# 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

**Theorem**

**1.**

**Remark**

**1.**

#### 4.2. Lagrange Multipliers Approach

**Theorem**

**2.**

## 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

**Theorem**

**3.**

## 8. Other Equations, Methods and Discussion

## Funding

## Conflicts of Interest

## Appendix A. Some Basic Notions of Stochastic Calculus

- (i)
- ${W}_{0}=x$;
- (ii)
- ${W}_{t}$ has independent increments;
- (iii)
- For $s<t$, ${W}_{t}-{W}_{s}$ has a normal distribution $\mathcal{N}(0,t-s)$.

- (i)
- ${M}_{t}$ is ${\mathcal{P}}_{t}$-measurable for all t (we also say that ${M}_{t}$ is adapted to ${\mathcal{P}}_{t}$);
- (ii)
- $E|{M}_{t}|<+\infty $;
- (iii)
- For $s<t$, ${E}_{s}({M}_{t})={M}_{s}$. Here E and ${E}_{t}$ denote respectively expectation and conditional expectation with respect to ${\mathcal{P}}_{t}$. 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.
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Cruzeiro, Ana Bela. 2020. "Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review" *Water* 12, no. 3: 864.
https://doi.org/10.3390/w12030864