Abstract
A time-dependent determining wavenumber was introduced in Cheskidov and Dai (Phys D Nonlinear Phenom 376–377:204–215, 2018) to estimate the number of determining modes for the surface quasi-geostrophic equation. In this paper we continue this investigation focusing on the subcritical case and study trajectories inside an absorbing set bounded in \(L^\infty \). Utilizing this bound we find a time-independent determining wavenumber that improves the estimate obtained in Cheskidov and Dai (Phys D Nonlinear Phenom 376–377:204–215, 2018). This classical approach is more direct, but it is contingent on the existence of the \(L^\infty \) absorbing set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahouri, H., Chemin, J., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehrender 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)
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.: Global attractors of evolutionary systems. J. Dyn. Differ. Equ. 21, 249–268 (2009)
Cheskidov, A., Dai, M.: Determining modes for the surface quasi-geostrophic equation. Phys. D Nonlinear Phenom. 376–377, 204–215 (2018)
Cheskidov, A., Dai, M.: The existence of a global attractor for the forced critical surface quasi-geostrophic equation in \(L^2\). J. Math. Fluid Mech. 20(1), 213–225 (2018)
Cheskidov, A., Dai, M., Kavlie, L.: Determining modes for the 3D Navier–Stokes equations. Phys. D Nonlinear Phenom. 374–375, 1–9 (2018)
Cheskidov, A., Kavlie, L.: Pullback attractors for generalized evolutionary systems. Discrete Contin. Dyn. Syst. B 20, 749–779 (2015)
Constantin, P., Zelati, M.C., Vicol, V.: Uniformly attracting limit sets for the critically dissipative SQG equation. Nonlinearity 29(2), 298 (2016)
Constantin, P., Foias, C., Manley, O.P., Temam, R.: Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427–440 (1985)
Constantin, P., Foias, C., Temam, R.: On the dimension of the attractors in two-dimensional turbulence. Physica D 30, 284–296 (1988)
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)
Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004)
Foias, C., Jolly, M., Kravchenko, R., Titi, E.: A determining form for the 2D Navier–Stokes equations—the Fourier modes case. J. Math. Phys. 53(11), 115623 (2012)
Foias, C., Jolly, M., Kravchenko, R., Titi, E.: A unified approach to determining forms for the 2D Navier–Stokes equations—the general interpolants case. Russ. Math. Surv. 69, 359 (2014)
Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications, vol. 83. Cambridge University Press, Cambridge (2001)
Foias, C., Manley, O.P., Temam, R., Tréve, Y.M.: Asymptotic analysis of the Navier–Stokes equations. Physica D 9(1–2), 157–188 (1983)
Foias, C., Prodi, G.: Sur le comportement global des solutions non-stationnaires des équations de Navier–Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39, 1–34 (1967)
Foias, C., Temam, R.: Some analytic and geometric properties of the solutions of the Navier–Stokes equations. J. Math. Pures Appl. 58, 339–368 (1979)
Foias, C., Titi, E.S.: Determining nodes, finite difference schemes and inertial manifolds. Nonlinearity 4, 135–153 (1991)
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)
Resnick, S.: Dynamical Problems in Nonlinear Advective Partial Differential Equations. Ph.D. Thesis, University of Chicago (1995)
Acknowledgements
The authors would wish to express their gratitude to the anonymous referee for the careful review and valuable suggestions which helped to improve the manuscript a lot.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work of A. Cheskidov was partially supported by NSF grants DMS–1108864 and DMS–1517583. The work of M. Dai was partially supported by NSF Grant DMS–1815069.
Rights and permissions
About this article
Cite this article
Cheskidov, A., Dai, M. On the Determining Wavenumber for the Nonautonomous Subcritical SQG Equation. J Dyn Diff Equat 32, 1511–1525 (2020). https://doi.org/10.1007/s10884-019-09794-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-019-09794-7