# PDE Geometric Analysis seminar

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

## Contents

- 1 Previous PDE/GA seminars
- 2 Seminar Schedule Fall 2014
- 3 Abstracts
- 3.1 Greg Drugan (U. of Washington)
- 3.2 Guo Luo (Caltech)
- 3.3 Xiaojie Wang(Stony Brook)
- 3.4 Roman Shterenberg(UAB)
- 3.5 Antonio Ache(Princeton)
- 3.6 Jean-Michel Roquejoffre (Toulouse)
- 3.7 Myoungjean Bae (POSTECH)
- 3.8 Changhui Tan (University of Maryland)
- 3.9 Hongjie Dong (Brown University)
- 3.10 Hao Jia (University of Chicago)
- 3.11 Alexander Pushnitski (King's College)
- 3.12 Ronghua Pan (Georgia Tech)

### Previous PDE/GA seminars

# Seminar Schedule Fall 2014

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

September 15 | Greg Kuperberg (UC-Davis) |
TBA |
Viaclovsky |

September 22 (joint with Analysis Seminar) | Steven Hofmann (U. of Missouri) |
TBA |
Seeger |

Oct 6th, | Xiangwen Zhang (Columbia University) |
TBA |
B.Wang |

# 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*

Abstract: Ricci flow is an important evolution equation of Riemannian metrics. 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.

### Roman Shterenberg(UAB)

*Recent progress in multidimensional periodic and almost-periodic spectral*
problems

Abstract: We present a review of the results in multidimensional periodic and almost-periodic spectral problems. We discuss some recent progress and old/new ideas used in the constructions. The talk is mostly based on the joint works with Yu. Karpeshina and L. Parnovski.

### Antonio Ache(Princeton)

*Ricci Curvature and the manifold learning problem*

Abstract: In the first half of this talk we will review several notions of coarse or weak Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as motivation for developing a method to estimate the Ricci curvature of a an embedded submaifold of Euclidean space from a point cloud which has applications to the Manifold Learning Problem. Our method is based on combining the notion of ``Carre du Champ" introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is possible to recover the rough laplacian of embedded submanifolds of the Euclidean space from point clouds. This is joint work with Micah Warren.

### Jean-Michel Roquejoffre (Toulouse)

*Front propagation in the presence of integral diffusion*

Abstract: In many reaction-diffusion equations, where diffusion is given by a second order elliptic operator, the solutions will exhibit spatial transitions whose velocity is asymptotically linear in time. The situation can be different when the diffusion is of the integral type, the most basic example being the fractional Laplacian: the velocity can be time-exponential. We will explain why, and discuss several situations where this type of fast propagation occurs.

### Myoungjean Bae (POSTECH)

*Free Boundary Problem related to Euler-Poisson system*

One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed, transonic shock can be represented as a monotone function of exit pressure. From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system when exit pressure is prescribed in a proper range. In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system, which is formulated as a free boundary problem with mixed type PDE system. This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) and Jingjing Xiao(CUHK).

### Changhui Tan (University of Maryland)

*Global classical solution and long time behavior of macroscopic flocking models*

Abstract: Self-organized behaviors are very common in nature and human societies. One widely discussed example is the flocking phenomenon which describes animal groups emerging towards the same direction. Several models such as Cucker-Smale and Motsch-Tadmor are very successful in characterizing flocking behaviors. In this talk, we will discuss macroscopic representation of flocking models. These systems can be interpreted as compressible Eulerian dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set of initial conditions will lead to a finite time break down of the system. This is a joint work with Eitan Tadmor.

### Hongjie Dong (Brown University)

*Parabolic equations in time-varying domains*

Abstract: I will present a recent result on the Dirichlet boundary value problem for parabolic equations in time-varying domains. The equations are in either divergence or non-divergence form with boundary blowup low-order coefficients. The domains satisfy an exterior measure condition.

### Hao Jia (University of Chicago)

*Long time dynamics of energy critical defocusing wave equation with*
radial potential in 3+1 dimensions.

Abstract: We consider the long term dynamics of radial solution to the above mentioned equation. For general potential, the equation can have a unique positive ground state and a number of excited states. One can expect that some solutions might stay for very long time near excited states before settling down to an excited state of lower energy or the ground state. Thus the detailed dynamics can be extremely complicated. However using the ``channel of energy" inequality discovered by T.Duyckaerts, C.Kenig and F.Merle, we can show for generic potential, any radial solution is asymptotically the sum of a free radiation and a steady state as time goes to infinity. This provides another example of the power of ``channel of energy" inequality and the method of profile decompositions. I will explain the basic tools in some detail. Joint work with Baoping Liu and Guixiang Xu.

### Alexander Pushnitski (King's College)

*An inverse spectral problem for Hankel operators*

Abstract: I will discuss an inverse spectral problem for a certain class of Hankel operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a step towards description of evolution in a model integrable non-dispersive equation. Several features of this inverse problem make it strikingly (and somewhat mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will describe the available results for Hankel operators, emphasizing this similarity. This is joint work with Patrick Gerard (Orsay).

### Ronghua Pan (Georgia Tech)

*Compressible Navier-Stokes-Fourier system with temperature dependent dissipation*

Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and heat conductivity coefficients in compressible fluids depend on absolute temperature through power laws. The mathematical theory on the well-posedness and regularity on this setting is widely open. I will report some recent progress on this direction, with emphasis on the lower bound of temperature, and global existence of solutions in one or multiple dimensions. The relation between thermodynamics laws and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works with Junxiong Jia and Weizhe Zhang.