Local existence for inhomogeneous Schrödinger flow into Kähler manifolds
Let \((M,g)\) be a Riemannian manifold and \((N,J)\) a symplectic manifold. The inhomogeneous Schrödinger flow from \(M\) to \(N\) is defined by \(\partial_t=\sigma(x)J(u)\tau(u)\nabla\sigma(x)\cdot J(u) du\), where \(\tau\) is the tension field and \(\sigma\) is a positive real function on \(M\). Recently, H. Y. Wang and Y. D. Wang studied the existence of a global solution for the Cauchy problem in some particular cases. Motivated also by the known results on Schrödinger flow and nonlinear Schrödinger equation the authors proved that there exists a local solution for the Cauchy problem of the inhomogeneous Schrödinger flow from ~an \(n\)-dimensional (\(n\leq 3\)) compact Riemannian manifold into a compact Kähler manifold with nonpositive sectional curvature. The main result is: Theorem. Let \(M\) be a closed Riemannian manifold with \(\dim(M)\leq 3\) and let \((N,J)\) be a Kähler manifold with nonpositive sectional curvature. Assume that \(\sigma(x)\in C^4(M)\) and \(\min_{x\in M}|\sigma(x)|> 0\). Then, given the initial map \(u_0\in H^{5,2}(M)\), the Cauchy problem of the inhomogeneous Schrödinger flow from \(M\) into \((N,J)\) admits a unique local solution \(u\in L^{\infty}([0,T), H^{5,2}(M))\).
scientific article; zbMATH DE number 1530093
scientific article