# PDE Geometric Analysis seminar: Difference between revisions

Line 104: | Line 104: | ||

suggests that the blowing-up solution develops a self-similar structure near the point of | suggests that the blowing-up solution develops a self-similar structure near the point of | ||

the singularity, as the singularity time is approached. | the singularity, as the singularity time is approached. | ||

===Xiaojie Wang(Stony Brook)=== | |||

"Uniqueness of Ricci flow solutions on noncompact manifolds" | |||

Ricci flow is the following differential equation satisfied by | |||

a family of riemannian metrics $g(t)$ | |||

\begin{equation*} | |||

\partial_tg(t)=-2\mathrm{rc}_t, | |||

\end{equation*} | |||

where $\mathrm{rc}_t$ is the ricci curvature of $g(t)$. Since it was | |||

introduced by R. Hamilton in 1982, it has greatly changed the | |||

landscape of riemannian geometry. | |||

One of the fundamental question about ricci flow is when is its | |||

solution to initial value problem unique. On compact manifold, | |||

with arbitrary initial metric, it was confirmed by Hamilton. On | |||

noncompact manifold, we only know this is true when | |||

further restrictions are imposed to the solution. | |||

In this talk, we will discuss various conditions that guarantee the | |||

uniqueness. In particular, we will discuss in | |||

details with the following uniqueness result. | |||

Let $(M,g)$ be a complete noncompact non-collapsing | |||

$n$-dimensional riemannian | |||

manifold, whose complex sectional curvature is bounded from below | |||

and scalar curvature is bounded from above. Then ricci | |||

flow with above | |||

as its initial data, on $M\times [0,\epsilon]$ for some | |||

$\epsilon>0$, has at most one solution in the class of | |||

complete riemannian metric with complex sectional curvature bounded | |||

from below. |

## Revision as of 17:45, 14 October 2013

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

### Previous PDE/GA seminars

# Seminar Schedule Fall 2013

date | speaker | title | host(s) |
---|---|---|---|

September 9 | Greg Drugan (U. of Washington) |
Construction of immersed self-shrinkers |
Angenent |

October 7 | Guo Luo (Caltech) |
Potentially Singular Solutions of the 3D Incompressible Euler Equations. |
Kiselev |

November 18 | Roman Shterenberg (UAB) | Kiselev | |

December 2 | Xiaojie Wang | Wang |

# Seminar Schedule Spring 2014

date | speaker | title | host(s) |
---|---|---|---|

January 14 at 4pm in B139 (TUESDAY), joint with Analysis | Jean-Michel Roquejoffre (Toulouse) | Zlatos | |

March 3 | Hongjie Dong (Brown) | Kiselev | |

April 7 | Zoran Grujic (University of Virginia) | Kiselev |

# Abstracts

### Greg Drugan (U. of Washington)

*Construction of immersed self-shrinkers*

Abstract: We describe a procedure for constructing immersed self-shrinking solutions to mean curvature flow. The self-shrinkers we construct have a rotational symmetry, and the construction involves a detailed study of geodesics in the upper-half plane with a conformal metric. This is a joint work with Stephen Kleene.

### Guo Luo (Caltech)

*Potentially Singular Solutions of the 3D Incompressible Euler Equations*

Abstract: Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also suggests that the blowing-up solution develops a self-similar structure near the point of the singularity, as the singularity time is approached.

### Xiaojie Wang(Stony Brook)

"Uniqueness of Ricci flow solutions on noncompact manifolds"

Ricci flow is the following differential equation satisfied by

a family of riemannian metrics $g(t)$ \begin{equation*} \partial_tg(t)=-2\mathrm{rc}_t, \end{equation*} where $\mathrm{rc}_t$ is the ricci curvature of $g(t)$. Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution.

In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result.

Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.