Local existence for inhomogeneous Schrödinger flow into Kähler manifolds
type: Item
instance of: scholarly article
title: Local existence for inhomogeneous Schrödinger flow into Kähler manifolds
zbMATH Open document ID: 0974.58034
author: Peter Y. H. Pang
author: Hongyu Wang
author: Youde Wang
publication date: 2001-11-30T00:00:00Z
review text: 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 \(u\in L^{\infty}([0,T), H^{5,2}(M))\).
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 1530093
zbMATH Keywords: inhomogeneous Schrödinger flow
zbMATH Keywords: Kähler manifold
zbMATH Keywords: local existence
reviewed by: Novica Blažić
MaRDI profile type: Publication
full work available at URL: s101140000060
OpenAlex ID: W1533709525
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DOI: 10.1007/S101140000060
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1530093