| Date | SPEAKER | Host |
|---|---|---|
| September 4 | Yuan Gao, Purdue University TITLE: Optimal Transport in Inhomogeneous Media: Convergence of Gradient Flows and the Effective Wasserstein Metric ABSTRACT: The Fokker--Planck equation with rapidly oscillating coefficients can be formulated as a gradient flow in Wasserstein space, involving both inhomogeneous dissipation and oscillatory free energy. Using an evolutionary Gamma-convergence approach, we derive the homogenized dynamics, which retain the gradient flow structure in a limiting homogenized Wasserstein space. We will also discuss a comparison between the limiting Wasserstein distance induced by the gradient flow and the direct Gromov--Hausdorff limit of the Wasserstein distance. |
Wang |
| September 11 | (No Seminar This Week)
TITLE : ABSTRACT : |
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| September 18 | Dallas Albritton, University of Wisconsin - Madison
TITLE : Non-uniqueness and vanishing viscosity ABSTRACT: The forced 2D Euler equations exhibit non-unique solutions with vorticity in L^p, p > 1, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity and consider epsilon-size perturbations of his initial datum. We discover a uniqueness threshold below which the vanishing viscosity solution is unique and radial, and at which certain vanishing viscosity solutions converge to non-unique, non-radial solutions. |
Novack |
| September 19 (Special Colloquium) | Kevin Zumbrun, Indiana University, Bloomington
TITLE : ABSTRACT: |
Novack |
| September 25 | (No Seminar This Week)
TITLE: ABSTRACT: |
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| October 2 | Nicholas McCleerey, Purdue University TITLE : Lines in the space of K\"ahler metrics ABSTRACT : In recent years, the metric space of Kahler metrics has played an important role in variational problems in Kahler geometry. We report on some recent investigations (joint with Tamas Darvas) into the existence of geodesic lines in this space, which can be characterized as special solutions to the homogeneous complex Monge-Ampere equation. In particular, we construct a wide range of these lines on any projective K\"ahler manifold, disproving a folklore conjecture popularized by Berndtsson. In the case of Riemann surfaces, we additionally classify all such lines which are locally bounded. Finally, we investigate the validity of Euclid's fifth postulate for the space of K\"ahler metrics. | Wang |
| October 9 | Cheng Yu, University of Florida
TITLE: Universality in the Low Mach Number Limit via a Convex Integration Framework ABSTRACT : In this talk, I will present a result on the low Mach number limit of the compressible Euler equations through the lens of convex integration. For any $L^2$-bounded solution of the incompressible Euler equations, we construct a corresponding family of global weak solutions to the compressible Euler equations using convex integration. We then prove that, as the Mach number tends to zero, this family of solutions converges to the given incompressible solution in $L^p$. This approach highlights a form of universality: any incompressible weak solution can be realized as the asymptotic limit of compressible solutions. Our results provide a rigorous framework for understanding the incompressible limit from a new perspective. This talk is based on the joint work with Ming Chen, Alexis Vasseur and Dehua Wang. |
Dixi Wang |
| October 16 | Antonio Segatti, Universita Di Pavia (Virtual)
TITLE: On a variational approximation of the Heat flow of Harmonic Maps ABSTRACT: In this talk I will discuss a variational approximation of the heat flow of harmonic maps. This approach is based on the Weighted Energy Dissipation (WED) scheme, which is featured by a functional defined on entire trajectories and depending on a small parameter $\varepsilon$. The minimizers of this functional are shown to converge, when $\varepsilon\searrow 0$, to the (weak) solutions of the heat flow. In particular for smooth target manifolds with non positive sectional curvature we recover, via a variational argument, the well known theorem of Eells and Sampson. This is a joint work with Fang-Hua Lin, Yannick Sire and Changyou Wang. | Wang |
| October 23 | Po Chun Kuo, Purdue University
TITLE: On the well-posedness of immersed interface problems in Stokes flow ABSTRACT: Immersed interface problems in Stokes flow represent a class of fluid-structure interaction problems. One of the simplest examples is the two-dimensional Peskin problem, where a one-dimensional closed elastic structure is immersed in a two-dimensional Stokes fluid. This problem has been extensively studied both computationally and analytically. In this work, we extend the Peskin problem to three dimensions. Using the boundary integral formulation, the three-dimensional Peskin problem can be expressed as an evolution equation on a unit sphere for the elastic interface. We identify a principal linear operator so that , where is smoother. We then show that generates an analytic semigroup , allowing the evolution equation to be expressed in Duhamel's form. Based on this formulation, we establish the well posedness of the problem using a fixed-point argument. To handle the geometry of the two sphere, we employ more than one local chart and demonstrate well posedness in low-regularity H\"older spaces. Furthermore, we prove that the elastic membrane becomes instantly smooth for any positive time. |
Wang |
| October 30 | Changyu Guo, Shangdong University, China and University of Eastern Finland (Virtual) TITLE: Regularity of biharmonic maps and applications ABSTRACT: Biharmonic maps are higher order extensions of classical harmonic maps. Important partial regularity theory was established by Chang et al in 1999 for spherical target and by Wang in 2002 for general target. Based on the new method of Naber-Valtorta for harmonic maps, we establish the global regularity theory for biharmonic maps. We also apply the regularity theory to study energy quantization problems for biharmonic map type equations. |
Wang |
| November 6 | Xuenan Li, Columbia University TITLE : Soft modes in mechanism-based mechanical metamaterials: modeling, analysis, and applications ABSTRACT : Mechanism-based mechanical metamaterials are synthetic materials that exhibit unusual microscale buckling in response to mechanical deformations. These artificial materials are like elastic composites but sometimes more degenerate since they can deform with zero elastic energy. We call such zero energy deformations mechanisms. Origami and Kirigami are typical examples of these mechanism-based mechanical metamaterials. Other than mechanisms, these metamaterials also have "soft modes" -- macroscopic deformations with very little elastic energy, some but not all of which resemble modulated mechanisms. A key question is to identify all the soft modes for a given mechanism-based metamaterial. In this talk, I will address the two-fold challenge in identifying the soft modes and our treatments: first, we establish the existence of an effective energy for a broad class of lattice metamaterials; and second, we identify soft modes as macroscopic deformations where this energy vanishes, including a complete characterization of the zero sets of the effective energy in some conformal metamaterials. Together, these results provide a rigorous link between mechanisms and soft modes, laying a mathematical foundation for future analysis and design of mechanical metamaterials. This is joint work with Robert V. Kohn. | Wang |
| November 13 | Jiuyi Zhu, LSU TITLE: Spectral inequalities for Schrodinger equations with power growth potentials ABSTRACT : We obtain some sharp spectral inequalities for Schrodinger equations with power growth potentials. The spectral inequalities are concerned with a control estimate for a linear combination of eigenfunctions of Schrodinger operators. These sharp spectral inequalities depend on the thickness of the sensor sets, the growth rate of the potentials and eigenvalues of Schrodinger operators. We will also show the connection of these spectral inequalities with null controllability for parabolic equations in control theory. The proof relies on the study of quantitative Cauchy propagation of smallness |
Geng |
| November 20 | Sean McCurdy, Universidad Nacional Autonoma de Mexico (Virtual) TITLE: Some Recent Developments in Varifold Regularity ABSTRACT : Varifolds are a good class of "generalized surfaces" which allow us to solve many geometric variational problems, such as the Plateau problem (the geometric analog of the Dirichlet problem for Elliptic PDEs). Because of this, there is a deep connection between Elliptic PDEs and Varifold Regularity. Therefore, since the Sobolev embedding theorems give us a very well-understood picture of the regularity for solutions to the Poisson problem, the connection between Elliptic PDEs and varifold regularity begs for a similar picture to be completed for varifolds. This talk will focus on the history of work completing that picture, including recent work which gives the sharp regularity in the critical case in all dimensions. Varifolds have a deserved reputation for being extremely technical, but this talk will focus on the history and the ideas. It will begin with a semi-historical motivation for varifolds, include lots of pictures, and, if there is time, we will conclude with some open problems. |
Yeh |
| November 27 | ( No Seminar, Thanksgiving Holiday )
TITLE: ABSTRACT : |
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| December 4 | Razvan-Octavian Radu, Princeton University TITLE: Desingularization of V-states ABSTRACT : V-states are uniformly rotating vortex patch solutions to the 2D Euler equations. Namely, the vorticity is given by the characteristic function of a domain which rotates around the origin with constant angular velocity. Examples of V-states include Kirchhoff ellipses and m-fold symmetric patches which bifurcate from the disk. I will describe how, for any V-state satisfying a certain non-degeneracy condition, there exist smooth rigidly rotating solutions to the 2D Euler equations approximating it arbitrarily well in the natural Hölder spaces. I will then argue that all but countably many Kirchhoff ellipses, as well as all m-fold symmetric V-states near the disk satisfy this non-degeneracy. | Novack |
| December 11 | Nicholas Alikakos, University of Athens, Greece (Virtual)
TITLE: On triple junction networks: sharp and diffuse ABSTRACT : We consider two-triple junction tree-type networks on the disc under Dirichlet conditions and discuss the linking of the vector Allen-Cahn with minimal partitions. We also examine aspects of the associated dynamics. This is in collaboration with Giorgio Fusco, and Zhiyuan Geng. | Geng |