Preface
Contents
1 The Navier–Stokes Equations with Deterministic Rough Force
1.1 The Deterministic Navier–Stokes Equations
1.1.1 The Newtonian Equations
1.1.2 A Rigorous Deterministic Theorem in d=2
1.2 Well–Posedness of the Model with Rough Force
1.2.1 The Stokes Problem
1.2.2 Auxiliary Navier–Stokes Type Equations
1.2.3 Final Main Result on the Equation with Rough Force
1.3 Summary
2 Stochastic Navier–Stokes Equations and State-Dependent Noise
2.1 Introduction
2.1.1 Filtered Probability Space
2.2 Additive Noise Under the View of Stochastic Calculus
2.2.1 Consequences
2.3 2D Stochastic Navier–Stokes Equations
2.3.1 Proof of Uniqueness
2.4 Proof of Existence
2.4.1 Introduction
2.4.2 Gyongy–Krylov Convergence Criterion
2.4.3 Compactness Criteria
Deterministic Ascoli–Arzelà Theorem
Deterministic Aubin–Lions Type Theorems
Stochastic Theory
2.4.4 Application to Galerkin Approximations: 2D Case
Estimates and Compactness
Application of Gyongy–Krylov Criterion and Conclusion of the Proof of Existence
2.4.5 3D Navier–Stokes Equations with Additive Noise
The Problem of Uniqueness
Estimates on Galerkin and Tightness
Definition of Solution and Convergence
2.5 Summary
3 Transport Noise in the Heat Equation
3.1 Introduction: Stochastic Heat Transport
3.1.1 Divergence Form of the Operator
3.2 Existence and Uniqueness for the Heat Equation with Transport Noise
3.2.1 Variational Method
A Priori Estimates Using Stratonovich Formulation
A Priori Estimates Using Itô Formulation
Maximum Principle a Priori Estimates
3.2.2 Semigroup Method
Notions of Solution and Main Result
General Parabolic Equations with Itô–Type Transport Noise
Auxiliary Variables and End of the Proof
Super-Parabolicity Condition and Stratonovich Formulation
3.2.3 The Equation for the Average
3.3 When θ Is Close to
3.3.1 Main Assumption and Result
3.3.2 When Q,κ Is Small (and L Is Not Small)
The Case When Q( x,x) Is Degenerate
The Case When Q( x,x) Is Non-degenerate
3.3.3 The Result for Long Times
3.4 The Action of Transport Noise on Vector Fields
3.4.1 Passive Magnetic Field
The Corrector
The Difficulty
The Purely Transport Case
4 Transport Noise in the Navier–Stokes Equations
4.1 Well-Posedness for the Vorticity Formulation
4.1.1 Variational Method: Plan of Work
4.1.2 Functional Setting and Assumptions
4.1.3 Galerkin Approximation and Limit Equations
4.1.4 Existence, Uniqueness and Further Results
4.2 Eddy Viscosity for the Vorticity Equation
4.2.1 Some Analytical Lemmas
4.2.2 The Stochastic Convolution
4.2.3 Proof of Theorem 4.11
4.2.4 The Result for Long Times
4.3 Velocity Formulation
4.3.1 Functional Setting and Assumptions
4.3.2 Galerkin Approximation and Limit Equations
4.3.3 Existence, Uniqueness and Further Results
4.4 The 3D Navier–Stokes Equations with Transport Noise
4.4.1 The Result in the Case of Only Transport
4.5 Summary
5 From Small-Scale Turbulence to Eddy Viscosity and Dissipation
5.1 Introduction: The Global Heuristic Scheme
5.1.1 Large and Small Space Scales
5.2 Small-Scale Turbulence and Additive Noise
5.3 Action of Small-Scale Turbulence on Large-Scales: Transport Noise Under Scale Separation
5.4 Eddy Viscosity and Eddy Diffusion
5.5 More on Additive Noise at Small Scales: Vortex Production at Boundaries
5.5.1 Generation of Vortices Near Obstacles
The Brownian Limit
5.5.2 Scaling the Previous Example
5.5.3 Example of State-Dependent Noise
5.6 The Wong–Zakai Corrector and Stratonovich Integrals
5.6.1 A One-Dimensional Example
5.6.2 The Case of the Heat Equation
5.7 Summary
Bibliography