scholarly journals Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions

2007 ◽  
Vol 48 (10) ◽  
pp. 102105 ◽  
Author(s):  
Chengchun Hao ◽  
Ling Hsiao ◽  
Hai-Liang Li
Keyword(s):  
Angular Momentum ◽  
Three Dimensions ◽  
Pitaevskii Equation ◽  
Well Posedness ◽  
Rotational Term

Phase separation in spin-orbit-angular-momentum coupling Bose–Einstein condensate systems

2020 ◽  
Vol 34 (23) ◽  
pp. 2050241
Author(s):  
Jin Xu ◽  
Jinbin Li
Keyword(s):  
Angular Momentum ◽  
Phase Separation ◽  
Coupling Strength ◽  
Interaction Strength ◽  
Einstein Condensate ◽  
Three Dimensions ◽  
Pitaevskii Equation ◽  
Spin Orbit ◽  
Bose Einstein Condensate ◽  
Bose Einstein

We study the phase separation in three-component spin-orbit-angular-momentum coupled Bose–Einstein condensate with spin-1 in three dimensions. Different types of phase-separation are acquired upon an increase of the coupling strength, magnetic gradient strength, spin-dependent interaction strength and particle number above a critical value. Increasing the value of coupling strength and other related parameters shows distinct behaviors which are produced by repulsion for large strengths of spin-orbit angular-momentum (SOAM) coupling. The present investigation is carried out through a numerical Crank–Nicolson method of the underlying mean-field Gross–Pitaevskii equation.

scholarly journals Global Dispersive Solutions for the Gross–Pitaevskii Equation in Two and Three Dimensions

2007 ◽  
Vol 8 (7) ◽  
pp. 1303-1331 ◽  
Author(s):  
Stephen Gustafson ◽  
Kenji Nakanishi ◽  
Tai-Peng Tsai
Keyword(s):  
Three Dimensions ◽  
Pitaevskii Equation

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

2018 ◽  
Vol 2019 (21) ◽  
pp. 6797-6817
Author(s):  
Benjamin Dodson
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Three Dimensions ◽  
Well Posedness ◽  
Critical Space ◽  
Initial Value ◽  
Long Time ◽  
Well Posed

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s> \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

scholarly journals Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Xin Liu ◽  
Edriss S. Titi
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Physical Vacuum ◽  
Well Posedness ◽  
Small Initial Data ◽  
Anelastic Equations

Abstract We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

scholarly journals Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis

2017 ◽  
Vol 29 (4) ◽  
pp. 595-644 ◽  
Author(s):  
KEI FONG LAM ◽  
HAO WU
Keyword(s):  
Mass Transfer ◽  
Chemical Potential ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Fluid Viscosity ◽  
Well Posedness ◽  
Global Weak Solutions ◽  
Two Phase ◽  
Model Variant

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

scholarly journals SCATTERING THEORY FOR THE GROSS–PITAEVSKII EQUATION IN THREE DIMENSIONS

2009 ◽  
Vol 11 (04) ◽  
pp. 657-707 ◽  
Author(s):  
STEPHEN GUSTAFSON ◽  
KENJI NAKANISHI ◽  
TAI-PENG TSAI
Keyword(s):  
Scattering Theory ◽  
Fourier Multiplier ◽  
Large Time ◽  
Finite Energy ◽  
Three Dimensions ◽  
Pitaevskii Equation ◽  
Global Behavior ◽  
Fourier Space ◽  
Resonance Property ◽  
Nonlinear Energy

We study the global behavior of small solutions of the Gross–Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.

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