Partial differential equations
1. Postdoctoral fellow, Mathematical Sciences Research Institute in Berkeley,
Aug 2005-Dec 2005
2. Instructorship, University of Texas at Austin. Aug 2005-Aug 2008.
3. Postdoctoral fellow, Mathematical Sciences Research Institute in Berkeley,
July 20 2007-Aug 13 2007.
 High Degeneracy of Effective Hamiltonian in Two Dimensions, submitted.
 Effective fronts of polytope shapes (With Wenjia Jing, Hung V. Tran), to appear in Minimax Theory and its Applications.
 Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form (with X. Guo, H. V. Tran), SN Partial Differential Equations and Applications, 1, Article number: 15 (2020). Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize.
 Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting (with H. Mitake, H. Tran), Arch. Rational Mech. Analysis, 2019, Volume 233, Issue 2, pp 901–934
 A rigidity result for effective Hamiltonians with 3-mode periodic potentials (with Hung V. Tran), submitted, Advances in Mathematics, Volume 334, 20 2018, Pages 300-321.
 Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians (with Jianliang Qian, Hung Tran), Mathematische Annalen, 2018, Volume 372, Issue 1–2, pp 91–123.
 Curvature effect in shear flow: slowdown of turbulent flame speeds with Markstein number (with Jiancheng Lyu, Jack Xin), Communications in Math Physics, 2018, Volume 359, Issue 2, pp 515–533.
 Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method (with Jiancheng Lyu, Jack Xin), Numerical Mathematics: Theory, Methods & Applications, 2017, to appear.
 Performance analysis and signal design for a stationary signalized ring road (joint with Wenlong Jin), submitted.
 Spiral Waves, Edge Transport, and Front Speed Enhancement for ABC Flows (joint with Tyler McMillen, Jack Xin and Andrej Zlatos), SIAM J. Appl. Dyn. Syst., 15(3), 1753–1782.
 Inverse problems, non-roundness and flat pieces of the effective burning velocity from an inviscid quadratic Hamilton-Jacobi model (joint with Wenjia Jing and Hung Tran), Nonlinearity, Volume 30, Number 5, 2017.
 Periodic orbits of the ABC flow with $A=B=C=1$ (Joint with Jack Xin, Andrej Zlatos), SIAM J. Math. Anal., 48(6), 4087–4093.
 Some inverse problems in periodic homogenization of Hamilton-Jacobi equations (joint with Songting Luo, Hung Tran), Arch. Rational Mech. Analysis, 221 (3), pp 1585–1617, 2016.
 Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension (joint with Scott Armstrong, Hung Tran), J. Differential Equations, 261(5), 2016, pp. 2702–2737.
 Asymptotic solution and effective Hamiltonian of a Hamilton-Jacobi equation in the modeling of traffic flow on a homogeneous signalized road (joint with Wenlong Jin), J. Math. Pure Appl., 104(5), 2015, Pages 982-1004.
 Stochastic homogenization of a nonconvex Hamilton-Jacobi equation (joint with Scott Armstrong, Hung Tran), Calc. Var. Partial Differential Equations, October 2015, Volume 54, Issue 2, pp 1507–1524.
 Asymptotic growth rates and strong bending of turbulent flame speeds of G-equation in steady two dimensional incompressible periodic flows (joint with Jack Xin), SIAM J. Math Analysis, 46(4), pp. 2444–2467, 2014.
 Front Quenching in G-equation Model Induced by Straining of Cellular Flow (joint with Jack Xin), Arch. Rational. Mechanics. Anal, 214(2014), pp. 1–34.
 Turbulent Flame Speeds of G-equation Models in Unsteady Cellular Flows (with Y. Liu, Jack Xin), Math Model. Natural Phenom., 8(3), pp. 198-205, 2013.
 Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton-Jacobi models, (with Jack Xin), Annales I’Institut H. Poincare- Analyse non lineaire. 30(6), pp. 1049-1068, 2013.
 A Numerical Study of Turbulent Flame Speeds of Curvature and Strain G-equations in Cellular Flows, (with Y. Liu, Y. Yu), Physica D, 243(1), pp. 20-31, 2013. DOI:10.1016/j.physd.2012.09.008.
 Nonuniqueness of infinity ground states (with Ryan Hynd and Charles Smart), Calc. Var. Partial Differential Equations, November 2013, Volume 48, Issue 3, pp 545–554.
 $C^1$-boundary regularity of planar infinity harmonic functions (with Changyou Wang), Math. Res. Lett. 19 (2012), no. 4, 823–835.
 A new approximation for effective Hamiltonians for
homogenization of a class of Hamilton-Jacobi equations (joint with Songting Luo, Hongkai Zhao), Multiscale Modeling and Simulation (MMS), Vol. 9 Issue 2, 711-734 (2011).
 Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular
and shear flows (joint with Jack Xin, Yu-yu Liu), Arch. Rational. Mechanics. Anal, 199(2), pp 527-561 (2011).
 Periodic homogenization of G-equations and viscosity effects (with Yu-yu Liu, Jack Xin), Nonlinearity,23(2010), pp 2351-2367..
 Homogenization of generalized characteristics associated to solutions of time-dependent Hamilton-Jacobi equations, Asymptotic Analysis, Volume 69, Number 1-2/2010, page 99-116.
 Periodic Homogenization of Inviscid G-equation for Incompressible Flows (with J. Xin), Comm Math Sciences, Vol.8, No. 4, pp1067-1078, 2010.
 Bounds on Front Speeds for Inviscid and Viscous G-equations (with J. Nolen, J. Xin), Methods and Applications of Analysis, Vol 16, No.4, pp 507-520, 2009.
 Mather measures selected by an approximation scheme, (with Diogo Gomes, Renato Iturriaga, Hector Sanchez-Morgado), Proceedings of AMS, Volume 138, Number 10, October 2010, Pages 3591–3601.
 Maximal and minimal solutions of an Aronsson equation: L variational problems versus the game theory, Calc. Var. Partial Differential Equations 37 (2010), no. 1-2, 63--74.
 Dynamics of propagation of singularities of semiconcave functions, (with P. Cannarsa), J. Eur. Math. Soc (JEMS), 11 (2009), no. 5, 999-1024.
 Uniqueness of values of Aronsson operators and applications to game theory, Annales I’Institut H. Poincare- Analyse non lineaire, 26 (2009), no. 4, 1299-1308.
 Derivation of Aronsson equation for C 1 Hamiltonian, (with M.G. Crandall, Changyou Wang), Trans. Amer. Math. Soc. 361 (2009), no. 1, 103–124.
 Uniqueness and Nonuniqueness of viscosity solutions of Aronsson equations, (with Robert Jensen, Changyou Wang), Arch. Rational. Mechanics. Anal, 2008, no. 2, 347–370.
 Aronsson’s equations on Carnot-Caratheodory spaces, (with Changyou Wang), Illinois J. Math. 52 (2008), no. 3, 757--772.
 Asymptotic behavior of infinity harmonic functions near isolated singularity, (with Ovidius Savin, Changyou Wang), Int. Math. Res. Not, Vol. 2008. no. 6, Art. ID rnm163, 23 pp.
 C1-regularity of the Aronsson equation in R2, (with Changyou wang), Annales I’Institut H. Poincare- Analyse non lineaire, 25 (2008) 659-678.
 Singular set of a convex potential in two dimensions, Comm. Partial Differential Equations, 32 (2007), no. 10-12, 1883–1894.
 L variational problems and weak KAM theory, Comm. Pure Appl. Math, Vol. 60 (2007), no. 8, 1111–1147.
 A remark on the semiclassical measure, Proc. Amer. Math. Soc. 135 (2007), no. 5, 1449-1454.
 Some properties of the infinity ground state, Indiana University Mathematics Journal. 56 No. 2 (2007), 947-964.
 A remark on the C 2 solution of the infinity Laplacian equation, Electron. J. Diff. Eqns, Vol. 2006, No. 122, pp. 1-4.
 A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa. Cl. Sci. (5). 2006, 439-444.
 Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers, (with Ron Gariepy, Changyou Wang), Comm. Partial Differential Equations, 31 (2006), no. 7-9, 1027-1046.
 L variational problems and Aronsson equations, Arch. Rational. Mechanics. Anal, 182 (2006), no. 1, 153-180.
 Various properties of solutions to the infinity Laplacian equation, (with L.C. Evans). Comm. Partial Differential Equations, 30 (2005), no. 7-9, 1401–1428.
 Tangent lines of contact for the Infinity Laplacian, Calc. Var. Partial Differential Equations, 21 (2004), no. 4, 349–355
NSF collaborative research grant 0601403, 07/01/2006-06/30/2009. (PI, $78,000).
Title: L-infinity variational problems and the Aronsson equation.
NSF grant 0901460, 06/01/2009-05/31/2013. (PI, $332,771).
Title: Problems related to the infinity Laplacian operator, the weak KAM theory
and singularities of solutions of Monge-Ampere equations.
NSF CAREER award 1151919, July 2012-Sep 2018, $400,003, Analysis of G-equations in the modeling of turbulent flame speed and comparison with other math models.
NSF grant 2000191, 2020-2023 (PI, $302,742), Analysis of Properties of Effective Hamiltonians with Applications.