Connor Mooney, Yifeng Yu (UC Irvine)
Hongjie Dong (Brown University)
Daniela De Silva, Ovidiu Savin (Columbia University)
Andrzej Swiech, Chongchun Zeng (Georgia Tech University)
Hung V. Tran (University of Wisconsin at Madison)
Changyou Wang (Purdue University)
Inwon Kim (UCLA)
Andrej Zlatos (UCSD)
Stefania Patrizi (UT Austin)
1. Oct 6th, 12-1pm (PST) Hosted by UC Irvine
Speaker: Alessio Figalli, ETH Zurich
Title: Complete classification of global solutions to the obstacle problem
Abstract: The characterization of global solutions to the obstacle problems in R^n, or equivalently of null quadrature domains, has been studied for more than 90 years. In this talk, I will discuss a recent result with Eberle and Weiss, where we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
2. Nov 3rd, 12-1pm (PST) Hosted by UC San Diego
Speaker: Thomas Hou, Caltech
Title: A constructive proof of finite time blowup of 3D incompressible Euler equations with smooth data
Abstract: Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, we will present a new exciting result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essential difficulties in establishing such blowup results. We overcome these difficulties by establishing a constructive proof strategy. We first construct an approximate self-similar blowup profile using the dynamic rescaling formulation. To establish the stability of the approximate blowup profile, we decompose the linearized operator into a leading order operator plus a finite rank perturbation operator. We use sharp functional inequalities and optimal transport to establish the stability of the leading order operator. To estimate the finite rank operator, we use energy estimates and space-time numerical solutions with rigorous error control. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data. This provides the first rigorous justification of the Hou-Luo blowup scenario.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
3. Dec 8th, 12-1pm (PST) Hosted by UW Madison
Speaker: Tuoc Phan, The University of Tennessee, Knoxville
Title: Some recent results on $L_p$-estimates of solutions to linear elliptic and parabolic equations with singular or degenerate coefficients
Abstract: We discuss several classes of linear elliptic and parabolic equations in which the coefficients can singular or degenerate. Generic weighted and mixed-normed Sobolev spaces will be derived in which the existence, uniqueness, and regularity estimates of solutions are proved under some optimal regularity conditions on the leading coefficients. Some key ideas in the proofs will be pointed out and several future research directions will be mentioned.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
4. Jan 5th, 12-1pm (PST) Hosted by Brown University
Speaker: Yu Yuan, University of Washington, Seattle
Title: A monotonicity approach to Pogorelov’s Hessian estimates for Monge-Ampere equation
Abstract: We present an integral approach to the classical Hessian estimates for the Monge-Ampere equation, originally obtained via a pointwise argument by Pogorelov. The monotonicity employed here results from a maximal surface interpretation of “gradient” graph of solutions in pseudo-Euclidean space.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
5. Feb 16th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Fabio Pusateri, University of Toronto
Title: Recent results on the stability of solitons, kinks, and radiation damping
Abstract: This talk will give an overview of some recent results on nonlinear evolution equations with potentials, and applications to the stability of solitons and kinks, as well as to the phenomenon of “radiation damping”. Our approach to this class of problems is based on the use of the distorted Fourier transform and the development of multilinear harmonic analysis in this setting; in particular, we use these tools to understand the interaction of localized waves on the background of a large potential, and to handle the singularities in (distorted) Fourier space that naturally arise in many situations. This talk is based on joint works with P. Germain (Imperial), A. Soffer (Rutgers), T. Leger (Princeton), Gong Chen (Georgia Tech), Zhiyuan Zhang (NYU) and Adilbek Kairzhan (U of Toronto).
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
6. March 2nd, 12-1pm (PST) Hosted by Columbia University
Speaker: Nicola Garofalo, University of Padova
Title: Some strong unique continuation results for nonlocal parabolic equations
Abstract: I will discuss two results of strong unique continuation for nonlocal parabolic equations. The former establishes strong uniqueness backwards for global solutions, and is inspired to a well-known result by Poon. The latter is a space-like uniqueness theorem for local solutions which extends previous works of Escauriaza, Fernandez and Vessella. The material presented is taken from joint works with A. Banerjee, and with V. Arya, A. Banerjee and D. Danielli.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
7. April 27th, 12-1pm (PST) Hosted by UCLA
Speaker: Antonio De Rosa，University of Maryland
Title: Min-max construction of anisotropic CMC surfaces
Abstract: We prove the existence of nontrivial closed surfaces with
constant anisotropic mean curvature with respect to elliptic integrands
in closed smooth 3-dimensional Riemannian manifolds. The constructed
min-max surfaces are smooth with at most one singular point. The
constant anisotropic mean curvature can be fixed to be any real number.
In particular, we partially solve a conjecture of Allard [Invent.
Math.,1983] in dimension 3. Joint work with G. De Philippis.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
8. May 4th, 12-1pm (PST) Hosted by Purdue University
Speaker: Dallas Albritton, Princeton University
Title: Kinetic shock profiles for the Landau equation
Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Jacob Bedrossian (UCLA) and Matthew Novack (Purdue University).
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
9. June 8th, 12-1pm (PST) Hosted by UT Austin
Speaker: Serena Dipierro, University of Western Australia
Title: Long-range phase transitions
Abstract: Phase transitions are a classical topic of investigation. They represent a complex phenomenon which needs to be attacked with different methodologies and different perspectives. I will discuss some rigidity and symmetry results for a long-range phase coexistence equation, their close relation with surfaces of minimal perimeter and a famous conjecture by Ennio De Giorgi.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
1. Oct 14th, 12-1pm (PST) Hosted by UC Irvine
Speaker: Zihui Zhao, University of Chicago
Title: Boundary unique continuation of Dini domains and the estimate
of the singular set
Abstract: Let u be a harmonic function in a domain D in R^d. It is known that in the interior, the singular set S(u)={u=0=|\nabla u|} is (d-2)-dimensional, and moreover S(u) is (d-2)-rectifiable and its Minkowski content is bounded (depending on the frequency of u). We prove the analogue near the boundary for C^1-Dini domains: If the harmonic function u vanishes on an open subset E of the boundary, then near E the singular set S(u) \cap \overline{D} is (d-2)-rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which u is continuously differentiable towards the boundary, and in particular every C^{1,\alpha} domain is Dini. The main difficulty is the lack of the monotonicity formula for the frequency function near the boundary of a Dini domain. This is joint work with Carlos Kenig.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
2. Oct 28th, 12-1pm (PST) Hosted by UC Irvine
Speaker: Guido De Philippis, Courant Institute
Title: Michael-Simon inequality for anisotropic stationary varifolds and multilinear Kakeya inequality
Abstract: Michael Simon inequality is a fundamental tool in geometric analysis and geometric measure theory. Its extension to anisotropic integrands will allow to extend to anisotropic integrands a series of results which are currently known only for the area functional.
In this talk I will present an anistropic version of the Michael-Simon inequality, for for two-dimensional varifolds in R3, provided that the integrand is close to the area in the C1-topology. The proof is deeply inspired by posthumous notes by Almgren, devoted to the same result. Although our arguments overlap with Almgren’s, some parts are greatly simplified and rely on a nonlinear version of the planar multilinear Kakaeya inequality.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
3. Nov 4th, 12-1pm (PST) Hosted by UC San Diego
Speaker: Hao Jia, University of Minnesota
Title: Linear vortex symmetrization: the spectral density function approach and Gevrey regularity
Abstract: The two-dimensional incompressible Euler equation is globally well-posed but the long-time behavior is very difficult to understand due to the lack of global relaxation mechanism. Numerical simulations and physical experiments show that coherent vortices often become a dominant feature in two-dimensional fluid dynamics for a long time. The mathematical analysis of vortices, especially in connection to the so-called vortex symmetrization problem, has attracted a lot of attention in recent years.
In this talk, after a quick review of recent developments in the study of nonlinear asymptotic stability of shear flows and the symmetrization problem for (the special case of) point vortices, we turn to the general vortex symmetrization problem and report a recent result with A. Ionescu for the linearized flow. The linearized problem has been analyzed before by Bedrossian-Coti Zelati-Vicol who proved the optimal rate of decay for the stream function (as well as the so-called vortex depletion phenomenon) and obtained control on the profile of the vorticity field in Sobolev spaces with limited regularity.
Our main new discovery is that in the vortex problem, unlike the shear flow case, it is no longer possible to obtain smooth control uniformly in time on a single modulated profile for the vorticity field. Rather, there are two such profiles. To address this issue (for future nonlinear applications), we propose instead to control a new object, the so-called spectral density function, which is naturally associated with the linearized flow and can be bounded, for the linearized flow at least, in the same Gevrey space as the initial data.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
4. Nov 18th, 12-1pm (PST) Hosted by UC San Diego
Speaker: Alexander Kiselev, Duke University
Title: Reaction enhancement by chemotaxis
Abstract: Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions. One example is reproduction, where eggs release chemicals that attract sperm. Another example are infected tissues secreting chemokines, attracting monocytes to fight invading bacteria. I will talk about a basic model that consists of the system of two equations for two densities set in two dimensions. Mathematically, the problem is linked with the analysis of Fokker-Planck operators with logarithmic potential, and in particular the rate of convergence to ground state. There is no spectral gap in this case, and new weighted Poincare inequalities will be needed to derive sufficiently sharp estimates. The talks are based on works joint with Yishu Gong, Lenya Ryzhik, Fedja Nazarov and Yao Yao.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
5. Dec 2nd, 12-1pm (PST) Hosted by UW Madison
Speaker: Huy Nguyen, University of Maryland
Title: Some recent results on well-posedness and regularity for the Muskat problem
Abstract: The Muskat problem concerns the evolution of the interface between two fluids or between a fluid and vacuum in porous media. The dynamics is governed by a degenerate quasilinear parabolic PDE. I will discuss some recent results on (local and global) well-posedness and regularity for the Muskat problem.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
6. Dec 16th, 12-1pm (PST) Hosted by UW Madison
Speaker: Mikhail Feldman, UW Madison
Title: Existence and stability of solutions to the semigeostrophic system
Abstract: The semigeostrophic (SG) system is a model of large scale atmosphere/ocean flows. Solutions of this system are expected to contain singularities corresponding to the atmospheric fronts, and need to be understood in the appropriate weak sense. Most of known results were obtained for the SG system with constant Coriolis parameter, by rewriting the problem in the “dual variables” and using Monge-Kantorovich mass transport techniques. We will survey the results on existence of weak solutions, and describe recent results on weak-strong uniqueness, and on convergence of smooth solutions of incompressible Euler system with Coriolis force to a sufficiently regular solution of SG system in 2D and 3D. Both results are obtained by the relative entropy techniques.
A more physically realistic SG model has variable Coriolis parameter. Dual space is not available in this case. We work directly in the original “physical” coordinates, and show existence of smooth solutions for short time on two-dimensional torus. The solution is obtained by a time-stepping procedure which involves solving Monge-Ampere type equations on each step.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
7. Jan 13th, 12-1pm (PST) Hosted by Brown University
Speaker: Timur Yastrzhembskiy, Brown University
Title: Global $L_p$-estimates for kinetic Kolmogorov-Fokker-Planck equation with application to the initial boundary-value problem for the Landau equation.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
8. Jan 27th, 12-1pm (PST) Hosted by Brown University
Speaker: Zhongwei Sheng, University of Kentucky
Title: Quantitative Results for Darcy’s Law
Abstract: In this talk I will discuss some recent work on quantitative results for Darcy’s law. Consider the stationary Stokes equations in a periodically perforated domain with Dirichlet conditions on the boundaries of solid obstacles, assuming the size of obstacles is compatible to the period. As the period goes to zero, the limiting equations are governed by Darcy’s law. Here we shall be interested in the sharp convergence rates and large-scale regularity estimates for solutions.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
9. Feb 3rd, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Zaher Hani, University of Michigan
Title: The Mathematical Theory of Wave Turbulence
Abstract: The kinetic theory of waves, also known as wave turbulence theory, has been formulated in several fields of physics to describe the statistical behavior of various interacting wave systems. This started early in the past century with the pioneering works of Peierls, Hasselman, Zakharov, and others, and developed into a highly successful and informative paradigm widely employed nowadays, both in physical theory and practice. However, for the longest time, the mathematical foundation of the theory has not been established, with all its derivations based on formal manipulations and unproven postulates. The central objects here are the “wave kinetic equation” which describes the effective dynamics of an interacting wave system in the thermodynamic limit, and the “propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacked mathematical justification.
This problem of providing a rigorous justification and derivation of wave turbulence theory (Hilbert’s Sixth Problem for waves) has attracted considerable interest in the mathematical community over the past decade or so. In this talk, we shall discuss this research effort, which culminated in recent joint works with Yu Deng (University of Southern California), in which we provided the first rigorous derivation of the wave kinetic equation, and justified the propagation of chaos hypothesis in the same setting. The proof features a nice interplay of analysis, probability theory, combinatorics, and analytic number theory.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
10. Feb 10th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Nicolai V. Krylov, University of Minnesota
Title: On Aleksandrov type estimate for elliptic and parabolic equations with irregular drift terms
Abstract: We discuss estimates of the maximum of solutions of elliptic and parabolic second order equations through the L_p (elliptic case) or the L_pL_q (parabolic case) norms of the free term. The main emphasis will be on singularities of the first-order coefficients, which will be characterized by belonging either to Lebesgue or Morrey classes.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
11. March 10th, 12-1pm (PST) Hosted by Columbia University
Speaker: Changfeng Gui, University of Texas at San Antonio
Title: Some New Inequalities in Analysis and Geometry
Abstract: The classical Moser-Trudinger inequality is a borderline
case of Sobolev inequalities and plays an important role in geometric
analysis and PDEs in general. Aubin in 1979 showed that the best
constant in the Moser-Trudinger inequality can be improved by reducing
to one half if the functions are restricted to the complement of a
three dimensional subspace of the Sobolev space $H^1$, while Onofri
in 1982 discovered an elegant optimal form of Moser-Trudinger
inequality on sphere. In this talk, I will present new sharp
inequalities which are variants of Aubin and Onofri inequalities
on the sphere with or without mass center constraints.
One such inequality, for example, incorporates the mass center
deviation (from the origin) into the optimal inequality of Aubin on
the sphere, which is for functions with mass centered at the
origin. The main ingredient leading to the above inequalities is a
novel geometric inequality: Sphere Covering Inequality.
Efforts have also been made to show similar inequalities in higher
dimensions. Among the preliminary results, we have improved
Beckner’s inequality for axially symmetric functions when the
dimension $n=4, 6, 8$. Many questions remain open.
The talk is based on several joint papers with Amir Moradifam,
Sun-Yung Alice Chang, Yeyao Hu and Weihong Xie.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
12. March 23rd, 12-1pm (PST) Hosted by Columbia University
Speaker: Mihaela Ifrim, University of Wisconsin at Madison
Title: The time-like minimal surface equation in Minkowski space: low
regularity solutions
Abstract: It has long been conjectured that for nonlinear wave
equations which satisfy a nonlinear form of the null condition, the
low regularity well-posedness theory can be significantly improved
compared to the sharp results of Smith-Tataru for the generic case.
The aim of this article is to prove the first result in this
direction, namely for the time-like minimal surface equation in the
Minkowski space-time. Further, our improvement is substantial, namely
by 3/8 derivatives in two space dimensions and by 1/4 derivatives in
higher dimensions. This work is joint with Albert Ai and Daniel
Tataru.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
13. April 7th, 12-1pm (PST) Hosted by UCLA
Speaker: Fanghua Lin, Courant Institute, NYU
Title: Critical Point Sets of Solutions in Elliptic Homogenization.
Abstract: In this talk, I shall outline a proof for bounds on H^(n-2)–Hausdorff
measure of critical point sets of solutions in elliptic homogenization.
The method works for solutions of elliptic equations with Lipschitz coefficients also.
This is a joint work with Zhongwei Shen.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
14. April 28th, 12-1pm (PST) Hosted by UCLA
Speaker: Antoine Mellet,University of Maryland
Title: Free boundary problems for cell motility
Abstract: The crawling motion of cells on a substrate is often
explained by the formation of protrusions along the membrane of the
cell. In this talk, I will present some free boundary problems of
Hele-Shaw type, which describe this phenomena by combining the
(regularizing) effects of surface tension with the (destabilizing)
effects of a repulsive potential. We will discuss the derivation of
these models as singular limits of a diffuse interface approximation
(Cahn-Hilliard type equation) and study their properties. We will
focus in particular on symmetry breaking, hysteresis and
self-polarization, which are three important aspects of cell motility
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
15. May 5th, 12-1pm (PST) Hosted by Purdue University
Speaker: Dehua Wang, U of Pittsburgh
Title: Euler equations and transonic flows
Abstract: In this talk, we will consider the Euler equations of gas
dynamics and applications in transonic flows. First the basic theory
of Euler equations will be reviewed. Then we will present the results
on the transonic flows past obstacles, transonic flows in the fluid
dynamic formulation of isometric embeddings, and the transonic flows in nozzles. We will discuss global solutions and stability obtained through various techniques and approaches.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
16. May 12th, 12-1pm (PST) Hosted by Purdue University
Speaker: Matt Novack, IAS, Princeton University and Purdue University
Title: An Intermittent Onsager Theorem
Abstract: In 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 Kolmogorov’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.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
17. June 2nd, 12-1pm (PST) Hosted by UT Austin
Speaker: Francesco Maggi, UT Austin
Title: A mesoscale flatness criterion and its application to exterior isoperimetry
Abstract: We introduce a “mesoscale flatness criterion” for hypersurfaces with bounded mean curvature, discussing its relation and its differences with classical blow-up and blow-down theorems, and then we exploit this tool for a complete resolution of relative isoperimetric sets with large volume in the exterior of a compact obstacle. This is joint work with Michael Novack at UT Austin.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
18. June 16th, 12-1pm (PST) Hosted by UT Austin
Speaker: Pablo Raúl Stinga, Iowa State University
Title: Harnack inequality for fractional nondivergence form elliptic equations
Abstract: Fractional elliptic equations in nondivergence form come from several applications to elasticity and finance, and from the analysis of fractional Monge–Amp`ere equations. We prove the interior Harnack inequality for nonnegative solutions to nonlocal equations driven by fractional powers of nondivergence form elliptic operators. This is joint work with Mary Vaughan (UT Austin).
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
1. Oct 22nd, 12-1pm (PST) Hosted by UC Irvine
Speaker: Ovidiu Savin, Columbia University
Title: Free boundary regularity for the 3 membranes problem
Abstract: For a positive integer N, the N-membranes problem describes
the equilibrium position of N ordered elastic membranes subject
to forcing and boundary conditions. If the heights of the membranes are
described by real functions u_1, u_2,…,u_N, then the problem can be
understood as a system of N-1 coupled obstacle problems with interacting
free boundaries which can cross each other. When N=2 there is only one
free boundary and the problem is equivalent to the classical obstacle problem. I will review some of the regularity theory for the standard obstacle problem, and then discuss some recent work in collaboration with Hui Yu about the case when N=3 and there are two interacting free boundaries.
Youtube Video: https://www.youtube.com/watch?v=tELU7hMXR0M&t=2783s
Talk’s Slide: https://drive.google.com/drive/folders/1iXgVWcggI_jOSpa3fx16B6cGIcR87qgN
2. Oct 29th, 12-1pm (PST) Hosted by UC Irvine
Speaker: Luca Spolaor, UC San Diego
Title: Isolated singularities of minimal hypersurfaces
Abstract: In this talk I will discuss some old and new results about a class of isolated singularities of minimal surfaces arising from minimization or Min-Max.
Youtube Video: https://www.youtube.com/watch?v=xhpccTx04Kg
3. Nov 5th, 12-1pm (PST) Hosted by UT Austin
Speaker: Riccardo Montalto, Università Statale di Milano
Title: Quasi-periodic incompressible Euler flows in 3D
Abstract: In this talk I will present a recent result concerning the existence of time quasi-periodic solutions (invariant tori) for the Euler equation on the three-dimensional torus, with an external force which is “small” and quasi-periodic in time. If the forcing term is zero, then constant velocity fields are solutions of the Euler equation with zero pressure. We will show that (under suitable assumptions on the external force), the forced equation admits “many” quasi-periodic solutions bifurcating from constant velocity fields. This is a small “divisor problem”, hence we use a Nash-Moser scheme to construct the invariant torus. The key step is to solve the linearized PDE at any approximate solution and this is done by combining techniques coming from pseudo-differential operators theory and perturbation theory. The most difficult technical point in the procedure is to deal with pseudo-differential operators whose symbols are “matrix-valued”. This implies for instance that, unlike in the scalar case, the commutator of two operators of these form does not gain regularity, which is quite a crucial ingredient in the analysis of the linearized equation by using normal form methods.
Youtube Video: https://youtu.be/2NeFo4T5zR4
4. Nov 12th, 12-1pm (PST) Hosted by UT Austin
Speaker: Mikaela Iacobelli, ETH Zurich.
Title: Quantization of measures to ultrafast diffusion equations
Abstract: In this talk I will discuss some recent results on the asymptotic behaviour of a family of weighted ultrafast diffusion PDEs. These equations are motivated by the gradient flow approach to the problem of quantization of measures, introduced in a series of joint papers with Emanuele Caglioti and François Golse. In this presentation I will focus on a recent result with Francesco Saverio Patacchini and Filippo Santambrogio, where we use the JKO scheme to obtain existence, uniqueness, and exponential convergence to equilibrium under minimal assumptions on the data.
5. Nov 19th, 12-1pm (PST) Hosted by UT Austin
Speaker: Yannick Sire, Johns Hopkins University.
Title: Blow-up solutions via parabolic gluing
Abstract: We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow, Yamabe flow and Yang-Mills flow. I will also present some open questions.
Youtube Video: https://youtu.be/vhfwgPMPS-M
6. Dec 3rd, 12-1pm (PST) Hosted by Columbia University
Speaker: Max Engelstein, U. Minnesota
Title: Winding for Wave Maps
Abstract: Wave maps are harmonic maps from a Lorentzian domain to a
Riemannian target. Like solutions to many energy critical PDE, wave maps
can develop singularities where the energy concentrates on arbitrary
small scales but the norm stays bounded. Zooming in on these
singularities yields a harmonic map (called a soliton or bubble) in the
weak limit. One fundamental question is whether this weak limit is
unique, that is to say, whether different bubbles may appear as the
limit of different sequences of rescalings.
We show by example that uniqueness may not hold if the target manifold
is not analytic. Our construction is heavily inspired by Peter
Topping’s analogous example of a “winding” bubble in harmonic map heat flow. However, the Hamiltonian nature of the wave maps will occasionally
necessitate different arguments. This is joint work with Dana Mendelson
(U Chicago).
Youtube video: https://youtu.be/apfXS_S_clM
7. Dec 10th, 12-1pm (PST) Hosted by Columbia University
Speaker: Salvatore Stuvard, UT Austin
Title: Brakke flow of surfaces with prescribed boundary: a dynamical approach to Plateau’s problem
Abstract: Brakke flow is a measure-theoretic generalization of the mean curvature flow which describes the evolution by (generalized) mean curvature of surfaces with singularities. In the first part of the talk, I am going to discuss global existence and large time asymptotics of solutions to the Brakke flow with fixed boundary when the initial datum is given by any arbitrary rectifiable closed subset of a convex domain which disconnects the domain into finitely many “grains”. Such flow represents the motion of material interfaces constrained at the boundary of the domain, and evolving towards a configuration of mechanical equilibrium according to the gradient of their potential energy due to surface tension.
In the second part, I will focus on the case when the initial datum is already in equilibrium (a generalized minimal surface): I will prove that the presence of certain singularity types in the initial datum guarantees the existence of non-constant solutions to the Brakke flow. This suggests that the class of dynamically stable minimal surfaces, that is minimal surfaces which cannot be moved by Brakke flow, may be worthy of further study within the investigation on the regularity properties of generalized minimal surfaces.
Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology)
Youtube link: https://youtu.be/WXduyQcKHp4
8. Dec 17th, 12-1pm (PST) Hosted by Columbia University
Speaker: Arshak Petrosyan, Purdue University
Title: Almost minimizers for the thin obstacle problem
Abstract: In this talk, we will consider Anzellotti-type almost
minimizers for the thin obstacle (or Signorini) problem with zero thin
obstacle. We will discuss the regularity properties of the almost
minimizers, as well as the structure of their free boundaries. The
analysis of the free boundary is based on a successful adaptation of
energy methods such as a family of Weiss-type monotonicity formulas,
Almgren-type frequency formula, and the epiperimetric and logarithmic
epiperimetric inequalities for the solutions of the thin obstacle
problem. This is a joint work with Seongmin Jeon.
Youtube link: https://www.youtube.com/watch?v=2jIMc9tSsmI
9. Jan 14th, 12-1pm (PST) Hosted by Purdue University
Speaker: Tim Laux, HCM, University of Bonn
Title: Sharp-interface limits in the dynamics of phase transitions: from the Allen-Cahn equation to liquid crystals
Abstract: The large-scale behavior of phase transitions has a long history. In this talk, I want to present two recent projects which establish convergence results based on a new relative entropy for phase-field models. With Julian Fischer and Theresa Simon, we prove optimal convergence rates for the Allen-Cahn equation to mean curvature flow before the onset of singularities. The proof does not rely on the maximum principle and does not require to understand the spectral properties of the linearized Allen-Cahn operator. With Yuning Liu, we consider the dynamics in the Landau-de Gennes theory of liquid crystals. We show that at the critical temperature, a scaling limit can be derived: The interface between the isotropic and nematic phases moves by mean curvature flow. Furthermore, in the nematic phase, the director field is a harmonic map heat flow with homogeneous Neumann boundary conditions. To derive the equations, we combine the relative entropy method with weak convergence methods.
Youtube link: https://youtu.be/SsR6_JrNBLo
10. Jan 21st, 12-1pm (PST) Hosted by Purdue University
Speaker: Zongyuan Li, Rutgers University
Title: Mixed Dirichlet-conormal problems for parabolic equations
Abstract: In this talk, we discuss the mixed Dirichlet-conormal problem for parabolic equations. Under very weak assumptions, we prove the solvability in both $L_p$-based and $L_{q,p}$-based mixed-norm Sobolev spaces. In particular, the domain is allowed to be a cylinder with a Reifenberg-flat base, and the interfacial boundary $\Gamma$ between two types of boundary conditions can be time-dependent and locally close to a Lipschitz graph with respect to the Hausdorff distance. The solution space here is optimal, even for heat equations on $(0,T)\times\mathbb{R}^d_+$ with $\Gamma=\{x_1=x_2=0\}$. This is based on recent joint works with Jongkeun Choi (Pusan) and Hongjie Dong (Brown).
Youtube link: https://youtu.be/pBoHJn65wDE
11. Jan 28th, 12-1pm (PST) Hosted by Purdue University
Speaker: Yanyan Li, Rutgers University
Title: On the \sigma_2-Nirenberg problem in dimension two
Abstract: We will present a result on the existence and compactness of solutions of the \sigma_2-Nirenberg problem in dimension two. We will first recall some previous results on the Nirenberg-problem and the \sigma_k-Nirenberg problem in dimension greater than two. Then we present some ingredients which are used in the proof of the existence and compactness result: a Liouville theorem and a Bocher theorem for Mobius invariant equations, gradient and second derivative estimates, and one point blow up phenomena and C^0 estimates. This is joint work with Han Lu and Siyuan Lu.
12. Feb 4th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Albert Fathi, Gerogia Tech
Title: Singularities of solutions of the Hamilton-Jacobi equation. A toy model: distance to a closed subset
Abstract: https://drive.google.com/drive/u/0/folders/1jt9QINT_QMCSDcIiYw04QHlvKeFEskGr
Youtube link: https://youtu.be/jyLyypC8-xw
13. Feb 11th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Yao Yao, Gerogia Tech
Title: Small scale formations in the incompressible porous media equation
Abstract: The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finite-time blow-up remains open for smooth initial data, although numerical evidences suggest that small scale formation can happen as time goes to infinity. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infinite-in-time growth of Sobolev norms, provided that they remain globally smooth in time. As an application, this allows us to obtain nonlinear instability of certain stratified steady states of IPM. This is a joint work with Alexander Kiselev.
Youtube link: https://youtu.be/4z1v5lxGCfU
14. Feb 18th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Mahir Hadzic, University College London
Title: Instability of self-gravitating galaxies and stars
Abstract: Radial finite mass and compactly supported steady states of the asymptotically flat Einstein-Vlasov and Einstein-Euler systems represent isolated self-gravitating stationary galaxies and stars respectively. Upon the specification of the equation of state, such steady states are naturally embedded in 1-parameter families of solutions parametrised by the size of their central redshift. In the first part of the talk we prove that highly relativistic galaxies/stars (the ones with high central redshift) are linearly unstable (joint work with Zhiwu Lin and Gerhard Rein). This is consistent with an instability scenario suggested in 1960s by Zeldovich et al. in the Vlasov case, and Wheeler et al. in the Euler case. In the second part of the talk we explain and prove the Turning Point Principle for the Einstein-Euler system, proposed by Wheeler et al. (joint work with Zhiwu Lin).
Youtube link: https://youtu.be/j4XAA9b7-kk
15. Feb 25th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Boyan Sirakov, PUC-Rio, Brazil
Title: The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients
Abstract: In this joint work with P. Souplet we develop a new, unified approach to the following two classical questions on elliptic PDE: (i) the strong maximum principle for equations with non-Lipschitz nonlinearities; and (ii) the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.
Youtube Link: https://youtu.be/kmfWmh5_Diw
Mar 4th, 12-1pm (PST) Hosted by Brown University
16. Speaker: Yan Guo, Brown University
Title: Dynamics of Contact Line
Abstract: Contact lines (e.g, where coffee meets the coffee cup) appear generically between a free surface and a fixed boundary. Even though the steady contact line and contact angle was studied by people like Gauss and Young, even the modelling of dynamic contact lines has been an active research area in physics. In a joint research program with Ian Tice, global well-posedness and stability of contact lines is established for a recent viscous fluid model in 2D.
Youtube Link: https://youtu.be/Utt7f8RJ6Ro
17. Mar 11th, 12-1pm (PST) Hosted by Brown University
Speaker: Justin Holmer, Brown University
Title: Quantitative Derivation and Scattering of the 3D Cubic NLS
Abstract: We consider the derivation of the cubic defocusing nonlinear Schrodinger equation from quantum N-body dynamics. Previous approaches first prove the convergence of the BBGKY hierarchy to GP hierarchy as a weak limit and then upgrade this to a strong limit by minimality and convexity. We reformat the argument using Klainerman-Machedon theory to directly prove the strong limit in the energy space and obtain an explicit rate estimate. This rate estimate is nearly optimal and there is no gap in regularity between the space of initial data and the space in which the limit is proved. We also discuss in detail a nonlinear scattering lemma involving the comparison between the nonlinear Hartree evolution and the nonlinear Schrodinger solution. This is joint work with Xuwen Chen (University of Rochester).
Youtube Link: https://youtu.be/UPEEJg4d
18. Mar 18th, 12-1pm (PST) Hosted by Brown University
Speaker: Tai-Peng Tsai, UBC
Title: The Green tensor of the nonstationary Stokes system in the half space
Abstract: We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier-Stokes equations in Lq, pointwise decay, and uniformly local Lq spaces in the half-space. We also show the existence of Navier-Stokes flows with finite global energy and unbounded velocity derivative near the boundary, caused by Holder continuous boundary fluxes with compact support. This talk is based on joint work with Kyungkeun Kang, Baishun Lai and Chen-Chih Lai (arXiv:2011.00134 and work in progress).
Youtube Link: https://youtu.be/iNoEJTvykjw
19. Mar 25th, 12-1pm (PST) Hosted by Brown University
Speaker: Juraj Foldes, University of Virginia
Title: Statistical solutions for equations of fluid dynamics
Abstract: Two dimensional turbulent flows for large Reynold’s numbers can be approximated by solutions of incompressible Euler’s equation. As time increases, the solutions of Euler’s equation are increasing their disorder; however, at the same time, they are limited by the existence of infinitely many invariants. Hence, it is natural to assume that the limit profiles are functions which maximize an entropy given the values of conserved quantities. These profiles, described by methods of Statistical Mechanics, are solutions of non-usual variational problems with infinite number of constraints. We will show how to analyze the problem and we will derive symmetry properties of entropy maximizers on symmetric domains. This is a joint work with Vladimir Sverak (University of Minnesota).
Youtube link: https://youtu.be/-i05Nh4sKoQ. (slides 8 and 9 are missing in the video due to internet connection problems in the process of recording.)
20. April 1st, 12-1pm (PST) Hosted by UCSD
Speaker: Tristan Buckmaster, Princeton University
Title: Stable shock wave formation for the compressible Euler equations
Abstract: I will talk about recent work with Steve Shkoller, and Vlad Vicol, regarding shock wave formation for the compressible Euler equations.
Youtube link: https://youtu.be/8WRv9auWZNw
21. April 15th, 12-1pm (PST) Hosted by UCSD
Speaker: Lenya Ryzhik, Stanford University
Title: Fisher-KPP equation with small data and the extremal process of branching Brownian motion
Abstract: The Fisher-KPP equation was introduced by Fisher, and Kolmogorov, Petrvoskii and Piskunov in 1937 as a basic reaction-diffusion spreading model. In 1975, H. Mc Kean discovered a direct connection between this PDE and the branching Brownian motion. M. Bramson in the early 1980’s used this connection to establish convergence of the solutions to the FKPP equation to a shift of a traveling wave. I will discuss how the “Bramson shift” for some particular (asymptotically small) initial conditions for the FKPP equation encodes a wealth of information about the limiting extremal process of BBM seen from the tip, and how this PDE approach can be used to understand the fluctuations of this process. It is natural to conjecture that similar results hold for other log-correlated random processes where the PDE techniques are not available. No evidence will be presented to support this conjecture. This is a joint work with L. Mytnik and J.-M. Roquejoffre.
22. April 22th, 12-1pm (PST) Hosted by UCSD
Speaker: Gianluca Crippa, University of Basel
Title: On the local limit for nonlocal conservation laws
Abstract: Consider a family of continuity equations where the velocity field is given by the convolution of the solution with a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? This question was posed by P. Amorim, R. Colombo and A. Teixeira and a positive answer was suggested by numerical simulations. In the talk we will exhibit counterexamples showing that in general convergence of the solutions does not hold. We will also show that the answer to the above question is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. We will also discuss the possible role of numerical viscosity in numerical simulations, as well as some more recent results dealing with the case of an anisotropic convolution kernel, a more realistic case for applications to traffic modelling. The talk will be based on some joint works with Maria Colombo, Marie Graff, Elio Marconi, and Laura Spinolo.
Youtube link: https://youtu.be/vFVfmluoQ9o
23. April 29th, 12-1pm (PST) Hosted by UCSD
Speaker: Henri Berestycki , EHESS, Paris
Title: Segregation in predator-prey models and the emergence of territoriality
Abstract: I report here on a series of joint works with Alessandro Zilio (Université de Paris) about systems of competing predators interacting with a single prey. We focus on the analysis of stationary states, stability issues, and the asymptotic behavior when the competition parameter becomes unbounded. Existence of solutions is obtained by a bifurcation theory type approach and the segregation analysis rests on a priori estimates and a free boundary problem. We discuss the classification of solutions by using spectral properties of the limiting system. Our results shed light on the conditions under which predators segregate into packs, on whether there is an advantage to have such hostile packs, and on comparing the various territory configurations that arise in this context. These questions lead us to nonstandard optimization problems.
24. May 6th, 12-1pm (PST) Hosted by UW Madison
Speaker: Chanwoo Kim, UW Madison
Title: Damping of kinetic transport equation with diffuse boundary condition
Abstract: We will discuss a quantitative study of the mixing effects by the stochastic boundary in the kinetic theory. We consider solutions of the kinetic transport equation in convex domains satisfying a stochastic boundary condition. We prove that the moments of a fluctuation decay pointwisely almost fast as $t^{-3}$ as $t\rightarrow\infty$. Two key ingredients are (1) establishing a local lower bound with an unreachable defect (similar to the Doeblin condition); and (2) developing an $L^1-L^\infty$ bootstrap argument using the stochastic characteristics. This talk is based on a recent joint work with Jiaxin Jin.
Youtube Link: https://youtu.be/0zWL_AhjNHM
25. Speaker: Nam Le, Indiana University, Bloomington
Title: Solvability of a class of singular fourth order equations of Monge-Ampere type
Abstract: We will discuss the solvability of a natural boundary value problem for a class of highly singular fourth order equations of Monge-Ampere type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu type equations that appear in geometric contexts.
In two dimensions, we establish global solutions to the second boundary value problem for highly singular Abreu equations where the right-hand sides are of q-Laplacian type. We show that minimizers of variational problems with a convexity constraint in two dimensions that arise from the Rochet-Chone model in the monopolist’s problem in economics with q-power cost can be approximated in the uniform norm by solutions of the Abreu equation for a full range of q. Both the Legendre and partial Legendre transforms are used in our analysis. This talk is based on joint work with Bin Zhou (Peking University).
Youtube Link: https://youtu.be/Ab-idECJFvw
26. Speaker: Luis Caffarelli, UT Austin
Title: REGULARITY FOR C ^1,α INTERFACE TRANSMISSION PROBLEMS ́
Youtube Link: https://youtu.be/kmAA6d_bqLg