Heat flow of harmonic maps whose gradients belong to \(L^{n}_{x}L^{\infty}_{t}\)
The author studies the \(L^n_x L^\infty_t\) norm estimate of the gradient of the heat flow from a compact Riemannian manifold \((M, g)\) without boundary to another compact Riemannian manifold \(N\subset\mathbb R^k\) without boundary. Let \(T\in(0,+\infty]\) be the maximal existence time, which depends on the initial data and is scaling invariant. Similar to the Navier-Stokes equation, the scaling invariant norm estimates play an important role in the regularity theory of the solutions. The author proves that for \(n\geq 4\), if \(u: M\times [0,T]\to N\) is a weak solution to the heat flow of harmonic maps satisfying \(\nabla u\in L^n_x L^\infty_t(M\times [0,T])\), then \(u\in C^\infty(M\times (0,T],N)\). As a consequence, he shows that for \(n\geq 3\), if \(0< T<\infty\) is the maximal time for the unique smooth solution \(u\in C^\infty(M\times(0, T], N)\) of the heat flow, then \(\|\nabla u(t)\|_n\) blows up as \(T\to\infty\).
scientific article; zbMATH DE number 5294554