Talk page
Title:
Quantitative convergence to equilibrium and some partial dissipation results for Euler-like SDE
Speaker:
Abstract:
In this talk I will present some results on the existence of invariant measures and convergence rate to equilibrium for a class of nonlinear SDEs that include damped-driven truncations of the 2d/3d Euler equations and Sabra shell model. If the diffusion is hypoelliptic and the damping acts on every degree of freedom, then it is well known that the system admits a unique invariant measure that attracts all initial conditions exponentially fast. A natural problem is then to study how the convergence rate to equilibrium scales in the zero-dissipation limit, which in the fluid setting amounts to understanding the convergence rate to equilibrium in the limit of infinite Reynolds number. In the first part of the talk I will discuss results in which we obtain an optimal estimate on the exponential convergence to equilibrium in the limit of zero dissipation. The proof is based on a PDE approach that hinges on quantitative hypoelliptic estimates for the stationary density as well as the associated time-dependent Kolmogorov equation. In the second part of the talk I will discuss the situation where the damping only acts on a proper subset of the degrees of freedom in the system. In this setting the existence of an invariant measure in a difficult question in general, as one must quantify carefully how the nonlinearity transfers energy from the undamped modes to the damped modes. I will discuss an approach for proving existence of an invariant measure based on time-averaged coercivity estimates and its application to some examples. This is joint work with Jacob Bedrossian.
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