Time: Every Wednesday afternoon, 14:00-15:30 (from 09/09/2013 to 30/11/2013)

Location: VIASM Lecture Hall (C2).

Abstract:

In this mini-course, I will present classical results on existence, uniqueness and regularity of weak/strong solutions to Navier-Stokes equations in both dimension two and three as well as results on the long-time behavior of these solutions in terms of global attractors and stability of stationary solutions. I will also present some recent significant results on control theory of Navier-Stokes equations, including controllability, optimal control and stabilization. Some related open problems will be also given.

Outline of the course (intended)

Chapter 1. Navier-Stokes equations and mathematical tools

1.1. Derivation of Navier-Stokes equations

1.2. Function spaces and operators. Inequalities for the nonlinear terms

Chapter 2. Existence, uniqueness and regularity of solutions

2.1. Weak solutions

2.2. Strong solutions

2.3. Analyticity and backward uniqueness

2.4. Regularity of solutions

2.5. Some open problems

Chapter 3. Long-time behavior of solutions

3.1. Existence and stability of stationary solutions

3.2. Existence and dimension estimates of global attractors

3.3. Long-time finite dimensional approximation

3.4. Determining modes. Data assimilation problem

3.5. Some open problems

Chapter 4. Optimal control of Navier-Stokes equations

4.1. Setting of the problem and existence of solutions

4.2. Necessary optimality conditions

4.3. Second-order sufficient optimality conditions

4.4. Stability of optimal controls. Convergence of the SQP-method

4.5. Some open problems

Chapter 5. Controllability of Navier-Stokes equations

5.1. Setting of the problem and Lions’ conjecture

5.2. Local exact controllability to the trajectory

5.3. Local approximate controllability

5.4. Some open problems

Chapter 6. Stabilization of Navier-Stokes equations

6.1. Internal stabilization

6.2. Boundary stabilization

6.3. Some open problems