Publications

 

 

 

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Partial Differential Equations

·        Hessian equations with elementary symmetric functions.

(22 pages, arXiv:math/0509487, Comm. Partial Differential Equations. 31 (2006) no. 7, 1005--1025.)

·        On the Local Smoothness of Solutions of the Navier-Stokes Equations.

(joint work with Dapeng Du, 14 pages, arXiv:math/0502104, J. Math. Fluid Mech. 9 (2007), no. 2, 139--152.)

·        Partial regularity of weak solutions of the Navier-Stokes equations in R^4 at the first blow up time.

(joint work with Dapeng Du, 16 pages, Comm. Math. Phys., arXiv:math/0601113, 273, (2007), no. 3, 785--801.)

·        On unique continuation for the schrodinger equation with gradient vector potentials.

(joint work with Wolfgang Staubach, Proc. Amer. Math .Soc. 135 (2007), 2141-2149, arXiv:math/0603443.)

·        On uniqueness of boundary blow-up solutions of a class of nonlinear elliptic equations.

(Joint work with Seick Kim and Mikhail V. Safonov, arXiv:0705.2287, Comm. Partial Differential Equations, 33 (2008), no. 2, 177--188.)

·        Spatial analyticity of the solutions to the sub-critical dissipative quasi-geostrophic equations.

(Joint work with Dong Li, Arch. Rational Mech. Anal., 189 (2008) no. 1, 131-158.)

·        Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness.

(submitted, 20 pages, arXiv:math/0701826, 2006, to appear in Discrete Contin. Dyn. Syst.)

·        Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations.

(Joint work with Dong Li, 14 pages, Comm. Math. Sci., 7 (2009) no. 1, 67—80, arXiv:math/0703864.)

·        Global well-posedness and a decay estimate for the critical quasi-geostrophic equation.

(Joint work with Dapeng Du, arXiv:math/0701828, Discrete Contin. Dyn. Syst., 21 (2008) no. 4, 1095-1101.)

·        On the Green’s matrices of strongly parabolic systems of second order.

(Joint work with Sungwon Cho and Seick Kim, arXiv:0705.1855, Indiana Univ. Math. J., 57 (2008) no. 4, 1633-1678.)

·        On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces.

(Joint work with Dong Li, preprint, 2007.)

·        Finite time singularities for a class of generalized surface quasi-geostrophic equations.

(Joint work with Dong Li, Proc. Amer. Math. Soc., 136 (2008), 2555-2563.)

·        Well-posedness for a transport equation with nonlocal velocity. (J. Funct. Anal., 255 (2008), no. 11, 3070-3097.)

·        Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains.

(Joint work Seick Kim, Trans. Amer. Math. Soc., 361 (2009), 3303-3323.)

·        A regularity criterion for the dissipative quasi-geostrophic equations.

(Joint work Natasa Pavlovic, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1607-1619.)

·        Finite time singularities and global well-posedness for fractal Burgers' equation.

(Joint work Dapeng Du and Dong Li, Indiana Univ. Math. J., 58 No 2 (2009), 807-822.)

·        Parabolic and elliptic systems with VMO coefficients.

(Joint work Doyoon Kim, Methods Appl. Anal., to appear, 2009.)

·        Regularity criteria for the dissipative quasi-geostrophic equations in Holder spaces.

(Joint work Natasa Pavlovic, Comm. Math. Phys., 290 No 3 (2009), 801-812.)

·        Solvability of parabolic equations in divergence form with partially VMO coefficients.

(submitted, 2008, preprint.)

·        Second-order elliptic and parabolic equations with B(R^2; VMO) coefficients.

(Joint work N. V. Krylov, Trans. Amer. Math. Soc., to appear, 2009.)

·        Elliptic equations in divergence form with partially BMO coefficients.

(Joint work Doyoon Kim, to appear, Arch. Rational Mech. Anal., 2009.)

·        L_p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients.

(Joint work Doyoon Kim, submitted, 2008.)

·        Parabolic equations with variably partially VMO coefficients. (submitted, 2008.)

·        The Navier-Stokes equations in the critical Lebesgue space.

(Joint work Dapeng Du, Comm. Math. Phys., 292 No. 3 (2009), 811-827.)

·        Parabolic and elliptic systems in divergence form with variably partially BMO coefficients.

(Joint work Doyoon Kim, submitted, 2009.)

·        Partial Schauder estimates for second-order elliptic and parabolic equations.

(Joint work Seick Kim, submitted, 2009.)

·        On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients.

(Joint work Doyoon Kim, submitted, 2009.)

Probability Theory

·        About Smoothness of Solutions of the Heat Equations in Closed Smooth Space-time Domains.

(22 pages, Comm. Pure Appl. Math. 58 (2005) no. 6, 799-820, link.)

·        On time inhomogeneous controlled diffusion processes in domains.

(joint work with Nicolai V. Krylov, 22 pages, Annal. Prob, 35, no. 1, 206-227, (2007), arXiv:math/0512200.)

·        Regularity of a degenerate parabolic equation appearing in Vecer's unified pricing of Asian options.

(Joint work Seick Kim, submitted, 2009.)

Finite-difference Approximations

·        On the Rate of Convergence of Finite-difference 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, 108-132;

translation in St. Petersburg Math. J. 17 (2006), no. 2, 295-313.)

·        Rate of convergence of finite-difference 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, 1-25, link.)

·        On the rate of convergence of finite-difference 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, 37--66.)

Thesis

·        On Some Problems Related to the Regularity Theory for Second-order Elliptic-Parabolic 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 DMS-0800129 and a start-up funding from the Division of Applied Mathematics of Brown University.