Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps (Q1566323)

The object of study is the harmonic map equation with additional right-hand side. More precisely, let \(\Omega\subset\mathbb{R}^n\) be an open domain and \(N\subset\mathbb{R}^k\) be a closed compact manifold. The equation under consideration is \(\Delta u+A(u)(\nabla u,\nabla u)=f\), for \(u:\Omega\to N\), where \(A\) is the second fundamental form of \(N\) and \(f:\Omega\to\mathbb{R}^k\) is of class \(L^p\) with \(p\geq n/2\). For \(p>n/2\) it is proved that weak solutions of the equation are Hölder continuous provided they satisfy a smallness condition for the gradient in some Morrey space. In the case \(n\geq 3\) and \(p=n/2\), one cannot expect continuity of weak solutions. However, in this case an energy inequality (roughly resembling Caccioppoli type estimates) for weak solutions can be proved. In some cases, it can be used to prove partial regularity. One particular application of this inequality is the harmonic map heat flow, i.e.\ we consider the equation with \(f=u_t\) which describes the heat flow on a time slice. If \(n=3\) or \(n=4\), it can be proved that \(u_t\) is in \(L^{n/2}\), and the energy inequality can be exploited to prove that any solution to the harmonic map heat flow satisfying some stability hypothesis is smooth outside a singular set of vanishing \(n\)-dimensional parabolic Hausdorff measure.