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We observe that for a unit tangent vector field u ∈ TM on a 3-dimensional Riemannian manifold M, there is a unique unit cotangent vector field A ∈ T∗M associated to u such that we can define the curl of u by dA. Through a unit cotangent vector field A ∈ T∗M, we define the Oseen–Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen–Frank energy on 3-dimensional Riemannian manifolds.

The Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants $L_1$, $L_2$, $L_3$, $L_4$. In the case of $L_4\neq 0$, Ball and Majumdar (Mol. Cryst. Liq. Cryst., 2010) found an example that the original Landau-de Gennes energy functional in physics does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. At first, we introduce a new Landau-de Gennes energy density with $L_4\neq 0$, which is equivalent to the original Landau-de Gennes density for uniaxial tensors and satisfies the coercivity condition for all $Q$-tensors. Secondly, we prove that solutions of the Landau-de Gennes system can approach a solution of the $Q$-tensor Oseen-Frank system without using energy minimizers. Thirdly, we develop a new approach to generalize the Nguyen and Zarnescu (Calc. Var. PDEs, 2013) convergence result to the case of non-zero elastic constants $L_2$, $L_3$, $L_4$., Comment: Revised vision. To appear in Calculus of Variations and Partial Differential Equations. arXiv admin note: text overlap with arXiv:2007.11144

We prove that the Yang–Mills $$\alpha $$ -functional satisfies the Palais–Smale condition, implying the existence of critical points, which are called Yang–Mills $$\alpha $$ -connections. It was shown in Hong et al. (Comment Math Helv 90:75–120, 2015) that as $$\alpha \rightarrow 1$$ , a sequence of Yang–Mills $$\alpha $$ -connections converges to a Yang–Mills connection away from finitely many points. We prove an energy identity for such a sequence of Yang–Mills $$\alpha $$ -connections. As an application, we also prove an energy identity for the Yang–Mills flow at the maximal existence time.

A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that any biharmonic submanifold in a Euclidean space is minimal. In the case of a hypersurface M n in R n + 1 , Chen's conjecture was settled in the case of n = 2 by Chen and Jiang around 1987 independently. Hasanis and Vlachos in 1995 settled Chen's conjecture for a hypersurface with n = 3 . However, the general Chen's conjecture on a hypersurface M n remains open for n > 3 . In this paper, we settle Chen's conjecture for hypersurfaces in R 5 for n = 4 .

We investigate the Ericksen–Leslie system with the Oseen–Frank energy for unequal Frank elastic constants in $${\mathbb R}^3$$ . Motivated by the results of Kato (Math Z 187:471–480, 1984) and Hineman–Wang (Arch Ration Mech Anal 210:177–218, 2013), we prove existence of solutions to the Ericksen–Leslie system with initial data having small $$L^3_{uloc}$$ -norm. In particular, we obtain a local $$L^3$$ -estimate by using interpolation inequalities and a covering argument, which is different from the one in Hineman–Wang (2013) of using local $$L^3$$ decay estimates for the Laplace operator. Moreover, we prove the uniqueness of the $$L^3_{uloc}$$ -solutions of the Ericksen–Leslie system without a closing condition of Frank constants $$k_1$$ , $$k_2$$ , $$k_3$$ by utilizing the method of Li–Titi–Xin (Math Models Methods Appl Sci 26:803–822, 2016).

In this paper we introduce an α-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 α-flow with smooth initial value. We prove that the limit of the solutions of the α-flow as α→1 is a weak solution to the Yang-Mills flow. By an application of the α-flow, we then follow the idea of Sacks and Uhlenbeck [22] to prove some existence results for Yang-Mills connections and improve the minimizing result of the Yang-Mills functional of Sedlacek [26]

In this paper, we generalize the no-neck result of Qing and Tian (in Commun Pure Appl Math 50:295–310, 1997) to show that there is no neck during blowing up for the n-harmonic flow as $$t\rightarrow \infty $$ . As an application of the no-neck result, we settle a conjecture of Hungerbuhler (in Ann Scuola Norm Sup Pisa Cl Sci 4:593–631, 1997) by constructing an example to show that the n-harmonic map flow on an n-dimensional Riemannian manifold blows up in finite time for $$n\ge 3$$ .

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.

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 note, we study the curvature flow to the Nirenberg problem on S2 with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problem has a solution provided the prescribed non-negative Gaussian curvature f has at least two positive local maxima of f, and at all positive valued saddle points q of f there holds \({\Delta_{S^2}f(q) >0 }\).

In this paper we consider the heat flow for the H-system with variable function H (·) in higher dimensions, and establish global existence and uniqueness.

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in C(infinity) up to gauge to a critical point of the Seiberg-Witten functional.

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.

In this paper, we establish the existence of the maximal time for a smooth solution to the anti-self-dual (ASD) flow in vector bundles over a 4-dimensional compact Riemannian manifold M and present a different proof of the Taubes’ existence theorem on anti-self-dual connections on 4-manifolds.

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

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 Kahler 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.

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.

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.

(1999). On the hausdorff dimension of the singular set of stable-stationary harmonic maps. Communications in Partial Differential Equations: Vol. 24, No. 11-12, pp. 1967-1985.

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.

We study Maxwell's equations in a quasi-static electromagnetic field, where the electrical conductivity of the material depends on the temperature. By establishing the reverse HAolder inequality, we prove partial regularity of weak solutions to the non-linear elliptic system and the non-linear parabolic system in a quasi-static electromagnetic field.

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

We describe the behavior as $\varepsilon\to 0$ of minimizers for a Ginzburg-Landau functional $$ E_{\varepsilon}(u;\Omega )=\int_{\Omega}\bigl [ \frac {|\nabla u|^n}n +\frac 1{4\varepsilon^n}(1-|u|^2)^2\bigr ]\,dx $$ in the space $H^{1,n}_g(\Omega ;\Bbb R^n)$, where $\Omega\subset\Bbb R^n$ and the boundary data $g:\partial \Omega\to S^{n-1}$ has a nonzero topological degree. Some recent results of Bethuel, Brezis and H\'elein, and of Struwe on the two-dimensional problem, are extended to higher-dimensional cases. New proofs for their results are also presented in this paper.

In the author's joint paper [HJS] with Jest and Struwe, we discuss asymtotic limits of a self-dual Ginzburg-Landau functional involving a section of a line bundle over a closed Riemann surface and a connection on this bundle. In this paper, the author generalizes the above results [HJS] to the case of bounded domains.

We prove two asymptotical estimates for minimizers of a Ginzburg-Landau functional of the form .