A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

Stephen Montgomery-Smith; Milan Pokorný

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 1, page 61-75
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Montgomery-Smith, Stephen, and Pokorný, Milan. "A counterexample to the smoothness of the solution to an equation arising in fluid mechanics." Commentationes Mathematicae Universitatis Carolinae 43.1 (2002): 61-75. <http://eudml.org/doc/248980>.

@article{Montgomery2002,
abstract = {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.},
author = {Montgomery-Smith, Stephen, Pokorný, Milan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Navier-Stokes equations; Euler equations; regularity of systems of PDE's; Eulerian-Lagrangian description of viscous fluids; Navier-Stokes equations; Euler equations; regularity of systems of PDE's; Eulerian-Lagrangian description of viscous fluids},
language = {eng},
number = {1},
pages = {61-75},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A counterexample to the smoothness of the solution to an equation arising in fluid mechanics},
url = {http://eudml.org/doc/248980},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Montgomery-Smith, Stephen
AU - Pokorný, Milan
TI - A counterexample to the smoothness of the solution to an equation arising in fluid mechanics
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 1
SP - 61
EP - 75
AB - 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.
LA - eng
KW - Navier-Stokes equations; Euler equations; regularity of systems of PDE's; Eulerian-Lagrangian description of viscous fluids; Navier-Stokes equations; Euler equations; regularity of systems of PDE's; Eulerian-Lagrangian description of viscous fluids
UR - http://eudml.org/doc/248980
ER -

References

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  1. Chorin A.J., Vorticity and Turbulence, Applied Mathematical Sciences 103, Springer-Verlag, New York, 1994. Zbl0795.76002MR1281384
  2. Constantin P., An Eulerian-Lagrangian approach to the Navier-Stokes equations, preprint, 2000. Zbl0988.76020MR1815721
  3. DiPerna R.J., Majda A., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), 667-212. (1987) Zbl0626.35059MR0877643
  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. (1957) MR0100448
  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. MR0241822
  6. Leray J., Sur le mouvement d'un liquide visqueux emplisant l'espace, Acta Math. 63 (1934), 193-248. (1934) MR1555394
  7. Lions P.-L., Mathematical Topics in Fluid Mechanics, Vol. 1, Clarendon Press, Oxford, 1996. Zbl0908.76004MR1422251
  8. Maunder C.R.F., Algebraic Topology, Cambridge University Press, Cambridge-New York, 1980. Zbl0435.55001MR0694843
  9. Oru F., Rôle des oscillation dans quelques problèmes d'analyse non-linéaire, Ph.D. Thesis, ENS Cachan, 1998. 

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