Fri, 29 Nov 2019
11:30
L5

Oscillations and Spirals in Two Problems of Global Analysis

Siran Li
(Rice University)
Abstract

We present our works on two problems in global analysis (i.e.,analysis on manifolds): One concerns the compactness of the space of smooth $d$-dimensional immersed hypersurfaces with uniformly $L^d$-bounded second fundamental forms, and the other concerns the validity of W^{2,p}$-elliptic estimates for the Laplace--Beltrami operator on open manifolds. We construct explicit counterexamples to both problems. The onstructions involve rapid oscillations and wild spirals, with motivations derived from physical phenomena.

Fri, 16 Nov 2018

12:00 - 13:00
L5

Some Problems On Harmonic Maps from $\mathbb{B}^3$ to $\mathbb{S}^2$

Siran Li
(Rice University)
Abstract

Harmonic map equations are an elliptic PDE system arising from the  
minimisation problem of Dirichlet energies between two manifolds. In  
this talk we present some some recent works concerning the symmetry  
and stability of harmonic maps. We construct a new family of  
''twisting'' examples of harmonic maps and discuss the existence,  
uniqueness and regularity issues. In particular, we characterise of  
singularities of minimising general axially symmetric harmonic maps,  
and construct non-minimising general axially symmetric harmonic maps  
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.  
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$  
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for  
$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and  
Mazowiecka--Strzelecki.

(Joint work with Prof. Robert M. Hardt.)

Fri, 02 Mar 2018

12:00 - 13:00
C3

On the Existence of $C^{1,1}$ Isometric Immersions of Some Negatively Curved Surfaces

Siran Li
(Rice University)
Abstract

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

Wed, 28 Feb 2018

12:00 - 13:00
L4

On the Geometric Regularity Criteria for Incompressible Navier--Stokes Equations

Siran Li
(Rice University)
Abstract

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$.  In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of  vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.

Thu, 25 May 2017

14:00 - 15:00
L4

An efficient and high order accurate direct solution technique for variable coefficient elliptic partial differential equations

Prof. Adrianna Gillman
(Rice University)
Abstract

 

For many applications in science and engineering, the ability to efficiently and accurately approximate solutions to elliptic PDEs dictates what physical phenomena can be simulated numerically.  In this seminar, we present a high-order accurate discretization technique for variable coefficient PDEs with smooth coefficients.  The technique comes with a nested dissection inspired direct solver that scales linearly or nearly linearly with respect to the number of unknowns.  Unlike the application of nested dissection methods to classic discretization techniques, the constant prefactors do not grow with the order of the discretization.  The discretization is robust even for problems with highly oscillatory solutions.  For example, a problem 100 wavelengths in size can be solved to 9 digits of accuracy with 3.7 million unknowns on a desktop computer.  The precomputation of the direct solver takes 6 minutes on a desktop computer.  Then applying the computed solver takes 3 seconds.  The recent application of the algorithm to inverse media scattering also will be presented.
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