##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Alexander Kiselev

#### Duke University

## Small scale creation in ideal fluids

##### Abstract:

The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of its solutions remain poorly understood. In particular, global regularity vs finite time blow up question for 3D Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of solutions has been double exponential in time. I will describe a construction (based on a work joint with Vladimir Sverak) that shows that such fast generation of small scales can actually happen, so that the double exponential upper bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions to 3D Euler equation. The scenario is axi-symmetric, and the geometry of the scenario involves hyperbolic points of the flow located at the boundary. If time permits, I will also discuss some models that attempt to gain insight into this scenario.

Host: Andrej Zlatos

### April 26, 2018

### 4:00 PM

### AP&M 6402

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