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A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  21 May 2020

JOSÉ A. CARRILLO Open the ORCID record for JOSÉ A. CARRILLO [Opens in a new window] ,
KATRIN GRUNERT Open the ORCID record for KATRIN GRUNERT [Opens in a new window]  and
HELGE HOLDEN Open the ORCID record for HELGE HOLDEN [Opens in a new window]
JOSÉ A. CARRILLO
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK; carrillo@maths.ox.ac.uk
KATRIN GRUNERT
Affiliation:
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491Trondheim, Norway; katrin.grunert@ntnu.no, helge.holden@ntnu.no
HELGE HOLDEN
Affiliation:
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491Trondheim, Norway; katrin.grunert@ntnu.no, helge.holden@ntnu.no

Abstract

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We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) 35B35: Stability
Secondary: 35B60: Continuation and prolongation of solutions
Type
Differential Equations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e27
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Ambrosio, L., Gigli, N. and Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich (Birkhäuser Verlag, Basel, 2008).Google Scholar
Blanchet, A., Calvez, V. and Carrillo, J. A., ‘Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model’, SIAM J. Numer. Anal. 46 (2008), 691–721.CrossRefGoogle Scholar
Bolley, F., Cañizo, J. A. and Carrillo, J. A., ‘Stochastic mean-field limit: non-Lipschitz forces and swarming’, Math. Models Methods Appl. Sci. 21 (2011), 2179–2210.CrossRefGoogle Scholar
Bonaschi, G. A., Carrillo, J. A., Di Francesco, M. and Peletier, M. A., ‘Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D’, ESAIM Control Optim. Calc. Var. 21 (2015), 414–441.CrossRefGoogle Scholar
Braun, W. and Hepp, K., ‘The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles’, Comm. Math. Phys. 56 (1977), 101–113.CrossRefGoogle Scholar
Bressan, A. and Constantin, A., ‘Global conservative solutions of the Camassa–Holm equation’, Arch. Ration. Mech. Anal. 183 (2007), 215–239.CrossRefGoogle Scholar
Bressan, A. and Constantin, A., ‘Global dissipative solutions of the Camassa–Holm equation’, Anal. Appl. (Singap.) 5 (2007), 1–27.CrossRefGoogle Scholar
Bressan, A. and Fonte, M., ‘An optimal transportation metric for solutions of the Camassa–Holm equation’, Methods Appl. Anal. 12 (2005), 191–219.Google Scholar
Bressan, A., Holden, H. and Raynaud, X., ‘Lipschitz metric for the Hunter–Saxton equation’, J. Math. Pures Appl. (9) 94 (2010), 68–92.CrossRefGoogle Scholar
Camassa, R. and Holm, D. D., ‘An integrable shallow water equation with peaked solutions’, Phys. Rev. Lett. 71 (1993), 1661–1664.CrossRefGoogle Scholar
Camassa, R., Holm, D. D. and Hyman, J., ‘A new integrable shallow water equation’, Adv. Appl. Mech. 31 (1994), 1–33.CrossRefGoogle Scholar
Cañizo, J. A., Carrillo, J. A. and Rosado, J., ‘A well-posedness theory in measures for some kinetic models of collective motion’, Math. Models Methods Appl. Sci. 21 (2011), 515–539.CrossRefGoogle Scholar
Carrillo, J. A., Choi, Y.-P., Hauray, M. and Salem, S., ‘Mean-field limit for collective behavior models with sharp sensitivity regions’, J. Eur. Math. Soc. (JEMS) 21 (2019), 121–161.CrossRefGoogle Scholar
Carrillo, J. A., Di Francesco, M., Figalli, A., Laurent, T. and Slepčev, D., ‘Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations’, Duke Math. J. 156 (2011), 229–271.CrossRefGoogle Scholar
Carrillo, J. A., Di Francesco, M. and Toscani, G., ‘Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering’, Proc. Amer. Math. Soc. 135 (2007), 353–363.CrossRefGoogle Scholar
Carrillo, J. A., Grunert, K. and Holden, H., ‘A Lipschitz metric for the Hunter–Saxton equation’, Comm. Partial Differential Equations 44 (2019), 309–334.CrossRefGoogle Scholar
Carrillo, J. A. and Moll, J. S., ‘Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms’, SIAM J. Sci. Comput. 31 (2009/10), 4305–4329.CrossRefGoogle Scholar
Carrillo, J. A., Ranetbauer, H. and Wolfram, M.-T., ‘Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms’, J. Comput. Phys. 327 (2016), 186–202.CrossRefGoogle Scholar
Carrillo, J. A. and Toscani, G., ‘Wasserstein metric and large–time asymptotics of non-linear diffusion equations’, inNew Trends in Mathematical Physics (World Sci. Publ., Hackensack, NJ, 2004), 234–244.Google Scholar
Carrillo, J. A. and Toscani, G., ‘Contractive probability metrics and asymptotic behavior of dissipative kinetic equations’, Riv. Mat. Univ. Parma (7) 6 (2007), 75–198.Google Scholar
Chen, R. M. and Liu, Y., ‘Wave breaking and global existence for a generalized two-component Camassa–Holm system’, Int. Math. Res. Not. IMRN 2011(6) (2011), 1381–1416.Google Scholar
Chen, M., Liu, S.-Q. and Zhang, Y., ‘A two-component generalization of the Camassa–Holm equation and its solutions’, Lett. Math. Phys. 75 (2006), 1–15.CrossRefGoogle Scholar
Chertock, A., Liu, J.-G. and Pendleton, T., ‘Convergence of a particle method and global weak solutions for a family of evolutionary PDEs’, SIAM J. Numer. Anal. 50 (2012), 1–21.CrossRefGoogle Scholar
Coclite, G. M., Holden, H. and Karlsen, K. H., ‘Global weak solutions to a generalized hyperelastic-rod wave equation’, SIAM J. Math. Anal. 37 (2005), 1044–1069.CrossRefGoogle Scholar
Constantin, A. and Escher, J., ‘Wave breaking for nonlinear nonlocal shallow water equations’, Acta Math. 181 (1998), 229–243.CrossRefGoogle Scholar
Constantin, A. and Kolev, B., ‘Geodesic flow on the diffeomorphism group of the circle’, Comment. Math. Helv. 78 (2003), 787–804.CrossRefGoogle Scholar
Constantin, A. and Lannes, D., ‘The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations’, Arch. Ration. Mech. Anal. 192 (2009), 165–186.CrossRefGoogle Scholar
Dai, H.-H. and Huo, Y., ‘Solitary shock waves and other travelling waves in a general compressible hyperelastic rod’, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 456 (2000), 331–363.CrossRefGoogle Scholar
Dobrushin, R. L., ‘Vlasov equations’, Funct. Anal. Appl. 13 (1979), 115–123.CrossRefGoogle Scholar
Düring, B., Matthes, D. and Milišić, J. P., ‘A gradient flow scheme for nonlinear fourth order equations’, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), 935–959.Google Scholar
Escher, J. and Kolev, B., ‘Geodesic completeness for Sobolev H s-metrics on the diffeomorphism group of the circle’, J. Evol. Equ. 14 (2014), 949–968.Google Scholar
Escher, J., Lechtenfeld, O. and Yin, Z., ‘Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation’, Discrete Contin. Dyn. Syst. 19 (2007), 493–513.CrossRefGoogle Scholar
Fu, Y. and Qu, C., ‘Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons’, J. Math. Phys. 50 (2009), 012906.CrossRefGoogle Scholar
Gallouët, T. and Vialard, F.-X., ‘The Camassa–Holm equation as an incompressible Euler equations: a geometric point of view’, J. Differential Equations 264 (2018), 4199–4234.CrossRefGoogle Scholar
Golse, F., ‘The mean-field limit for the dynamics of large particle systems’, J. Équ. aux Dérivées Partielles, 9, Univ. Nantes, Nantes, 2003, 47pp.CrossRefGoogle Scholar
Golse, F., ‘On the dynamics of large particle systems in the mean-field limit’, inMacroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lecture Notes in Applied and Mathematics and Mechanics, 3 (Springer, Cham, 2016), 1–144.Google Scholar
Gosse, L. and Toscani, G., ‘Lagrangian numerical approximations to one-dimensional convolution-diffusion equations’, SIAM J. Sci. Comput. 28 (2006), 1203–1227.CrossRefGoogle Scholar
Gosse, L. and Toscani, G., ‘Identification of asymptotic decay to self-similarity for one-dimensional filtration equations’, SIAM J. Numer. Anal. 43 (2006), 2590–2606.CrossRefGoogle Scholar
Grasmair, M., Grunert, K. and Holden, H., ‘On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa–Holm system’, inCurrent Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg, (ed. Rassias, Th. M.) Springer Optimization and Its Applications, 135 (Springer, Cham, Switzerland, 2018), 157–201.CrossRefGoogle Scholar
Grunert, K., ‘Solutions of the Camassa–Holm equation with accumulating breaking times’, Dyn. Partial Differ. Equ. 13 (2016), 91–105.CrossRefGoogle Scholar
Grunert, K. and Holden, H., ‘The general peakon-antipeakon solution for the Camassa–Holm equation’, J. Hyperbolic Differ. Equ. 13 (2016), 353–380.CrossRefGoogle Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Lipschitz metric for the periodic Camassa–Holm equation’, J. Differential Equations 250 (2011), 1460–1492.CrossRefGoogle Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Lipschitz metric for the Camassa–Holm equation on the line’, Discrete Contin. Dyn. Syst. 33 (2013), 2809–2827.CrossRefGoogle Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Global conservative solutions to the Camassa–Holm equation for initial data with nonvanishing asymptotics’, Discrete Contin. Dyn. Syst. 32 (2012), 4209–4227.Google Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Global dissipative solutions of the two-component Camassa–Holm system for initial data with nonvanishing asymptotics’, Nonlinear Anal. Real World Appl. 17 (2014), 203–244.CrossRefGoogle Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘Global solutions for the two-component Camassa–Holm system’, Comm. Partial Differential Equations 37 (2012), 2245–2271.CrossRefGoogle Scholar
Grunert, K., Holden, H. and Raynaud, X., ‘A continuous interpolation between conservative and dissipative solutions for the two-component Camassa–Holm system’, Forum Math. Sigma 3 (2014), e1, 73 pages.Google Scholar
Guan, C., Karlsen, K. H. and Yin, Z., ‘Well-posedness and blow-up phenomenal for a modified two-component Camassa–Holm equation’, inNonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, (eds. Holden, H. and Karlsen, K. H.) Contemporary Mathematics, 526 (American Mathematical Society, 2010), 199–220.CrossRefGoogle Scholar
Guan, C. and Yin, Z., ‘Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system’, J. Differential Equations 248 (2010), 2003–2014.CrossRefGoogle Scholar
Guan, C. and Yin, Z., ‘Global weak solutions for a two-component Camassa–Holm shallow water system’, J. Funct. Anal. 260 (2011), 1132–1154.CrossRefGoogle Scholar
Gui, G. and Liu, Y., ‘On the global existence and wave breaking criteria for the two-component Camassa–Holm system’, J. Funct. Anal. 258 (2010), 4251–4278.CrossRefGoogle Scholar
Gui, G. and Liu, Y., ‘On the Cauchy problem for the two-component Camassa–Holm system’, Math. Z. 268 (2011), 45–66.CrossRefGoogle Scholar
Guo, Z. and Zhou, Y., ‘On solutions to a two-component generalized Camassa–Holm equation’, Stud. Appl. Math. 124 (2010), 307–322.CrossRefGoogle Scholar
Holden, H. and Raynaud, X., ‘Global conservative solutions for the Camassa–Holm equation — a Lagrangian point of view’, Comm. Partial Differential Equations 32 (2007), 1511–1549.CrossRefGoogle Scholar
Holden, H. and Raynaud, X., ‘Global conservative solutions of the generalized hyperelastic-rod wave equation’, J. Differential Equations 233 (2007), 448–484.CrossRefGoogle Scholar
Holden, H. and Raynaud, X., ‘Dissipative solutions for the Camassa–Holm equation’, Discrete Contin. Dyn. Syst. 24 (2009), 1047–1112.CrossRefGoogle Scholar
Holden, H. and Raynaud, X., ‘Global dissipative multipeakon solutions of the Camassa–Holm equation’, Comm. Partial Differential Equations 33 (2008), 2040–2063.CrossRefGoogle Scholar
Johnson, R. S., ‘Camassa–Holm, Korteweg–de Vries and related models for water waves’, J. Fluid Mech. 455 (2002), 63–82.CrossRefGoogle Scholar
Junge, O., Matthes, D. and Osberger, H., ‘A fully discrete variational scheme for solving nonlinear Fokker–Planck equations in multiple space dimensions’, SIAM J. Numer. Anal. 55 (2017), 419–443.CrossRefGoogle Scholar
Kouranbaeva, S., ‘The Camassa–Holm equation as a geodesic flow on the diffeomorphism group’, J. Math. Phys. 40 (1999), 857–868.CrossRefGoogle Scholar
Li, H. and Toscani, G., ‘Long-time asymptotics of kinetic models of granular flows’, Arch. Ration. Mech. Anal. 172 (2004), 407–428.CrossRefGoogle Scholar
Matthes, D. and Osberger, H., ‘Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation’, ESAIM Math. Model. Numer. Anal. 48 (2014), 697–726.CrossRefGoogle Scholar
Neunzert, H., ‘An introduction to the nonlinear Boltzmann–Vlasov equation’, inKinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Mathematics, 1048 (Springer, Berlin, 1984), 60–110.CrossRefGoogle Scholar
Spohn, H., Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics (Springer, Berlin, 1991).CrossRefGoogle Scholar
Sznitman, A.-S., ‘Topics in propagation of chaos’, inEcole d’Eté de Probabilités de Saint-Flour XIX 1989, Lecture Notes in Mathematics, 1464 (Springer, Berlin, 1991), 165–251.CrossRefGoogle Scholar
Toscani, G., ‘One-dimensional kinetic models of granular flows’, M2AN Math. Model. Numer. Anal. 34 (2000), 1277–1291.CrossRefGoogle Scholar
Villani, C., Topics in Optimal Transportation, Graduate Studies in Mathematics, 58 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
Westdickenberg, M. and Wilkening, J., ‘Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations’, M2AN Math. Model. Numer. Anal. 44 (2010), 133–166.CrossRefGoogle Scholar