Evolution equations and Brockett-Wegner flow

DATES: 06-10 February 2023 (5 sessions)
TIME: 10:00-12:00 (a total of 10 hours)
LOCATION: Maryam Mirzakhani Seminar Room at BCAM and Online

CONTENT
ABSTRACT. The main goal of this course is to present the so-called Brockett-Wegner flow. It is an operator-valued differential equation presented for the first time in the nineties. Brockett intially introduced this evolution equation [1] in order to diagonalize matrices, it reads as

(1) (∂tHt = [Ht, [Ht,A]]
Ht=0 = H0,

where H0 is a symmetric matrix we want to diagonalize, and A a well-chosen matrix. More recently, this evolution equation turned out to have applications in mathematical physics which is what has drawn our attention to this equation in particular.
The goal of this course is to present a study of the evolution equation (1) with H0 being a bounded operator as initial condition [2], and that will ultimately lead us to show that the solution converges to the diagonal form of the initial operator. Along the way, it will be the occasion to present or remind the general theory of evolution equation[3], in particular of non-autonomous evolution equation. Indeed, the Brockett-Wegner flow is a non-autonomous equation, and it is what makes its study interesting and challenging.
If time permits, some results for unbouded operators could be discussed.

1. GENERAL THEORY OF EVOLUTION EQUATIONS
1.1. Reminder on autonomous evolution equation.
1.2. Non autonomous evolution equation and Dyson expansion.

2. BROCKETT-WEGNER FLOW FOR BOUNDED OPERATORS
2.1. Local existence and uniqueness.
2.2. Structure of the solution and global existence.
2.3. Convergence.

3. RESULTS FOR UNBOUNDED OPERATORS (IF TIME PERMITS)

REFERENCES
[1] R. W. Brockett. "Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems". In: Linear Algebra Appl. (1991).
[2] Jean-Bernard Bru and Volker Bach. "Rigorous foundations of the Brockett-Wegner flow for operators". In: Journal of evolution equation 10.2 (2010).
[3] A. Pazy. 2Semigroups of Linear Operators and Applications to Partial Differential Equations". In: Springer New York. (1983).

*Registration is free, but mandatory before 1 February 2023. To sign-up go to https://forms.gle/bb6D246bQvUGTjnx5 and fill the registration form.

• BCAM Course 2022-2023 Jean-Bernard Bru And Natha Metraud