Well-posedness of the Cauchy problem for the Maxwell–Dirac system in one space dimension

2019 ◽  
Author(s):  
Mamoru Okamoto
Keyword(s):  
Cauchy Problem ◽  
Space Dimension ◽  
Well Posedness ◽  
Dirac System ◽  
The Cauchy Problem ◽  
One Space Dimension

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.

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.

Large Time Behavior for the Cubic Nonlinear Schrödinger Equation

2002 ◽  
Vol 54 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Space Dimension ◽  
Nonlinear Schrodinger Equation ◽  
The Cauchy Problem ◽  
Cubic Nonlinear Schrödinger Equation ◽  
One Space Dimension

AbstractWe consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimensionCubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data are small and such that for some n ∈ Z, and , then the solution has an additional logarithmic timedecay in the short range region . In the far region the asymptotics have a quasilinear character.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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