Professor Vladimir Sverak

The research focus on open problems in the theory of partials differential equations (PDEs) and the Calculus of Variations. The topics will include:

• Computer assisted proofs of the existence of singularities for certain complex Ginsburg-Landau equations and non-linear Schroedinger equations.
• Explorations of regularity questions for equations of Fluid Mechanics and their simplified models. Both the fundamental equations of incompressible fluid mechanics (the Euler and the Navier-Stokes equations) and their simplified models (such as the incompressible porous medium equation and other models where singularity formation remain open) will be studied.
• Exploration of Morrey's quasi-convexity in the calculus of variation and the long-standing conjectures about its validity in certain particular cases. (This is essentially about investigating a certain type of functional inequalities that can be approximated by certain (fairly large) linear programming problems. The hope is that the insight obtained through the linear programming optimization will lead to a resolution of a long-standing conjecture.)