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Series: PDE Seminar

Series: PDE Seminar

Series: PDE Seminar

Series: PDE Seminar

Series: PDE Seminar

In this talk we will introduce two models for the movement of a small droplet over a substrate: the thin film equation and the quasi static approximation. By tracking the motion of the apparent support of solutions to the thin film equation, we connect these two models. This connection was expected from Tanner's law: the edge velocity of a spreading thin film on a pre-wetted solid is approximately proportional to the cube of the slope at the inflection. This is joint work with Prof. Antoine Mellet.

Series: PDE Seminar

A rotating star may be modeled as gas under self gravity with a given total mass and prescribed angular velocity. Mathematically this leads to the Euler-Poisson system. In this talk, we present an existence theorem for such stars that are rapidly rotating, depending continuously on the speed of rotation. No previous results using continuation methods allowed rapid rotation. The key tool for the result is global continuation theory via topological degree, combined with a delicate limiting process. The solutions form a connected set $\mathcal K$ in an appropriate function space. Take an equation of state of the form $p = \rho^\gamma$; $6/5 < \gamma < 2$, $\gamma\ne 4/3$. As the speed of rotation increases, we prove that either the density somewhere within the stars becomes unbounded, or the supports of the stars in $\mathcal K$ become unbounded. Moreover, the latter alternative must occur if $\frac43<\gamma<2$. This result is joint work with Walter Strauss.

Series: PDE Seminar

We discuss bilateral obstacle problems for fully nonlinear second order
uniformly elliptic partial differential equations (PDE for short) with
merely continuous obstacles. Obstacle problems arise not only in
minimization of energy functionals under restriction by obstacles but
also stopping time problems in stochastic optimal control theory. When
the main PDE part is of divergence type, huge amount of works have been
done. However, less is known when it is of non-divergence type.
Recently, Duque showed that the Holder continuity of viscosity solutions
of bilateral obstacle problems, whose PDE part is of non-divergence
type, and obstacles are supposed to be Holder continuous. Our purpose is
to extend his result to enable us to apply a much wider class of PDE.
This is a joint work with Shota Tateyama (Tohoku University).

Series: PDE Seminar

The two dimensional Euler equation is globally wellposed, but the long time behavior of solutions is not well understood. Generically, it is conjectured that the vorticity, due to mixing, should weakly but not strongly converge as $t\to\infty$. In an important work, Bedrossian and Masmoudi studied the perturbative regime near Couette flow $(y,0)$ on an infinite cylinder, and proved small perturbation in the Gevrey space relaxes to a nearby shear flow. In this talk, we will explain a recent extension to the case of a finite cylinder (i.e. a periodic channel) with perturbations in a critical Gevrey space for this problem. The main interest of this extension is to consider the natural boundary effects, and to ensure that the Couette flow in our domain has finite energy. Joint work with Alex Ionescu.

Series: PDE Seminar

We give derivative estimates for solutions to divergence form elliptic equations with piecewise
smooth coefficients. The novelty of these estimates is that, even though they depend on the shape
and on the size of the surfaces of discontinuity of the coefficients, they are independent of the
distance between these surfaces.

Series: PDE Seminar

In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). The forward-forward models arise in the study of numerical schemes to approximate stationary MFGs. We establish a link between these models and a class of hyperbolic conservation laws. Furthermore, we investigate the existence of solutions and examine long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.