Hongjie Dong
Hongjie Dong
Professor of Applied Mathematics
Overview
Brown University: 2007.7–
Professor: 2017.6–
Associate Professor: 2012.7–2017.6
Assistant Professor: 2007.7–2012.6
Member of the Lefschetz Center for Dynamical Systems.
University of Pennsylvania: 2010.9–2010.12
Visiting scholar (Sabbatical).
Institute for Advanced Study: 2006.9–2007.6, 2008.9–2008.12, 2021.9-2022.5
Member & Visiting Scholar.
Mathematical Sciences Research Institute: 2007.7–2007.8
Member of ”Summer Microprogram on Nonlinear Partial Differential Equations”.
University of Chicago: 2005.9–2006.8
L.E. Dickson Instructor & Research post-doctor
University of Minnesota: 2001.9–2005.8.
Research Assistant & Teaching Assistant
Brown Affiliations
Research Areas
elliptic and parabolic equations | local and global regularity | Naiver-Stokes equations | quasi-geostrophic equations | rate of convergence | stochastic processes
On the Web
Publications
| Dong, Hongjie, Xu, Longjuan. "Hessian Estimates for Non-divergence form Elliptic Equations Arising from Composite Materials." Potential Analysis, 2020. |
| Dong, Hongjie, Kim, Doyoon. "L-estimates for time fractional parabolic equations in divergence form with measurable coefficients." Journal of Functional Analysis, vol. 278, no. 3, 2020, pp. 108338. |
| Dong, Hongjie, Yang, Tong, Zhong, Mingying. "Exterior Problem of the Linear Vlasov--Poisson--Boltzmann System." SIAM Journal on Mathematical Analysis, vol. 51, no. 3, 2019, pp. 1792-1823. |
| Dong, Hongjie, Xu, Longjuan. "Gradient Estimates for Divergence Form Elliptic Systems Arising from Composite Material." SIAM Journal on Mathematical Analysis, vol. 51, no. 3, 2019, pp. 2444-2478. |
| Choi, Jongkeun, Dong, Hongjie. "Gradient estimates for Stokes systems in domains." Dynamics of Partial Differential Equations, vol. 16, no. 1, 2019, pp. 1-24. |
| Choi, Jongkeun, Dong, Hongjie. "Gradient estimates for Stokes systems with Dini mean oscillation coefficients." Journal of Differential Equations, vol. 266, no. 8, 2019, pp. 4451-4509. |
| Dong, Hongjie, Zhang, Hong. "On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations." Calculus of Variations and Partial Differential Equations, vol. 58, no. 2, 2019. |
| Choi, Jongkeun, None, None, Dong, Hongjie, Kim, Doyoon, None, None. "Conormal derivative problems for stationary Stokes system in Sobolev spaces." Discrete & Continuous Dynamical Systems - A, vol. 38, no. 5, 2018, pp. 2349-2374. |
| Dong, Hongjie, Jin, Tianling, Zhang, Hong. "Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts." Analysis & PDE, vol. 11, no. 6, 2018, pp. 1487-1534. |
| Dong, Hongjie, Zhang, Hong. "Dini estimates for nonlocal fully nonlinear elliptic equations." Annales de l'Institut Henri Poincaré C, Analyse non linéaire, vol. 35, no. 4, 2018, pp. 971-992. |
| Dong, Hongjie, Kim, Seick. "Fundamental solutions for second-order parabolic systems with drift terms." Proceedings of the American Mathematical Society, vol. 146, no. 7, 2018, pp. 3019-3029. |
| Choi, Jongkeun, Dong, Hongjie, Kim, Doyoon. "Green Functions of Conormal Derivative Problems for Stationary Stokes System." Journal of Mathematical Fluid Mechanics, vol. 20, no. 4, 2018, pp. 1745-1769. |
| Dong, Hongjie, Gallarati, Chiara. "Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions." Journal of Functional Analysis, vol. 274, no. 7, 2018, pp. 1993-2038. |
| Dong, Hongjie, Kim, Doyoon. "On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights." Transactions of the American Mathematical Society, vol. 370, no. 7, 2018, pp. 5081-5130. |
| Dong, Hongjie, Escauriaza, Luis, Kim, Seick. "On $$C^1$$ C 1 , $$C^2$$ C 2 , and weak type-(1, 1) estimates for linear elliptic operators: part II." Mathematische Annalen, vol. 370, no. 1-2, 2017, pp. 447-489. |
| Dong, Hongjie, Kim, Seick. "Partial Schauder Estimates for Second-Order Elliptic and Parabolic Equations: A Revisit." International Mathematics Research Notices, vol. 2019, no. 7, 2017, pp. 2085-2136. |
| Kim, Doyoon, Dong, Hongjie, Zhang, Hong. "Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces." Discrete and Continuous Dynamical Systems, vol. 36, no. 9, 2016, pp. 4895-4914. |
| Cho, Sungwon, Dong, Hongjie, Kim, Doyoon. "Boundary Value Problems for Parabolic Operators in a Time-Varying Domain." Communications in Partial Differential Equations, vol. 40, no. 7, 2015, pp. 1282-1313. |
| Dong, Hongjie, Zhang, Hong. "Conormal problem of higher-order parabolic systems with time irregular coefficients." Trans. Amer. Math. Soc., vol. 368, no. 10, 2015, pp. 7413-7460. |
| Dong, Hongjie, Kim, Doyoon. "Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces." Advances in Mathematics, vol. 274, 2015, pp. 681-735. |
| Ammari, H., Dong, H., Kang, H., Kim, S. "On An Elliptic Equation Arising from Photo-Acoustic Imaging in Inhomogeneous Media." International Mathematics Research Notices, 2015. |
| Dong, Hongjie, Xiong, Jingang. "Boundary Gradient Estimates for Parabolic and Elliptic Systems from Linear Laminates." International Mathematics Research Notices, vol. 2015, no. 17, 2014, pp. 7734-7756. |
| Dong, Hongjie, Gu, Xumin. "Boundary partial regularity for the high dimensional Navier–Stokes equations." Journal of Functional Analysis, vol. 267, no. 8, 2014, pp. 2606-2637. |
| Dong, Hongjie, Li, Dong. "Global $${{dot{H}^1 cap dot{H}^{-1}}}$$ H ˙ 1 ∩ H ˙ - 1 Solutions to a Logarithmically Regularized 2D Euler Equation." J. math. fluid mech., vol. 17, no. 1, 2014, pp. 1-7. |
| Dong, Hongjie, Kim, Seick. "Green's functions for parabolic systems of second order in time-varying domains." Communications on Pure and Applied Analysis, vol. 13, no. 4, 2014, pp. 1407-1433. |
| Li, Dong, Dong, Hongjie. "On a generalized maximum principle for a transport-diffusion model with $log$-modulated fractional dissipation." Discrete and Continuous Dynamical Systems, vol. 34, no. 9, 2014, pp. 3437-3454. |
| Dong, Hongjie. "On a multi-dimensional transport equation with nonlocal velocity." Advances in Mathematics, vol. 264, 2014, pp. 747-761. |
| Dong, Hongjie, Kim, Doyoon. "On the impossibility of Wp2 estimates for elliptic equations with piecewise constant coefficients." Journal of Functional Analysis, vol. 267, no. 10, 2014, pp. 3963-3974. |
| Dong, Hongjie, Kim, Doyoon. "Parabolic Equations in Simple Convex Polytopes with Time Irregular Coefficients." SIAM Journal on Mathematical Analysis, vol. 46, no. 3, 2014, pp. 1789-1819. |
| Dong, Hongjie, Gu, Xumin. "Partial regularity of solutions to the four-dimensional Navier-Stokes equations." Dynamics of Partial Differential Equations, vol. 11, no. 1, 2014, pp. 53-69. |
| Dong, Hongjie, Zhang, Hong. "Schauder estimates for higher-order parabolic systems with time irregular coefficients." Calculus of Variations and Partial Differential Equations, vol. 54, no. 1, 2014, pp. 47-74. |
| Dong, Hongjie, Li, Dong. "On a one-dimensional $alpha $-patch model with nonlocal drift and fractional dissipation." Trans. Amer. Math. Soc., vol. 366, no. 4, 2013, pp. 2041-2061. |
| Dong, Hongjie. "On elliptic equations in a half space or in convex wedges with irregular coefficients." Advances in Mathematics, vol. 238, 2013, pp. 24-49. |
| Dong, Hongjie, Krylov, N. V. "On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions." Communications in Partial Differential Equations, vol. 38, no. 6, 2013, pp. 1038-1068. |
| Dong, Hongjie. "Gradient Estimates for Parabolic and Elliptic Systems from Linear Laminates." Archive for Rational Mechanics and Analysis, vol. 205, no. 1, 2012, pp. 119-149. |
| Dong, Hongjie, Kim, Doyoon. "On -estimates for a class of non-local elliptic equations." Journal of Functional Analysis, vol. 262, no. 3, 2012, pp. 1166-1199. |
| Dong, Hongjie, Krylov, N. V., Li, Xu. "On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains." St. Petersburg Math. J., vol. 24, no. 1, 2012, pp. 39-69. |
| Dong, Hongjie, Strain, Robert M. "On partial regularity of steady-state solutions to the 6D Navier-Stokes equations." Indiana University Mathematics Journal, vol. 61, no. 6, 2012, pp. 2211-2229. |
| Dong, H. "Parabolic equations with variably partially VMO coefficients." St. Petersburg Math. J., vol. 23, no. 3, 2012, pp. 521-539. |
| Dong, Hongjie, Kim, Doyoon. "Schauder estimates for a class of non-local elliptic equations." Discrete and Continuous Dynamical Systems, vol. 33, no. 6, 2012, pp. 2319-2347. |
| Dong, Hongjie. "Solvability of second-order equations with hierarchically partially BMO coefficients." Trans. Amer. Math. Soc., vol. 364, no. 1, 2012, pp. 493-517. |
| Cho, Sungwon, Dong, Hongjie, Kim, Seick. "Global Estimates for Green’s Matrix of Second Order Parabolic Systems with Application to Elliptic Systems in Two Dimensional Domains." Potential Analysis, vol. 36, no. 2, 2011, pp. 339-372. |
| Dong, Hongjie, Kim, Doyoon. "Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth." Communications in Partial Differential Equations, vol. 36, no. 10, 2011, pp. 1750-1777. |
| Dong, Hongjie, Kim, Doyoon. "Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains." Journal of Functional Analysis, vol. 261, no. 11, 2011, pp. 3279-3327. |
| Dong, Hongjie. "On Similarity Solutions to the Multidimensional Aggregation Equation." SIAM Journal on Mathematical Analysis, vol. 43, no. 4, 2011, pp. 1995-2008. |
| Dong, Hongjie, Kim, Doyoon. "Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients." SIAM Journal on Mathematical Analysis, vol. 43, no. 3, 2011, pp. 1075-1098. |
| Dong, Hongjie. "The Aggregation Equation with Power-Law Kernels: Ill-Posedness, Mass Concentration and Similarity Solutions." Commun. Math. Phys., vol. 304, no. 3, 2011, pp. 649-664. |
| Dong, Hongjie, Kim, Doyoon. "L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients." Calculus of Variations and Partial Differential Equations, vol. 40, no. 3-4, 2010, pp. 357-389. |
| Dong, Hongjie, Li, Dong. "On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces." Journal of Differential Equations, vol. 248, no. 11, 2010, pp. 2684-2702. |
| Dong, Hongjie, Kim, Doyoon. "On the L p -Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients." Archive for Rational Mechanics and Analysis, vol. 199, no. 3, 2010, pp. 889-941. |
| Dong, Hongjie, Kim, Seick. "Partial Schauder estimates for second-order elliptic and parabolic equations." Calculus of Variations and Partial Differential Equations, vol. 40, no. 3-4, 2010, pp. 481-500. |
| Dong, Hongjie, Krylov, N. V. "Second-order elliptic and parabolic equations with $B(mathbb{R}^{2}, VMO)$ coefficients." Trans. Amer. Math. Soc., vol. 362, no. 12, 2010, pp. 6477-6477. |
| Dong, Hongjie. "Solvability of parabolic equations in divergence form with partially BMO coefficients." Journal of Functional Analysis, vol. 258, no. 7, 2010, pp. 2145-2172. |
| Dong, Hongjie, Pavlović, Nataša. "A regularity criterion for the dissipative quasi-geostrophic equations." Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 26, no. 5, 2009, pp. 1607-1619. |
| Dong, Hongjie. "Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness." Discrete and Continuous Dynamical Systems, vol. 26, no. 4, 2009, pp. 1197-1211. |
| Dong, Hongjie, Kim, Doyoon. "Elliptic Equations in Divergence Form with Partially BMO Coefficients." Archive for Rational Mechanics and Analysis, vol. 196, no. 1, 2009, pp. 25-70. |
| Dong, Hongjie, Du, Dapeng, Li, Dong. "Finite time singularities and global well-posedness for fractal Burgers equations." Indiana University Mathematics Journal, vol. 58, no. 2, 2009, pp. 807-822. |
| Dong, Hongjie, Kim, Seick. "Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains." Trans. Amer. Math. Soc., vol. 361, no. 06, 2009, pp. 3303-3303. |
| Dong, Hongjie, Li, Dong. "Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations." Communications in Mathematical Sciences, vol. 7, no. 1, 2009, pp. 67-80. |
| Dong, Hongjie, Kim, Doyoon. "Parabolic and Elliptic Systems with VMO Coefficients." Methods and Applications of Analysis, vol. 16, no. 3, 2009, pp. 365-388. |
| Dong, Hongjie, Pavlović, Nataša. "Regularity Criteria for the Dissipative Quasi-Geostrophic Equations in Hölder Spaces." Commun. Math. Phys., vol. 290, no. 3, 2009, pp. 801-812. |
| Dong, Hongjie, Du, Dapeng. "The Navier-Stokes Equations in the Critical Lebesgue Space." Commun. Math. Phys., vol. 292, no. 3, 2009, pp. 811-827. |
| Dong, Hongjie, Li, Dong. "Finite time singularities for a class of generalized surface quasi-geostrophic equations." Proceedings of the American Mathematical Society, vol. 136, no. 07, 2008, pp. 2555-2563. |
| Du, Dapeng, Dong, Hongjie. "Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space." Discrete and Continuous Dynamical Systems, vol. 21, no. 4, 2008, pp. 1095-1101. |
| Cho, Sungwon, Dong, Hongjie, Kim, Seick. "On the Green's matrices of strongly parabolic systems of second order." Indiana University Mathematics Journal, vol. 57, no. 4, 2008, pp. 1633-1678. |
| Dong, Hongjie, Kim, Seick, Safonov, Mikhail. "On Uniqueness of Boundary Blow-Up Solutions of a Class of Nonlinear Elliptic Equations." Communications in Partial Differential Equations, vol. 33, no. 2, 2008, pp. 177-188. |
| Dong, Hongjie, Li, Dong. "Spatial Analyticity of the Solutions to the Subcritical Dissipative Quasi-geostrophic Equations." Archive for Rational Mechanics and Analysis, vol. 189, no. 1, 2008, pp. 131-158. |
| Dong, Hongjie. "Well-posedness for a transport equation with nonlocal velocity." Journal of Functional Analysis, vol. 255, no. 11, 2008, pp. 3070-3097. |
| Dong, Hongjie, Krylov, N. V. "On time-inhomogeneous controlled diffusion processes in domains." The Annals of Probability, vol. 35, no. 1, 2007, pp. 206-227. |
| Dong, Hongjie, Du, Dapeng. "Partial Regularity of Solutions to the Four-Dimensional Navier-Stokes Equations at the First Blow-up Time." Commun. Math. Phys., vol. 273, no. 3, 2007, pp. 785-801. |
| Dong, Hongjie, Krylov, Nicolai V. "The Rate of Convergence of Finite-Difference Approximations for Parabolic Bellman Equations with Lipschitz Coefficients in Cylindrical Domains." Applied Mathematics and Optimization, vol. 56, no. 1, 2007, pp. 37-66. |
| Dong, Hongjie, Staubach, Wolfgang. "Unique continuation for the Schrödinger equation with gradient vector potentials." Proceedings of the American Mathematical Society, vol. 135, no. 07, 2007, pp. 2141-2150. |
| Dong, Hongjie. "Hessian Equations with Elementary Symmetric Functions." Communications in Partial Differential Equations, vol. 31, no. 7, 2006, pp. 1005-1025. |
| Dong, Hongjie, Du, Dapeng. "On the Local Smoothness of Solutions of the Navier–Stokes Equations." J. math. fluid mech., vol. 9, no. 2, 2006, pp. 139-152. |
| Dong, Hongjie. "About smoothness of solutions of the heat equations in closed, smooth space-time domains." Communications on Pure and Applied Mathematics, vol. 58, no. 6, 2005, pp. 799-820. |
Research
Research Overview
My main research interest is in partial differential equations, including nonlinear elliptic and parabolic equations, the Navier-Stokes equations, the quasi-geostrophic equations, the reaction diffusion equations, the probability approaches of PDEs, and rates of convergence of finite difference approximations for elliptic and parabolic Bellman's equations. I am also interested in the probability theory and stochastic partial differential equations.
Research Statement
My main research interest is in partial differential equations, including nonlinear elliptic and parabolic equations, the Navier-Stokes equations, the quasi-geostrophic equations, the reaction diffusion equations, the probability approaches of PDEs, and rates of convergence of finite difference approximations for elliptic and parabolic Bellman's equations. I am also interested in the probability theory and stochastic partial differential equations.
Funded Research
I was partially supported by the National Science Foundation under agreement No. DMS-0111298, DMS-1056737, DMS-1600593, DMS-2055244, t he Simons Foundation grant no. 709545, a Simons fellowship grant no. 007638, and a start-up funding from the Division of Applied Mathematics of Brown University.
Background
Education and Training
| Year | Degree | Institution |
|---|---|---|
| 2005 | PhD | University of Minnesota |
| 2001 | BS | Fudan University |
Honors and Awards
NSF Grant: DMS-1600593 (P.I.), 2016.7 to 2019.6.
NSF CAREER Award: DMS-1056737 (P.I.), 2011.7 to 2016.6.
NSF Grant: DMS-0800129 (P.I.), 2008.7 to 2012.6.
MSRI Membership: 2007.7 to 2007.8.
IAS Membership: 2006 to 2007, 2008.9 to 2008.12.
Travel Grant from AMS for ICM 2006, Madrid, Spain: August 22 to 30, 2006.
Outstanding Graduate: Fudan University, 2001.
The First, Second Prizes of People's Scholarship: Fudan University, 1998 to 2001.
Meritorious Winner of Mathematical Contest in Modeling: The Consortium for Mathematics and its Applications, 2000.
Perfect Paper on the American Invitational Mathematics Examination: The Mathematical Association of America, 1997.
The First Prize of Chinese Mathematical Olympiad: rank top 4 in P.R.China, 1997.
The First, Second Prizes: in contests of Mathematics, Physics and Chemistry, 1991 to 1997.
Affiliations
Collaborators
| Name | Title |
|---|---|
| Guo, Yan | L. Herbert Ballou University Professor of Applied Mathematics |
| Yastrzhembskiy, Timur | Prager Assistant Professor of Applied Mathematics |
Affiliations
Journal Refereed: Appl. Math. Optim., Arch. Rational Mech. Anal., Canad. J.
Math., Comm. Partial Differential Equations, Discrete Contin. Dyn. Syst., Duke
Math. J., Dyn. Partial Differ. Equ., Int. Math. Res. Not., J. Funct. Anal., J. Math.
Anal. Appl., Math. Comp., Methods Appl. Anal., NoDEA. Nonlinear Differential
Equations and Applications Nonlinearity, Numer. Math., Selecta Math., SIAM J.
Control Optim., SIAM J. Math. Anal., Trans. AMS,
NSF Panelist: 2012, 2013.
Organizer of PDE Seminar: 2008/09, 2009/10, 2012, 2013 (spring).
Committee Member of Ph.D. Thesis Defence for: Yi Cai (2012)
Committee Member of Preliminary Exam for: Zhen Chen, Yang Yang (2010),
Hong Zhang (2012), Carey Caginalp (2012, 2013), Tong Qin (2014), Cengke Shi,
Sameer Iyer (2015).
Member of Undergraduate Program Advisors: 2009/10, 2011/2012, 2013 (spring), 2014.
Member of Graduate Program Advisors: 2013/2014.
Director of Undergraduate Studies at Division of Applied Mathematics: 2014–2017.
Member of Professional Organizations: American Mathematical Society (AMS),
Society for Industrial and Applied Mathematics (SIAM).
Reviewer of Mathematical Reviews: since 2007
Math., Comm. Partial Differential Equations, Discrete Contin. Dyn. Syst., Duke
Math. J., Dyn. Partial Differ. Equ., Int. Math. Res. Not., J. Funct. Anal., J. Math.
Anal. Appl., Math. Comp., Methods Appl. Anal., NoDEA. Nonlinear Differential
Equations and Applications Nonlinearity, Numer. Math., Selecta Math., SIAM J.
Control Optim., SIAM J. Math. Anal., Trans. AMS,
NSF Panelist: 2012, 2013.
Organizer of PDE Seminar: 2008/09, 2009/10, 2012, 2013 (spring).
Committee Member of Ph.D. Thesis Defence for: Yi Cai (2012)
Committee Member of Preliminary Exam for: Zhen Chen, Yang Yang (2010),
Hong Zhang (2012), Carey Caginalp (2012, 2013), Tong Qin (2014), Cengke Shi,
Sameer Iyer (2015).
Member of Undergraduate Program Advisors: 2009/10, 2011/2012, 2013 (spring), 2014.
Member of Graduate Program Advisors: 2013/2014.
Director of Undergraduate Studies at Division of Applied Mathematics: 2014–2017.
Member of Professional Organizations: American Mathematical Society (AMS),
Society for Industrial and Applied Mathematics (SIAM).
Reviewer of Mathematical Reviews: since 2007
Teaching
Teaching Overview
(APMA 2240) Partial Differential Equations
Teaching
| APMA 0360 - Applied Partial Differential Equations I |
| APMA 1330 - Methods of Applied Mathematics |
| APMA 2110 - Real Analysis |
| APMA 2120 - Hilbert Spaces and Their Applications |
| APMA 2230 - Partial Differential Equations |
| APMA 2230 - Partial Differential Equations I |
| APMA 2240 - Partial Differential Equations |
| APMA 2240 - Partial Differential Equations II |
| APMA 2812I - Topics in the Regularity Theory of PDEs |
| APMA 2822I - Topics in PDEs |