Seminar Analysis and Mathematical Physics: Mickaël Latocca (ENS Paris)

High-regularity Invariant Measures for 2d and 3d Euler Equations and Growth of the Sobolev Norms

The possible growth of the Sobolev norms of the solution to the 2d (and 3d) Euler equations and its quantification remains ill-known. The only general bound is double exponential. Conversely, such a double exponential growth scenario occurs for specific initial data in the setting of the disc (Kiselev-Sverak). In the setting of the torus, only an exponentially growing scenario has been exhibited (Zlatos). Could the double exponential scenario occur on the torus? What is the typical behaviour that could be expected? It is highly possible that on the torus, Sobolev norms generically do not grow fast.

In this talk, I will present some results obtained in this direction. We will construct invariant measures for the 2d Euler equation at high regularity ($H^s$, $s>2$) and prove that on the support of the measure, Sobolev norms do not grow faster than polynomially.

Refining the method allows to construct an invariant measure to the 3d Euler equations at high regularity ($H^s$, $s>7/2$) and thus construct
global dynamics on the support of the measure, exhibiting at most polynomial growth.

Finally, it time permits we will discuss the properties of the measures constructed.

Veranstaltung übernehmen als iCal