PDE SEMINARS

• Fall 2020
• Spring 2021
• Fall 2022
• Spring 2023
• Fall 2023
• Spring 2024
• Fall 2024

Honors and Awards

• Alfred P. Sloan Dissertation Fellowship, 1995-1996.
• American Mathematical Society Centennial Fellowship, 1999-2000.
• IMA New Directions Research Professorship, 2009-2010.
• Simons Fellowship in Mathematics, 2012-2013.

Grant Supports

• NSF DMS 2101224, July 1, 2021-June 30, 2024.
• NSF DMS 1764417, July 1, 2018-June 30, 2022.
• NSF DMS 1265574, July 1, 2013-June 30, 2017.
• NSF DMS 1001115, August 1, 2010-July, 31, 2013.
• NSF DMS 0601162, July 1, 2006-June 30, 2010.
• NSF-NSFC CMR 0819742, July, 1 2008-June 30, 2011.
• NSF DMS 0400718, July 1, 2004-June 30 2007.
• NSF DMS 9972043, August 1, 1999-July 31, 2002 (Co-PI, R. Jensen and N. Barron=PI).
• NSF DMS 9970549, August 1, 1999-July 31, 2002.
• NSF DMS 9706855, August 1, 1997-July 31, 1999.

List of Publications

• The list from AMS MathSciNet (License to MathSciNet is required)

• Click here to access another list of publications (partially supported by NSF grants)

Professional Services

• Member of Editoral Board of Journal of Elliptic and Parabolic Equations
• Member of Editoral Board of The Open Mathematics Journal, an online mathematics journal from Bentham Science Publishers
• Member of Editoral Board of Journal of Advanced Research in Differential Equations from Institute of Advanced Scientific Research
• Member of Editoral Board of ISRN Mathematical Analysis

A Family Picture

• Amy (Visiting Centre College, Danville, KY)

Some Previous Conferences

• 63th Midwest PDE seminar, November 7-8, 2009, Purdue University.
• Iowa PDE workshop, April 30-May 2, 2010, University of Iowa.
• International Conference on Geometric Nonlinear PDEs, June 14-18, 2010, Beijing International Center for Mathematics Research, Beijing, China.
• Mini-workshop on Geometric Analysis, June 23-24, 2010, University of Warwick, England.
• Workshop on Geometric Analysis, July 5-9, 2010, East China Normal University, Shanghai, P.R. China.
• The Second Oklahoma PDE Workshop, October 9-10, 2010, University of Oklahoma.
• International Conference on Partial Differential Equations in Fluid Mechanics and Material Sciences, Jan. 8-10, 2011, South China Normal Univ., Guangzhou, P.R. China.
• International Confernece on Recent Developments in L-infinity Variational Problems and Associated Nonlinear Partial Differential Equations
• Second Ohio River Analysis Meeting, April 21-22, 2012

Current Teachings at Purdue University

Spring 2024

• MA 303-Section 255: Differential Equations and Partial Differential Equations for Engineering and The Sciences
• MA 643: Methods Of Partial Differential Equations II

Fall 2023

• MA 642: Methods Of Partial Differential Equations I

Spring 2023

• MA 362-Section 024: Topics in Vector Calculus

Fall 2022

• MA 303-Section 318: Differential Equations and Partial Differential Equations for Engineering and The Sciences
• MA 523: Introduction to Partial Differential Equations

Spring 2022

• MA 341-Section 067: Foundations of Analysis
• MA 520: Boundary Value Problems of Differential Equations

Fall 2020

• MA 341-Section 100: Foundations of Analysis
• MA 362-Section 023: Topics in Vector Calculus

Spring 2020

• MA416-Section 168: Probability
• MA693: Topics in Analysis

Fall 2019

• MA416-Sections 156 and 157, Probability

Fall 2018

• MA266-Sections 052, 071, and 075, Ordinary Differential Equations

Spring 2018

• MA341-Sections 031 and 052, Foundations Of Analysis

Fall 2017

• MA266: Ordinary Differential Equations

Spring 2017

• MA341-052, Foundations Of Analysis
• MA523, Introduction to Partial Differential Equations

Fall 2016

• MA266, Ordinary Differential Equations

Fall 2015

• MA266, Ordinary Differential Equations
• MA523, Introduction to Partial Differential Equations

Fall 2014

• MA266, Ordinary Differential Equations

Spring 2015

• MA266, Ordinary Differential Equations Section 063 and Section 151

Some Courses Taught at University of Kentucky

• Spring 2007-Spring 2014

Analysis/PDE Seminars Organized at University of Kentucky

• Spring 2009
• Fall 2008
• Spring 2008
• Fall 2007
• Spring 2007
• Fall 2006

Recent Publications

• Stable stationary harmonic maps into spheres. Acta Math. Sinica (Engl. Ser.) 22 (2006), no. 2, 319-330 (with FangHua Lin). pdf file.
• On Landau-Lifshitz equation in dimensions at most four. Indiana Univ. Math. J. Vol. 55, no.6 (2006) 1615-1644. dvi file.
• Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers. Comm. P.D.E. V. 31, No. 7 (2006) 1027-1046 (with R. Gariepy, Y. Yu). dvi file .
• The Aronsson equation for absolute minimizers of $L^\infty$ functionals associated with vector fields satisfying H\"ormander's conditions. Trans. AMS. 359 (2007), 91-113. dvi file.
• Heat flow of biharmonic maps in dimension four and its application. Pure and Applied Math. Quarterly, Vol. 3, No. 2 (2007) 595-613 (Special issue: In honor of L. Simon, Part 1 of 2). pdf file.
• Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int. Math. Reas. Notices, Vol. 2007 (with Shijin Ding). pdf file
• Asymptotic behavior of infinity harmonic functions near an isolated singularity. Int. Math. Reas. Notices, Vol. 2008, article ID rnm163, 23 pages (with O. Savin, Y. Yu). pdf file .
• Heat flow of harmonic maps whose gradients belong to $L^n_xL^\infty_t$. Arch. Rational Mech. Anal. 188 (2008), 309-349. pdf file
• Harmonic maps from manifolds of $L^\infty$-Riemannian metrics. Calc. Var. & PDE (2008), Vol 32, 387-405 (with W. Ishizuka). pdf file .
• $C^1$-regularity of the Aronsson equation in $\R^2$, Annales I'Institut H. Poincar\'e- Analyse non lin\'eaire, 25 (2008) 659-678 (with Y. Yu). dvi file .
• Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation. Arch. Ration Mech. Anal. (with R. Jensen, Y. Yu), Vol. 190, no. 2 (2008) 347-370. pdf file .
• Derivation of the Aronsson equation for C-1 Hamiltonians. Trans. AMS 361 (2009) 103-124(with M. Crandall, Y. Yu). pdf file .
• Aronsson's equation on Carnot-Carath\'eodory spaces. Illinois Journal of Mathematics, Vol. 52, No. 3, Fall 2008, 757-772 (with Y. Yu). pdf file .
• Boundary regularity for polyharmonic maps in the critical dimension. Advances in Calculus of Variations 2, 2009, No. 1, 1-16(with T. Lamm). pdf file .
• Regularity of Dirac-harmonic maps. Int. Math. Reas. Notices, 2009, no. 20, 3759-3792 (with D. L. Xu). pdf file .
• Landau-Lifshitz-Maxwell equation in dimension three. Pacific J. Math., Vol. 243, No. 2, 2009, 243-276(with S.J. Ding, X.G. Liu). pdf file .
• Liquid crystall flows in two dimensions. Arch. Ration. Mech. Anal., 197 (2010) 297-336 (with F. H. Lin, J. Y. Lin) pdf file .
• Notes on the regularity of harmonic map systems. Proceedings of American Mathematical Society, 138 (2010) no. 6, 2015-2023 (with T. Huang) pdf file .
• A Remark on Nonlinear Dirac Equations. Proceedings of American Mathematical Society, 138 (2010), 3753-3758. pdf file .
• Weak solution to compressible hydrodynamic flow of liquid crystals in 1D. DCDS-B,15 (2011), no. 2, 357-371(with S. J. Ding, Y. H. Wen). pdf file .
• Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal., 200 (2011), no. 1, 1-19. pdf file .
• On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chinese Annals of Mathematics, Ser. B 31 (2010), no. 6, 921-938(with F. H. Lin). pdf file .
• Well-posedness for the heat flow of polyharmonic maps with rough initial data. Advances in Calculus of Variations, 4 (2011), no. 2, 175-193. (with T. Huang). pdf file .
• On the heat flow of equation of surfaces of constant mean curvature. Manuscripta Math., 134 (2011), no. 1-2, 259-271 (with T. Huang, Z. Tan). pdf file .
• Well-posedness for the heat flow of biharmonic maps with rough initial data. J. Geom. Anal., 22 (2012), no. 1, 223-243. pdf file .
• Compressible hydrodynamic flow of liquid crystals in 1D. DCDS-A, 32 (2012), no. 2, 539-563(with S. J. Ding, J. Y. Lin, Y. H. Wen). pdf file .
• Phase transitions for potentials of high dimensional wells. CPAM, 65 (2012), no. 6, 833-888 (with F. Lin, and X. Pan). pdf file .
• Strong solutions of the compressible nematic liquid crystal flow. JDE, 252 (2012), no. 3, 2222-2265 (with T. Huang, H. Y. Wen). pdf file .
• Boundary regularity of stationary biharmonic maps. CVPDE, 45 (2012), no. 1-2, 165-191 (with Huajun Gong, Tobias Lamm). pdf file .
• Blow up criterion for compressible nematic liquid crystal flows in dimension three. ARMA, 204 (2012), no. 1, 285-311(with T. Huang, H. Y. Wen). pdf file .
• Blow up criterion for nematic liquid crystal flows, CPDE, 37 (2012), no. 5, 875-884 (with T. Huang). pdf file .
• Energy identity for a class of approximate biharmonic maps into sphere in dimension four, DCDS-A, 33 (2013), no. 2, 861-878 (with Shenzhou Zheng).arXiv:1110.6536 pdf file .
• Energy identity for approximate biharmonic maps to Riemannian manifolds and its application, Journal of Functional Analysis, 263 (2012), no. 4, 960-987 (with Shenzhou Zheng). arXiv:1112.6362 pdf file .
• $C^1$-boundary regularity of planar infinity harmonic functions, Math. Res. Lett., 19 (2012), no. 4, 823-835(with Yifeng Yu). pdf file .
• Well-posedness of nematic liquid crystal flow in $L^3_uloc(R^3)$, Arch. Ration. Mech. Anal, 210 (2013), no.1, 177-218 (with J. Hineman) pdf file .
• Harmonic maps on domains with piecewise Lipschitz continuous metrics. Pacific J. Math., 264 (2013), no.1, 125-149 (with Haigang Li). pdf file .
• Regularity and uniqueness of the heat flow of biharmonic maps. CVPDE, 50 (2014), no. 3-4, 491-524 (with J. Hineman, T. Huang). pdf file .
• On the rigidity of nematic liquid crystal flow on $S^2$. J. Funct. Anal., 266 (2014), no. 8, 5360-5376 (with Xiang Xu) pdf file .
• Regularity and existence of global solutions to the Ericksen-Leslie system in $R^2$. Comm. Math. Phys., 331 (2014), no. 2, 805-850 (with J. R. Huang, F. H. Lin). pdf file .
• Recent developments of analysis for hydrodynamic flow of nematic liquid crystals (with Fanghua Lin), Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2029, 20130361, 18 pp. pdf file.
• Global well-posedness of the Landau-Lifshitz-Gilbert equation for initial data in Morrey speace (with J. Y. Lin, B. S. Lai), CVPDE, 54 (2015), no. 1, 665-692, pdf file .
• Global existence of weak solutions of the nematic liquid crystal flow in dimension three (with Fanghua Lin), CPAM, 69 (2016), no. 8, 1532-1571 pdf file.
• MULTIPLE VALUED MAPS INTO A SEPARABLE HILBERT SPACE THAT ALMOST MINIMIZE THEIR p DIRICHLET ENERGY OR ARE SQUEEZE AND SQUASH STATIONARY (with PHILIPPE BOUAFIA, THIERRY DE PAUW), CVPDE, 54 (2015), no. 2, 2167-2196.pdf file .
• On uniqueness of heat flow of harmonic maps (with T. Huang), IUMJ, Vol. 65, No. 5 (2016), 1525-1546. pdf file.
• Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three (with Junyu Lin and Baishun Lai), SIAM Jour. Math. Anal. 47 (2015), no. 4, 2952-2983. pdf file.
• Everywhere differentiability of viscosity solutions to a class of Aronsson's equations (with Juhana Siljander and Yuan Zhou), Annales IHP-ANL 34 (2017), no. 1, 119-138 pdf file.
• Finite time singularity of the nematic liquid crystal flow in dimension three (with Tao Huang, Fanghua Lin, and Chun Liu), ARMA, 221(3) (2016), 1223-1254. pdf file.
• Heat flow of extrinsic biharmonic maps from a four dimensional manifold with boundary (with T. Huang, L. Liu, and Yong Luo), J. Elliptic and Parabolic Equations, Vol 2 (2016), 1-26, pdf file.
• Uniqueness of absolute minimizers for L-infinity functionals involving Hamiltonians H(x,p) (with Qianyun Miao, Yuan Zhou), ARMA, 223 (2017), no. 1, 141-198 pdf file.
• Weak solutions of geometric flows associated to integro-differential harmonic maps (with Armin Scikorra, Yannick Sire), Manuscripta Math., in press, pdf file.
• Forward self-similar solutions to the viscoelastic Navier-Stokes equation with damping (with B. S. Lai, J. Y. Lin), SIAM J. Math. Anal., 49 (2017), no. 1, 501-529. pdf file.
• Remarks on approximate harmonic maps in dimension two, CVPDE, 56 (2017), no. 2, 56:23.pdf file.
• Heat flow of Yang-Mills-Higgs functionals in dimension two (with C. Song), Journal of Functional Analysis, in press pdf file.
• Boundary bubbling analysis of approximate harmonic maps under either weak or strong anchoring conditions in dimension two (with T. Huang), IMRN, to appearpdf file.

Recent Preprints

• Partial regularity of suitable weak solutions to the liquid crystal flow in three dimensions. In preparation (with F. H. Lin).
• Boundary regularity for a new class of minimizing harmonic maps. In preparation (with F. H. Lin).

Monography and Lecture Notes

• The Analysis of Harmonic Maps and Their Heat Flows, 267 pages (with F. H. Lin), by World Scientific Publisher Inc, ISBN-109812779523.
  
• An Introduction of Infinity Harmonic Functions. Preprint, 2008. pdf file .
• Lecture note of the Naiver-Stokes equation. pdf file .
• An open question on regularity of Aronsson's equation. Communications in Mathematical Analysis, Proceeding of the 2006 Midwest Geometry Conference, pp. 80-83 (2008), edited by E. B. Lin, W. Wei. pdf file