FUNCTIONAL AND SPECTRAL INEQUALITIES IN MATHEMATICAL HYDRODYNAMICS

FUNCTIONAL AND SPECTRAL INEQUALITIES IN MATHEMATICAL HYDRODYNAMICS

Time:

Venue/Location: C101, VIASM

Prof. Alexei Ilyin (Keldysh Institute of Applied mathematics, Russian Academy of Sciences Email: ilyin@keldysh.ru)

  • Lecture 1 (14h30 ngày 18/10): Sobolev and spectral inequalities for orthonormal systems in the theory of attractors I: Berezin--Li-Yau inequalities.
  • Lecture 2 (14h30 ngày 25/10): Sobolev and spectral inequalities for orthonormal systems in the theory of attractors II: Lieb--Thirring inequalities.
  • Lecture 3 (14h30 ngày 1/11): Sobolev and spectral inequalities for orthonormal systems in the theory of attractors III: Inequalities for systems with orthonormal derivatives and applications.

Short bio: Prof. Alexei Ilyin is a principal scientific researcher at the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences in Moscow. He received PhD in 1990 at the Moscow State University and Doctor of science in 2006 at the Steklov Institute of Mathematics. His domain of research consists of nonlinear partial differential equations, mathematical hydrodynamics, attractors, averaging, spectral theory, integral and spectral inequalities, Sobolev spaces. He has published about 70 scientific papers.

Abstract: Sobolev inequalities provide an indispensable tool in the PDE theory. In the theory of attractors upper and lower bounds for orthonormal systems are required for good estimates of the N-traces of the linearized operators on the attractor. We shall discuss in reasonable detail lower bounds of Berezin and Li–Yau-type for the eigenvalues of the Laplace and Stokes operators (including the operators on the sphere) and the corresponding inequalities with correction terms. Upper bounds for the Lp-norms of orthonormal systems systems are called the Lieb–Thirring inequalities and have important applications in mathematical physics, analysis, dynamical systems and attractors, to mention a few. We shall discuss and prove Lieb–Thirring inequalities in the dual form for various boundary conditions including the case of orthonormal divergence free vector functions. We shall also discuss Sobolev inequalities for systems with orthonormal derivatives and their applications to the attractors of certain regularized models in hydrodynamics as well as applications to classical interpolation inequalities of Gagliardo–Nirenberg type.

The autumn lecture of Prof. Alexei Ilyin consists of three following parts:

• Part 1: Sobolev and spectral inequalities for orthonormal systems in the theory of attractors I. Berezin–Li-Yau inequalities.
• Part 2: Sobolev and spectral inequalities for orthonormal systems in the theory of attractors II. Lieb–Thirring inequalities.
• Part 3: Sobolev and spectral inequalities for orthonormal systems in the theory of attractors III. Inequalities for systems with orthonormal derivatives and application

Link Zoom:
https://zoom.us/j/8948173518?pwd=RlI4QkdnQndiWDlPekFkMlZqZVMwZz09

ID: 894 817 3518
Passcode: 2023