Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review
Abstract
:1. Introduction
2. Results
3. The Non Viscous Case
4. The Deterministic Navier–Stokes Equation
4.1. Stochastic Geometric Mechanics Approach
4.2. Lagrange Multipliers Approach
5. Constructing Solutions of Navier–Stokes Equation by Probabilistic Methods
5.1. Forward–Backward Stochastic Differential Systems
5.2. Entropy Methods
5.3. A Weak Notion of Navier–Stokes Solutions
6. Stability Properties
7. A Stochastic Navier–Stokes Equation
8. Other Equations, Methods and Discussion
Funding
Conflicts of Interest
Appendix A. Some Basic Notions of Stochastic Calculus
- (i)
- ;
- (ii)
- has independent increments;
- (iii)
- For , has a normal distribution .
- (i)
- is -measurable for all t (we also say that is adapted to );
- (ii)
- ;
- (iii)
- For , . Here E and denote respectively expectation and conditional expectation with respect to . Multidimensional martingales are defined analogously.
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Cruzeiro, A.B. Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review. Water 2020, 12, 864. https://doi.org/10.3390/w12030864
Cruzeiro AB. Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review. Water. 2020; 12(3):864. https://doi.org/10.3390/w12030864
Chicago/Turabian StyleCruzeiro, Ana Bela. 2020. "Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review" Water 12, no. 3: 864. https://doi.org/10.3390/w12030864
APA StyleCruzeiro, A. B. (2020). Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review. Water, 12(3), 864. https://doi.org/10.3390/w12030864