scholarly journals Ill-posedness of the Cauchy problem for the Chern–Simons–Dirac system in one dimension

2015 ◽  
Vol 258 (4) ◽  
pp. 1356-1394 ◽  
Author(s):  
Shuji Machihara ◽  
Mamoru Okamoto
Keyword(s):  
Cauchy Problem ◽  
One Dimension ◽  
Dirac System ◽  
The Cauchy Problem ◽  
Chern Simons

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.

THE IVP FOR THE SCHRÖDINGER–POISSON-Xα EQUATION IN ONE DIMENSION

2005 ◽  
Vol 15 (08) ◽  
pp. 1169-1180 ◽  
Author(s):  
H. P. STIMMING
Keyword(s):  
Cauchy Problem ◽  
Quantum System ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Semiclassical Limit ◽  
Semigroup Theory ◽  
One Dimension ◽  
The Cauchy Problem ◽  
Electron Quantum ◽  
Particle Approximation

The Schrödinger–Poisson-Xα equation is an effective one-particle approximation of a many-electron quantum system. In space dimension d<3, existence analysis for this equation is not contained in standard results for nonlinear Schrödinger equations. We obtain existence and uniqueness of the Cauchy problem in d = 1 using semigroup theory. Furthermore, we discuss the semiclassical limit.

On the Cauchy problem for one dimension generalized Boussinesq equation

2015 ◽  
Vol 26 (03) ◽  
pp. 1550023 ◽  
Author(s):  
Yinxia Wang
Keyword(s):  
Cauchy Problem ◽  
Boussinesq Equation ◽  
Nonlinear Approximation ◽  
Global Solutions ◽  
One Dimension ◽  
Diffusion Waves ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Nonlinear Diffusion Waves ◽  
Generalized Boussinesq Equation

In this paper, we study the Cauchy problem for one dimension generalized damped Boussinesq equation. First, global existence and decay estimate of solutions to this problem are established. Second, according to the detail analysis for solution operator the generalized damped Boussinesq equation, the nonlinear approximation to global solutions is established. Finally, we prove that the global solution u to our problem is asymptotic to the superposition of nonlinear diffusion waves expressed in terms of the self-similar solution of the viscous Burgers equation as time tends to infinity.

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.

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