| Date | SPEAKER | Host |
|---|---|---|
| August 29 | (NO SEMINAR) TITLE: ABSTRACT: |
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| September 5 | Zhiyuan Geng (Purdue University)
TITLE : Uniqueness of the blow-down limit for the Allen-Cahn solution with a triple junction structure ABSTRACT : In this talk, we explore the vector-valued Allen-Cahn system with a potential that vanishes at three energy wells. Recent independent studies by Sandier-Sternberg and Alikakos-Geng have established the existence of a minimizing entire solution exhibiting a triple junction structure at infinity along some subsequences. We will demonstrate the uniqueness of the blow-down limit for such a solution through a variational method. The key idea involves estimating the location and size of the diffuse interface by deriving tight energy upper and lower bounds. Furthermore, I will present new findings on the asymptotic flatness of the diffuse interface at infinity, which is closely related to a vector version of the De Giorgi conjecture for energy minimizers. |
Wang |
| September 12 | (NO SEMINAR)
TITLE: ABSTRACT: |
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| September 19 | Dixi Wang (Purdue University)
TITLE: Inviscid limit of the inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in $\mathbb{R}^3$ ABSTRACT: we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in $\mathbb{R}^3$. In particular, this limit is a weak solution of the corresponding Euler equations. We first deduce the Kolmogorov-type hypothesis in $\mathbb{R}^3$, which yields the uniform bounds of $\alpha^{th}$-order fractional derivatives of $\sqrt{\rho^\mu}{\bf u}^\mu $ in $L^2_x$ for some $\alpha>0$, independent of the viscosity. The uniform bounds can provide strong convergence of $\sqrt{\rho^{\mu}}\bf u^{\mu}$ in $L^2$ space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations. | Wang |
| September 26 | Yuxi Han (Purdue University) TITLE : Quantitative homogenization of state-constraint Hamilton–Jacobi equations on perforated domains ABSTRACT : Homogenization is commonly applied in the analysis of heterogeneous materials with small-scale variations, where the microscale behavior is averaged out to focus on the macroscopic properties. In this talk, we present the rate of convergence for the periodic homogenization of state-constraint Hamilton–Jacobi equations on perforated domains in a convex setting. Specifically, we study domains containing 'holes' and then explore a dilute scenario where the holes' diameters are much smaller than the microscale. Finally, we consider domain defects where some holes are missing. Additionally, the convergence rates are essentially optimal. | Gao |
| October 3 | Kuan-Ting Yeh (Purdue University) TITLE: Symmetrization Methods and Perimeter Inequalities ABSTRACT : In this talk, we introduce the basic concepts of symmetrization methods and explore their connection to isoperimetric-type inequalities. Additionally, we present several perimeter inequalities involving symmetrization with different weights within the framework of sets of locally finite perimeter. We further establish that, within a certain class of weights, the isotropic Gaussian weight is the only one whose perimeter decreases under Ehrhard symmetrization. |
Torres |
| October 10 | Jay Bang (WestLake University, China)
TITLE: Self-Similar Solutions to the Stationary Navier-Stokes Equations in a Two-Dimensional Sector. ABSTRACT: Self-similar solutions of the stationary Navier-Stokes equations are useful to study the asymptotic behavior of a general solution at infinity. We investigated self-similar solutions in a two-dimensional sector with the no-slip boundary condition. We found necessary and sufficient conditions for the existence of a self-similar solution in terms of the angle of the sector and the flux. In addition, we established the uniqueness and non-uniqueness of flows with a given type. As an application, we identified the leading order term of a solution to the stationary Navier-Stokes equations in an aperture domain when the flux is small. The main idea is to study the reduced ODE system and use properties of both complete and incomplete elliptic functions. This is a joint work with Changfeng Gui, Chunjing Xie, Yun Wang and Hao Liu. | Wang |
| October 17 | Alexey Cheskidov (WestLake University, China)
TITLE: Energy cascade in fluids: from convex integration to mixing ABSTRACT: In the past couple of decades, mathematical fluid dynamics has made significant strides with numerous constructions of solutions to fluid equations that exhibit pathological or wild behaviors. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations. I will focus on dissipation anomaly for viscous fluid flows as well as anomalous dissipation for the limiting inviscid flows. These two intrinsically linked laws of turbulence are postulated by Kolmogorov and Onsager’s empirical theories built on the persistence of the energy flux through the inertial range. I will first analyze these phenomena on a finite time interval and prove the existence of various scenarios in the limit of vanishing viscosity, ranging from the total dissipation anomaly to a pathological one where inviscid anomalous dissipation occurs without viscous dissipation anomaly, as well as the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. Finally, I will show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias upper bound is sharp. |
Novack |
| October 24 | Tao Huang (Wayne State University) TITLE: Poiseuille flow of hyperbolic Ericksen-Leslie system in dimension two ABSTRACT In this talk, we consider the Poiseuille flow of hyperbolic Ericksen-Leslie system modeling hydrodynamics of nematic liquid crystals with simplified coefficients in dimension two. We first construct the global weak solutions depending only on radius and time by the Ginzburg-Landau approximation and the fixed-point arguments. An $\varepsilon$-regularity is proved by using the local energy inequality. At the first possible blowup time, there are blowup sequences which converge to a non-constant time-independent (axisymmetric) harmonic map. As a direct result, there is no singularity if the initial energy of the weak solution is less than the energy of single blowup. |
Wang |
| October 31 | Po-Chun Kuo (Purdue University) TITLE : Dynamics of immersed interface problems in Stokes flow ABSTRACT :Immersed interface problems in Stokes flow are a fluid structure interaction problem. One of the simplest of such problems is the 2D Peskin problem, in which a 1D closed elastic structure is immersed in a 2D Stokes fluid. This has been studied computationally and analytically. We extend the 2D Peskin problem into two different cases: (1) 2D inextensible interface problem. (2) 3D Peskin problem. In the 2D inextensible interface problem, we assume that the interface is inextensible. Through the boundary integral method, we reformulate the problem into two contour equations, an evolution equation and a tension determination equation. We first study the well-posedness and the regularity of the generalized tension determination problem in Hölder spaces. Next, we use a suitable time-weighted Hölder space to study the well-posedness and the regularity of the dynamic problem. We also study the Peskin problem in the 3D case. With the boundary integral method, the 3D Peskin may be reformulated to an evolution equation on a unit sphere for the elastic interface. We use more than one local chart to prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time. | Wang and Ning |
| November 7 | Michael Novack (LSU) TITLE: A variational model for 3D features in soap films ABSTRACT : Area minimization among a suitable class of 2D surfaces is the basic variational model describing the interfaces in soap films. In this talk we will discuss a modification of this paradigm in which surfaces are replaced with regions of small but positive volume. The model captures features of real films, such as Plateau borders, that cannot be described by zero volume objects. We will also discuss the PDE approximation of this problem via an Allen-Cahn free boundary problem and its relation to Plateau's laws. |
Wang and Yip |
| November 14 | Will Golding (University of Texas, Austin) TITLE: Recent progress on global solutions to the homogeneous Landau equation ABSTRACT : The Landau equation is a fundamental kinetic model that describes the evolution of collisional plasmas. The equation includes a quadratic, non-local term that models the effects of binary collisions mediated by a Coulomb force. This collision term introduces substantial mathematical challenges, leaving many fundamental questions--such as the existence of global-in-time smooth solutions--largely open. In this talk, I will explore recent progress made in understanding a simplified model, the homogeneous Landau equation, which retains the complex collision term. In a recent breakthrough work, Luis Silvestre and Nestor Guillen showed the existence of a new monotone functional ---the Fisher information---which is used to construct global-in-time solutions for smooth rapidly decaying initial data. I will discuss joint work with Maria Gualdani and Amelie Loher, where we extend these results to general initial data and obtain new results on global-in-time existence and various forms of uniqueness. I will conclude with a discussion of how these results inform future research of the full model. |
Novack |
| November 21 | Monica Torres (Purdue University)
TITLE: Extended divergence-measure fields, the Gauss-Green formula and Cauchy fluxes ABSTRACT : We establish the Gauss-Green formula for extended divergence-measure fields ({\it i.e.}, vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for {\it almost every open set}, the normal trace is a measure supported on the boundary of the set. Moreover, for any open set, we provide a representation of the normal trace of the field over the boundary of the open set as the limit of measure-valued normal traces over the boundaries of approximating sets. Furthermore, using this theory, we extend the balance law from classical continuum physics to a general framework in which the production on any open set is measured with a Radon measure and the associated Cauchy flux is bounded by a Radon measure concentrated on the boundary of the set. We prove that there exists an extended divergence-measure field such that the Cauchy flux can be recovered through the field, locally on almost every open set and globally on every open set. Our results generalize the classical Cauchy's Theorem (that is only valid for continuous vector fields) and extend the previous formulations of the Cauchy flux (that generate vector fields within L^p). Thereby, we establish the equivalence between entropy solutions of the multidimensional nonlinear partial differential equations of divergence form and of the mathematical formulation of physical balance laws via the Cauchy flux through the constitutive relations in the axiomatic foundation of Continuum Physics. This is a joint work with Gui-Qiang Chen (University of Oxford) and Christopher Irving (Technical University Dortmund). |
Wang |
| November 28 (No Seminar, Thanksgiving holiday) | (NO SEMINAR) TITLE: ABSTRACT : | |
| December 5 | Nick Alikakos (University of Athens, Greece) [Postponed to Spring 2025] TITLE: ABSTRACT : |
Geng and Wang |