# Regularity and Blow up for Active Scalars

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 4, page 225-255
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topKiselev, A.. "Regularity and Blow up for Active Scalars." Mathematical Modelling of Natural Phenomena 5.4 (2010): 225-255. <http://eudml.org/doc/197710>.

@article{Kiselev2010,

abstract = {We review some recent results for a class of fluid mechanics equations called active
scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic
equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss
nonlocal maximum principle methods which allow to prove existence of global regular
solutions for the critical dissipation. We also recall what is known about the possibility
of finite time blow up in the supercritical regime.},

author = {Kiselev, A.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {active scalars; global regularity; finite time blow up; nonlocal maximum principle},

language = {eng},

month = {5},

number = {4},

pages = {225-255},

publisher = {EDP Sciences},

title = {Regularity and Blow up for Active Scalars},

url = {http://eudml.org/doc/197710},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Kiselev, A.

TI - Regularity and Blow up for Active Scalars

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/5//

PB - EDP Sciences

VL - 5

IS - 4

SP - 225

EP - 255

AB - We review some recent results for a class of fluid mechanics equations called active
scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic
equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss
nonlocal maximum principle methods which allow to prove existence of global regular
solutions for the critical dissipation. We also recall what is known about the possibility
of finite time blow up in the supercritical regime.

LA - eng

KW - active scalars; global regularity; finite time blow up; nonlocal maximum principle

UR - http://eudml.org/doc/197710

ER -

## References

top- G.R. Baker, X. Li and A.C. MorletAnalytic structure of 1D-transport equations with nonlocal fluxes. Physica D, 91 (1996), 349–375.
- A. Bertozzi and A. Majda. Vorticity and Incompressible Flow. Cambridge University Press, 2002.
- K. Bogdan, A. Stoś and P. Sztonyk. Harnack inequality for stable processes on d-sets, Studia Math., 158 (2003), 163–198.
- L. Caffarelli and A. Vasseur. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Preprint arXiv:math / 0608447.
- J. Carrillo and L. Ferreira. The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations. Nonlinearity, 21, (2008), 1001–1018.
- D. Chae and J. Lee. Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Comm. Math. Phys.233 (2003), 297–311.
- Q. Chen, C. Miao and Z. Zhang. A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Comm. Math. Phys., 271, (2007), 821–838.
- P. Constantin. Active scalars and the Euler equation. Tatra Mountains Math. Publ., 4 (1994), 25–38.
- P. Constantin. Energy spectrum of quasigeostrophic turbulence. Phys. Rev. Lett., 89, (2002), 184501.
- P. Constantin, D. Cordoba and J. Wu. 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.
- P. Constantin, A. Majda and E. Tabak. Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity, 7, (1994), 1495–1533.
- P. Constantin, G. Iyer and J. Wu. Global regularity for a modified critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J., 57, (2008), 2681–2692.
- P. Constantin and J. Wu. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal., 30, (1999), 937–948.
- 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.
- D. Cordoba. Nonexistence of simple hyperbolic blow up for the quasi-geostrophic equation. Ann. of Math., 148, (1998), 1135–1152.
- A. Cordoba and D. Cordoba. A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys., 249, (2004), 511–528.
- A. Cordoba, D. Cordoba and M. Fontelos. Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. (2), 162, (2005), 1377–1389.
- S. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete Contin. Dyn. Syst., 23, (2009), 755-764.
- 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.
- H. Dong, D. Du and D. Li. Finite time singularities and global well-posedness for fractal Burgers equations. Indiana Univ. Math. J., 58, (2009), 807–821.
- I. Held, R. Pierrehumbert, S. Garner and K. Swanson. Surface quasi-geostrophic dynamics. J. Fluid Mech., 282, (1995), 1–20.
- S. Friedlander, N. Pavlovic and V. Vicol. Nonlinear instability for critically dissipative quasi-geostrophic equation. Preprint.
- N. Ju. The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys., 255, (2005), 161–181.
- N. Ju. Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation. Math. Ann., 334, (2006), 627–642.
- N. Ju. Geometric constrains for global regularity of 2D quasi-geostrophic flows. J. Differential Equations, 226, (2006), 54–79.
- N. Ju. Dissipative 2D quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions. Indiana Univ. Math. J., 56, (2007), 187–206.
- W. Feller. Introduction to Probability Theory and Applications. Vol. 2, Wiley, 1971.
- A. Kiselev, F. Nazarov and R. Shterenberg. On blow up and regularity in dissipative Burgers equation. Dynamics of PDE, 5, (2008), 211–240.
- A. Kiselev, F. Nazarov and A. Volberg. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Inventiones Math., 167, (2007), 445–453.
- 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.
- D. Li and J. Rodrigo. Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation. Adv. Math., 217, (2008), 2563–2568.
- 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).
- H. Miura. Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space. Comm. Math. Phys., 267, (2006), 141–157.
- A.C. Morlet. Further properties of a continuum of model equations ith globally defined flux. J. Math. Anal. Appl., 22, (1998), 132–160.
- N. S. Nadirashvili. 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).
- 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.
- J. Wu. The quasi-geostrophic equation and its two regularizations. Comm. Partial Differential Equations, 27, (2002), 1161–1181.
- J. Wu. Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. Nonlinear Anal., 67, (2007), 3013–3036.
- J. Wu. Solutions of the 2D quasi-geostrophic equation in Hölder spaces. Nonlinear Anal., 62, (2005), 579–594.
- J. Wu. The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity, 18, (2005), 139–154.
- V.I. Yudovich. The loss of smoothness of the solutions of Euler equations with time. (Russian) Dinamika Sploshn. Sredy Vyp. 16, Nestacionarnye Problemy Gidrodinamiki121 (1974), 71–78.
- V.I. Yudovich. 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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.