Title: Entrainment dominated effects in the long residence times of a sphere settling in stratified fluids
Abstract: We study the delayed settling dynamics of spheres going through stably stratified sharp density variations in viscous-dominated regimes. In particular, we focus on cases with long residence times at the interface rivaling the ones observed under similar configurations for marine snow aggregates in the ocean. To capture the most significant aspects of the system, we solve a first principle model that reduces to a highly coupled system. Taking the far field asymptotic approach speeds up the numerics in the appropriate region of validity. The entrainment dominated regime, however, requires the exact solution for the sphere exterior problem of the Stokes equations. In this regime, we discuss the numerical challenges in computing a convolution integral and the effects of diffusion.

Title: Strong Ill-Posedness in L∞ for the Riesz Transform Problem
Abstract: We prove strong ill-posedness in L∞ for linear perturbations of the 2d Euler equations of the form: ∂tω+u⋅∇ω=R(ω), where R is any non-trivial second order Riesz transform. Namely, we prove that there exist smooth solutions that are initially small in L∞ but become arbitrarily large in short time. Previous works in this direction relied on the strong ill-posedness of the linear problem, viewing the transport term perturbatively, which only led to mild growth. In this work we derive a nonlinear model taking all of the leading order effects into account to determine the precise pointwise growth of solutions for short time. Interestingly, the Euler transport term does counteract the linear growth so that the full nonlinear equation grows an order of magnitude less than the linear one.

Title: An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension
Abstract: In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume. We extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $\alpha\ge 0$. Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $\alpha > 0$. In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $\alpha\ge 0$ and shallow radially symmetric containers with m azimuthal nodal lines, $m=0,1$. Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $\alpha = 0$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension. This is joint work with Christel Hohenegger and Braxton Osting.

Title: Global well-posedness and scattering in nonlinear wave equations
Abstract: In this talk we will consider the wave equation in odd space dimensions with energy-supercritical nonlinearity, in the radial setting. We will review the concentration compactness and rigidity arguments starting from the earlier work by Kenig-Merle and Duyckaerts-Kenig-Merle in the energy-critical and energy-supercritical cases, and outline the key ideas behind the proof of global well-posedness and scattering results in 3d and higher odd dimensions