Local Solutions for Stochastic Navier Stokes Equations

Alain Bensoussan; Jens Frehse

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 2, page 241-273
  • ISSN: 0764-583X

Abstract

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In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations.

How to cite

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Bensoussan, Alain, and Frehse, Jens. "Local Solutions for Stochastic Navier Stokes Equations." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 241-273. <http://eudml.org/doc/197447>.

@article{Bensoussan2010,
abstract = { In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations. },
author = {Bensoussan, Alain, Frehse, Jens},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = { Navier Stokes equations; stochastic equations; abstract parabolic equations; Ito integral; local solution; Ito equation; Stokes operator; Functional equation; Mild solution; Random time.; deterministic Navier-Stokes equations; stochastic Navier Stokes equations; smoothness; random parameter; stochastic differential equations},
language = {eng},
month = {3},
number = {2},
pages = {241-273},
publisher = {EDP Sciences},
title = {Local Solutions for Stochastic Navier Stokes Equations},
url = {http://eudml.org/doc/197447},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Bensoussan, Alain
AU - Frehse, Jens
TI - Local Solutions for Stochastic Navier Stokes Equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 2
SP - 241
EP - 273
AB - In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations.
LA - eng
KW - Navier Stokes equations; stochastic equations; abstract parabolic equations; Ito integral; local solution; Ito equation; Stokes operator; Functional equation; Mild solution; Random time.; deterministic Navier-Stokes equations; stochastic Navier Stokes equations; smoothness; random parameter; stochastic differential equations
UR - http://eudml.org/doc/197447
ER -

References

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  1. A. Bensoussan, Stochastic Navier Stokes Equations. Acta Appl. Math.38 (1995) 267-304.  
  2. A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes. J. Func. Anal.13 (1973) 195-222.  
  3. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge (1992).  
  4. F. Flandoli and D. Gatarek, Martingale and Stationary Solutions for Navier-Stokes Equations, Preprints di Matematica - n° 14 (1994).  
  5. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (1981).  
  6. I. Karatzas and S.E. Shereve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York (1988).  
  7. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris (1969).  
  8. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North Holland (1977).  
  9. W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Aspects of Mathematics, Fr. Viewig & Sohn, Braunschweig/Wiesbaden (1985).  

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