scholarly journals Large data well-posedness in the energy space of the Chern–Simons–Schrödinger system

2018 ◽  
Vol 264 (4) ◽  
pp. 2553-2597 ◽  
Author(s):  
Zhuo Min Lim
Keyword(s):  
Large Data ◽  
Energy Space ◽  
Well Posedness ◽  
Chern Simons ◽  
Schrödinger System

scholarly journals Global well-posedness in energy space for the Chern–Simons–Higgs system in temporal gauge

2016 ◽  
Vol 13 (02) ◽  
pp. 331-351 ◽  
Author(s):  
Hartmut Pecher
Keyword(s):  
Cauchy Problem ◽  
Energy Space ◽  
Well Posedness ◽  
Null Condition ◽  
Temporal Gauge ◽  
Time Estimates ◽  
Dimensional Minkowski Space ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

The Cauchy problem for the Chern–Simons–Higgs system in the [Formula: see text]-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d’Ancona, Foschi and Selberg, an [Formula: see text]-estimate for solutions of the wave equation, and also takes advantage of a null condition.

scholarly journals ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 549-575 ◽  
Author(s):  
J. COLLIANDER ◽  
S. IBRAHIM ◽  
M. MAJDOUB ◽  
N. MASMOUDI
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Supercritical Case ◽  
Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

scholarly journals ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRÖDINGER SYSTEM

2009 ◽  
Vol 51 (3) ◽  
pp. 499-511 ◽  
Author(s):  
LI MA ◽  
XIANFA SONG ◽  
LIN ZHAO
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Nonlinear Schrödinger ◽  
Non Linear ◽  
Schrödinger Systems ◽  
Schrödinger System

AbstractThe non-linear Schrödinger systems arise from many important physical branches. In this paper, employing the ‘I-method’, we prove the global-in-time well-posedness for a coupled non-linear Schrödinger system in Hs(n) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math Res. Lett. 9, 2002, 659–682) to the system.

Global Well-Posedness and Finite-Time Blow-Up of Some Heat-Type Equations

2016 ◽  
Vol 60 (2) ◽  
pp. 481-497 ◽  
Author(s):  
Tarek Saanouni
Keyword(s):  
Heat Equation ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
High Order ◽  
Energy Space ◽  
Well Posedness ◽  
Exponential Nonlinearity

AbstractWe study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

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 well-posedness of the Chern-Simons-Higgs equations with finite energy

2013 ◽  
Vol 33 (6) ◽  
pp. 2531-2546 ◽  
Author(s):  
Sigmund Selberg ◽  
◽  
Achenef Tesfahun
Keyword(s):  
Finite Energy ◽  
Well Posedness ◽  
Chern Simons
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