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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.

Series: PDE Seminar

I will discuss the limit shapes for local minimizers of the Alt-Caffarelli energy. Fine properties of the associated pinning intervals, continuity/discontinuity in the normal direction, determine the formation of facets in an associated quasi-static motion. The talk is partially based on joint work with Charles Smart.

Series: PDE Seminar

We consider a class of nonlinear, degenerate drift-diffusion equations in R^d. By a scaling argument, it is widely believed that solutions are uniformly Holder continuous given L^p-bound on the drift vector field for p>d. We show the loss of such regularity in finite time for p≤d, by a series of examples with divergence free vector fields. We use a barriers argument.

Series: PDE Seminar

The boundary value and mixed value problems for linear and
nonlinear degenerate abstract elliptic and parabolic equations are
studied. Linear problems involve some parameters. The uniform
L_{p}-separability properties of linear problems and the optimal
regularity results for nonlinear problems are obtained. The equations
include linear operators defined in Banach spaces, in which by choosing
the spaces and operators we can obtain numerous classes of problems for
singular degenerate differential equations which occur in a wide variety
of physical systems.
In this talk, the classes of boundary value problems (BVPs) and
mixed value problems (MVPs) for regular and singular degenerate
differential operator equations (DOEs) are considered. The main
objective of the present talk is to discuss the maximal regularity
properties of the BVP for the degenerate abstract elliptic and parabolic
equation
We prove that for f∈L_{p} the elliptic problem has a unique
solution u∈ W_{p,α}² satisfying the uniform coercive estimate
∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)}
where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and
W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be
defined later. We also prove that the differential operator generated by
this elliptic problem is R-positive and also is a generator of an
analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with
p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for
the corresponding degenerate parabolic problem. This fact is used to
obtain the existence and uniqueness of maximal regular solution of the
MVP for the nonlinear parabolic equation.

Series: PDE Seminar

The ground state solution to the nonlinear Schrödinger equation (NLS) is a global, non-scattering solution that often provides a threshold between scattering and blowup. In this talk, we will discuss new, simplified proofs of scattering below the ground state threshold (joint with B. Dodson) in both the radial and non-radial settings.

Series: PDE Seminar

We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.