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

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.