Date | SPEAKER and TITLE | Host |
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September 15 (Virtual) | Giorgio Saracco, Universita di Trento, Italy TITLE: Full characterization of isoperimetric sets in 2d domains without necks ABSTRACT: Let \Omega \subset R^N be an open, bounded set. The isoperimetric profile is the function defined on [0, \Omega] given by J: V -> min { P(E) : E \subset\Omega , |E| = V } that is, it associates to every volume the least perimeter needed to enclose the given volume among subsets of \Omega. During this talk we shall characterize all isoperimetric sets of a Jordan domain \Omega, that is, a 2d set whose boundary is a Jordan curve, satisfying some geometric assumptions. As a byproduct, we shall see that there exists a threshold such that the isoperimetric profile is concave below it and convex above it. Based on joint works with Leonardi and Neumayer. |
Indrei |
September 22 (Moved to November 17) | Matt Novack, Purdue University (postponed to November 17) TITLE: ABSTRACT : |
Wang |
September 29 | Yuan Gao, Purdue University TITLE:A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures ABSTRACT : We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. Freidlin-Wentzell's variational formulas for both a self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the Freidlin-Wentzell's rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed. |
Wang |
October 6 | Bingyang Hu, Purdue University TITLE: Suppression of Epitaxial Thin Film Growth by Mixing ABSTRACT: In this talk, we consider the fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus, with or without an advection term. We will show that in the absence of advection, there exists initial data which will make the solution blow up in finite time; while in the advective case, if the imposed advection is sufficiently mixing, the global existence of the solutions can be achieved. Finally, we will make some further remarks on the general framework on how advection can guarantee the global existence of certain non-linear equations. This talk is based on several joint work with Yu Feng, Xiaoqian Xu and Yeyu Zhang. |
Wang |
October 13 | Monica Torres, Purdue University TITLE: Divergence-measure Fields: Gauss-green formulas and normal traces ABSTRACT : The Gauss-Green formula is a fundamental tool in analysis. In this talk we present new Gauss-Green formulas for divergence-measure fields(i.e., vector fields F:\Omega ->R^n in L^p whose distributional divergence can be represented by a Radon measure), and for extended divergence-measure fields(i.e., vectors fields that are Rn-valued Radon measures and whose distributional divergence can be represented by a Radon measure). These formulas hold on sets with low regularity, thus allowing the integration by parts on very rough domains. This is a joint project with Christopher Irving (University of Oxford), Gui-Qiang Chen (University of Oxford), Giovanni Comi (University of Hamburg) and Qinfeng Li (Hunan University). | Wang |
October 20 (Virtual) | Diogo Gomes, KAUST, SAUDI ARABIA TITLE: Mean-field game price formation models ABSTRACT: In this talk, we discuss a mean-field game price formation model. This model describes a large number of rational agents that can trade a commodity with an exogenous supply. The price is determined by a balance condition between supply and demand. We discuss the well-posedness of the model, and the uniqueness and regularity of the price function. Then, we examine two explicit models - the linear-quadratic problem and a model with finitely many agents. Time permitting, we will examine the connections between this problem and optimal transport with constraints. |
Gao |
October 27 | Yifei Pan, Purdue University, Fort Wayne TITLE: Examples of twice differentiable functions with continuous Laplacian ABSTRACT : In this talk I shall discuss how to construct twice differentiable (everywhere, but not C^2) functions with continuous Laplacian and either bounded Hessian or unbounded Hessian. A simple argument proves that any function with these properties cannot be radially symmetric. The same method of construction turns out to be also applicable to the Monge-Ampere operator. |
Indrei and Wang |
November 3 (Virtual) | Yuning Liu, New York University at Shanghai TITLE: Isotropic-nematic phase transition in liquid crystals ABSTRACT: In this talk, we shall study the co-dimensional one interface limit of an anisotropic Ginzburgâ€“Landau equation under parabolic scalings. This is a semi-linear parabolic system of a planar vector field which depends on a small parameter characterizing the width of the transition layer. We shall show rigorous derivations of effective geometric motions of the system using modulated energy methods and weak convergence methods. |
Wang |
November 10 (SPECIAL TIME/LOCATION: 2:30-3:30 pm, BRNG 2290) | Matt Jacobs, Purdue University TITLE: Lagrangian solutions to the Porous Media Equation ABSTRACT : In this talk, I will sketch the construction of the global-in-time forward and backward Lagrangian flow maps along the pressure gradient generated by weak solutions of the Porous Media Equation. The main difficulty is that when the initial data has compact support, it is well-known that the pressure gradient is not a BV function. Thus, the theory of regular Lagrangian flows cannot be applied to construct the flow maps. To overcome this difficulty, we develop a new argument that combines Aronson-Benilan type estimates for the porous media equation with the quantitative Lagrangian flow theory of Crippa and De Lellis to provide a stability estimate for the flow maps that guarantees compactness. The arguments are flexible enough to handle the Hele-Shaw limit and a multiphase generalization of the Porous Media Equation where the equation is replaced by a coupled hyperbolic-parabolic system of reaction diffusion equations. As one application of the flow maps, we are able to construct solutions where different phases cannot mix together if they were separated at initial time. |
Wang |
November 16, 3-4pm (Special date and time; Virtual) | Alessandro Audrito, ETH TITLE: The parabolic obstacle problem with fully nonlinear diffusion ABSTRACT : In this talk I will present some recent results concerning the free boundary regularity in the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is smooth in space and time, while the set of singular points is locally covered by a Lipschitz manifold of dimension n-1. This is a joint work with T. Kukuljan. |
Indrei and Wang |
November 17 | Matt Novack, Purdue University TITLE: An Intermittent Onsager Theorem ABSTRACT : n this talk, we will motivate and outline a construction of non-conservative weak solutions to the 3D incompressible Euler equations with regularity which simultaneously approaches the thresholds C^0_t H^{1/2}_x and C^0_t L^{\infty}_x. By interpolation, such solutions possess nearly 1/3 of a derivative in L^3. Hence this result provides a new proof of the flexible side of the Onsager conjecture which is independent from that of Isett. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to deviate from the scaling predicted by Kolmogorovâ€™s 1941 theory of turbulence. This talk is based on a recent joint work with Vlad Vicol and an earlier joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol. |
Wang |