Chemotactic blow up in the context of the Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that when the Keller-Segel equation is coupled with passive advection, blow-up can be prevented if the flow possesses mixing or diffusion-enhancing properties, and its amplitude is sufficiently strong. In this talk, we consider the Keller-Segel equation coupled with an active advection, which is an incompressible flow obeying Darcy's law for incompressible porous media equation and driven by buoyancy force. We prove that in contrast with passive advection, this active advection coupling is capable of suppressing chemotactic blow up at arbitrary small coupling strength: namely, the system always has globally regular solutions. (Joint work with Zhongtian Hu and Alexander Kiselev).[-]

The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).[-]

The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).[-]

In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows. In a joint work with László, Tristan Buckmaster and Vlad Vicol we improve Isett's theorem to show the existence of dissipative solutions of the incompressible Euler equations below the Onsager's threshold.[-]

In many physical fluid systems, the constituent particles present a parity-breaking intrinsic angular momentum: this is the case, for instance, of quantum fluids and super-fluids, polyatomic gases, chiral active matter and vortex dynamics. In such situations, only the skew-symmetric component of the total viscous stress tensor, often dubbed odd viscosity, is non-zero, implying that the viscosity becomes non-dissipative.
At the level of the mathematical model, the odd viscosity term is responsible for a loss of regularity, as it involves higher order space derivatives of the velocity field and, in the case of non-homogeneous fluids, of the density.
In this talk we consider the dynamics of non-homogeneous incompressible fluids having odd viscosity and we set up a well-posedness theory in Sobolev spaces for the related system of equations. The proof is based on the introduction of a set of suitable 'good unknowns' for the system, which allow to put in evidence an underlying hyperbolic structure and to circumvent, in this way, the loss of derivatives created by the odd viscosity term.
The talk is based on a joint work with Rafael Granero-Belinchón (Universidad de Cantabria) and Stefano Scrobogna (Università degli Studi di Trieste).[-]

The patch solutions of the 2D Euler and (modified) SQG equations have form $\omega(x, t)=\chi_{\Omega(t)}(x)$ of a characteristic function of a domain $\Omega(t)$ evolving in time according to the Biot-Savart law $u=\nabla^{\perp}(-\Delta)^{-1+\alpha} \omega$, here $\alpha=0$ corresponds to the Euler case and $0<\alpha<1$ to the modified SQG family. For the Euler case, the first proof of global regularity for pathes was given by Chemin in Hölder spaces $C^{k, \beta}, 0<\beta<1$. For the modified SQG family, the problem remains largely open - with the only finite time singularity formation result available in the presence of boundary and for small $\alpha[5,2]$. I will discuss some recent conditional results on the possible scenarios for finite time blow up. Also, for the Euler patch case, I will describe a construction of patches that are $C^{2}$ at the initial and all integer times, but lack this regularity for all other times - without being time periodic. This result is based on the analysis of the curvature evolution equation, which may also be useful for other applications.[-]

It is well known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.
In recent joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov's h-principle.
In this lecture we explain this connection and outline recent progress towards Onsager's conjecture.[-]