Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-17T20:16:10.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularity and Blow up for Active Scalars

Published online by Cambridge University Press:  12 May 2010

A. Kiselev
A. Kiselev*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, USA
*
1 E-mail: kiselev@math.wisc.edu
Get access

Abstract

We review some recent results for a class of fluid mechanics equations called activescalars, with fractional dissipation. Our main examples are the surface quasi-geostrophicequation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discussnonlocal maximum principle methods which allow to prove existence of global regularsolutions for the critical dissipation. We also recall what is known about the possibilityof finite time blow up in the supercritical regime.

Keywords

active scalarsglobal regularityfinite time blow upnonlocal maximum principle
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 225 - 255
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, G.R., Li, X. and Morlet, A.C. Analytic structure of 1D-transport equations with nonlocal fluxes . Physica D, 91 (1996), 349–375.CrossRefGoogle Scholar
A. Bertozzi and A. Majda. Vorticity and Incompressible Flow. Cambridge University Press, 2002.
Bogdan, K., Stoś, A. and Sztonyk, P.. Harnack inequality for stable processes on d -sets , Studia Math., 158 (2003), 163–198.CrossRefGoogle Scholar
L. Caffarelli and A. Vasseur. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Preprint arXiv:math / 0608447.
Carrillo, J. and Ferreira, L.. The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations . Nonlinearity, 21, (2008), 1001–1018. CrossRefGoogle Scholar
Chae, D. and Lee, J.. Global well-posedness in the super-critical dissipative quasi-geostrophic equations . Comm. Math. Phys. 233 (2003), 297–311.CrossRefGoogle Scholar
Chen, Q., Miao, C. and Zhang, Z.. A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation . Comm. Math. Phys., 271, (2007), 821–838. CrossRefGoogle Scholar
Constantin, P.. Active scalars and the Euler equation . Tatra Mountains Math. Publ., 4 (1994), 25–38.Google Scholar
Constantin, P.. Energy spectrum of quasigeostrophic turbulence . Phys. Rev. Lett., 89, (2002), 184501. CrossRefGoogle ScholarPubMed
Constantin, P., Cordoba, D. and Wu, J.. On the critical dissipative quasi-geostrophic equation . Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). Indiana Univ. Math. J., 50, (2001), 97–107. Google Scholar
Constantin, P., Majda, A. and Tabak, E.. Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar . Nonlinearity, 7, (1994), 1495–1533. CrossRefGoogle Scholar
Constantin, P., Iyer, G. and Wu, J.. Global regularity for a modified critical dissipative quasi-geostrophic equation . Indiana Univ. Math. J., 57, (2008), 2681–2692. CrossRefGoogle Scholar
Constantin, P. and Wu, J.. Behavior of solutions of 2D quasi-geostrophic equations . SIAM J. Math. Anal., 30, (1999), 937–948. CrossRefGoogle Scholar
P. Constantin and J. Wu. Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Preprint, arXiv:math / 0701592.
P. Constantin and J. Wu. Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. Preprint, arXiv:math / 0701594.
Cordoba, D.. Nonexistence of simple hyperbolic blow up for the quasi-geostrophic equation . Ann. of Math., 148, (1998), 1135–1152. CrossRefGoogle Scholar
Cordoba, A. and Cordoba, D.. A maximum principle applied to quasi-geostrophic equations . Commun. Math. Phys., 249, (2004), 511–528. CrossRefGoogle Scholar
Cordoba, A., Cordoba, D. and Fontelos, M.. Formation of singularities for a transport equation with nonlocal velocity . Ann. of Math. (2), 162, (2005), 1377–1389. CrossRefGoogle Scholar
Denisov, S.. Infinite superlinear growth of the gradient for the two-dimensional Euler equation . Discrete Contin. Dyn. Syst., 23, (2009), 755-764. CrossRefGoogle Scholar
H. Dong. Higher regularity for the critical and super-critical dissipative quasi-geostrophic equations. Preprint arXiv:math / 0701826.
H. Dong and D. Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Preprint arXiv:math / 0701828.
H. Dong and N. Pavlovic. A regularity criterion for the dissipative quasi-geostrophic equations. Preprint arXiv:math / 07105201.
Dong, H., Du, D. and Li, D.. Finite time singularities and global well-posedness for fractal Burgers equations . Indiana Univ. Math. J., 58, (2009), 807–821. CrossRefGoogle Scholar
Held, I., Pierrehumbert, R., Garner, S. and Swanson, K.. Surface quasi-geostrophic dynamics . J. Fluid Mech., 282, (1995), 1–20. CrossRefGoogle Scholar
S. Friedlander, N. Pavlovic and V. Vicol. Nonlinear instability for critically dissipative quasi-geostrophic equation. Preprint.
Ju, N.. The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations . Comm. Math. Phys., 255, (2005), 161–181. CrossRefGoogle Scholar
Ju, N.. Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation . Math. Ann., 334, (2006), 627–642. CrossRefGoogle Scholar
Ju, N.. Geometric constrains for global regularity of 2D quasi-geostrophic flows . J. Differential Equations, 226, (2006), 54–79. CrossRefGoogle Scholar
Ju, N.. Dissipative 2D quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions . Indiana Univ. Math. J., 56, (2007), 187–206. CrossRefGoogle Scholar
W. Feller. Introduction to Probability Theory and Applications. Vol. 2, Wiley, 1971.
Kiselev, A., Nazarov, F. and Shterenberg, R.. On blow up and regularity in dissipative Burgers equation . Dynamics of PDE, 5, (2008), 211–240. Google Scholar
Kiselev, A., Nazarov, F. and Volberg, A.. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation . Inventiones Math., 167, (2007), 445–453. CrossRefGoogle Scholar
A. Kiselev and F. Nazarov. A variation on a theme of Caffarelli and Vasseur. to appear at Zapiski Nauchn. Sem. POMI.
A. Kiselev and F. Nazarov. Nonlocal maximum principles for active scalars, title tentative, in preparation.
Li, D. and Rodrigo, J.. Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation . Adv. Math., 217, (2008), 2563–2568. CrossRefGoogle Scholar
C. Marchioro and M. Pulvirenti. Mathematical Theory of Incompressible Nonviscous Fluids. Springer-Verlag, New York 1994.
C. Miao and L. Xue. Global wellposedness for a modified critical dissipative quasi-geostrophic equation, arXiv:math / 0901.1368 (2009).
Miura, H.. Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space . Comm. Math. Phys., 267, (2006), 141–157. CrossRefGoogle Scholar
Morlet, A.C.. Further properties of a continuum of model equations ith globally defined flux . J. Math. Anal. Appl., 22, (1998), 132–160. CrossRefGoogle Scholar
Nadirashvili, N. S.. Wandering solutions of the two-dimensional Euler equation . (Russian) Funkcional. Anal. i Prilozh., 25, (1991), 70–71; translation in Funct. Anal. Appl. 25, (1991), 220–221 (1992). Google Scholar
S. Resnick. Dynamical problems in nonlinear advective partial differential equations. Ph.D. Thesis, University of Chicago, 1995.
L. Smith and J. Sukhatme. Eddies and waves in a family of dispersive dynamically active scalars. Preprint arXiv:0709.2897.
L. Sylvestre. Eventual regularization for the slightly supercritical quasi-geostrophic equation. Preprint arXiv:math / 0812.4901.
M. Taylor. Partial Differential Equations III: Nonlinear Equations. Springer-Verlag, New York, 1997.
Wu, J.. The quasi-geostrophic equation and its two regularizations . Comm. Partial Differential Equations, 27, (2002), 1161–1181. CrossRefGoogle Scholar
Wu, J.. Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation . Nonlinear Anal., 67, (2007), 3013–3036. CrossRefGoogle Scholar
Wu, J.. Solutions of the 2D quasi-geostrophic equation in Hölder spaces . Nonlinear Anal., 62, (2005), 579–594. CrossRefGoogle Scholar
Wu, J.. The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation . Nonlinearity, 18, (2005), 139–154. CrossRefGoogle Scholar
Yudovich, V.I.. The loss of smoothness of the solutions of Euler equations with time . (Russian) Dinamika Sploshn. Sredy Vyp. 16, Nestacionarnye Problemy Gidrodinamiki 121 (1974), 71–78. Google Scholar
Yudovich, V.I.. On the loss of smothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid . Chaos, 10, (2000), 705–719. CrossRefGoogle ScholarPubMed