| Abstract: |
| In the 1960s, Robert Kraichnan [Kraichnan 1968] proposed a synthetic model for passive scalar turbulence, consisting of a scalar advected by a random Gaussian velocity field that is white in time and H\{o}lder continuous in space. Despite its simplicity, this linear SPDE exhibits key features of realistic turbulent flows, such as anomalous dissipation. Renewed interest in this model followed [Coghi, Maurelli 2023], where it was proved that the same transport-type noise restores well-posedness in regimes where the deterministic 2D Euler equations admit non-unique weak solutions.
In this talk, we further develop this line of research by investigating additional properties of the solutions constructed in [Coghi, Maurelli 2023]. In particular, we present new results on anomalous fractional Sobolev regularity and anomalous dissipation of the mean enstrophy for solutions to the 2D Euler equations with rough Kraichnan noise. Time permitting, we will also discuss implications for the well-posedness theory of more singular nonlinear advection models, such as the Surface Quasi-Geostrophic and Incompressible Porous Media equations.
This talk is based on ongoing joint work with L. Galeati and U. Pappalettera. |
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