Learning Seminar on PDE and Applications ---- Spring 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:

    In this min-course, we will talk about a few basic results concerning the Incompressible Navier-Stokes equations.

    1 Leray weak solutions

    2 Well-posedness in Sobolev spaces

    3 Well-posedness in $L^d$

    4 Well-posedness in Besov spaces

    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:

    In this min-course, we will talk about a few basic results concerning the Incompressible Navier-Stokes equations.

    1 Leray weak solutions

    2 Well-posedness in Sobolev spaces

    3 Well-posedness in $L^d$

    4 Well-posedness in Besov spaces

    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''