scholarly journals Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Hyungjin Huh ◽  
Bora Moon
Keyword(s):  
Cauchy Problem ◽  
Total Energy ◽  
Global Solution ◽  
Gauge Condition ◽  
Global Energy ◽  
Nirenberg Inequality ◽  
The Cauchy Problem ◽  
Neutral Field ◽  
Chern Simons ◽  
Uniqueness Of A Solution

We study the Cauchy problem of the Chern-Simons-Schrödinger equations with a neutral field, under the Coulomb gauge condition, in energy space H1(R2). We prove the uniqueness of a solution by using the Gagliardo-Nirenberg inequality with the specific constant. To obtain a global solution, we show the conservation of total energy and find a bound for the nondefinite term.

scholarly journals The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Hyungjin Huh ◽  
Jihyun Yim
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Space Dimension ◽  
Space Time ◽  
Gauge Condition ◽  
Existence Of Solution ◽  
Temporal Gauge ◽  
The Cauchy Problem ◽  
Temporal And Spatial ◽  
One Space Dimension

We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition A1=0 and the temporal gauge condition A0=0.

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE CHERN–SIMONS–DIRAC SYSTEM IN TWO DIMENSIONS

2013 ◽  
Vol 10 (04) ◽  
pp. 735-771 ◽  
Author(s):  
MAMORU OKAMOTO
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Sobolev Space ◽  
Gauge Invariance ◽  
Two Dimensions ◽  
Well Posedness ◽  
Dirac System ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.

scholarly journals Global Analysis for Rough Solutions to the Davey-Stewartson System

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Han Yang ◽  
Xiaoming Fan ◽  
Shihui Zhu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Equation ◽  
Global Solution ◽  
Global Analysis ◽  
Space Time ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
New Space ◽  
Time Bounds

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

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