Hypocoercivity is a mathematical concept used to describe the behavior of certain dynamical systems and partial differential equations (PDEs) that do not exhibit traditional coercivity properties but still exhibit convergence to equilibrium over time. In the study of kinetic equations and systems where traditional coercive methods (like those used in parabolic PDEs) do not apply, we may resort to hypocoercivity.

The concept often involves a mix of dissipative and conservative components. For example, in kinetic equations, there may be a combination of relaxation mechanisms (which are dissipative) and transport mechanisms (which are conservative).

Proving hypocoercivity typically involves sophisticated techniques such as constructing Lyapunov functions, using spectral gap inequalities, or employing entropy methods. The goal is to find a functional setting in which the dissipative effects dominate over time, leading to eventual relaxation to equilibrium.

Coercive

In more formal terms, if $\cL$ is an operator describing the system, coercivity would mean that

A Theorectical Framework for Bounding the Mixing Time of Lindbladian Dynamics

Notation

Intuition