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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.

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 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 .

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 \({\Delta_{S^2}f(q) >0 }\).

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.

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 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