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