PDE seminars

Time	Speaker	Title	Zoom
23 Feb 2024, 9:30am Beijing time.	Luis Silvestre (University of Chicago)	The Landau equation does not blow up Abstract Abstract: The Landau equation is one of the main equations in kinetic theory. It models the evolution of the density of particles when they are assumed to repel each other by Coulomb potentials. It is a limit case of the Boltzmann equation with very soft potentials. In the space-homogeneous case, we show that the Fisher information is monotone decreasing in time. As a consequence, we deduce that for any initial data the solutions stay smooth and never blow up, closing a well-known open problem in the area.	ID: 967 1736 3318 Passcode: 679489
29 Feb 2024, 4pm Beijing time.	Antonio Fernandez (Universidad Autónoma de Madrid)	A Schiffer-type problem with applications to stationary Euler flows Abstract Abstract: If on a smooth bounded domain \(\Omega\subset\mathbb{R}^2\) there is a nonconstant Neumann eigenfunction \(u\) that is locally constant on the boundary, must \(\Omega\) be a disk or an annulus? This question can be understood as a weaker analog of the well-known Schiffer conjecture, in that the function \(u\) is here allowed to take a different constant value on each connected component of \(\partial \Omega\) yet many of the known rigidity properties of the original problem are essentially preserved. In this talk we provide a negative answer by constructing a family of nontrivial doubly connected domains \(\Omega\) with the above property. Then, we will show how our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. The talk is based on a joint work with Alberto Enciso, David Ruiz and Pieralberto Sicbaldi.	ID: 975 0876 4296 Passcode: 583722
7 March 2024, 4pm Beijing time.	Christian Seis (Universität Münster, Germany)	Steady vortex rings with surface tension Abstract Abstract: The existence of steady vortex rings for the two-phase Euler equations with surface tension is studied, describing the evolution of a perfect bubble air ring in water. Such objects are created in nature by cetaceans such as dolphins or beluga whales, and they appear to be surprisingly stable configurations. The mathematical model features a vortex sheet on the surface of the air bubble. We construct such vortex rings with small cross sections with the help of an implicit function theorem and derive the asymptotics of various quantities for small cross sections. Joint work with David Meyer (Münster).	ID: 989 5275 7926 Passcode: 340496
14 March 2024, 4pm Beijing time.	Simon Noah Nowak (Universitat Bielefeld, Germany)	Nonlinear nonlocal potential theory at the gradient level Abstract Abstract: We present pointwise gradient potential estimates for a class of nonlinear nonlocal equations related to strongly convex nonlocal energy functionals. Our pointwise estimates imply that the first-order regularity properties of such general nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian. The talk is based on joint work with Lars Diening, Kyeongbae Kim and Ho-Sik Lee.	ID: 996 9739 6030 Passcode: 231607
11 April 2024, 4pm Beijing time.	Martin Mayer (Scuola Superiore Meridionale, Italia)	Classical arguments and their limitations in conformal geometry Abstract Abstract: We consider the classical conformally prescribed scalar curvature problem on a closed Riemannian manifold. In the most generic and simple situations we review the relevant formulations, strategies and arguments to show solvability, in particular exhibiting some recent existence results. Complementarily we will see, that the theory as of today is far from being complete, as in most situations and for generic curvature candidate functions the underlying equation has trivial degree, but is solvable, while only a few obstructions are explicitly known.	ID: 942 9965 6235 Passcode: 154986
18 April 2024, 4pm Beijing time.	Xavier Fernandez-Real (EPFL)	On the one-phase problem Abstract Abstract: In this talk we present an introduction to elliptic free boundary problems, with a particular emphasis on the one-phase problem. We will explain the basic regularity theory for it, and compare it with other classical free boundary problems. At the end of the talk, we will introduce some recent results in collaboration with Max Engelstein and Hui Yu on both the Bernstein problem for one-phase free boundaries, and generic uniqueness and regularity for both the one-phase and Alt- Phillips problem.	ID: 927 1816 1052 Passcode: 657193
25 April 2024, 4pm Beijing time. (This is the last seminar of this semester.)	Xavier Lamy (Universite Paul Sabatier, Toulouse)	Generation of vortices in the Ginzburg Landau heat flow Abstract Abstract: In the two-dimensional Ginzburg-Landau heat flow, a configuration with a logarithmic energy bound has well-formed vortices and their motion is well-understood. For an initial condition with a finite number of nondegenerate zeros, but possibly very high energy, we show that the initial zeros are conserved and the flow rapidly enters the logarithmic energy regime. This is joint work with M.Kowalczyk.	ID: 980 2447 4585 Passcode: 367891

Time	Speaker	Title	Zoom
8 Sept 2023, 9:30am Beijing time.	Matthew Gursky (University of Notre Dame)	Deformation of ASD four-manifolds and a nonlinear PDE from spectral theory Abstract Abstract: In this talk I will give a brief overview of the deformation theory for anti-self-dual 4-manifolds, and introduce a geometric differential operator that plays a key role. I will then report on some recent progress toward understanding when the cokernel of this operator vanishes, and the connection to a nonlinear PDE from spectral geometry. This is joint work with Rod Gover.	ID: 952 9345 5527 Passcode: 002473
14 Sept 2023, 4pm Beijing time.	Sunghan Kim (Uppsala University, Sweden)	Constraint maps with free boundaries Abstract Abstract: In this talk, we shall consider maps that minimize the Dirichlet energy subject to constraints on their image. We shall these map (minimizing) constraint maps. In the manifold level, these maps are precisely harmonic maps into manifolds-with-boundary. On the other hand, the minimization problem can also be considered as canonical extension of the classical obstacle problem to the vectorial setting. The constraint maps were considered several decades ago, mainly by F. Duzaar and M. Fuchs, who established the optimal partial regularity theory. The observations were aligned with the development of the theory for harmonic maps. What differentiates the constraint maps from the (usual) harmonic maps (into manifolds without boundary) is the presence of free boundaries. Although they were studied many years ago, only basic properties of free boundaries were studied for the constraint maps. Recently, together with A. Figalli and H. Shahgholian, I took a closer look, from the perspective of free boundary problems, at the behavior of the mappings in the vicinity of their free boundaries. Our result shows some interesting (vectorial) features, which do not (and cannot) arise in the scalar obstacle problems. In this talk, I will give a brief overview on the development and the characters of the constraint maps, and present the recent result, and if time allows, some interesting, new problems in this direction. The talk will be based on the joint works by A. Figalli, A. Guerra and H. Shahgholian.	ID: 920 4113 2301 Passcode: 393276
21 Sept 2023, 4pm Beijing time.	Angela Pistoia (Sapienza Universita di Roma)	Conformal metrics with prescribed curvatures Abstract Abstract: I will present some recent results concerning the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a n-dimensional compact Riemannian manifold.	ID: 972 7880 3385 Passcode: 524995
28 Sept 2023, 4pm Beijing time.	Joaquim Serra (ETH)	Nonlocal approximation of minimal surfaces: optimal estimates from stability Abstract Abstract: Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. In two recent papers --- with Chan, Dipierro and Valdinoci, and with Caselli and Florit--- we set the ground for a new approximation based on nonlocal minimal surfaces. In the first paper, we prove that stable s-minimal surfaces in the unit ball of \(\mathbb{R}^3\) satisfy curvature estimates that are robust as s approaches 1 (i.e. as the energy approaches that of classical minimal surfaces). Moreover, we obtain optimal sheet separation estimates and show that critical interactions are encoded by nontrivial solutions to a (local) "Toda type" system. As a nontrivial application, we establish that hyperplanes are the only stable s-minimal hypersurfaces in \(\mathbb{R}^4\), for s sufficiently close to 1. In the second paper, we establish the existence of infinitely many nonlocal minimal surfaces in every closed manifold (i.e., a version of Yau's conjecture).	ID: 942 8487 7910 Passcode: 675302
20 Oct 2023, 9:30am Beijing time.	Ao Sun (Lehigh University)	Genus one singularities in mean curvature flow Abstract Abstract: We show that for certain one-parameter families of initial conditions in \(\mathbb{R}^3\), when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. Among applications, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut, and we proved that the fourth lowest entropy self-shrinker in \(\mathbb{R}^3\) can not be rotationally symmetric. Based on joint work with Adrian Chu.	ID: 968 7788 7622 Passcode: 107767
26 Oct 2023, 4pm Beijing time.	Jose M. Espinar (Universidad de Granada, Spain)	Frankel property and Maximum Principle at Infinity for complete minimal hypersurfaces Abstract Abstract: We extend Mazet's Maximum Principle at infinity for parabolic, two-sided, properly embedded minimal hypersurfaces, up to ambient dimension seven. Parabolicity is a necessary condition in dimension \(n \geq 4\), even in Euclidean space, as the example of the higher-dimensional catenoid shows. Next, inspired by the Tubular Neighborhood Theorem of Meeks-Rosenberg in Euclidean three-space we focus on the existence of an embedded \(\epsilon -\)tube when the ambient manifold \(\mathcal{M}\) has non-negative Ricci curvature. These results will allow us to establish Frankel-type properties and to extend the Anderson-Rodriguez Splitting Theorem under the existence of an area-minimizing \({\rm mod}(2)\) hypersurface \(\Sigma\) in these manifolds \(\mathcal{M}\) (up to dimension seven), the universal covering space of \(\mathcal{M}\) is isometric to \(\Sigma \times \mathbb{R}\) with the product metric.	ID: 941 3577 6085 Passcode: 794539
3 Nov 2023, 9:30am Beijing time.	Yilun Wu (University of Oklahoma)	Existence of rotating stars with variable Entropy Abstract Abstract: Rotating stars can be modeled by steady solutions to the Euler-Poisson equations. An extensive literature has established the existence of rotating stars for given differentially rotating angular velocity profiles. However, all of the existing results require the angular velocity to depend on the distance to the rotation axis, but not on the distance to the equatorial plane. Incidentally, all of these solutions have constant entropy within the star. In this talk, I will present a recent result which is the first that allows a general rotation profile, without restrictions. It is also the first result that allows genuinely changing entropy within the star. The variation of entropy causes the previous methods used to construct steady solutions inapplicable. We discover a div-curl reformulation of the problem and perform analysis on the resulting elliptic-hyperbolic system. This is joint work with Juhi Jang and Walter Strauss.	ID: 985 3186 6133 Passcode: 045613
10 Nov 2023, 9:30am Beijing time.	Jeaheang Bang (The University of Texas at San Antonio)	Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications Abstract Abstract: Solutions with scaling-invariant bounds, such as self-similar solutions, play an important role in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. We recently proved that any steady solution satisfying \(|u(x)|\leq C/|x|\) for any constant C in \(\mathbb{R}^n\setminus \{0\}\) with \(n\geq 4\), must be zero without imposing a smallness or self-similarity assumption. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates with suitable multipliers, and our proof is pretty elementary and short. These results not only give the Liouville-type theorem for steady solutions in higher dimensions but also help to remove a class of singularities of solutions and give the optimal asymptotic behaviors of solutions at infinity in the exterior domains. This is a joint work with Changfeng Gui, Hao Liu, Yun Wang and Chunjing Xie.	ID: 952 0206 9493 Passcode: 305143
23 Nov 2023, 4pm Beijing time. (This is the last seminar of this semester.)	Christophe Prange (Cergy Paris University)	Some recent trends for the regularity of the 3D Navier-Stokes equations Abstract Abstract: In this talk, I will review some recent developments in collaboration with Dallas Albritton (UW Madison), Tobias Barker (U Bath) and Jin Tan (Cergy Paris U) on the regularity for solutions of the three-dimensional Navier-Stokes equations.	ID: 967 2621 5463 Passcode: 887262

Time	Speaker	Title	Zoom
3 Mar 2023, 9am Beijing time.	Sigurd Angenent (University of Wisconsin at Madison)	Ancient solutions to space curve shortening and where they appear Abstract Abstract: Ancient solutions appear as blow-up limits in the study of singularities of geometric flows. In the case of planar and space curve shortening I will give an overview of various ancient solutions that have been constructed, and discuss ways in which they can appear as blow-ups.	ID: 968 2493 3414 Passcode: 542678
10 Mar 2023, 9am Beijing time.	Connor Mooney (University of California at Irvine)	The anisotropic Bernstein problem Abstract Abstract: The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). This is joint work with Y. Yang.	ID: 953 3934 3887 Passcode: 232558
17 March 2023, 9am Beijing time.	Robin Neumayer (Carnegie Mellon University)	Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit Abstract Abstract: The Alt-Caffarelli-Friedman (ACF) monotonicity formula is an important tool in the study of free boundary problems. More generally, given any pair of nonnegative subharmonic functions with disjoint positivity sets, the ACF formula provides information about the interface between the supports. In this talk we'll show that on the portion of the interface where the ACF formula is asymptotically positive forms an H^{n-1}-rectifiable set, and that the two functions have unique blowups at H^{n-1} almost every such point. This talk is based on joint work with Mark Allen and Dennis Kriventsov.	ID: 979 3708 6485 Passcode: 336205
24 March 2023, 5pm Beijing time.	María del Mar González (Universidad Autónoma de Madrid)	Spectral properties of Levy Fokker-Planck equations Abstract Abstract: We study the spectrum of a fractional Laplacian equation with drift in suitable weighted spaces. This operator arises when studying the fractional heat equation in self-similar variables. We show, in the radially symmetric case, compactness, and then calculate the eigenfunctions in terms of Laguerre polynomials. The proofs involve Mellin transform and complex analysis methods. This is joint work with H. Chan, M. Fontelos and J. Wei.	ID: 934 7331 0348 Passcode: 692453
31 March 2023, 9am Beijing time.	Zhuolun Yang (Brown University)	Optimal estimates for the conductivity problems with closely spaced inclusions of high contrast Abstract Abstract: In this talk, I will describe an elliptic PDE that models electric conduction, and the electric field concentration phenomenon between closely spaced inclusions of high contrast. In the first part, I will present some results on the insulated conductivity problem (jointly with Hongjie Dong and Yanyan Li). We obtained an optimal gradient estimate in terms of the distance between inclusions. This solved one of the major open problems in this area. In the second part, I will present a recent work regarding optimal estimates for higher derivatives of the conductivity problem with circular inclusions in 2D, when the relative conductivities of inclusions have different signs (Jointly with Hongjie Dong). This improves a recent result of Ji and Kang.	ID: 984 5920 0811 Passcode: 834124
14 April 2023, 9am Beijing time.	Tai-Peng Tsai (University of British Columbia)	Growth rates for axisymmetric Euler flows Abstract Abstract: We consider axisymmetric, swirl-free solutions of the Euler equations in three and higher dimensions, of generalized anti-parallel-vortex-tube-pair-type: the initial scalar vorticity has a sign in the half-space, is odd under reflection across the plane, is bounded and decays sufficiently rapidly at the axis and at spatial infinity. We prove lower bounds on the growth of such solutions in all dimensions, improving a lower bound proved by Choi and Jeong in three dimensions. This is based on joint work with Stephen Gustafson and Evan Miller, arXiv:2303.12043.	ID: 910 4275 9581 Passcode: 154270
21 April 2023, 9am Beijing time.	Yifu Zhou (Johns Hopkins University)	Singularity formation for the Landau-Lifshitz-Gilbert equation in dimension two Abstract Abstract: Landau-Lifshitz-Gilbert equation (LLG), which models the evolution of spin fields in continuum ferromagnetism, can be viewed as a coupling between the harmonic map heat flow and the Schrodinger map flow. In this talk, we shall report some recent gluing construction of finite-time singularities for LLG in dimension two. To overcome the difficulties caused by the dispersion, technical ingredients such as distorted Fourier transform and sub-Gaussian estimates are employed. This is based on a joint work with J. Wei and Q. Zhang.	ID: 968 5118 7301 Passcode: 166862
28 April 2023, 9am Beijing time.	Antonio De Rosa (University of Maryland)	Min-max construction of anisotropic CMC surfaces Abstract 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.	ID: 974 9316 6138 Passcode: 421146
5 May 2023, 4pm Beijing time. (This is the last seminar of this semester.)	Monica Musso (University of Bath)	Leapfrogging for Euler equations Abstract Abstract: We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).	ID: 917 3528 8155 Passcode: 158116

Time	Speaker	Title	Zoom
16 Sept 2022, 4pm Beijing time.	Matteo Bonforte (Universidad Autónoma de Madrid)	Stability in Gagliardo-Nirenberg-Sobolev inequalities: nonlinear flows, regularity and the entropy method Abstract Abstract: We discuss stability results in Gagliardo-Nirenberg-Sobolev inequalities, a joint project with J. Dolbeault, B. Nazaret and N. Simonov. We have developped a new quantitative and costructive "flow method", based on entropy methods and sharp regularity estimates for solutions to the fast diffusion equation (FDE). This allows to study refined versions of the Gagliardo-Nirenberg-Sobolev inequalities that are nothing but explicit stability estimates. Using the quantitative regularity estimates, we go beyond the variational results and provide fully constructive estimates, to the price of a small restriction of the functional space which is inherent to the method.	ID: 943 4890 8003 Passcode: 257490
23 Sept 2022, 4pm Beijing time.	Tobias König (Goethe University Frankfurt)	Reverse conformally invariant Sobolev inequalities on the sphere Abstract Abstract: In this talk I will present a result about the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-dimensional sphere S^n involving an operator of order 2s > n. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of s. In particular, we obtain the sharp threshold value of s for the validity of a corresponding Sobolev inequality in all dimensions n >= 2. This is joint work with Rupert L. Frank (LMU Munich and Caltech) and Hanli Tang (Beijing Normal University).	ID: 948 0616 1456 Passcode: 826283
30 Sept 2022, 4pm Beijing time.	Juan Dávila (University of Bath, UK)	Infinite time blow-up for the Keller-Segel system in the plane Abstract Abstract: We study the Keller-Segel system in the plane with an initial condition with sufficient decay and critical mass 8 pi. We find a function u0 with mass 8 pi such that for any initial condition sufficiently close to u0 and mass 8 pi, the solution is globally defined and blows up in infinite time. This proves the non-radial stability of the infinite-time blow up for some initial conditions, answering a question by Ghoul and Masmoudi (2018). This is joint work with Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).	ID: 993 1965 1034 Passcode: 711831
7 Oct 2022, 4pm Beijing time.	Xavier Ros-Oton (Universitat de Barcelona)	The singular set in the Stefan problem Abstract Abstract: The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points. The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.	ID: 996 5119 3566 Passcode: 615823
14 Oct 2022, 9am Beijing time.	Beomjun Choi (POSTECH, South Korea)	Higher order asymptotics for fast diffusions on bounded domains Abstract Abstract: Sobolev-subcritical fast diffusion with vanishing boundary condition leads to finite-time extinction, with a vanishing profile selected by the initial datum. In a joint work with R. McCann and C. Seis, we quantify the rate of convergence to this profile for general smooth bounded domains. In rescaled time variable, the solution either converges exponentially fast or algebraically slow. In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes, giving a higher order asymptotics. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes.	ID: 952 3554 4779 Passcode: 991961
21 Oct 2022, 9am Beijing time.	Xiangwen Zhang (University of California, Irvine)	Geometric flows in symplectic geometry Abstract Abstract: Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. Aiming to study the equations from the flux compactifications of Type IIA superstrings, we introduce the so-called Type IIA flow, which is a flow of closed and primitive 3-forms on a symplectic Calabi-Yau 6-manifold. Remarkably, the Type IIA flow can also be viewed as a flow as a coupling of the Ricci flow with a scalar field. In this talk, we will discuss the recent progress on this flow. This is based on a joint project with Fei, Phong and Picard.	ID: 913 6112 7888 Passcode: 877592
28 Oct 2022, 9am Beijing time.	Changyou Wang (Purdue University)	Analysis on isotropic-nematic phase transition and liquid crystal droplet Abstract Abstract: In this talk, I will discuss the phase transition phenomena between the isotropic and nematic states within the framework of Ericksen theory of liquid crystals with variable degrees of orientations. Treating it as the singular perturbation problems within the Gamma convergence theory, we will show that the sharp interface formed between isotropic and nematic states is an area minimizing surface. Under suitable assumptions either on the strong anchoring boundary values on the boundary of a bounded domain or the volume constraint of nematic regions in the entire space, we also show that the limiting nematic liquid configuration in the nematic region is a minimizer of the corresponding Oseen-Frank energy with either homeotropic or planar anchoring on the free sharp interface pending on the relative sizes of leading Frank elasticity coefficients. This is a joint work with Fanghua Lin.	ID: 956 3307 8331 Passcode: 648553
4 Nov 2022, 9am Beijing time.	Moon-Jin Kang (KAIST)	Stability of Riemann solution containing a shock under physically admissible perturbations Abstract Abstract: I will present the so-called "a-contraction with shifts” method, which is basically energy based. This method is quite useful in studying the stability of Riemann solution containing a shock under physically admissible perturbations. First, based on the method, we prove the long-time behavior of compressible Navier-Stokes flows perturbed from Riemann solution composed of a shock and other waves of different kinds. This resolves the long-standing problem since the 1980s. On the other hand, since the method can handle large perturbations of a shock, we can also prove the uniform stability of a shock w.r.t. the strength of viscosity, which provides the result on uniqueness of Riemann solution composed of a shock in the class of inviscid limits of solutions to the compressible Navier-Stokes system.	ID: 952 4020 0735 Passcode: 611512
11 Nov 2022, 4pm Beijing time. (This is joint with Jingshi Distinguished Lecture at Beijing Normal University. This is the last seminar of this semester.)	Juan Luis Vázquez (Universidad Autónoma de Madrid)	Nonlinear Diffusion Equations driven by p-Laplacian operators of nonlocal type Abstract Abstract: We will first review some basic models of nonlinear diffusion. The usual p-Laplacian is a model of strongly nonlinear operator with a well-developed elliptic and parabolic theory. In recent years, interest has focused on versions of this model that include nonlocal effects of fractional type. We describe recent results obtained for the nonlocal parabolic theory for p finite. Finally, we discuss a nonlocal infinity Laplacian model motivated by a tug-of-war process introduced by Bjorland, Caffarelli and Figalli.	ID: 836 7562 6006 Passcode: 221111

Time	Speaker	Title	Zoom
25 Feb 2022, 9am Beijing time.	Sun-Sig Byun (Seoul National University)	Elliptic and parabolic equations with matrix weights and measurable nonlinearities Abstract Abstract: This talk is concerned with general elliptic and parabolic equations with matrix weights and measurable nonlinearities leading to Calderon-Zygmund type estimates	ID: 921 7807 0034 Passcode: 990108
4 Mar 2022, 9am Beijing time.	Hui Yu (National University of Singapore)	Rate of blow up in the thin obstacle problem Abstract Abstract: The thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lower-dimensional obstacle. Concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. In this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones' at contact points with integer frequencies. This talk is based on a recent joint work with Ovidiu Savin (Columbia).	ID: 980 1551 7417 Passcode: 446612
11 Mar 2022, 9am Beijing time.	Marcello Lucia (City University of New York, CSI and Graduate Center)	A variational approach to describe the moduli space of minimal immersions in hyperbolic manifolds Abstract Abstract: In a seminal paper, Uhlenbeck studied the set of first and second fundamental form that arise from minimal immersions of a given surface into a three dimensional hyperbolic manifold. However, as will be explained in this talk, such data are not suitable for describing the moduli space of minimal immersions. Following the ideas of Goncalves-Uhlenbeck, one must instead consider the “dual data” given by the cohomology class of (0,1) forms. To justify that approach, one is led to justify the existence and uniqueness of critical points of a functional whose lack of regularity and compactness do not allow to apply classical variational results. This is a joint work with Z. Huang and G.Tarantello.	ID: 968 8518 2169 Passcode: 184804
25 Mar 2022, 9am Beijing time.	Xiaodan Zhou (Okinawa Institute of Science and Technology Graduate University)	Horizontally quasiconvex envelope in the Heisenberg group Abstract Abstract: Various notions of convexity of sets and functions in the Heisenberg group have been studied in the past two decades. In this talk, we focus on the horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group. Inspired by the first-order characterization and construction of quasiconvex envelope by Barron, Goebel and Jensen in the Euclidean space, we obtain a PDE approach to construct the h-quasiconvex envelope for a given function f in the Heisenberg group. In particular, we show the uniqueness and existence of viscosity solutions to a non-local Hamilton-Jacobi equation and iterate the equation to obtain the h-quasiconvex envelope. Relations between h-convex hull of a set and the h-quasiconvex envelopes are also investigated. This is joint work with Antoni Kijowski (OIST) and Qing Liu (Fukuoka University/OIST).	ID: 980 4965 4261 Passcode: 495913
1 Apr 2022, 9am Beijing time.	Kyeongsu Choi (Korea Institute for Advanced Study )	Translating solitons to the power-of-Gauss curvature flow Abstract Abstract : The evolution of convex surfaces by powers of the Gauss curvature is a fully nonlinear parabolic equation. In particular, its translating solitons are complete convex graphs of solutions to a Monge-Ampere type equation. Hence, the classification of translators is a Liouville type theory for a Monge-Ampere type equation. In this talk, we address the existence and classification of translating surfaces by sub-affine-critical powers of the Gauss curvature.	ID: 992 7314 9445 Passcode: 309990
8 Apr 2022, 4pm Beijing time.	Daniele Bartolucci (University of Rome "Tor Vergata", Italy)	On the uniqueness and monotonicity of solutions of Grad-Shafranov type equations Abstract Abstract: We report about a series of results concerning a Grad-Shafranov type equation, which in dimension 2 describes the equilibrium configurations of a plasma in a Tokamak. In a neat interval depending only by the Sobolev constant of the domain we deduce uniqueness and monotonicity of the boundary density and of a suitably defined energy. Interestingly enough, in dimension 2 we derive the sharp values of the positivity threshold and of the energy upper bound. We also answer open questions about the lack of free boundary and generic properties of the global bifurcation diagram. This is part of a joint research project with A. Jevnikar (Udine), Y. Hu (Changsha), W. Yang (Wuhan).	ID: 910 4994 6922 Passcode: 399051
22 Apr 2022, 4pm Beijing time.	Jose Miguel Urbano (University of Coimbra, Portugal)	Regularity for singular and degenerate PDEs: qualitative vs. sharp estimates Abstract Abstract: Singular and degenerate partial differential equations are unavoidable in the modelling of several phenomena, like phase transitions and chemotaxis, and are also used in machine learning in the context of semi-supervised learning. They encompass a crucial issue in the analysis of PDEs, namely wether we can still derive analytical estimates when the crucial algebraic assumption of ellipticity collapses. We provide a broad overview of qualitative versus quantitative regularity estimates for solutions of these equations, introducing the method of intrinsic scaling and deriving sharp estimates by means of geometric tangential analysis. We discuss, in particular, recent results concerning the Stefan problem and the parabolic p-Poisson equation.	ID: 931 7473 3026 Passcode: 766088
29 Apr 2022, 9am Beijing time.	Xinliang An (National University of Singapore)	Anisotropic Dynamical Horizons Arising in Gravitational Collapse Abstract Abstract: Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou's monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price's law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.	ID: 975 4285 8380 Passcode: 897811
6 May 2022, 4pm Beijing time. (This is the last seminar of this semester.)	Manuel del Pino (University of Bath, UK)	Dynamics of concentrated vorticities in 2D and 3D Euler flows Abstract Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices. We rigorously establish the law of motion of "leapfrogging vortex rings", initially conjectured by Helmholtz in 1858.	ID: 967 6138 4440 Passcode: 085839

Time	Speaker	Title	Zoom
17 Sept 2021, 9am Beijing time.	Seunghyeok Kim (Hanyang University)	Infinite-time blowing-up solutions to small perturbations of the Yamabe flow Abstract Abstract: In this talk, we will examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold $M$ exists for all time $t$ and uniformly converges to a solution to the Yamabe problem on $M$ as $t \to \infty$. We will observe that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on $M$ in the infinite time. We also concern the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points. This is joint work with Monica Musso (University of Bath, UK).	ID: 974 4590 7096 Passcode: 875622
24 Sept 2021, 9am Beijing time.	Yao Yao (National University of Singapore)	Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states Abstract Abstract: The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modeled by nonlinear diffusion, and long-range attraction modeled by nonlocal interaction. In this talk, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint work with Carrillo, Hittmeir and Volzone). In a recent work, we further investigate whether they are unique within the radial class, and show that for a given mass, uniqueness/non-uniqueness of steady states are determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint work with Delgadino and Yan.)	ID: 919 4845 1738 Passcode: 774559
8 Oct 2021, 9am Beijing time.	Yannick Sire (Johns Hopkins University)	Some results on harmonic maps with free boundary and beyond Abstract Abstract: The theory of harmonic maps with free boundary is an old topic in geometric analysis. I will report on recent results on their Ginzburg-Landau approximation, regularity theory, and their heat flow. I will also describe several models in the theory of liquid crystals where the heat flow of those maps appears, emphasizing on some well-posedness issues and some hints on the construction of blow-up solutions. Several important results in geometric analysis such as extremal metrics for the Steklov eigenvalues for instance make a crucial use of such maps. I’ll give some open problems and will try to explain how to attack few open questions in the field using tools recently developed.	ID: 939 4965 3207 Passcode: 405755
15 Oct 2021, 9am Beijing time.	Goro Akagi (Tohoku University)	Energy method for quantitative analysis of rates of convergence to asymptotic profiles for fast diffusion Abstract Abstract: In this talk, we shall discuss quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles will be revealed via an energy method. The sharp rate of convergence to positive asymptotic profiles was recently discussed by Bonforte and Figalli (2021, CPAM) based on an entropy method. An alternative proof for their result will also be provided.	ID: 924 9007 6141 Passcode: 639787
22 Oct 2021, 9am Beijing time.	Hiroyoshi Mitake (University of Tokyo)	On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation Abstract Abstract: In this talk, we consider an initial-boundary value problem for the time-fractional diffusion equation. We show the equivalence of two notions of weak solutions, viscosity solutions and distributional solutions. It is worth emphasizing that in general the notion of viscosity solutions is based on the comparison principle, while the notion of distributional solutions is based on the variational principle. Since two notions of weak solutions are introduced in totally different manners, it is highly nontrivial whether two notions are same or not. In our approach, we use the discrete scheme for time-fractional diffusion equations which was introduced by Giga-Liu-Mitake (Asymptot. Anal. 2020). A main difficulty is in proving that the error term which comes from the approximated solution and the distributional solution converges to zero in a suitable weak sense. The idea to overcome this difficulty is to introduce an approximation of kernel in consideration of the discrete scheme. Due to the discrete scheme and kernel approximation, we can get the precise error estimate which enables us to get our main theorem. This is a joint work with Y. Giga (U. Tokyo) and S. Sato (U. Tokyo).	ID: 936 8185 8116 Passcode: 302432
29 Oct 2021, 9am Beijing time.	Dennis Kriventsov (Rutgers University)	Stability for Faber-Krahn inequalities and the ACF formula Abstract Abstract: The Faber-Krahn inequality states that the first Dirichlet eigenvalue of the Laplacian on a domain is greater than or equal to that of a ball of the same volume (and if equality holds, then the domain is a translate of a ball). Similar inequalities are available on other manifolds where balls minimize perimeter over sets of a given volume. I will present a new sharp stability theorem for such inequalities: if the eigenvalue of a set is close to a ball, then the first eigenfunction of that set must be close to the first eigenfunction of a ball, with the closeness quantified in an optimal way. I will also explain an application of this to the behavior of the Alt-Caffarelli-Friedman monotonicity formula, which has implications for free boundary problems with multiple phases. This is based on recent joint work with Mark Allen and Robin Neumayer.	ID: 937 3422 9393 Passcode: 936334
5 Nov 2021, 3pm Beijing time.	Naian Liao (Universität Salzburg)	Continuity of solutions to doubly nonlinear parabolic equations Abstract Abstract: I will report some recent advances on the local continuity of weak solutions to some doubly nonlinear parabolic equations, which include the parabolic p-Laplacian, the porous medium equation and the Stefan problem as particular cases. Various moduli of continuity are obtained depending upon the type of nonlinearity.	ID: 965 2222 8911 Passcode: 630202
12 Nov 2021, 9am Beijing time.	Qing Liu (Fukuoka University)	Principal eigenvalue problem for infinity Laplacian in metric spaces Abstract Abstract: In this talk, we discuss the Dirichlet eigenvalue problem associated to the infinity Laplacian in metric spaces. We provide a direct PDE approach to find the principal eigenvalue and eigenfunctions for a bounded domain in a proper geodesic space with no measure structure. We give an appropriate notion of solutions to the infinity eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process, introduced by Juutinen-Lindqvist-Manfredi (ARMA,1999), via the variational eigenvalue formulation for $p$-Laplacian in the Euclidean space. Several further results and concrete examples will be given in the case of finite metric graphs. This talk is based on joint work with Ayato Mitsuishi at Fukuoka University.	ID: 986 8580 0264 Passcode: 440023
19 Nov 2021, 9am Beijing time. (This is the last seminar of this semester.)	Juncheng Wei (University of British Columbia)	Title: Stability of Sobolev Inequalities and Related Topics Abstract Abstract: In this talk I will discuss the stability of Sobolev inequalities from critical point theory, thereby solving a conjecture of Figalli-Glaudo. I will also report related surprising results on harmonic map inequalities and half-harmonic map inequalities.	ID: 942 6222 3505 Passcode: 176052