Regularity and stability of finite energy weak solutions for the Camassa–Holm equations with nonlocal viscosity


We consider the n-dimensional (\(n=2,3\)) Camassa–Holm equations with nonlocal diffusion of type \((-\,\Delta )^{s}, \ \frac{n}{4}\le s<1\). In Gan et al. (Discrete Contin Dyn Syst 40(6):3427–3450, 2020), the global-in-time existence and uniqueness of finite energy weak solutions is established. In this paper, we show that with regular initial data, the finite energy weak solutions are indeed regular for all time. Moreover, the weak solutions are stable with respect to the initial data. The main difficulty lies in establishing higher order uniform estimates with the presence of the fractional Laplacian diffusion. To achieve this, we need to explore suitable fractional Sobolev type inequalities and bilinear estimates for fractional derivatives. The critical case \(s=\frac{n}{4}\) contains extra difficulties and a smallness assumption on the initial data is imposed.

This is a preview of subscription content, access via your institution.


  1. 1.

    Bjorland, C., Schonbek, M.E.: On questions of decay and existence for the viscous Camassa–Holm equations. Ann. I. H. Poincaré Anal. Non Linéaire I(25), 907–936 (2008)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bobaru, F., Duanpanya, M.: The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Conduct 53, 4047–4059 (2010)

    Article  Google Scholar 

  3. 3.

    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Eqs. 32, 1245–1260 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, W., Holm, S.: Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115, 1424–1430 (2004)

    Article  Google Scholar 

  5. 5.

    Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen, W.: A speculative study of \(2/3\)-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. Chaos 16, 023126 (2006)

    Article  Google Scholar 

  7. 7.

    Chen, R., Liu, Y., Qu, C., Zhang, S.: Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion. Adv. Math. 272, 225–251 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Colombo, M., De Lellis, C., De Rosa, L.: Ill-posedness of Leray solutions for the hyperdissipative Navier–Stokes equations. Commun. Math. Phys. 362, 659–688 (2018)

    Article  Google Scholar 

  9. 9.

    Constantin, P., Vicol, V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22, 1289–1321 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Córdoba, A., Córdoba, C., Fontelos, M.A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. Math. 162, 1377–1389 (2005)

    MathSciNet  Article  Google Scholar 

  12. 12.

    de Lellis, C., Kappeler, T., Topalov, P.: Low-regularity solutions of the periodic Camassa–Holm equation. Commun. Partial Differ. Eqs. 32, 87–126 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    D’Elia, M., Gunzaburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66, 1245–1260 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Escher, J., Yin, Z.: Initial boundary value problems of the Camassa–Holm equation. Commun. Partial Differ. Eqs. 33, 377–395 (2008)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Fujiwara, K., Georgiev, V., Ozawa, T.: Higher order fractional Leibniz rule. J. Fourier Anal. Appl. 24(3), 650–665 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gan, Z., Lin, F., Tong, J.: On the viscous Camassa–Holm equations with fractional diffusion. Discrete Contin. Dyn. Syst. 40(6), 3427–3450 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hakkaev, S., Kirchev, K.: Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation. Commun. Partial Differ. Eqs. 30, 761–781 (2005)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hart, J.,Torres, R.H., Wu, X.: Smooth properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces. arXiv:1701.02631v1 (2017)

  19. 19.

    Holm, D.D., Marsden, J.E., Ratiu, T.S.: Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Katz, N.H., Pavlović, N.: A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation. Geom. Funct. Anal. 12, 355–379 (2002)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Marsden, J.E., Shkoller, S.: Global well-posedness for the Lagrangian averaged Navier–Stokes (LANS-\(\alpha \)) equations on bounded domains. Philos. Trans. R. Soc. Lond. A 359, 1449–1468 (2001)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Ladyzhenskaya, O.A., Shkoller, S.: Mathematical Problems of the Dynamics of Viscous Incompressible Fluid, p. 203. GIFML, Moscow (1961)

    Google Scholar 

  25. 25.

    Ladyzhenskaya, O.A.: On global existence of weak solutions to some 2-dimensional initial-boundary value problems for Maxwell fluids. Appl. Anal. 65, 251–255 (1997)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Ladyzhenskaya, O.A., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Ladyzhenskaya, O.A.: Sixth problem of the millennium Navier–Stokes equations, existences and smoothness. Russ. Math. Surv. 58(2), 251–286 (2003)

    Article  Google Scholar 

  28. 28.

    Landkof, N.S.: Foundations of Modern Potential, Die Grundlehren der Mathematischen Wissenschaften, vol. 180. Springer, New York (1972)

    Google Scholar 

  29. 29.

    Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Misiolek, G.: Classical solutions of the periodic Camassa–Holm equation. Geom. Funct. Anal. 12, 1080–1104 (2002)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Mongiovi, M.S., Zingales, M.: A non-local model of thermal energy transport: the fractional temperature equation. Int. J. Heat Mass Transf. 67, 593–601 (2013)

    Article  Google Scholar 

  32. 32.

    Musina, R., Nazarov, A.I.: On fractional Laplacian. Commun. Partial Diff. Eqs. 39, 1780–1790 (2014)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Nezza, E.D., Palarucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Pang, G., Chen, W., Fu, Z.: Space-fractional advection–dispersion equations by the Kansa method. J. Comput. Phys. 293, 280–296 (2015)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Perrollaz, V.: Initial boundary value problem and asymptotic stabilization of the Camassa–Holm equation on an interval. J. Funct. Anal. 259, 2333–2365 (2010)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Tan, W., Yin, Z.: Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation. J. Funct. Anal. 261, 1204–1226 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Tarasov, V.E.: Electromagnetic fields on fractals. Mod. Phys. Lett. A 21, 1587–1600 (2006)

    Article  Google Scholar 

  38. 38.

    Wu, S., Yin, Z.: Global existence and blow-up phenomena for the weakly dissipative Camassa–Holm equation. J. Differ. Equ. 246, 4309–4321 (2009)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the generalized Camassa–Holm equation in Besov space. J. Differ. Equ. 256, 2876–2901 (2014)

    MathSciNet  Article  Google Scholar 

Download references


Zaihui Gan is partially supported by the National Science Foundation of China under Grant (No. 11571254). Qing Guo is partially supported by the National Science Foundation of China under Grant (No. 11771469).

Author information



Corresponding author

Correspondence to Yong Lu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by F.-H. Lin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gan, Z., Guo, Q. & Lu, Y. Regularity and stability of finite energy weak solutions for the Camassa–Holm equations with nonlocal viscosity. Calc. Var. 60, 26 (2021).

Download citation

Mathematics Subject Classification

  • 35G25
  • 35K30
  • 35A01