Talks

We consider the following topics related to singularity formation in incompressible Euler equations.
1) Local well-posedness in Hoelder spaces and global well-posedness in 2D flows and 3D axisymmetric flows without swirl.
2) Self-similar singularity analysis for Riccati ODE and Constantin-Lax-Madja model. 
3) Singularity formation for 3D axisymmetric flows using self-similar analysis by Tarek Elgindi.
4) Different proof of singularity formation by Córdoba, Martinez-Zoroa, Zheng.

Singular stochastic partial differential equations (PDEs) informally refer to PDEs forced by random noise that are so rough that the nonlinear term becomes ill-defined in the classical sense, for example as illustrated by Bony's paraproduct estimates. Convex integration is a technique that was originally developed in differential geometry but has been adapted in the past few decades to deterministic PDEs and stochastic PDEs to construct non-unique solutions at a very low regularity level. We discuss global-in-time uniqueness theory of the solutions to singular stochastic PDEs, as well as applications of convex integration to such equations.

Stirring and mixing in fluids, specifically incompressible fluids, impact many physical and biological processes, from dispersal of pollutants to transport of nutrients. From a mathematical point of view, mixing can be studied in different contexts, from ergodic theory to homogenization. In this talk, I will present a quantitative approach to mixing that arises in the analysis of partial differential equations. I I will discuss examples of incompressible flows that mix optimally in time and consequences of mixing on systems modeled by partial differential equations. In particular, I will show how these lead to loss of regularity for linear transport equations.I will discuss the combined effect of transport and diffusion, leading to enhanced and anomalous dissipation. Enhanced dissipation has important effects on dissipative systems, from preventing blow-up to phase separation. I will discuss an application to the Kuramoto-Sivashinsky equation in two space dimensions, a model for flame front propagation in combustion.Enhanced dissipation can occur in the presence of advection by flows that are not mixing, such as shear flows. I will discuss examples of advection-diffusion operators that exhibit enhanced dissipation for well prepared data. The proof relies on certain resolvent estimates. I will then give an application to the phenomenon of striation in the two-dimensional Cahn-Hilliard equation, a model for binary fluids, with advection by a background shear flow.

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