Min-Chun Hong

The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps

We prove a global existence of solutions for the Landau-Lifshitz equation of the ferromagnetic spin chain from am-dimensional manifoldM into the unit sphereS2 of ℝ3 and establish some new links between harmonic maps and the solutions of the Landau-Lifshitz equation.

Heat-Flow of P-Harmonic Maps with Values Into Spheres

The global existence of weak solutions for the heat ow of p-harmonic maps of closed Riemannian manifolds into spheres is proved in this paper.

Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3 for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.

Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric

For a parameter λ > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section φ of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the λ-stability of (E, φ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.

On the Jager-Kaul theorem concerning harmonic maps

In 1983, Jager and Kaul proved that the equator map u*(x) = (x/\x\,0) : B-n --> S-n is unstable for 3 less than or equal to n less than or equal to 6 and a minimizer for the energy functional E(u, B-n) = integral B-n \del u\(2) dx in the class H-1,H-2(B-n, S-n) with u = u* on partial derivative B-n when n greater than or equal to 7. In this paper, we give a new and elementary proof of this Jager-Kaul result. We also generalize the Jager-Kaul result to the case of p-harmonic maps.

The Yang–Mills

In this paper, we introduce an \alpha -flow for the Yang-Mills functional in vector bundles over four dimensional Riemannian manifolds, and establish global existence of a unique smooth solution to the \alpha -flow with smooth initial value. We prove that the limit of solutions of the \alpha -flow as \alpha\to 1 is a weak solution to the Yang-Mills flow. By an application of the \alpha -flow, we then follow the idea of Sacks and Uhlenbeck to prove some existence results for Yang-Mills connections and improve the minimizing result of the Yang-Mills functional of Sedlacek.

\alpha

The Yang-Mills-Higgs field generalizes the Yang-Mills field. The authors establish the local existence and uniqueness of the weak solution to the heat flow for the Yang-Mills-Higgs field in a vector bundle over a compact Riemannian 4-manifold, and show that the weak solution is gauge-equivalent to a smooth solution and there are at most finite singularities at the maximum existing time.

-flow in vector bundles over four manifolds and its applications

In this paper we show that the equator map is a minimizer of the Hessian energy H(u) = integral Omega vertical bar Delta u vertical bar(2) dx in H-2(Omega;S-n) for n >= 10 and is unstable for 5 <= n <= 9.

Heat Flow for Yang-Mills-Higgs Fields, Part II

We prove two asymptotical estimates for minimizers of a Ginzburg-Landau functional of the form integral(Omega) [1/2 \del u\(2) + 1/4 epsilon(2) (1 - \u\(2))(2) W (x)] dx.

Stability of the equator map for the Hessian energy

In this paper we prove partial regularity for a weakly stable p-harmonic map from Omega into S-k when k > 2p - 1.