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