| Date | SPEAKER | Host |
|---|---|---|
| January 16 (Speical time, SCHM 117) | Yannick Sire (Johns Hopkins University) TITLE: Harmonic maps between singular spaces ABSTRACT: After reviewing briefly the classical theory of harmonic maps between smooth manifolds, I will describe some recent results related to harmonic maps with free boundary, emphasizing on two different approaches based on recent developments by Da Lio and Riviere. This latter approach allows in particular to give another formulation which is well-suited for such maps between singular spaces. After the works of Gromov, Korevaar and Schoen, harmonic maps between singular spaces have been instrumental to investigate super-rigidity in geometry. I will report on recent results where we introduce a new energy between singular spaces and prove a version of Takahashi's theorem (related to minimal immersions by eigenfunctions) on RCD spaces. |
Wang |
| January 25 | Alex Waldron (University of Wisconsin, Madison)
TITLE: Lojasiewicz inequalities for maps of the 2-sphere ABSTRACT: Infinite-time convergence of geometric flows, as even for finite-dimensional gradient flows, is a notoriously subtle problem. The best (or only) bet is to get a ``Lojasiewicz(-Simon) inequality'' stating that a power of the gradient dominates the distance to the critical energy value. I'll discuss the recent proof of a Łojasiewicz inequality between the tension field and the Dirichlet energy of a map from the 2-sphere to itself, removing virtually all assumptions from an estimate of Topping (Annals '04). This gives us convergence of weak solutions of harmonic map flow from S^2 to S^2 assuming only that the body map is nonconstant. |
Wang |
| February 1 | Changyou Wang (Purdue University)
TITLE: Global existence and compactness for axisymmetric Ericksen-Leslie system in dimension three ABSTRACT: In this talk, I will discuss the Ericksen-Leslie system, which is the governing equation for the hydrodynamics of nematic liquid crystals, in dimension three. Mathematically, this is a dissipative system strongly coupling between the Navier-Stokes equation for the underlying fluid velocity field and the transported harmonic flow into the unit sphere for the macroscopic orientation field of liquid crystal molecules. Because of the super-critical nonlinearities induced by Ericksen stress tensors, it has been an outstanding open question to establish Leray-Hopf type global solutions for any initial data with finite energy. I will describe a recent work, joint with Joshua Kortum, that proves the existence of such a global solution in the axisymmetric setting. |
Wang |
| February 8 | Yifeng Yu (University of California, Irvine)[Virtual]
TITLE: Existence and nonexistence of effective burning velocity under the curvature G-equation model ABSTRACT: G-equation is a well known level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when the curvature effect is considered: G_t + \left(1-d\, \Div{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0. In this talk, I will show the existence of effective burning velocity under the above curvature G-equation model when $V$ is a two dimensional cellular flow, which can be extended to more general two dimensional incompressible periodic flows. Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation based on the two dimensional structures. In three dimensions, the effective burning velocity will cease to exist even for simple periodic shear flows when the flow intensity surpasses a bifurcation value. | Wang |
| February 15 | Peijun Li (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) TITLE : Stochastic inverse problems for the biharmonic wave equation ABSTRACT : Stochastic inverse problems refer to inverse problems that involve uncertainties. Compared to deterministic counterparts, stochastic inverse problems are substantially more challenging due to the additional difficulties of randomness and uncertainties. In this talk, our recent progress will be discussed on some stochastic inverse problems for the biharmonic wave equation, which plays an important role in thin plate elasticity. I will present new models for the inverse random source and potential problems. Given the random source or potential, the direct problem is to determine the wave field; the inverse problem is to recover the unknown source or potential that generates the prescribed wave field. The well-posedness and regularity of the solutions will be addressed for the direct problems. For the inverse problems, I will show that the micro-correlation strength of the random source or potential can be uniquely determined by the high frequency limit of the wave field at a single realization. I will also highlight ongoing projects on some direct and inverse scattering problems for biharmonic waves | Yip and Wang |
| February 22 | Jiahong Wu (University of Notre Dame)
TITLE: Stabilizing phenomenon for incompressible fluids ABSTRACT: This talk presents recent stability results on several PDE systems modeling fluid flows. These results reflect a seemingly universal stabilizing phenomenon exhibited in quite different fluids. The 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor in the Oldroyd-B model, small smooth data always lead to global and stable solutions. Solutions of the 2D Navier-Stokes in R^2 with dissipation in only one direction are not known to be stable, but the Boussinesq system involving this Navier-Stokes is always stable near the hydrostatic equilibrium. The buoyancy forcing helps stabilize the fluid. The 3D incompressible Navier-Stokes equation with dissipation in only one direction is not known to always have global solutions even when the initial data are small. However, when this Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamic system, solutions near a background magnetic field are shown to be always global in time. In all these examples the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect. | Wang |
| February 29 | Nick McCleerey (Purdue University) TITLE: The Eigenvalue Problem for the Complex Hessian Operator on m-Pseudoconvex Manifolds ABSTRACT : We establish $C^{1,1}$-regularity and uniqueness of the first eigenfunction of the complex Hessian operator on strongly $m$-pseudoconvex manifolds, along with a variational formula for the first eigenvalue. From these results, we derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue. This is joint work with Jianchun Chu and Yaxiong Liu |
Wang |
| March 7 | Zhuolun Yang (Brown University) [Virtual]
TITLE: Recent progress on gradient characterization for conductivity problems from high-contrast composite materials ABSTRACT: In this talk, I will describe the conductivity problem from composite materials. The electric field, represented by the gradient of solutions, may blow up as the distance between inclusions approaches to 0. When the current-electric field relation obeys the Ohm's law, we obtained an optimal gradient estimate of solutions in terms of the distance between two insulators, which settled down a major open problem in this area. I will also present our recent results on both perfect and insulated conductivity problems when the current-electric field relation is a power law. The talk is based on joint work with Hongjie Dong (Brown), Yanyan Li (Rutgers), and Hanye Zhu (Brown). |
Bang and Wang |
| March 14 | NO SEMINAR (Spring Break): |
|
| March 21 | No Seminar | |
| March 28 | Alexis Vasseur (University of Texas, Austin) TITLE: Viscous perturbations to discontinuous solutions of the compressible Euler equation ABSTRACT : The compressible Euler equation can lead to the emergence of shocks-discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities.The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, we will present the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent a-contraction with shifts method. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation. |
Novack |
| April 4 | TITLE: (No Seminar) ABSTRACT : |
|
| April 11 | Matt Rosenzweig (Carnegie Mellon University)
TITLE: The attractive log gas: stability, uniqueness, and propagation of chaos ABSTRACT : We consider the dynamics of a system of particles with logarithmic attractive interaction on the torus at inverse temperature beta. We show phase transitions for the stability and uniqueness of the uniform distribution. Investigating the mean-field convergence of the system by the modulated free energy method, we show uniform-in-time convergence is true for small beta, while false for large beta. In the process, we identify interesting questions concerning functional inequalities of logarithmic Hardy-Littlewood-Sobolev type and uniqueness for Kazhdan-Warner type equations. Based on joint work with Antonin Chodron de Courcel (IHES) and Sylvia Serfaty (NYU). |
Novack |
| April 18 | Can Cui (Purdue University) TITLE: Non-parametric anisotropic analogues of mean curvature flow with contact angle ABSTRACT : Anisotropic analogues of mean curvature flow have been considered as models for crystal growth and other phenomena involving the motions of interfaces. They are gradient flows for natural area-like functionals: $I(u) = \int F(\nu)\sqrt{1+|Du|^2} dx$, where F is homogeneous of degree 1. In this talk, I will introduce the contact angle boundart value problem to anisotropic mean curvature flow over a convex domain. Our main results include a prior gradient estimate and long time behavior in $\mathbb{R}^2$. The solutions converge to ones moving by translation. Recently, we apply a new technique and generalize this result to arbitrary dimensions. This talk is based on joint work with my advisor, Aaron Yip. | Yip and Wang |
| April 25 | Siming He (University of South Carolina) TITLE: Application of the Enhanced Dissipation Phenomenon ABSTRACT : The dissipation effect can be amplified when the fluid transport effect is apparent. This amplification is known as the enhanced dissipation phenomenon. In specific parameter regimes, this fluid mechanism can contribute to suppressing potential singularities in nonlinear chemotaxis systems, ensuring hydrodynamic stability of shear flows, and facilitating communication among agents in collective dynamics. In this talk, I will survey several results related to the enhanced dissipation phenomenon. |
Gao |
| May 2 | Han Lu (University of Notre Dame) TITLE: A constructive proof for the existence of $\sigma_2$-Yamabe problem for $5\leq n\leq 8$ ABSTRACT : Consider the compact Riemannian manifold $(M,g)$ of dimension $n\geq 5$. The $\sigma_2$ Yamabe problem seeks to find a conformal metric to $g$ that has a constant $\sigma_2$ curvature. Sheng-Trudinger-Wang have previously proved the existence of such a metric when $n\geq 5$, while Ge-Wang have provided a constructive proof for $n\geq 9$. In this talk, we will present a constructive proof for the case $n=8$ and discuss the obstruction for such construction when $5\leq n\leq 7$, which is tightly related to the Green's function of the $\sigma_2$-Yamabe problem. The proof is a joint work with Bin Deng and Juncheng Wei. | Bang and Wang |