# Learning Seminar on PDE and Applications ---- Spring 2021 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

The talks will be held virtually:

Zoom Link   Meeting ID: 924 8937 5526   Passcode: 565238

• May 12, 2021, Wed. 9:30am-11:30am, (Eastern Time)

Speaker: Qing Han, Notre Dame University

Title: The isometric embedding of abstract surfaces in the 3-dim Euclidean space

Abstract: A surface in the 3-dim Euclidean space can be viewed as the image of a map from a planar domain to the 3-dim Euclidean space, at least locally. The standard metric in the Euclidean space induces a metric on the surface. The induced metric on the surface can be transformed to an abstract metric by the abovementioned map. Now, we consider the converse question. Given an abstract metric on a planar domain, can we find a surface in the 3-dim Euclidean space whose induced metric is the given abstract metric? This is the isometric embedding problem we will discuss. It started with a conjecture by Schlaefli in 1873 that this can always be achieved near any given point. This conjecture is widely open and there are only a few results under various conditions. The question can be reformulated in terms of partial differential equations and is essentially a PDE problem. Despite the technical description, the underlying equation has a simple form.

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• Jan. 26, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Freid Tong, Columbia University

Title: Moser-Trudinger inequality for complex Monge-Ampere and applications. I

Abstract: We will discuss some recent works of Wang-Wang-Zhou on a proof of Moser-Trudinger inequalities for complex Monge ampere equation. We will then discuss its applications to establish apriori estimates for the complex Monge-Ampere equation which previously can only be obtained using the methods of pluripotential theory.

• Feb. 2, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Freid Tong, Columbia University

Title: Moser-Trudinger inequality for complex Monge-Ampere and applications. II

Abstract: We will discuss some recent works of Wang-Wang-Zhou on a proof of Moser-Trudinger inequalities for complex Monge ampere equation. We will then discuss its applications to establish apriori estimates for the complex Monge-Ampere equation which previously can only be obtained using the methods of pluripotential theory.

• Feb. 9, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Jean Van Schaftingen, UCLouvain, Belgium

Title: Marcinkiewicz-Gagliardo weak-type formulas for Sobolev norms

Abstract: First-order Sobolev spaces have their norm defined in terms of weak derivative whereas fractional Sobolev spaces are endowed with the Gagliardo semi-norm controlling some integral of a differential quotient; this results in a qualitative jump at the endpoint. These two perspectives were reconciled through the introduction of a suitable corrective factor by Jean Bourgain, Haïm Brezis and Petru Mironescu have reconciled. In a recent work in collaboration with Haïm Brezis (Rutgers, Technion Haifa and Sorbonne) and Po-Lam Yung (Chinese University of Hong Kong and Australian National University), we are proving that Sobolev norms are equivalent to a Marcinkiewicz weak-type quantity associated to the Gagliardo seminorm. I will present how these estimates can be obtained through various tools of real analysis.

• Feb. 16, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Wanxing Liu, Northwestern University

Title: Convergence of cscK metrics on smooth minimal models of general type

Abstract: On a smooth minimal model of general type, it is known that there exists a unique sequence of cscK metrics in the neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed Kahler metric. In this talk, we will consider the local smooth convergence behavior of such a sequence. In particular, we will discuss how to establish uniform zeroth order estimate and uniform upper entropy bound for it.

• Feb. 23, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Jian Song, Rutgers University

Title: Constant scalar curvature Kahler metrics

Abstract: We will review some a priori estimates in the important work of Chen-Cheng on the existence of constant scalar curvature Kahler metrics.

• March 2, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Hui Yu, Columbia University

Title: Improvement of flatness in elliptic PDEs and free boundary problems

Abstract: In this talk, we discuss the classic technique of improvement of flatness and several of its recent applications to free boundary problems.

Speaker: Jian Song, Rutgers University

Title: Constant scalar curvature Kahler metrics. II

Abstract: We will continue to discuss a priori estimates in the work of Chen-Cheng on constant scalar curvature Kahler metrics. We will also apply their techniques for the Kahler-Ricci flow on Fano manifolds.

• March 23, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Bin Guo Rutgers University -- Newark

Title: A new proof of Evans-Krylov estimate for Monge-Ampere equations

Abstract: We'll discuss a recent proof of Wang and Wu on the C^{2,\alpha} estimate for solutions of real MA equations. The idea is to employ the Green function for the linearized operator and a contradiction argument.

• March 30, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Zongyuan Li Rutgers University

Title: Unique continuation for elliptic and parabolic equations. I

Abstract: In these two talks, we introduce the unique continuation for elliptic and parabolic equations. In the first talk, we will discuss the strong and weak unique continuation for elliptic equations. We will mention some open problems, exciting progress made by Logunov, et. al., and a proof of a simple unique continuation result using the Carleman estimate. In the second talk, we will discuss the application of a parabolic unique continuation result in the regularity criteria for 3D Navier-Stokes equations. The second talk will be based on Escauriaza-Seregin-Sverak's 2003 ARMA paper.

• April 6, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Zongyuan Li Rutgers University

Title: Unique continuation for elliptic and parabolic equations. II

Abstract: In these two talks, we introduce the unique continuation for elliptic and parabolic equations. In the first talk, we will discuss the strong and weak unique continuation for elliptic equations. We will mention some open problems, exciting progress made by Logunov, et. al., and a proof of a simple unique continuation result using the Carleman estimate. In the second talk, we will discuss the application of a parabolic unique continuation result in the regularity criteria for 3D Navier-Stokes equations. The second talk will be based on Escauriaza-Seregin-Sverak's 2003 ARMA paper.

• April 13, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Siyuan Lu, McMaster University

Title: Capacity in General Relativity

Abstract: The capacity estimate plays an important role in Bray's proof of Riemannian Penrose inequality. In this talk, we will review some capacity estimates in general relativity due to Bray, Bray-Miao and Mantoulidis-Miao-Tam.

• April 20, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Dennis Kriventsov, Rutgers University

Title: Boundary Harnack Principles

Abstract: Boundary Harnack inequalities are common tools in free boundary problems, potential theory, and harmonic analysis on domains. We will look at the proof of the boundary Harnack inequality on Lipschitz domains, discuss what other kinds of domains they are available for, and then consider some recent results on equations with right-hand side (based on joint work with Mark Allen and Henrik Shahgholian).

• May 4, 2021, Tuesday, 12:40-2:40pm. (Eastern Time)

Speaker: Jacob Sturm Rutgers University

Title: Singular Complex Monge Ampere equations

Abstract: The Complex Monge-Ampere equation has the form MA(phi) = mu, where mu is a given Borel measure on a domain in C^n (resp. complex manifold) X, phi is an unknown plurisubharmonic (resp. quasi-plurisubharmonic) function on X, and MA is the Monge-Ampere operator. The existence, uniqueness and regularity questions have been widely studied in the case where mu is absolutely continuous, but much less is known is known if mu is singular (e.g. if mu charges a point). I will discuss some partial results in the singular case and will also describe some examples.

# Learning Seminar on PDE and Applications ---- Fall 2020 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

The talks will be held virtually:

Zoom Link   Meeting ID: 948 4160 4914   Passcode: 654986

• Dec. 8, 2020, Tuesday, 1:40-3:40pm.

Speaker: Jian Song, Rutgers University

Title: Volume comparison and its applications for the Ricci flow. I

Abstract: We will review the recent work of Bamler on volume estimates for the Ricci flow. We also show such estimates can be applied to establish and improve the relative volume comparison of Tian-Zhang for the Ricci flow. Finally, we prove a uniform diameter bound for long time solutions of the Kahler-Ricci flow, which is a natural extension of Perelman's diameter estimate for the Fano Ricci flow.

• Dec. 15, 2020, Tuesday, 1:40-3:40pm.

Speaker: Jian Song, Rutgers University

Title: Volume comparison and its applications for the Ricci flow. II

Abstract: We will review the recent work of Bamler on volume estimates for the Ricci flow. We also show such estimates can be applied to establish and improve the relative volume comparison of Tian-Zhang for the Ricci flow. Finally, we prove a uniform diameter bound for long time solutions of the Kahler-Ricci flow, which is a natural extension of Perelman's diameter estimate for the Fano Ricci flow.

• Nov. 10, 2020, Tuesday, 1:40-3:40pm.

Speaker: Weiyong He, University of Oregon

Title: The Gursky-Streets equations

Abstract: Gursky and Streets introduced a family of fully nonlinear elliptic equations, as geodesic equations of the space of metrics in a fixed conformal class. The set-up of geometric structure was motivated by the Kahler geometry, and can be used to prove the uniqueness of sigma_2 Yamabe problem in dimension four. We discuss the Gursky-Streets equations, the regularity and the applications.

• Nov. 20, 2020, Friday, 2:00-4:00pm.

Speaker: Chao Li, Princeton University

Title: Generalized soap bubbles and the topology of manifolds with positive scalar curvature

Abstract: It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will first review some history of this question, and present some recent progress, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov), as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. Applications include a Schoen-Yau Liouville theorem for all locally conformally flat manifolds. The proofs of these results are based on analyzing generalized soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. This talk is based on joint work with O. Chodosh.

• Nov. 24, 2020, Tuesday, 1:40-3:40pm.

Speaker: Jiahong Wu, Oklahoma State University

Title: Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium

Abstract: This talk presents the global existence and stability result on solutions near a background magnetic field to the 3D incompressible magnetohydrodynamic (MHD) system with horizontal dissipation and vertical magnetic diffusion. Due to the lack of the vertical dissipation and the horizontal magnetic diffusion, the stability problem concerned here is difficult and classical approaches no longer apply. This work exploits the extra smoothing and stabilizing effect in the wave structure resulting from the coupling and interaction of the velocity and the magnetic fields. By constructing a suitable energy functional to incorporate this stabilizing effect and employing the bootstrapping argument, we are able to show the desired existence and stability result.

• Dec. 1, 2020, Tuesday, 1:40-3:40pm.

Speaker: Yannick Sire, Johns Hopkins University

Title: Rigidity in stationary Euler equation with free boundary

Abstract: We investigate a free boundary version of the results of Hamel and Nadirashvili on Liouville theorems for stationary Euler equations in 2 dimensions. I will first introduce stationary Euler with free boundary. Then, I will explain how to reduce the equations to an overdetermined semi-linear problem, for which a Serrin’s type result can be shown.

# Learning Seminar on PDE and Applications ---- Spring and Summer 2020 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

• August 25, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Zongyuan Li, Rutgers University

Title: Elliptic boundary value problems on irregular domains

Abstract: In this talk, we will discuss the elliptic boundary value problems on irregular domains, with non-smooth boundary data. We will first setup the problems with $L_p$ boundary data, where the boundary conditions are understood in the sense of the non-tangential limit''. Some interesting good and bad'' domains will be given in the process. For the second part, let us focus on problems on Lipschitz domains. We will introduce an interpolation method which is by proving the boundary reverse H\"older inequality via the theory of weak solutions. As a concrete example, we will present a recent joint work with H. Dong on the mixed Dirichlet-conormal problems.

• August 18, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jian Song, Rutgers University

Title: Complex hessian quotient equations. II.

Abstract: The J-flow is proposed by Donaldson to study symplectomorphisms of a Kaher manifold. The J-equation, critical equation of the J-flow, corresponds to a standard complex hessian quotient equation. The global J-equation does not always admit a smooth solution due to topological obstructions. We review both the analytic and topological conditions for solving the J-equation. These equivalent conditions are natural analogues of the Nakai criterion in algebraic geometry.

• August 11, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jian Song, Rutgers University

Title: Complex hessian quotient equations

Abstract: The J-flow is proposed by Donaldson to study symplectomorphisms of a Kaher manifold. The J-equation, critical equation of the J-flow, corresponds to a standard complex hessian quotient equation. The global J-equation does not always admit a smooth solution due to topological obstructions. We review both the analytic and topological conditions for solving the J-equation. These equivalent conditions are natural analogues of the Nakai criterion in algebraic geometry.

• July 28, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jeaheang Bang, Rutgers University

Title: A Liouville Theorem for the Euler Equations in the Plane. II

Abstract: In these two expository talks, I will talk about the paper by Hamel and Nadirashvili in 2019. They proved that any bounded solution to the stationary Euler Equations in the plane is necessarily a shear flow provided that the solution does not have stagnation points (not even at infinity). The proof is twofold. First, they studied the geometrical properties of the streamlines and of the gradient flows—first talk. Second, they derived some logarithmic estimates on the argument of the flow in large balls—second talk.

• July 21, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jeaheang Bang, Rutgers University

Title: A Liouville Theorem for the Euler Equations in the Plane. I

Abstract: In these two expository talks, I will talk about the paper by Hamel and Nadirashvili in 2019. They proved that any bounded solution to the stationary Euler Equations in the plane is necessarily a shear flow provided that the solution does not have stagnation points (not even at infinity). The proof is twofold. First, they studied the geometrical properties of the streamlines and of the gradient flows—first talk. Second, they derived some logarithmic estimates on the argument of the flow in large balls—second talk.

• July 14, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Weisong Dong, Tianjing University

Title: C-subsolution and its applications in a priori estimates for solutions of fully nonlinear equations

Abstract: In this expository talk, I will introduce the definition of C-subsolution given by G. Szekelyhidi in 2018 JDG. This is more powerful in a priori estimates than the usual subsolution. I will show the role of C-subsolution in the C^0 and C^2 estimates for solutions.

• July 7, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jingang Xiong, Beijing Normal University

Title: Isolated singularities of solutions to the Yamabe equation. II.

Abstract: I will talk about the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not locally conformally flat. I will show that when the dimension is less than or equal to 6, any solution is asymptotic to a radially symmetric Fowler solution. This is based on two papers: F. Marques 2008 for dimensions less than 6 and a recent preprint of L. Zhang and myself for dimension 6. When the metric is locally conformally flat, the above mentioned asymptotic behavior holds in all dimensions, which was proved in the classical paper of Caffarelli-Gidas and Spruck 1989. See also an important refinement due to Korevaar, Mazzeo, Pacard and Schoen 1999.

• June 30, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jingang Xiong, Beijing Normal University

Title: Isolated singularities of solutions to the Yamabe equation. I.

Abstract: I will talk about the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not locally conformally flat. I will show that when the dimension is less than or equal to 6, any solution is asymptotic to a radially symmetric Fowler solution. This is based on two papers: F. Marques 2008 for dimensions less than 6 and a recent preprint of L. Zhang and myself for dimension 6. When the metric is locally conformally flat, the above mentioned asymptotic behavior holds in all dimensions, which was proved in the classical paper of Caffarelli-Gidas and Spruck 1989. See also an important refinement due to Korevaar, Mazzeo, Pacard and Schoen 1999. 

• June 16, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jeaheang Bang, Rutgers University

Title: Construction of a Smooth, Compactly-Supported Solution to the Three-Dimensional Stationary Euler Equations

Abstract: It is an expository talk about the work of Constantin, La, and Vicol in 2019. In this paper, they constructed a smooth, compactly-supported solution to the stationary Euler equations in the three-dimensional Euclidean space. To do so, they seek for axisymmetric solutions and use some specific ansatz, which leads to Hicks equation (equivalent to Grad-Shafranov equation). A solution to Hicks equation have meaning as a stream function of a solution to the Euler equations. They try to construct a solution to Hicks equation, satisfying some additional condition called localization. To construct a solution to Hicks equation with the localization condition, they use some transformation called hodograph. Majority of it boils down to existence of a solution to some ODEs. This leads to a smooth solution to the Euler equation (with some nice properties deduced from the localization condition) but it does not necessarily have compact support. Thanks to the localization condition, finally they can 'localize' the solution that they have found. In other words, they can immediately generate another solution with compact support. (This paper is inspired by the work of A. V. Gavrilov in 2019 about the Euler equations.)

• May 26, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jian Song, Rutgers University

Title: Construction of cut-off functions

Abstract: We will talk about the construction of cut-off functions constructed by Cheeger-Colding on Riemannian manifolds with Ricci curvature bounded below.

• May 19, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jeaheang Bang, Rutgers University

Title: Shear Flows of Euler Equations in a Two-dimensional Strip.

Abstract: It is an expository talk about the work of François Hamel and Nikolai Nadirashvili in 2017. They proved that, in a two-dimensional strip, a steady flow of Euler equations with tangential boundary conditions and with no stationary point is a shear flow. The proof is based on the study of geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semi-linear elliptic equations.

• May 12, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Dennis Kriventsov, Rutgers University

Title: The Alt-Caffarelli functional with volume constraint.

Abstract: I will discuss an old, classic example from free boundary theory: minimizing the Alt-Caffarelli functional with volume constraint. The paper is based on an obvious idea: finding it difficult to deal with a constraint, we replace it with a penalization. The catch is that the penalized functional is not differentiable with respect to domain variations, while the estimates available for it all seem to depend on the penalization parameter. This example explains how sometimes, on good days, one may still obtain Euler-Lagrange equations for non-differentiable functionals, and sometimes if you penalize correctly, the penalized and constrained problems turn out to have identical minimizers. While the example is free boundary based, I think the insights are applicable to many calculus of variations problems.

• May 5, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jian Song, Rutgers University

Title: Hessian equations in complex geometry. II.

Abstract: We will review results on both smooth and weak solutions to complex hessian equations. There exists obstruction to the existence of solutions to global complex hessian equations. We will establish relations between the solvability of such equations and positivity in intersection theory of topological cycles and geometric structures of holomorphic fibration.

• April 28, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jian Song, Rutgers University

Title: Hessian equations in complex geometry. I.

Abstract: We will review results on both smooth and weak solutions to complex hessian equations. There exists obstruction to the existence of solutions to global complex hessian equations. We will establish relations between the solvability of such equations and positivity in intersection theory of topological cycles and geometric structures of holomorphic fibration.

• April 21, 2020, Tuesday, 1:40-3:10pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jingang Xiong, Beijing Normal University

Title: Existence of solutions of the steady Navier-Stokes equations in unbounded domains with pipe ends

Abstract: I will present a paper of Amick 1977, where he proved an existence result of the Navier-Stokes flow in unbounded domains with pipe ends. Those flows converge to Poiseuille flows at the ends. Amick assumed that the viscosity coefficient is greater than some positive constant $\nu_0$. Without this assumption, the existence is still OPEN.

• April 14, 2020, Tuesday, 1:40-3:10pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jingang Xiong, Beijing Normal University

Title: Energy decay estimates for steady Navier-Stokes equations in pipes

Abstract: I will present a paper by Horgan-Wheeler in 1978, where they established an exponential decay estimates of the Dirichlet energy when the viscosity coefficient is large. Their method is to derive an integro-differential inequality of the energy.

• April 7, 2020, Tuesday, 1:40-3:40pm.

This talk will be held virtually on WebEx, copy and paste the following web address:

https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6

Speaker: Jeaheang Bang, Rutgers University

Title: A Liouville Theorem for Weak Beltrami Flows

Abstract: Abstract: This is an expository talk about the work of Dongho Chae and Jörg Wolf in 2016. They established a Liouville theorem for weak Beltrami flows in three dimensional Euclidean space. The sufficient condition that they impose in order to ensure the triviality of such flows is that some sort of tangential part of the flow decays at infinity in some weak sense (with no assumption on the normal part). This theorem is a generalization of all the known results of this problem. The main idea is to test the Euler equations against some simple vector fields over a ball and over an annulus.

• March 20, 2020, Friday, 2-4pm, Hill 705.

Speaker: Jingang Xiong, Beijing Normal University

Title: Steady solutions of the Navier-Stokes equations in unbounded channels and pipes. V

Abstract: I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows was proved under certain assumptions.

• March 19, 2020, Thursday, 2-4pm, Hill 705.

Speaker: Jingang Xiong, Beijing Normal University

Title: Steady solutions of the Navier-Stokes equations in unbounded channels and pipes. IV

Abstract: I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows was proved under certain assumptions.

• March 18, 2020, Wednesday, 2-4pm, Hill 705.

Speaker: Jingang Xiong, Beijing Normal University

Title: Steady solutions of the Navier-Stokes equations in unbounded channels and pipes.III

Abstract: I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows was proved under certain assumptions.

• March 17, 2020, Tuesday, 2-4pm, Hill 705.

Speaker: Jingang Xiong, Beijing Normal University

Title: Steady solutions of the Navier-Stokes equations in unbounded channels and pipes. II

Abstract: I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows was proved under certain assumptions.

• March 16, 2020, Monday, 2-4pm, Hill 705.

Speaker: Jingang Xiong, Beijing Normal University

Title: Steady solutions of the Navier-Stokes equations in unbounded channels and pipes. I

Abstract: I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows was proved under certain assumptions.

• Feb. 20, 2020, Thursday, 11:45a-1:45pm, Hill 525,

Speaker: Ting Zhang, Zhejiang University

Title: Introduction to the Incompressible Navier-Stokes equations. II.

Abstract:

#### 5 Prodi-Serrin conditions and Partial regularity results for weak solutions

• Feb. 17, 2020, Monday, 3:10-4:50pm, Hill 425,

Speaker: Ting Zhang, Zhejiang University

Title: Introduction to the Incompressible Navier-Stokes equations. I.

Abstract:

#### 5 Prodi-Serrin conditions and Partial regularity results for weak solutions

• Feb. 5, 2020, Wed. 1:40pm, Hill 705,

Speaker: Jeaheang Bang, Rutgers University

Title: A Liouville-Type Theorem for Beltrami Flows

Abstract: This is an expository talk on the work of Dongho Chae and Peter Constantin in 2015. They presented a simple, short, and elementary proof that any three-dimensional Beltrami flow with a finite energy has to be trivial. The main idea of the proof is to use the continuity of the Fourier transform of functions belonging to L^1.

# Student Seminar on PDE and Applications ---- Spring 2018 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

• April 11, 2018, Wed. 10:30am, Hill 705,

Speaker: Hongjie Dong, Brown University

Title: Some partial regularity and regularity criteria results for the Navier-Stokes equations. I.

Abstract: I will first review some previous results on the conditional regularity of solutions to the incompressible Navier-Stokes equations (both stationary and time-dependent) in the critical Lebesgue spaces, and then discuss a recent work which mainly addressed the boundary regularity issue.

• April 13, 2018, 1:40-2:40pm, Hill 525

Speaker: Hongjie Dong, Brown University

Title: Some partial regularity and regularity criteria results for the Navier-Stokes equations. II.

Abstract: I will first review some previous results on the conditional regularity of solutions to the incompressible Navier-Stokes equations (both stationary and time-dependent) in the critical Lebesgue spaces, and then discuss a recent work which mainly addressed the boundary regularity issue.

• April 16, 2018, Monday, 3:20-4:40pm, Hill 705

Speaker: Jiaping Wang, University of Minnesota

Title: Structure at infinity for four dimensional shrinking Ricci solitons

Abstract: Ricci solitons, as self-similar solutions to the Ricci flows, play a prominent role in the study of singularity formations of Ricci flows. The primary concern of the talk is four dimensional shrinking Ricci solitons. We will discuss some joint work with Ovidiu Munteanu concerning their geometric structure at infinity.

# Student Seminar on PDE and Applications ---- Fall 2013 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

• Friday, Nov. 22, 1:40pm, Hill 525,

Speaker: Hao Jia, University of Chicago

Title: Reading report of Kato's pioneering paper on $L^p$ solutions to Navier Stokes equations

Abstract: In a paper published in 1984, Kato made an important observation that one can obtain global existence of solution for Navier Stokes if the initial data is small in $L^3(R^3)$, which is scale invariant for Navier Stokes. Various applications to long time decay of solutions were also given. I will explain these results and if time permits, review some recent progress on the Cauchy problem with initial data in scale invariant spaces. (Refence: Tosio Kato Strong L^p-solutions of the Navier-Stokes equation in R^m , with applications to weak solutions. Math. Z. 187 (1984), no. 4, 471-480.'')

• Friday, Nov. 15, 1:40pm, Hill 525,

Speaker: Bo Yang, Rutgers University

Title: Kaehler-Ricci solitons

Abstract: We introduce Kaehler-Ricci solitons and discuss applications in the uniformization problems on Kaehler manifolds with nonnegative curvature. This is an expository talk on the work of Chau-Tam. The reference is MR2488949.

• Friday, Oct. 25, 1:40pm, Hill 525,

Speaker: Natasa Sesum, Rutgers University

Title: Mean curvature flow of entire graphs. II

Abstract: We discuss the joint work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss the longtime existence and convergence of the class of entire g raphs that grow at most linearly. We also tackle the question on how to handle the previous problem once we drop the condition on a linear growth. We discuss the join t work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss the longtime existence and convergence of the class of entire graphs that grow at most linearly. We also tackle the question on how to handle the previous problem on ce we drop the condition on a linear growth.

• Friday, Oct. 18, 1:40pm, Hill 525,

Speaker: Natasa Sesum, Rutgers University

Title: Mean curvature flow of entire graphs

Abstract: We discuss the joint work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss the longtime existence and convergence of the class of entire graphs that grow at most linearly. We also tackle the question on how to handle the previous problem once we drop the condition on a linear growth. We discuss the joint work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss the longtime existence and convergence of the class of entire graphs that grow at most linearly. We also tackle the question on how to handle the previous problem once we drop the condition on a linear growth.

# Student Seminar on PDE and Applications ---- Fall 2012 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

• Friday, Nov. 30, 1:40pm, Hill 525,

Speaker: Natasa Sesum, Rutgers University

Title: Generic singularities of the mean curvature flow. II

Abstract: It has long been conjectured that starting at a generic smooth closed embedded surface in R^3 the mean curvature flow remains smooth until it arrives at a sing ularity in a neighborhood of which the flow looks like concentric spheres or cyl inders. That is, the only singularities of a generic flow are spherical or cylin drical. The key in showing this conjecture is to show that shrinking spheres, cy linders and planes are the only stable self-shrinkers under the mean curvature f low. That was proved by Colding and Minicozzi and we will discuss parts of their paper.

• Wednesday, Nov. 21, 1:40pm, Hill 525,

Speaker: Natasa Sesum, Rutgers University

Title: Generic singularities of the mean curvature flow

Abstract: It has long been conjectured that starting at a generic smooth closed embedded surface in R^3 the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. That was proved by Colding and Minicozzi and we will discuss parts of their paper.

• Wednesday, Nov. 7, 12pm, Hill 525,

Speaker: Jian Song, Rutgers University

Title: L^2 estimates and its applications in Riemannian and algebraic geometry. III

Abstract: We will discuss the partial C^0 estimates in the recent paper of Donaldson and Sun as a solution to a conjecture of Tian. This gives a compactness result for Kahler-Einstein manifolds with positive scalar curvature.

• Wednesday, October 24, 12pm, Hill 525,

Speaker: Jian Song, Rutgers University

Title: L^2 estimates and its applications in Riemannian and algebraic geometry. II

Abstract: We will discuss the partial C^0 estimates in the recent paper of Donaldson and Sun as a solution to a conjecture of Tian. This gives a compactness result for Kahler-Einstein manifolds with positive scalar curvature.

• Friday, October 19, 1:40pm, Hill 525,

Speaker: Jian Song, Rutgers University

Title: L^2 estimates and its applications in Riemannian and algebraic geometry

Abstract: Abstract: We will discuss the partial C^0 estimates in the recent paper of Donaldson and Sun as a solution to a conjecture of Tian. This gives a compactness result for Kahler-Einstein manifolds with positive scalar curvature.

• Friday, September 28, 1:40pm, Hill 525,

Speaker: Yanyan Li, Rutgers University

Title: A theorem of Leray on Navier-Stokes equations and an open problem, I

Abstract: This is the first of two expository talks. We will present a proof of a classic theorem of Jean Leray on the nonhomogeneous steady incompressible Navier-Stokes equations in a two dimensional domain. The theorem establishes the existence of a solution when the prescribed boundary velocity has zero flux through each boundary component. It remains up-to-date a challenging open problem whether the existence result holds under the (weaker) compatibility condition that the total velocity flux through the boundary is equal to zero. We will also give a brief survey on partial results on the open problem. Both talks are meant to be accessible to first year graduate students.

• Friday, October 5, 1:40pm, Hill 525,

Speaker: Yanyan Li, Rutgers University

Title: A theorem of Leray on Navier-Stokes equations and an open problem, II

Abstract: This is the first of two expository talks. We will present a proof of a classic theorem of Jean Lera y on the nonhomogeneous steady incompressible Navier-Stokes equations in a two dimensional domain. Th e theorem establishes the existence of a solution when the prescribed boundary velocity has zero flux through each boundary component. It remains up-to-date a challenging open problem whether the existe nce result holds under the (weaker) compatibility condition that the total velocity flux through the boundary is equal to zero. We will also give a brief survey on partial results on the open problem. Both talks are meant to be accessible to first year graduate students.

# Student Seminar on PDE and Applications ---- Fall 2011 (Organizers: Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song)

• Friday, October 28th Longzhi Lin, Rutgers University " On a regularity theorem of Riviere, I" Time: 1:50 PM Location: Hill 525

• Friday, November 4th Longzhi Lin, Rutgers University "On a regularity theorem of Riviere, II" Time: 1:50 PM Location: Hill 525

• Friday, November 11th Longzhi Lin, Rutgers University "On a regularity theorem of Riviere, III" Time: 1:50 PM Location: Hill 525

• Friday, November 18th Jian Song, Rutgers University " Hamilton's conjecture and the differential sphere theorem" Time: 1:50 PM Location: Hill 525

• Friday, December 2nd Jian Song, Rutgers University "Hamilton's conjecture and the differential sphere theorem, II" Time: 1:50 PM Location: Hill 525

• Friday, December 9th Jian Song, Rutgers University "Hamilton's conjecture and the differential sphere theorem, III" Time: 1:50 PM Location: Hill 525

# Student Seminar on PDE and Applications ---- Spring 2011 (Organizers: Yanyan Li, Zheng-Chao Han, Marta Lewicka, Natasa Sesum, Jian Song)

• Wednesday, March 30th Marta Lewicka, Rutgers University "Existence of solutions to the nonhomogeneous steady Navier-Stokes equations, I" Time: 12:00 PM Location: Hill 425

• Wednesday, April 6th Marta Lewicka, Rutgers University (NOTE: SPECIAL TIME!!!) "Existence of solutions to the nonhomogeneous steady Navier-Stokes equations, II" Time: 3:20 PM Location: Hill 423

• Wednesday, April 13th Marta Lewicka, Rutgers University (NOTE: SPECIAL TIME!!!) "Existence of solutions to the nonhomogeneous steady Navier-Stokes equations, III'" Time: 10:20 AM Location: Hill 425

• Wednesday, April 20th Marta Lewicka, Rutgers University (NOTE: SPECIAL TIME!!!) "Existence of solutions to the nonhomogeneous steady Navier-Stokes equations, IV" Time: 10:20 AM Location: Hill 423

• Wednesday, April 27th Natasa Sesum and Tianling Jin, Rutgers University (NOTE: DIFFERENT ROOM AND TIME!!!) "On the extinction profile of solutions to a parabolic equation " Time: 10:20 AM Location: Hill 705

• Thursday, May 5th Tianling Jin, Rutgers University (NOTE: CHANGE IN TIME AND PLACE!!!) "On the extinction profile of solutions to a parabolic equation. II" Time: 10:20 AM Location: SERC 216

• Wednesday, May 11th Tianling Jin, Rutgers University "On the extinction profile of solutions to a parabolic equation.III" Time: 10:20 AM Location: Hill 705

• Monday, May 23rd Ali Maalaoui, Rutgers University "Optimal rigidity estimates for nearly umbilical surfaces. I" Time: 10:20 AM Location: Hill 705

• Monday, May 23rd Ali Maalaoui, Rutgers University "Optimal rigidity estimates for nearly umbilical surfaces. II" Time: 1:40 PM Location: Hill 705

# Student PDE Seminar ---- Fall 2009 (Organizer: Yanyan Li)

• Dec. 16, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. VII

• Dec. 9, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. VI

• Dec. 2, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. V

• Nov. 18, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. IV

• Nov. 11, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. III

• October 21, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. II

• October 7, 2009, 12pm-1:20pm, Hill 525, Ali Maalaoui, Rutgers, The Yamabe problem. I

# Student Seminar ---- Spring 2007 (Organizer: Yanyan Li)

• Feb. 9, 2007, 1:40-3pm, Hill 525, Zheng-Chao Han, Rutgers, A brief survey of recent techniques for proving regularity in some geometric PDEs with critical nonlinearity. I

• Feb. 23, 2007, 1:40-3pm, Hill 525, Zheng-Chao Han, Rutgers, A brief survey of recent techniques for proving regularity in some geometric PDEs with critical nonlinearity. II

• March 2 , 2007, 1:40-3pm, Hill 525, Yanyan Li, Rutgers, An open problem concerning some linear elliptic equations from composite material.

• March 30, 2007, 1:40-3pm, Hill 525, Kurt Bryan, Rose-Hulman Institute of Technology, Inverse problems arising from nondestructive evaluation. I

• April 13, 2007, 1:40-3pm, Hill 525, Kurt Bryan, Rose-Hulman Institute of Technology, Inverse problems arising from nondestructive evaluation. II

• April 20, 2007, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, The obstacle problem. I

• April 27, 2007, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, The obstacle problem. II

• May 4, 2007, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, The obstacle problem. III

# Student Seminar ---- Fall 2006 (Organizer: Yanyan Li)

• Yanyan Li is on sabbatical this semester

# Student Seminar ---- Spring 2006 (Organizer: Yanyan Li)

• April 28, 2006, 1:40-3pm, Hill 525, Pablo Angulo Ardoy, Rutgers, Solution of the Ambrose problem in dimension two. V

• April 14, 2006, 1:40-3pm, Hill 525, Pablo Angulo Ardoy, Rutgers, Solution of the Ambrose problem in dimension two. IV

• April 7, 2006, 1:40-3pm, Hill 525, Pablo Angulo Ardoy, Rutgers, Solution of the Ambrose problem in dimension two. III

• March 24, 2006, 1:40-3pm, Hill 525, Pablo Angulo Ardoy, Rutgers, Solution of the Ambrose problem in dimension two. II

• March 10, 2006, 1:40-3pm, Hill 525, Pablo Angulo Ardoy, Rutgers, Solution of the Ambrose problem in dimension two. I

• February 24, 2006, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, A theorem of Morrey on minimizers of quadratic functionals. IV.

• February 17, 2006, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, A theorem of Morrey on minimizers of quadratic functionals. III.

• February 10, 2006, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, A theorem of Morrey on minimizers of quadratic functionals. II.

• February 3, 2006, 1:40-3pm, Hill 525, Luc Nguyen, Rutgers, A theorem of Morrey on minimizers of quadratic functionals. I.

# Student Seminar ---- Fall 2005 (Organizer: Yanyan Li)

• November 18, 2005, 1:40-3pm, Hill 525, Eduardo V. Teixeira, Rutgers, On a Carleman-type inequality.

• November 11, 2005, 1:40-3pm, Hill 525, Yuan Zhang, Rutgers, Some aspects of special Lagrangian equations. IV.

• October 7, 2005, 1:40-3pm, Hill 525, Yuan Zhang, Rutgers, Some aspects of special Lagrangian equations. III.

• Sept. 30, 2005, 1:40-3pm, Hill 525, Yuan Zhang, Rutgers, Some aspects of special Lagrangian equations. II.

• Sept. 23, 2005, 1:40-3pm, Hill 525, Yuan Zhang, Rutgers, Some aspects of special Lagrangian equations. I.

• Sept. 16, 2005, 1:40-3pm, Hill 525, Zhengchao Han, Rutgers, Classification of singular radial solutions to the $\sigma_k$ Yamabe equation on annular domains.

# Student Seminar ---- Spring 2005 (Organizer: Yanyan Li)

• April 29, 2005, 9:50am -- 11:10am, Hill 423, Pablo Angulo Ardoy, Rutgers, C^{2,\alpha} estimates of solutions to second order fully nonlinear elliptic equations. IV.''

• April 27, 11:30am -- 12:50pm, Hill 423, Pablo Angulo Ardoy, Rutgers, C^{2,\alpha} estimates of solutions to second order fully nonlinear elliptic equations. III.''

• April 26, 2005, 9:50am -- 11:10am, , Hill 423, Pablo Angulo Ardoy, Rutgers, C^{2,\alpha} estimates of solutions to second order fully nonlinear elliptic equations. II.''

• April 22, 2005, 9:50am -- 11:10am, , Hill 423, Yanyan Li, Rutgers, C^{2,\alpha} estimates of solutions to second order fully nonlinear elliptic equations. I.''

• April 20, 2005, 11:30am -- 12:50pm, Hill 423, Biao Yin, Rutgers, Introduction to a problem of Ambrose in Riemannian geometry. V''

• April 19, 2005, 9:50am -- 11:10am, , Hill 423, Biao Yin, Rutgers, Introduction to a problem of Ambrose in Riemannian geometry. IV''

• April 13, 2005, 11:30am -- 12:50pm, Hill 423, Biao Yin, Rutgers, Introduction to a problem of Ambrose in Riemannian geometry. III''

• April 6, 2005, 11:30am -- 12:50pm, Hill 423, Biao Yin, Rutgers, Introduction to a problem of Ambrose in Riemannian geometry. II''

• March 29, 2005, 9:50am -- 11:10am, Hill 423, Eduardo Teixeira, University of Texas, Austin, An introduction to the obstacle problem''

• March 23, 2005, 11:30am -- 12:50pm, Hill 423, Biao Yin, Rutgers, Introduction to a problem of Ambrose in Riemannian geometry. I''

• March 9, 2005, 11:30am -- 12:50pm, Xiaojun Huang, Rutgers, Hill 423, Aronszajn's Carleman estimates for elliptic operators with variable coefficients. III''

• Feb. 23, 2005, 11:30am -- 12:50pm, Xiaojun Huang, Rutgers, Hill 423, Aronszajn's Carleman estimates for elliptic operators with variable coefficients. II''

• Feb. 16, 2005, 11:30am -- 12:50pm, Hill 423, Xiaojun Huang, Rutgers, `Aronszajn's Carleman estimates for elliptic operators with variable coefficients. I''