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Convex integration for a class of active scalar equations
Author:
R. Shvydkoy
Journal:
J. Amer. Math. Soc. 24 (2011), 1159-1174
MSC (2010):
Primary 35Q35; Secondary 76W05
Posted:
June 6, 2011
MathSciNet review:
2813340
Full-text PDF
Abstract |
References |
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Abstract: We show that a general class of active scalar equations, including porous media and certain magnetostrophic turbulence models, admits non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics.
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ℎ-principle in geometry and topology. MR 1488424
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differential equations, Nonlinear analysis and mechanics: Heriot-Watt
Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston,
Mass., 1979, pp. 136–212. MR 584398
(81m:35014)
- 1.
- Peter Constantin.
Scaling exponents for active scalars.
J. Statist. Phys., 90(3-4):571-595, 1998. MR 1616902 (99c:76052)
- 2.
- Diego Córdoba, Daniel Faraco, and Francisco Gancedo.
Lack of uniqueness for weak solutions of the incompressible porous media equation. arXiv:0912.3210v1
- 3.
- Camillo De Lellis and László Székelyhidi, Jr.
The Euler equations as a differential inclusion.
Ann. of Math. (2), 170(3):1417-1436, 2009. MR 2600877 (2011e:35287)
- 4.
- Camillo De Lellis and László Székelyhidi, Jr.
On admissibility criteria for weak solutions of the Euler equations.
Arch. Ration. Mech. Anal., 195(1):225-260, 2010. MR 2564474 (2011d:35386)
- 5.
- Gregory L. Eyink and Katepalli R. Sreenivasan.
Onsager and the theory of hydrodynamic turbulence.
Rev. Modern Phys., 78(1):87-135, 2006. MR 2214822 (2007g:76108)
- 6.
- Susan Friedlander and Vlad Vicol.
Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics. arXiv:1007.1211v3
- 7.
- Mikhael Gromov.
Partial differential relations, volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)].
Springer-Verlag, Berlin, 1986. MR 864505 (90a:58201)
- 8.
- Bernd Kirchheim, Stefan Müller, and Vladimír Šverák.
Studying nonlinear pde by geometry in matrix space.
In Geometric analysis and nonlinear partial differential equations, pages 347-395. Springer, Berlin, 2003. MR 2008346 (2006f:35087)
- 9.
- A. N. Kolmogorov.
The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers.
Dokl. Akad. Nauk. SSSR, 1941.
- 10.
- R. H. Kraichnan.
Small-scale structure of a scalar field convected by turbulence.
Phys. of Fluids, II(5):945-953, 1968.
- 11.
- H.K. Moffatt.
Magnetostrophic turbulence and the geodynamo.
In IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, pages 339-346. Springer, Dordrecht, 2008. MR 2432632 (2009j:76227)
- 12.
- S. Müller and V. Šverák.
Convex integration for Lipschitz mappings and counterexamples to regularity.
Ann. of Math. (2), 157(3):715-742, 2003. MR 1983780 (2005i:35028)
- 13.
- L. Onsager.
Statistical hydrodynamics.
Nuovo Cimento (9), 6(Supplemento, 2(Convegno Internazionale di Meccanica Statistica)):279-287, 1949. MR 0036116 (12:60f)
- 14.
- V. Scheffer.
Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities.
1974.
Dissertation, Princeton University (unpublished). MR 2624766
- 15.
- Vladimir Scheffer.
An inviscid flow with compact support in space-time.
J. Geom. Anal., 3(4):343-401, 1993. MR 1231007 (94h:35215)
- 16.
- A. Shnirelman.
On the nonuniqueness of weak solution of the Euler equation.
Comm. Pure Appl. Math., 50(12):1261-1286, 1997. MR 1476315 (98j:35149)
- 17.
- A. Shnirelman.
Weak solutions with decreasing energy of incompressible Euler equations.
Comm. Math. Phys., 210(3):541-603, 2000. MR 1777341 (2002g:76009)
- 18.
- R. Shvydkoy.
Lectures on the Onsager conjecture.
Discrete and Continuous Dynamical Systems - Series S, 3(3):473-496, 2010. MR 2660721
- 19.
- David Spring.
Convex integration theory, volume 92 of Monographs in Mathematics.
Birkhäuser Verlag, Basel, 1998.
Solutions to the  -principle in geometry and topology. MR 1488424 (99e:58024)
- 20.
- L. Tartar.
Compensated compactness and applications to partial differential equations.
In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, volume 39 of Res. Notes in Math., pages 136-212. Pitman, Boston, Mass., 1979. MR 584398 (81m:35014)
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Additional Information
R. Shvydkoy
Affiliation:
Department of Mathematics, Statistics and Computer Science, 851 S. Morgan St., M/C 249, University of Illinois, Chicago, Illinois 60607
Email:
shvydkoy@math.uic.edu
DOI:
http://dx.doi.org/10.1090/S0894-0347-2011-00705-4
PII:
S 0894-0347(2011)00705-4
Received by editor(s):
October 25, 2010
Posted:
June 6, 2011
Additional Notes:
The work was partially supported by NSF grant DMS – 0907812
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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