| Title:
|
A counterexample to the smoothness of the solution to an equation arising in fluid mechanics (English) |
| Author:
|
Montgomery-Smith, Stephen |
| Author:
|
Pokorný, Milan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
1 |
| Year:
|
2002 |
| Pages:
|
61-75 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for the Navier-Stokes equations. (English) |
| Keyword:
|
Navier-Stokes equations |
| Keyword:
|
Euler equations |
| Keyword:
|
regularity of systems of PDE's |
| Keyword:
|
Eulerian-Lagrangian description of viscous fluids |
| MSC:
|
35Q35 |
| MSC:
|
55M25 |
| MSC:
|
60H15 |
| MSC:
|
60H30 |
| MSC:
|
76D03 |
| MSC:
|
76D05 |
| idZBL:
|
Zbl 1090.35146 |
| idMR:
|
MR1903307 |
| . |
| Date available:
|
2009-01-08T19:19:33Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119300 |
| . |
| Reference:
|
[1] Chorin A.J.: Vorticity and Turbulence.Applied Mathematical Sciences 103, Springer-Verlag, New York, 1994. Zbl 0795.76002, MR 1281384 |
| Reference:
|
[2] Constantin P.: An Eulerian-Lagrangian approach to the Navier-Stokes equations.preprint, 2000. Zbl 0988.76020, MR 1815721 |
| Reference:
|
[3] DiPerna R.J., Majda A.: Oscillations and concentrations in weak solutions of the incompressible fluid equations.Comm. Math. Phys. 108 (1987), 667-212. Zbl 0626.35059, MR 0877643 |
| Reference:
|
[4] Kiselev A.A., Ladyzhenskaya O.A.: On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid (in Russian).Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957), 655-680. MR 0100448 |
| Reference:
|
[5] Ladyzhenskaja O.A., Solonnikov V.A., Uralceva N.N.: Linear and quasilinear equations of parabolic type (in Russian).Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. MR 0241822 |
| Reference:
|
[6] Leray J.: Sur le mouvement d'un liquide visqueux emplisant l'espace.Acta Math. 63 (1934), 193-248. MR 1555394 |
| Reference:
|
[7] Lions P.-L.: Mathematical Topics in Fluid Mechanics.Vol. 1, Clarendon Press, Oxford, 1996. Zbl 0908.76004, MR 1422251 |
| Reference:
|
[8] Maunder C.R.F.: Algebraic Topology.Cambridge University Press, Cambridge-New York, 1980. Zbl 0435.55001, MR 0694843 |
| Reference:
|
[9] Oru F.: Rôle des oscillation dans quelques problèmes d'analyse non-linéaire.Ph.D. Thesis, ENS Cachan, 1998. |
| . |
