Abstract
We prove that for the 3D MHD equations with hyper-dissipations (-Δ)α (1 < α < 5/4) the Hausdorff dimension of singular set at the first blowing up time is at most 5 − 4α, by means of physical and frequency localization, Bony’s paraproduct and Littlewood-Paley theory.
Similar content being viewed by others
References
Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions to the Navier-Stokes equations. Comm. Pure. Appl. Math., 35, 771–831 (1982)
Cannone, M., Miao, C. X., Wu, G.: On the inviscid limit of the two-dimensional Navier-Stokes equations with fractional diffusion. Adv. Math. Sci. Appl., 18, 607–624 (2008)
Duvaut, D., Lions, J.: Inequalitys en thermoilasticiti et magnitohydrodymamique. Arch. Ration. Mech. Anal., 46, 241–279 (1972)
He, C., Xin, Z. P.: Partial regularity of suitable weak solutions to the incompressible magneto-hydrodymanic equations. J. Funct. Anal., 227, 113–152 (2005)
Jia, H., Sverák, V.: Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math., 196, 233–265 (2014)
Jiu, Q. S., Wang, Y. Q.: On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations. Dyn. Partial Differ. Equ., 11, 321–343 (2014)
Ladyzenskaja, O. A., Seregin, G.: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid. Mech., 1, 356–387 (1999)
Leray, J.: Sur le mouvement déun liquide visqueux emplissant léspace. Acta. Math., 63, 193–248 (1934)
Lin, F. H.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure. Appl. Math., 51, 241–257 (1998)
Katz, N., Pavlović, N.: A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Sotkes equation with hyper-dissipation. Geom. Funct. Anal., 12, 355–379 (2002)
Scheffer, V.: Partial regularity of solutions to the Navier-Stokes. Pacific J. Math., 66, 535–552 (1976)
Scheffer, V.: Hausdroff measure and the Navier-Stokes. Comm. Math. Phys., 55, 97–112 (1977)
Scheffer, V.: The Navier-Stokes equations in the space dimension four. Comm. Math. Phys., 61, 41–68 (1978)
Scheffer, V.: The Navier-Stokes equations on a bounded domain. Comm. Math. Phys., 73, 1–42 (1980)
Stein, E.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970
Tang, L., Yu, Y.: Partial reqularity of suitable weak solutions to the fractional Navier-Stokes equations. Comm. Math. Phys., 334, 1455–1482 (2015)
Tao, T.: Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation. Analysis & PDE, 3, 361–366 (2009)
Ukhovskii, M. R., Yudovich, V. I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. Prikl. Mat. Meh., 32, 59–69 (1968)
Wang, Y. Q., Wu, G.: A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations. J. Differential Equations, 256, 1224–1249 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author is supported by National Natural Science Foundation of China (Grant No. 11101405) and the President Fund of UCAS
Rights and permissions
About this article
Cite this article
Ren, W., Wu, G. Partial regularity for the 3D magneto-hydrodynamics system with hyper-dissipation. Acta. Math. Sin.-English Ser. 31, 1097–1112 (2015). https://doi.org/10.1007/s10114-015-4498-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-015-4498-8
Keywords
- 3D MHD system
- fractal dissipation
- partial regularity
- physical and frequency localization
- Bony’s paraproduct decomposition