Convex integration for a class of active scalar equations
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- by R. Shvydkoy;
- J. Amer. Math. Soc. 24 (2011), 1159-1174
- DOI: https://doi.org/10.1090/S0894-0347-2011-00705-4
- Published electronically: June 6, 2011
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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.References
- Peter Constantin, Scaling exponents for active scalars, J. Statist. Phys. 90 (1998), no. 3-4, 571–595. MR 1616902, DOI 10.1023/A:1023264617618
- Diego Córdoba, Daniel Faraco, and Francisco Gancedo. Lack of uniqueness for weak solutions of the incompressible porous media equation. arXiv:0912.3210v1
- Camillo De Lellis and László Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417–1436. MR 2600877, DOI 10.4007/annals.2009.170.1417
- Camillo De Lellis and László Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 225–260. MR 2564474, DOI 10.1007/s00205-008-0201-x
- Gregory L. Eyink and Katepalli R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys. 78 (2006), no. 1, 87–135. MR 2214822, DOI 10.1103/RevModPhys.78.87
- Susan Friedlander and Vlad Vicol. Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics. arXiv:1007.1211v3
- Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
- Bernd Kirchheim, Stefan Müller, and Vladimír Šverák, Studying nonlinear pde by geometry in matrix space, Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003, pp. 347–395. MR 2008346
- A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk. SSSR, 1941.
- R. H. Kraichnan. Small-scale structure of a scalar field convected by turbulence. Phys. of Fluids, II(5):945–953, 1968.
- H. Keith Moffatt, Magnetostrophic turbulence and the geodynamo, IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, IUTAM Bookser., vol. 4, Springer, Dordrecht, 2008, pp. 339–346. MR 2432632, DOI 10.1007/978-1-4020-6472-2_{5}1
- S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2) 157 (2003), no. 3, 715–742. MR 1983780, DOI 10.4007/annals.2003.157.715
- L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no. Supplemento, 2 (Convegno Internazionale di Meccanica Statistica), 279–287. MR 36116, DOI 10.1007/BF02780991
- Vladimir Scheffer, REGULARITY AND IRREGULARITY OF SOLUTIONS TO NONLINEAR SECOND-ORDER ELLIPTIC SYSTEMS OF PARTIAL DIFFERENTIAL-EQUATIONS AND INEQUALITIES, ProQuest LLC, Ann Arbor, MI, 1974. Thesis (Ph.D.)–Princeton University. MR 2624766
- Vladimir Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal. 3 (1993), no. 4, 343–401. MR 1231007, DOI 10.1007/BF02921318
- A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1261–1286. MR 1476315, DOI 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4
- A. Shnirelman, Weak solutions with decreasing energy of incompressible Euler equations, Comm. Math. Phys. 210 (2000), no. 3, 541–603. MR 1777341, DOI 10.1007/s002200050791
- Roman Shvydkoy, Lectures on the Onsager conjecture, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 3, 473–496. MR 2660721, DOI 10.3934/dcdss.2010.3.473
- David Spring, Convex integration theory, Monographs in Mathematics, vol. 92, Birkhäuser Verlag, Basel, 1998. Solutions to the $h$-principle in geometry and topology. MR 1488424, DOI 10.1007/978-3-0348-0060-0
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Bibliographic 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
- Received by editor(s): October 25, 2010
- Published electronically: June 6, 2011
- Additional Notes: The work was partially supported by NSF grant DMS – 0907812
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 24 (2011), 1159-1174
- MSC (2010): Primary 35Q35; Secondary 76W05
- DOI: https://doi.org/10.1090/S0894-0347-2011-00705-4
- MathSciNet review: 2813340