Nonuniqueness of the heat flow of director fields (Q969250)

The author considers a new example of nonuniqueness of the weak solution of the initial-boundary value problem for the heat flow of the harmonic maps from an infinitely long vertical cylinder in \(\mathbb{R}^3\) to the unit sphere \(S^2\) in \(\mathbb{R}^3\). He constructs a ``minimal solution'' \(h_{\min}\) of the equation: \[ h_t= h_{rr}+ h_{zz}+ {h_r\over r}- {\sin 2h\over 2r^2},\quad r\in (0,1),\;z\in\mathbb{R}^1,\;t\in \mathbb{R}_+, \] which satisfies the following: \(h_{\min}(0,z,t)\equiv 0\) if \(t< T\), \(\limsup_{t\to T-0}\|\nabla h_{\min}(\cdot,\cdot,t)\|_\infty= \infty\), and for \(t\geq T\), \(h_{\min}(0,z,t)= 0\) if \(|z|> \zeta(t)\), and \(= \pi\) if \(|z|< \zeta(t)\), where \(\zeta: [T,\infty)\to [0,\infty)\) is an increasing function. Also he constructs a family of nonnegative weak solutions \(h_M\), for \(M\geq 0\) satisfying the following: \(h_M\geq h_{\min}\), and for \(t> 0\), \(h_M(0,z,t)= 0\), if \(|z|> \zeta_M(t)\), and \(=\pi\) if \(|z|< \zeta_M(t)\), where \(\zeta_M: [0,\infty)\to [0,\infty)\) is an increasing function, which satisfies, for a suitable constant \(S> 0\), independent of \(M\), \[ M\leq\zeta_M(t)\leq S+ M+ t\quad\text{for }t> 0. \] This example confirms a connection between nonuniqueness of axially symmetric solutions for the heat flow of harmonic maps and the occurrence of singularities in the solution deeper than previous examples in [\textit{M. Bertsch}, \textit{R. Dal Passo} and \textit{A. Pisante}, Commun. Partial Differ. Equations 28, No. 5--6, 1135-1160 (2003; Zbl 1029.58008) and \textit{A. Pisante}, Calc. Var. Partial Differ. Equ. 19, No. 4, 337--378 (2004; Zbl 1059.58010)].