Abstract
This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker than all the Prodi–Serrin type regularity conditions. Moreover, we prove that any viscosity solution of the supercritical SQG in \(L^2(0,T; B^{1/2}_{2,c(\mathbb N)})\) satisfies energy equality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bahouri, H., Chemin, J-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 343. Springer, Heidelberg (2011)
Caffarelli, L.A., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171(3), 1903–1930 (2010)
Chae, D.: On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J. Math. Anal. 37(5), 1649–1656 (2006)
Chae, D.: The quasi-geostrophic equation in the Triebel–Lizorkin spaces. Nonlinearity 16(2), 479–495 (2003)
Chen, Q., Miao, C., Zhang, Z.: A new Bernstein inequality and the 2D dissipative quasi-geostrophic equation. Commun. Math. Phys. 271, 821–838 (2007)
Cheskidov, A., Constantin, P., Friedlander, S., Shvydkoy, R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21, 1233–1252 (2008)
Cheskidov, A., Dai, M.: Finite determining modes for the quasi-geostrophic equation. (2015). arXiv:1507.01075
Cheskidov, A., Dai, M.: The existence of a global attractor for the forced critical surface quasi-geostrophic equation in \(L^2\). (2014). arXiv:1402.4801
Cheskidov, A., Shvydkoy, R.: A unified approach to regularity problems for the 3D Navier–Stokes and Euler equations: the use of Kolmogorov’s dissipation range. J. Math. Fluid Mech. 16(2), 263–273 (2014)
Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994)
Constantin, P., Vicol, V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22(5), 1289–1321 (2012)
Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30(5), 937–948 (1999)
Constantin, P., Wu, J.: Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26, 159–180 (2009)
Constantin, P., Wu, J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 25, 1103–1110 (2008)
Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004)
Dabkowski, M.: Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation. Geom. Funct. Anal. 21(1), 1–13 (2011)
Dabkowski, M., Kiselev, A., Silvestre, L., Vicol, V.: Global well-posedness of slightly supercritical active scalar equations. Anal. PDE 7, 43–72 (2014)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Mathematiques 136(5), 521–573 (2012)
Dong, H., Pavlović, N.: A regularity criterion for the dissipative quasi-geostrophic equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26, 1607–1619 (2009)
Fan, J., Gao, H., Nakamura, G.: Regularity criteria for the generalized magneto-hydrodynamic equations and the quasi-geostrophic equations. Taiwan. J. Math. 15(3), 1059–1073 (2011)
Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 2nd edn. Springer, New York (2009)
Kiselev, A., Nazarov, F.: A variation on a theme of Caffarelli and Vasseur. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370(Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 40): 58–72, 220 (2009)
Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167(3), 445–453 (2007)
Lemarié-Rieusset, P.G.: Recent developments in the Navier–Stokes problem. In: Chapman and Hall/CRC Research Notes in Mathematics, vol. 431. Chapman and Hall/CRC, Boca Raton, FL (2002)
Onsager, L.: Statistical hydrodynamics. Nuovo Cimento (Supplemento) 6, 279–287 (1949)
Pedlosky, J.: Geostrophical Fluid Dynamics. Springer, New York (1987)
Silvestre, L.: Eventual regularization for the slightly supercritical quasi-geostrophic equation. Ann. Inst. Henri Poincare (C) Nonlinear Anal. 27(2), 693–704 (2010)
Silvestre, L., Vicol, V., Zlatos, A.: On the loss of continuity for super-critical drift-diffusion equations. Arch. Ration. Mech. Anal. 207, 845–877 (2013)
Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36, 1014–1030 (2004/2005)
Xiang, Z.: A regularity criterion for the critical and supercritical dissipative quasi-geostrophic equations. Appl. Math. Lett. 23, 1286–1290 (2010)
Yamazaki, K.: Regularity criteria of supercritical Beta-generalized quasi-geostrophic equations in terms of partial derivatives. J. Funct. Anal. 258(10), 3376–3387 (2010)
Yu, X.: Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation. J. Math. Anal. Appl. 339, 359–371 (2008)
Yuan, B.: Regularity condition of solutions to the quasi-geostrophic equations in Besov spaces with negative indices. Acta Math. Appl. Sin. 26, 381–386 (2010)
Yuan, J.: On regularity criterion for the dissipative quasi-geostrophic equations. J. Math. Anal. Appl. 340, 334–339 (2008)
Zelati, M.C., Vicol, V.: On the global regularity for the supercritical SQG equation. (2014). arXiv:1410.3186v1
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Friedlander
Rights and permissions
About this article
Cite this article
Dai, M. Regularity Criterion and Energy Conservation for the Supercritical Quasi-geostrophic Equation. J. Math. Fluid Mech. 19, 191–202 (2017). https://doi.org/10.1007/s00021-017-0320-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-017-0320-y