
Hongjie Dong 

Publications






Some of my papers can be found on arXiv and MathSciNet.
Fluid Equations
· On the Local Smoothness of Solutions of the NavierStokes Equations.
(joint work with Dapeng Du, 14 pages, arXiv, J. Math. Fluid Mech. 9 (2007), no. 2, 139152.)
· Partial regularity of weak solutions of the NavierStokes equations in R^4 at the first blow up time.
(joint work with Dapeng Du, 16 pages, Comm. Math. Phys., 273, (2007), no. 3, 785801.)
· Spatial analyticity of the solutions to the subcritical dissipative quasigeostrophic equations.
(Joint work with Dong Li, Arch. Rational Mech. Anal., 189 (2008) no. 1, 131158.)
· Dissipative quasigeostrophic equations in critical Sobolev spaces: smoothing effect and global wellposedness.
(submitted, 20 pages, arXiv, 2006, Discrete Contin. Dyn. Syst.,Vol 26, No 4 (2010), 11971211.)
· Optimal local smoothing and analyticity rate estimates for the generalized NavierStokes equations.
(Joint work with Dong Li, 14 pages, Comm. Math. Sci., 7 (2009) no. 1, 67—80, arXiv.)
· Global wellposedness and a decay estimate for the critical quasigeostrophic equation.
(Joint work with Dapeng Du, arXiv, Discrete Contin. Dyn. Syst., 21 (2008) no. 4, 10951101.)
· On the 2D critical and supercritical dissipative quasigeostrophic equation in Besov spaces.
(Joint work with Dong Li, J. Differential Equations, 248 (2010), 26842702.)
·
Finite time singularities for a class of
generalized surface quasigeostrophic equations.
(Joint
work with Dong Li, Proc.
Amer. Math.
Soc., 136 (2008), 25552563.)
· Wellposedness for a transport equation with nonlocal velocity. (J. Funct. Anal., 255 (2008), no. 11, 30703097.)
· A regularity criterion for the dissipative quasigeostrophic equations.
(Joint work with Natasa Pavlovic, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 16071619, arXiv.)
· Finite time singularities and global wellposedness for fractal Burgers' equation.
(Joint work with Dapeng Du and Dong Li, Indiana Univ. Math. J., 58 No 2 (2009), 807822.)
· Regularity criteria for the dissipative quasigeostrophic equations in Holder spaces.
(Joint work with Natasa Pavlovic, Comm. Math. Phys., 290 No 3 (2009), 801812.)
· The NavierStokes equations in the critical Lebesgue space.
(Joint work with Dapeng Du, Comm. Math. Phys., 292 No. 3 (2009), 811827, arXiv.)
· The aggregation equation with powerlaw kernels: illposedness, mass concentration and similarity solutions.
(Comm. Math. Phys., 304 (2011) no. 3, 649664., arXiv.)
· On similarity solutions to the multidimensional aggregation equation.
(SIAM J. Math. Anal. 43 (2011) no. 4, 1995–2008, arXiv)
· Partial regularity of steadystate solutions to the 6D NavierStokes equations.
(Joint with Robert Strain, to appear in Indiana Univ. Math. J. 61 (2012), no. 6, 22112229, arXiv)
· On a onedimensional alphapatch model with nonlocal drift and fractional dissipation.
(Joint with Dong Li, Trans. Amer. Math. Soc. 366 (2014), no. 4, 20412061, arXiv)
· Global $\dot H^1 \cap \dot H^{1}$ solutions to a logarithmically regularized 2D Euler equation.
(joint work with D. Li, J. Math. Fluid Mech. to appear (2012), arXiv)
· On a family of exact solutions to the incompressible liquid crystals in two dimensions.
(joint work with Z. Lei, submitted (2012), arXiv)
· On a generalized maximum principle for a transportdiffusion model with logmodulated fractional dissipation.
(Joint with Dong Li, Discrete Contin. Dyn. Syst.A 34 (2013), no.9, 34373454, arXiv)
· Partial regularity of solutions to the fourdimensional NavierStokes equations.
(Joint with X. Gu, Dyn. Partial Differ. Equ. (2014), arXiv)
· On a multidimensional transport equation with nonlocal velocity.
(Adv. Math., 264 (2014), 747–761, arXiv)
· Boundary partial regularity for the high dimensional NavierStokes equations.
(Joint with X. Gu, J. Funct. Anal. (2014), arXiv)
General Elliptic and Parabolic Equations
· Hessian equations with elementary symmetric functions.
(22 pages, arXiv, Comm. Partial Differential Equations. 31 (2006) no. 7, 10051025.)
· On uniqueness of boundary blowup solutions of a class of nonlinear elliptic equations.
(Joint work with Seick Kim and Mikhail V. Safonov, arXiv, Comm. Partial Differential Equations, 33 (2008), no. 2, 177188.)
· On the Green’s matrices of strongly parabolic systems of second order.
(Joint work with Sungwon Cho and Seick Kim, arXiv, Indiana Univ. Math. J., 57 (2008) no. 4, 16331678.)
· Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains.
(Joint work with Seick Kim, Trans. Amer. Math. Soc., 361 (2009), 33033323.)
· Parabolic and elliptic systems with VMO coefficients.
· Solvability of parabolic equations in divergence form with partially VMO coefficients.
(J. Funct. Anal., 258 (2010) 2145–2172.)
· Secondorder elliptic and parabolic equations with B(R^2; VMO) coefficients.
(Joint work with N. V. Krylov, Trans. Amer. Math. Soc., 362 (2009), no. 12, 6477–6494, arXiv.)
· Elliptic equations in divergence form with partially BMO coefficients.
(Joint work with Doyoon Kim, Arch. Rational Mech. Anal., 196 no. 1 (2010), 2570.)
· L_p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients.
(Joint work with Doyoon Kim, Calc. Var. Partial Differential Equations, 40 (2011) no. 34, 357389, PDF.)
· Parabolic equations with variably partially VMO coefficients.
(Algebra i Analis (St. Petersburg Math. J.), 23 (2011) no.3, 150–174., arXiv.)
· Parabolic and elliptic systems in divergence form with variably partially BMO coefficients.
(Joint work with Doyoon Kim, SIAM J. Math. Anal., 43 (2011) no. 3, 1075–1098, arXiv.)
· Partial Schauder estimates for secondorder elliptic and parabolic equations.
(Joint work with Seick Kim, Calc. Var. Partial Differential Equations, 40 (2011) no. 34, 481500, arXiv)
· On the L_psolvability of higher order parabolic and elliptic systems with BMO coefficients.
(Joint work with Doyoon Kim, Arch. Rational Mech. Anal., 199 (2010), no. 3, 889941, arXiv.)
· Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains.
(Joint work with Sungwon Cho and Seick Kim, Potential Anal., to appear, (2010), arXiv)
· Solvability of secondorder equations with hierarchically partially BMO coefficients.
(Trans. Amer. Math. Soc., 364 (2012) no. 1, 493–517, arXiv)
· Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controllable growth.
(Joint work with Doyoon Kim, Comm. Partial Differential Equations., 36 (2011) no. 10, 1750–1777, arXiv)
· On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients.
(Joint work with N. V. Krylov and Xu Li, Algebra i Analis (St. Petersburg Math. J.) 24 (2012) no. 1., arXiv)
· Gradient estimates for parabolic and elliptic systems from linear laminates.
(Arch. Rational Mech. Anal., 205 (2012) no. 1, 119–149.
· Higher order elliptic systems in Sobolev spaces with partially BMO coefficients.
(Joint work with Doyoon Kim, J. Funct. Anal., 261 (2011) no. 11, 3279–3327, arXiv)
· On L_pestimates for a class of nonlocal elliptic equations.
(Joint work with Doyoon Kim, J. Funct. Anal., 262 (2012) no 3, 1166–1199, arXiv)
· Schauder estimates for a class of nonlocal elliptic equations.
(Joint work with Doyoon Kim, Discrete Contin. Dyn Syst.A, 33 (2013) no. 6, 2319–2347, arXiv)
· The conormal derivative problem for higher order elliptic systems with irregular coefficients.
(Joint work with Doyoon Kim, Contemp. Math., 581 (2012), 69–97, arXiv)
· On the existence of smooth solutions for fully nonlinear parabolic equations with measurable “coefficients” without convexity assumptions,
(Joint work with N. V. Krylov, Comm. Partial Differential Equations, 38 (2013), no. 6, 10381068, arXiv).
· On elliptic equations in a half space or in convex wedges with irregular coefficients.
(Adv. Math., 238 (2013), 24–49, arXiv)
· Parabolic equations in convex polytopes with time irregular coefficients.
(Joint work with Doyoon Kim, SIAM J. Math. Anal., 46 (2014) no. 3, 17891819.
· Green's functions for parabolic systems of second order in timevarying domains.
(Joint work with Seick Kim, Commun. Pure Appl. Anal. 13 (2014), no. 4, 1407—1433.)
· Boundary value problem for parabolic operators in time varying domains.
(Joint work with Doyoon Kim and Sungwon Cho,
Comm. Partial Diﬀerential Equations, 40 (2015), no. 7, 1282–1313.)
· Schauder estimates for higherorder parabolic systems with
time irregular coefficients.
(Joint with H. Zhang,
Calc. Var. Partial Differential Equations, 54 (2015), no. 1, 47–74.)
· Boundary gradient estimates for parabolic and elliptic systems from linear laminates.
(Joint with J. Xiong, Int. Math. Res. Not. IMRN, (2015), no. 17, 7734–7756.)
· On the impossibility of $W_p^2$ estimates for elliptic equations with piecewise constant coefficients.
(Joint with D. Kim, J. Funct. Anal. 267 (2014), no. 8, 2606–2637, arXiv)
· Elliptic and parabolic equations with measurable coeﬃcients in weighted Sobolev spaces.
(Joint with D. Kim, Adv. Math, 274 (2015), 681–735.)
· On L_p and Schauder estimate for a cornormal problem of higherorder parabolic
systems.
(Joint with H. Zhang, Trans. Amer. Math. Soc., (2016), no. 10, 7413–7460.)
· On an elliptic equation arising from photoacoustic imaging in inhomogeneous media.
(Joint with H. Ammari, H. Kang, and S. Kim, Int. Math. Res. Not., 2015 (2015), no. 22, 12105–12113.)
· Neumann problem for nondivergence elliptic and parabolic equations with BMO x coef
ﬁcients in weighted Sobolev spaces.
(Joint with D. Kim and H. Zhang, Discrete Contin. Dyn.
Syst.A, to appear (2016).)
· On an elliptic equation arising from composite materials.
(Joint with H. Zhang, Arch. Rational
Mech. Anal., to appear (2016).)
Probability Theory
· About Smoothness of Solutions of the Heat Equations in Closed Smooth Spacetime Domains.
(22 pages, Comm. Pure Appl. Math. 58 (2005) no. 6, 799820, link.)
· On time inhomogeneous controlled diffusion processes in domains.
(joint work with Nicolai V. Krylov, 22 pages, Annal. Prob, 35, no. 1, 206227, (2007), arXiv.)
Finitedifference Approximations
· On the Rate of Convergence of Finitedifference Approximations for Bellman's Equations with Constant Coefficients.
(joint work with Nicolai V. Krylov, Algebra i Analis (St. Petersburg Math. J.) 17 (2005), no. 2, 108132; translation in St. Petersburg Math. J. 17 (2006), no. 2, 295313.)
· Rate of convergence of finitedifference approximations for degenerate linear parabolic equations with C^1 and C^2 coefficients.
(joint work with Nicolai V. Krylov, 25 pages, Electro. J. Differential Equations 2005 (2005), no. 102, 125, link.)
· On the rate of convergence of finitedifference approximations for bellman equations in a domain with lipschitz coefficients.
(joint work with Nicolai V. Krylov, 31 pages, Appl. Math. Optim., 56, (2007) no. 1, 3766.)
Others
· On Unique continuation for the schrodinger equation with gradient vector potentials.
(joint work with Wolfgang Staubach, Proc. Amer. Math .Soc. 135 (2007), 21412149, arXiv)
· OnRegularity of a degenerate parabolic equation appearing in Vecer's unified pricing of Asian options.
(joint work with Seick Kim, Bull. Korean Math. Soc., to appear (2014).)
Thesis
· On Some Problems Related to the Regularity Theory for
Secondorder EllipticParabolic Equations and Their Numerical Approximations.
(Ph.D thesis, University of Minnesota, 2005.8, thesis advisor: Nicolai V. Krylov.)
·
On the compatibility condition of a system of nonlinear
first order partial differential equations and its connection to the Frobenius theorem.
(B.S. thesis, Fudan University, 2001.5, thesis advisor: Jiaxing Hong.)
Research partially supported by NSF grants DMS0800129, DMS1056737 (CAREER award), DMS1600593, and a startup funding from the Division of Applied Mathematics of Brown University.