I will discuss on the existence and regularity results for the heat flow of the so called H-systems and for more general parabolic p-laplacian problems with critical growth.

# Past Partial Differential Equations Seminar

We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in an arbitrary open subset of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderón problem. This is a joint work with T. Ghosh (HKUST) and G. Uhlmann (Washington).

In this talk we consider particular solutions of the Boltzmann equation which have the form $f (x,v,t) = g (v − M (t)x,t)$ where $M (t) = A(I + tA)^{−1}$ with the matrix $A$ describing a shear ﬂow or a dilatation or a combination of both. These solutions are known as equidispersive solutions. We will show that, for diﬀerent choices for the matrix A and for diﬀerent homogeneities of the collision kernel, we obtain diﬀerent long time asymptotics for the corresponding equidispersive solutions. In particular we will focus on the case of simple shear ﬂow and prove rigorously the existence of self-similar solutions with exponentially increasing internal energy.

We consider the Beris-Edwards model of liquid crystal dynamics. We study a non-dimensionalisation and regime suited for the study of defect patterns, that amounts to a combined high Ericksen and high Reynolds number regime.

We identify some of the flow mechanisms responsible for the appearance of localized gradients that increase in time.

This is joint work with Hao Wu (Fudan).

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen.

In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.