In this talk, we develop an analysis of the compressible, isentropic Euler equations in two spatial dimensions for a generalized polytropic gas law. The main focus is rotational flows in the subsonic regimes, described through the framework of the Euler equations expressed in self-similar variables and pseudo-velocities. A Bernoulli type equation is derived, serving as a cornerstone for establishing a self-similar system tailored to rotational flows. At the end, the study extends to an analysis of a perturbed model, introducing the concept of quasi-potential flows, offering insights into their behavior and implications.

Weighted Fourier extension estimates are intimately connected to a wide range of problems of geometric and discrete nature. The main goal of the talk is to present a certain "ray bundle representation" of the Fourier extension operator in terms of the Wigner transform to investigate such estimates.

In joint work with Bennett, Gutierrez and Nakamura, we show how Sobolev estimates for the Wigner transform can be converted into "tomographic bounds" for the Fourier extension operator, which implies a variant of the (recently shown by H. Cairo to be false) Mizohata-Takeuchi conjecture. Together with Bez and the previous three authors, we employed our phase-space approach to study the orthonormal systems version of the Mizohata-Takeuchi conjecture, which allowed us, in particular, to give a direct proof of the orthonormal Strichartz estimates of Frank and Sabin in dimension 1. If time allows, we will make a further connection between our results and Flandrin's conjecture in signal processing through the study of certain singular integral operators similar to those studied by Lacey, Lie, Muscalu, Tao and Thiele.

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.

Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.

Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.

We consider the characterization of global attractors $A_f$ for semiflows generated by scalar one-dimensional semilinear parabolic equations of the form $u_t = u_{xx} + f(u,u_x)$, defined on the circle $x\in S^1$, for a class of reversible nonlinearities. We modify a proof developed for nonlinearities of simple type, making it simpler and amenable to generalization. We obtain a classification up to connection equivalence of global attractors for $S^1$-equivariant parabolic equations.