Low regularity solutions for the wave map equation into the 2-D sphere
The authors investigate the well posedeness of the Cauchy problem for the wave map equation \(\partial^{2}_{t}u- \Delta_{x}u+(| \partial_{t}u|^{2} - |\nabla_{x}u|^{2})u=F\) subject to the constraint \(| u |=1\) in two dimensions (\(u:{\mathbb{R}}\times{\mathbb{R}^{2}} \rightarrow {\mathbb{S}^{2}} \subset{\mathbb{R}^{3}}\)). They prove a nonuniqueness result and that in general the solutions may not depend continuously on \(F\). More precisely, using self-similar solutions, they construct two weak solutions of the homogeneous equation which coincide for \(t<0\) and which belong to \(C(\mathbb{R}, H^{1-\varepsilon}(\Omega))\) for all \(\varepsilon>0\) and all bounded \(\Omega\) and to \(L^{\infty}(\mathbb{R}, B^{1}_{2, \infty}(\Omega))\), \(B^{1}_{2, \infty}\) the Besov space, for all bounded \(\Omega\). They also construct functions \(F_{k}\) and weak solutions \(u_{k}\) which satisfy the same initial conditions such that the norm of \(F_{k}\) tends to \(0\) in \(L^{p}([0,T],L^{q}(\mathbb{R}^{2}))\), \(p\geq 1\), \(q\geq 2\), \({1\over{p}}+{2\over{q}}>2\), while the norm of \(u_{k}(T,x)-u(0,x)\) diverges to \(+\infty\) in \(H^{1}(\mathbb{R}^{2})\) for all \(T>0\).
scientific article; zbMATH DE number 2131979