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In the first part of this paper, we establish the global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen–Frank model. The liquid crystal flow is a prototype of equations from the Ericksen–Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen–Leslie system is a system of the Navier–Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove the global existence of solutions of the Ericksen–Leslie system for a general Oseen–Frank model in R 2 .

We propose a new approximation for the relaxed energy E of the Dirichlet energy and prove that the minimizers of the approximating functionals converge to a minimizer u of the relaxed energy, and that u is partially regular without using the concept of Cartesian currents. We also use the same approximation method to study the variational problem of the relaxed energy for the Faddeev model and prove the existence of minimizers for the relaxed energy \({\tilde{E}_F}\) in the class of maps with Hopf degree ±1.

In the 1960s, Ericksen and Leslie established the hydrodynamic theory for modelling liquid crystal flow. In this paper, we investigate a simplified model of the Ericksen–Leslie system, which is a system of the Navier–Stokes equations coupled with the harmonic map flow. We prove global existence of solutions to the Ericksen–Leslie system in $${\mathbb{R}^{2}}$$ with initial data, where the solutions are regular except for at a finite number of singular times.

In this paper we prove partial regularity for a weakly stablep-harmonic map from Ω intoSkwhenk> 2p– 1.

In 1986, Hardt, Kinderlehrer and Lin established the existence and partial regularity of minimizers of the liquid crystal energy with the Oseen-Frank density. Motivated by the earlier results of Bethuel-Brezis-Coron and Riviere on harmonic maps, we prove the existence of infinitely many equilibrium configurations of the liquid crystal energy prescribing the same nonconstant boundary data.

We discuss the partial regularity of minimizers of energy functionals such as $$ \frac{1} {p}\int_\Omega {[\sigma (u)|dA|^p + \frac{1} {2}|\nabla u|^{2p} ]\,dx,} $$ where u is a map from a domain % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey % icI4SaaeOuamaaCaaaleqabaGaamOBaaaaaaa!3AF5! $$\Omega \in {\text{R}}^n $$ into the m-dimensional unit sphere of % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOuamaaCa % aaleqabaGaamyBaiabgUcaRiaaigdaaaaaaa!3985! $${\mathbf{R}}^{m + 1} $$ and A is a differential one-form in Ω.

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

The use of non-linear parabolic equations (the heat flow method) to find solutions of corresponding elliptic equations goes back to Eells-Sampson in 1964. In their seminal paper [ES], Eells and Sampson introduced the heat flow for harmonic maps to establish the existence of smooth harmonic maps from a compact Riemmanian manifold into a Riemmanian manifold having non-positive section curvature. In general, the heat flow for harmonic maps even on two dimensional manifolds may develop singularity at finite time (cf. [CDY]). Struwe [St1] established the existence of the unique global weak solution, which is smooth with exception of at most finitely many points, to the heat flow for harmonic maps in two dimensions. The harmonic map flow in two dimensions is very similar to the Yang-Mills flow in four dimensions. It is desirable to have a similar picture for Yang-Mills flow. In this paper, we consider the Yang-Mills flow in a vector bundle over four dimensional manifolds. Let X be a compact 4-dimensional Riemannian manifold and let E → X be a vector bundle with a compact Lie group G. Let A be a connection on E. The Yang-Mills functional is

We introduce a rectified $n$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed Riemannian manifold. We prove existence of a global solution, which is regular except for a finite number of points, of the rectified n-harmonic map flow and establish an energy identity for the flow at each singular time. Finally, we present two applications of the rectified n-harmonic map flow to minimizing the n-energy functional and the Dirichlet energy functional in a homotopy class., Comment: 29 pages. A revised version

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.

We find some new examples to show nonuniquence for the heat flow of harmonic maps where weak solutions satisfy the same monotonicity property.

In this paper, we study an $\alpha$-flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the $\alpha$-flows is a weak solution to the harmonic map flow. By an application of the $\alpha$-flow, we present a simple proof of an energy identity of a minimizing sequence in each homotopy class., Comment: minor changes

In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $\Sigma$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $\Sigma$ is 1-rectifiable., Comment: 25 pages

We define a negative exponential harmonic map from the ballBn of ℝn into the sphereSn of ℝn+1. We prove that the equator map\(u^* = (\frac{x}{{\left| x \right|}},0)\) is a negative exponential harmonic map, but not stable for the negative exponential functional whenn≥2. Moreover, we consider maps from a ballBn into the unit sphereSm of ℝm+1 wherem≥2, and prove that no nonconstant, non surjective map can reach either the minimum or the maximum of the negative exponential functional.

For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $\Omega\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $\lambda\in [0,1]$ we introduce a $\lambda$-relaxed energy $\H_\lambda$ for the Hessian energy for maps in $W^{2,2}(\Omega,S^4)$ so that each minimizer $u_\lambda$ of $\H_\lambda$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_\lambda$ for $\lambda\in [0,1)$., Comment: 28 pages