GLOBAL INHOMOGENEOUS SCHRÖDINGER FLOW

2000 ◽  
Vol 11 (08) ◽  
pp. 1079-1114 ◽  
Author(s):  
HONG-YU WANG ◽  
YOU-DE WANG
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Spin System ◽  
Holomorphic Sectional Curvature ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Schrodinger Flow ◽  
The One ◽  
Complete Kähler Manifold

In this paper, we consider the global existence of one-dimensional nonautonomous inhomogeneous Schrödinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the one-dimensional nonautonomous inhomogeneous Schrödinger flow from S1 into a complete Kähler manifold with constant holomorphic sectional curvature admits a unique global smooth solution. As a corollary, we establish the global existence for the Cauchy problem of the inhomogeneous Heisenberg spin system.

Classical solutions to a dissipative hyperbolic geometry flow in two space variables

2019 ◽  
Vol 16 (02) ◽  
pp. 223-243
Author(s):  
De-Xing Kong ◽  
Qi Liu ◽  
Chang-Ming Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Scalar Curvature ◽  
Hyperbolic Geometry ◽  
Existence Of Solutions ◽  
Classical Solutions ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
The One ◽  
Uniformly Bounded

We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.

scholarly journals Global existence of small amplitude solutions to one-dimensional nonlinear Klein–Gordon systems with different masses

2015 ◽  
Vol 12 (04) ◽  
pp. 745-762 ◽  
Author(s):  
Donghyun Kim
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Small Amplitude ◽  
Space Dimension ◽  
Structural Condition ◽  
Cauchy Data ◽  
One Dimensional ◽  
Compactly Supported ◽  
The Cauchy Problem ◽  
Klein Gordon

We study the Cauchy problem for systems of cubic nonlinear Klein–Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the rate [Formula: see text] in [Formula: see text], [Formula: see text] as [Formula: see text] tends to infinity even in the case of mass resonance, if the Cauchy data are sufficiently small, smooth and compactly supported.

scholarly journals On non-resistive limit of 1D MHD equations with no vacuum at infinity

2021 ◽  
Vol 11 (1) ◽  
pp. 702-725
Author(s):  
Zilai Li ◽  
Huaqiao Wang ◽  
Yulin Ye
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Strong Solutions ◽  
Mhd Equations ◽  
Well Posedness ◽  
Large Initial Data ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Global Strong Solutions ◽  
The One

Abstract In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

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