Math., 53 (2000), 1411.   Google Scholar

[33]

Y. Xu and C.-W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation,, SIAM J. Numer. Anal., 46 (2008), 1998.  doi: 10.1137/070679764.  Google Scholar

[34]

W. P. Ziemer, Weakly Differentiable Functions,, Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

J. T. Beale and A. Majda, Vortex methods II: Higher order accuracy in two and three dimensions,, Math. Comput., 39 (1982), 29.  doi: 10.2307/2007618.  Google Scholar

[2]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem,, Oxford Lecture Ser. Math. Appl. 20, (2000).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

D. Chae and J.-G. Liu, Blow-up,Zero Alpha limit and the Liouville type theorem for the Euler-Poincaré equations,, Comm. Math. Phys., 314 (2012), 671.  doi: 10.1007/s00220-012-1534-8.  Google Scholar

[7]

A. Chertock, P. Du Toit and J. Marsden, Integration of the EPDiff equation by particle methods,, ESAIM Math. Model. Numer. Anal., 46 (2012), 515.  doi: 10.1051/m2an/2011054.  Google Scholar

[8]

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

[9]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60.  doi: 10.1016/j.jfa.2005.07.008.  Google Scholar

[10]

G. M. Coclite, K. H. Karlsen and N. H. Risebro, A convregent finite defference scheme for the Camassa-Holm equation with general $H^1$ initial data,, SIAM J. Numer. Anal., 46 (2008), 1554.  doi: 10.1137/060673242.  Google Scholar

[11]

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

[12]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[13]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[14]

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

[15]

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

[16]

G. H. Cottet and P. D. Koumoutsakos, Vortex Methods: Theory and Practice,, Cambridge University Press, (2000).  doi: 10.1017/CBO9780511526442.  Google Scholar

[17]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta. Mech., 127 (1998), 193.  doi: 10.1007/BF01170373.  Google Scholar

[18]

A. Degasperis and M. Procesi, Asymptotic Integrability,, Symmetry and perturbation theory (Rome, (1998), 23.   Google Scholar

[19]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, (Russian) Teoret. Mat. Fiz., 133 (2002), 170.  doi: 10.1023/A:1021186408422.  Google Scholar

[20]

K. E. Dika and L. Molinet, Stability of multipeakons,, Ann. I. H. Poincaré-AN., 26 (2009), 1517.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[21]

Y. Duan and J.-G. Liu, Error Estimate of the Particle Method For The B-Equation,, Submitted to SIAM J. Numer. Anal., ().   Google Scholar

[22]

J. Goodman, T. Y. Hou and J. Lowengrub, Convergence of the Point Vortex Method for the 2-D Euler Equations,, Comm. Pure Appl. Math., 43 (1990), 415.  doi: 10.1002/cpa.3160430305.  Google Scholar

[23]

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

[24]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323.  doi: 10.1137/S1111111102410943.  Google Scholar

[25]

D. D. Holm, J. T. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy,, Neuroimage, 23 (2004).  doi: 10.1016/j.neuroimage.2004.07.017.  Google Scholar

[26]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry,, From finite to infinite dimensions. With solutions to selected exercises by David C. P. Ellis. Oxford Texts in Applied and Engineering Mathematics, (2009).   Google Scholar

[27]

Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62 (2009), 125.   Google Scholar

[28]

J.-G. Liu and Z. Xin, Convergence of vortex methods for weak solutions to 2-D Euler equations with vortex sheets data,, Comm. Pure Appl. Math., 48 (1995), 611.  doi: 10.1002/cpa.3160480603.  Google Scholar

[29]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inv. Problems, 19 (2003), 1241.  doi: 10.1088/0266-5611/19/6/001.  Google Scholar

[30]

T. Matsuo and Y. Miyatake, Conservative finite difference schemes for Degasperis-Procesi equation,, J. Comput. Appl. Math, 236 (2012), 3728.  doi: 10.1016/j.cam.2011.09.004.  Google Scholar

[31]

G. D. Rocca, M. C. Lombrado, M. Sammartino and V. Sciacca, Singularity tracking for Camassa-Holm and Prandtl's equations,, Appl. Numer. Math, 56 (2006), 1108.  doi: 10.1016/j.apnum.2005.09.009.  Google Scholar

[32]

Z. Xin and P. Zhang, On the weak solutions to a shallow water wave equation,, Comm. Pure Appl. Math., 53 (2000), 1411.   Google Scholar

[33]

Y. Xu and C.-W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation,, SIAM J. Numer. Anal., 46 (2008), 1998.  doi: 10.1137/070679764.  Google Scholar

[34]

W. P. Ziemer, Weakly Differentiable Functions,, Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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