Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

2021 ◽  
pp. 1-13
Author(s):  
Kita Naoyasu ◽  
Sato Takuya
Keyword(s):  
Sobolev Space ◽  
Global Solution ◽  
Initial Value Problem ◽  
Trivial Solution ◽  
Decay Estimate ◽  
Weighted Sobolev Space ◽  
The Other ◽  
Initial Value ◽  
Decay Of Solutions ◽  
Dissipative Nonlinearity

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

Decay of solutions to a two-layer quasi-geostrophic model

2017 ◽  
Vol 15 (04) ◽  
pp. 595-606 ◽  
Author(s):  
Boling Guo ◽  
Daiwen Huang ◽  
Jingjun Zhang
Keyword(s):  
Fluid Dynamics ◽  
Galerkin Method ◽  
Decay Rate ◽  
Global Solution ◽  
Initial Value Problem ◽  
Splitting Method ◽  
Geophysical Fluid Dynamics ◽  
Initial Value ◽  
Decay Of Solutions

We consider a two-layer quasi-geostrophic model in geophysical fluid dynamics. By Faedo–Galerkin method and asymptotic argument, we prove the existence of the global solution to the initial value problem of this model in [Formula: see text]. Moreover, using the Fourier splitting method, we also obtain the decay rate of the solutions.

Stability of time-dependent rotational Couette flow Part 2. Stability analysis

1971 ◽  
Vol 48 (2) ◽  
pp. 365-384 ◽  
Author(s):  
C. F. Chen ◽  
R. P. Kirchner
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Kinetic Energy ◽  
Initial Value Problem ◽  
Basic Flow ◽  
The Other ◽  
Time Dependent ◽  
Stability Criteria ◽  
Initial Value ◽  
The Stability

The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results.

A Novel Method for Solving Consistency-Order Initial Value of Ordinary Differential Equations

2014 ◽  
Vol 543-547 ◽  
pp. 1844-1847
Author(s):  
Si Min Zhu ◽  
Hai Yun Deng ◽  
Kai Zheng ◽  
Hua Mei Li ◽  
Xiao Zhou Chen
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Integral Formula ◽  
Absolute Error ◽  
The Other ◽  
It Use ◽  
Initial Value ◽  
Novel Method ◽  
The One ◽  
Legendre Quadrature

It is known that the level of the consistency-order of initial value problem is an important standard to determine whether the constructed methods for solving initial value problem of ODEs is suitable or not. There are two methods to solve the consistency-order of initial value problem in general. The one is using the remainder of integral formula as local truncated error, and the other one is using absolute error as local truncated error. In the paper, we propose a novel method based on Gauss-Legendre quadrature formula. It use the method of the remainder of integral formula as local truncated error exists in most of the literatures, and it will be solved once again for the consistency-order of the constructed methods that exist in currently literatures by using absolute error as local truncated error, and then draw a conclusion that is differ from what has been proved correspondingly.

An iteration method for a nonlinear differential equation describing the convergence to equilibrium in a system of reacting polymers

1981 ◽  
Vol 375 (1763) ◽  
pp. 529-542
Keyword(s):  
Differential Equation ◽  
Initial Data ◽  
Global Solution ◽  
Nonlinear Differential Equation ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Constructive Method ◽  
Convergence To Equilibrium ◽  
Initial Value

A constructive method is presented to give the global solution to a nonlinear initial value problem describing the convergence to equilibrium in a system of reacting polymers. The solution is proved to be unique and continuous with respect to small variations in the initial data.

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