# 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''