Hong, Min-Chun - University of Queensland - School of Mechanical and Mining Engineering

个人简介

He has collaborated with top mathematicians such as Professor Mariano Giaquinta (SNS-Pisa), Professor Jurgen Jost (Germany), Professor Michael Struwe (Zurich), Professor Gang Tian (Princeton) and Professor Zhouping Xin (Hong Kong). Some highlights of his research after joining UQ in 2004 are: In the area of harmonic maps, collaborated with Giaquinta and Yin (Calc. Var. PDEs 2011), he developed a new approximation of the Dirichlet energy, yielding a new proof on partial regularity of minimizers of the relax energy for harmonic maps as well as for the Faddeev model. The method leads to solve an open problem on partial regularity in the relax energy of biharmonic maps by him and Hao Yin (J. Funct. Anal. 2012). Based on the well-known result of Sack and Uhlenbeck in 1981 (Uhlenbeck 2019 Abel Award Winner), with collaboration of Hao Yin in 2013, he introduced the Sack-Uhlenbeck flow to prove new existence results of the harmonic map flow in 2D and made new application to homotopy classes. More recently, collaborated with his PhD student L. Cheng (Calc. Var. PDEs 2018), he settled a conjecture of Hungerbuhler on the n-harmonic map flow. In the area of Yang-Mills equations, with Gang Tian (Math. Ann. 2004), he established asymptotic behaviour of the Yang-Mills flow to prove the existence of singular Hermitian-Yang-Mills connections, which was used to settle a well-known conjecture of Bando and Siu. Collaborated with Tian and Yin (Commun. Math. Helv. 2015), he extended the Sack-Uhlenbeck program to Yang-Mills equations and introduced the Yang-Mills alpha-flow to approximate the Yang-Mills flow in 4D. More recently, collaborated with his PhD student L. Schabrun (Calc. Var. PDEs 2019), we proved the energy identity for a sequence of Yang-Mills α-connections. In the area of liquid crystals, he (Calc. Var. PDEs 2011) resolved a long-standing open problem on the global existence of the simplified Ericksen-Leslie system in 2D (a highly cited paper in Web of Science). Collaborated with Zhouping Xin (Adv. Math. 2012), he solved the global existence problem on the Ericksen-Leslie system with unequal Frank constants in 2D. Collaborated with Li and Xin (CPDE 2014), he resolved a problem on converging of the approximate Ericksen-Leslie system in 3D.

研究领域

Dr Min-Chun Hong has solved a number of open problems and conjectures on harmonic maps, liquid crystals and Yang Mills equations in the areas of nonlinear partial differential equations and geometric analysis.

近期论文

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Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system Feng, Zhewen, Hong, Min-Chun and Mei, Yu (2020) Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system. SIAM Journal on Mathematical Analysis, 52 1: 481-523. doi:10.1137/18m1182887 The energy identity for a sequence of Yang–Mills α -connections Hong, Min-Chun and Schabrun, Lorenz (2019) The energy identity for a sequence of Yang–Mills α -connections. Calculus of Variations and Partial Differential Equations, 58 3: . doi:10.1007/s00526-019-1535-y Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in Luloc3(R3) Hong, Min-Chun and Mei, Yu (2019) Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in Luloc3(R3). Calculus of Variations and Partial Differential Equations, 58 1: . doi:10.1007/s00526-018-1453-4 Biharmonic hypersurfaces with constant scalar curvature in space forms Fu, Yu and Hong, Min-Chun (2018) Biharmonic hypersurfaces with constant scalar curvature in space forms. Pacific Journal of Mathematics, 294 2: 329-350. doi:10.2140/pjm.2018.294.329 The rectified n-harmonic map flow with applications to homotopy classes Hong, Min-Chun (2018) The rectified n-harmonic map flow with applications to homotopy classes. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 18 4: 1249-1283. doi:10.2422/2036-2145.201701_010 Finite time blowup of the n-harmonic flow on n-manifolds Cheung, Leslie Hon-Nam and Hong, Min-Chun (2017) Finite time blowup of the n-harmonic flow on n-manifolds. Calculus of Variations and Partial Differential Equations, 57 9: 1-24. doi:10.1007/s00526-017-1282-x The Yang-Mills α-flow in vector bundles over four manifolds and its applications Hong, Min-Chun, Tian, Gang and Yin, Hao (2015) The Yang-Mills α-flow in vector bundles over four manifolds and its applications. Commentarii Mathematici Helvetici, 90 1: 75-120. doi:10.4171/CMH/347 Blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3 Hong, Min-Chun, Li, Jinkai and Xin, Zhouping (2014) Blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3. Communications in Partial Differential Equations, 39 7: 1284-1328. doi:10.1080/03605302.2013.871026 Some results on harmonic maps Hong, Min-Chun (2014) Some results on harmonic maps. Bulletin of the Institute of Mathematics Academia Sinica New Series, 9 2: 187-221. On the sacks-uhlenbeck flow of Riemannian surfaces Hong, Min-Chun and Yin, Hao (2013) On the sacks-uhlenbeck flow of Riemannian surfaces. Communications in Analysis and Geometry, 21 5: 917-955. doi:10.4310/CAG.2013.v21.n5.a3 Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2 Hong, Min-Chun and Xin, Zhouping (2012) Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2. Advances in Mathematics, 231 3-4: 1364-1400. doi:10.1016/j.aim.2012.06.009 Partial regularity of a minimizer of the relaxed energy for biharmonic maps Hong, Min-Chun and Yin, Hao (2012) Partial regularity of a minimizer of the relaxed energy for biharmonic maps. Journal of Functional Analysis, 262 2: 682-718. doi:10.1016/j.jfa.2011.10.003 A new approximation of relaxed energies for harmonic maps and the Faddeev model Giaquinta, Mariano, Hong, Min-Chun and Yin, Hao (2011) A new approximation of relaxed energies for harmonic maps and the Faddeev model. Calculus of Variations and Partial Differential Equations, 41 1-2: 45-69. doi:10.1007/s00526-010-0353-z Global existence of solutions of the simplified Ericksen-Leslie system in dimension two Hong, Min-Chun (2011) Global existence of solutions of the simplified Ericksen-Leslie system in dimension two. Calculus of Variations And Partial Differential Equations, 40 1-2: 15-36. doi:10.1007/s00526-010-0331-5 Global existence for the Seiberg–Witten flow Hong, Min-Chun and Schabrun, Lorenz (2010) Global existence for the Seiberg–Witten flow. Communications In Analysis And Geometry, 18 3: 433-473. doi:10.4310/CAG.2010.v18.n3.a2 Curvature flow to the Nirenberg problem Ma, Li and Hong, Min-Chun (2010) Curvature flow to the Nirenberg problem. Archiv der Mathematik, 94 3: 277-289. doi:10.1007/s00013-010-0101-9 The heat flow for H-systems on higher dimensional manifolds Hong, Min-Chun and Hsu, Deliang (2010) The heat flow for H-systems on higher dimensional manifolds. Indiana University Mathematics Journal, 59 3: 761-790. doi:10.1512/iumj.2010.59.3917 Anti-self-dual connections and their related flow on 4-manifolds Hong, M.-C. and Yu, Z. (2008) Anti-self-dual connections and their related flow on 4-manifolds. Calculus of Variations, 31 3: 325-349. doi:10.1007/s00526-007-0114-9 Partial regularity of weak solutions to Maxwell's equations in a quasi-static electromagnetic field Hong, M.-C., Tonegawa, Y. and Alzubaidi, Y. (2008) Partial regularity of weak solutions to Maxwell's equations in a quasi-static electromagnetic field. Methods and Applications of Analysis, 15 2: 199-215. Stability of the equator map for the hessian energy Hong, M. C. and Thompson, B. (2007) Stability of the equator map for the hessian energy. Proceedings of the American Mathematical Society, 135 10: 3163-3170. doi:10.1090/S0002-9939-07-08950-2