Schrödinger-Poisson systems in dimension d ≦ 3: The whole-space case

1993 ◽  
Vol 123 (6) ◽  
pp. 1179-1201 ◽  
Author(s):  
F. Nier
Keyword(s):  
Variational Formulation ◽  
Uniqueness Result ◽  
Periodic Case ◽  
Small Data ◽  
Bounded Domains ◽  
Paper Briefly ◽  
Existence Of A Solution ◽  
Whole Space ◽  
Space Case

SynopsisAfter a study made for bounded domains and in the periodic case, we investigate the variational formulation of Schrödinger-Poisson systems set on the whole space ℝd,d≦ 3. This variational formulation leads to a uniqueness result, while the existence of a solution is proved only for ‘small data’ because of the lack of coerciveness. The end of this paper briefly presents the extension of this formalism to a physically relevant problem where the potential is periodic in one direction.

scholarly journals Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components

2019 ◽  
Vol 9 (1) ◽  
pp. 633-643
Author(s):  
Hugo Beirão da Veiga ◽  
Jiaqi Yang
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Whole Space ◽  
Space Case

Abstract H.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria for the incompressible Navier-Stokes equations in the whole space ℝ3 based on two velocity components. Recently, one of the present authors extended this result to the half-space case $\begin{array}{} \displaystyle \mathbb{R}^3_+ \end{array}$. Further, this author in collaboration with J. Bemelmans and J. Brand extended the result to cylindrical domains under physical slip boundary conditions. In this note we obtain a similar result in the case of smooth arbitrary boundaries, but under a distinct, apparently very similar, slip boundary condition. They coincide just on flat portions of the boundary. Otherwise, a reciprocal reduction between the two results looks not obvious, as shown in the last section below.

scholarly journals Parabolic elliptic type Keller-Segel system on the whole space case

2015 ◽  
Vol 36 (2) ◽  
pp. 1061-1084 ◽  
Author(s):  
Jinhuan Wang ◽  
Li Chen ◽  
Liang Hong
Keyword(s):  
Elliptic Type ◽  
Whole Space ◽  
Space Case

ON THE EQUATIONS GOVERNING THE FLOW OF MECHANICALLY INCOMPRESSIBLE, BUT THERMALLY EXPANSIBLE, VISCOUS FLUIDS

2008 ◽  
Vol 18 (06) ◽  
pp. 813-857 ◽  
Author(s):  
ANGIOLO FARINA ◽  
ANTONIO FASANO ◽  
ANDRO MIKELIĆ
Keyword(s):  
Stationary Flow ◽  
Boussinesq Approximation ◽  
Momentum Equation ◽  
Higher Order ◽  
Order Correction ◽  
Boussinesq System ◽  
Small Data ◽  
Heat Conducting ◽  
Existence Of A Solution ◽  
Advection Diffusion Equation

In this paper, the stationary flow of a heat conducting viscous fluid, which is mechanically incompressible but thermally expansible is studied. The flow takes place in a bounded domain and the discharge is prescribed. The thermodynamical modeling of this situation is discussed first. Then the stationary model with zero Eckert number and prove existence of a solution is studied. Using these results, the Oberbeck–Boussinesq system obtained in the zero expansivity limit is proved. Next, uniqueness for small data and the regularity of the weak solutions is proved. For the unique regular solution the higher order corrections for Boussinesq' approximation and we constructed the error estimate with respect to the thermal expansivity coefficient is proved. The next order correction in this limit is an Oseen type momentum equation coupled with a linear advection/diffusion equation for the temperature. Such higher order correction seems to be new in the literature.

Remarks on the Reynolds problem of elastohydrodynamic lubrication

1993 ◽  
Vol 4 (1) ◽  
pp. 83-96 ◽  
Author(s):  
José-Francisco Rodrigues
Keyword(s):  
Free Boundary ◽  
A Priori ◽  
Elastohydrodynamic Lubrication ◽  
Small Data ◽  
Existence Of A Solution ◽  
Pressure Dependent Viscosity ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Dependent Viscosity ◽  
The Mathematical Model

The mathematical model of the flow of a viscous lubricant between elastic bearings leads to the study of a highly non-linear and non-local elliptic variational inequality. We discuss the existence of a solution by using an a prioriL∞-estimate. This method allows us to solve a large class of problems, including those arising from the linear Hertzian theory, and yields new existence results for the cases of a pressure-dependent viscosity or the inclusion of a load constraint. For small data the uniqueness of the solution holds, and we show that in the cylindrical journal bearing problem with small eccentricity ratio, the free boundary is given by two disjoint differentiable arcs close to the free boundary of the first-order approximate solution.

scholarly journals A HISTORY-DEPENDENT FRICTIONAL CONTACT PROBLEM WITH WEAR FOR THERMOVISCOELASTIC MATERIALS

2019 ◽  
Vol 24 (3) ◽  
pp. 351-371
Author(s):  
Lamia Chouchane ◽  
Lynda Selmani
Keyword(s):  
Contact Problem ◽  
Thermal Effects ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Viscoelastic Body ◽  
Perturbation Parameter ◽  
Convergence Result ◽  
Uniqueness Result ◽  
Quasivariational Inequalities ◽  
Wear And Friction

In this manuscript we study a contact problem between a deformable viscoelastic body and a rigid foundation. Thermal effects, wear and friction between surfaces are taken into account. A variational formulation of the problem is supplied and an existence and uniqueness result is proved. The idea of the proof rested on a recent result on history-dependent quasivariational inequalities. Finally, a perturbation of the data is initiated and a convergence result is demonstrated when the perturbation parameter converges to zero.

Multiple Solitary Waves For a Non-homogeneous Schrödinger–Maxwell System in ℝ3

2006 ◽  
Vol 6 (2) ◽  
Author(s):  
A. Salvatore
Keyword(s):  
Eigenvalue Problem ◽  
Solitary Waves ◽  
Variational Formulation ◽  
Maxwell’S Equations ◽  
Schrodinger Equation ◽  
Maxwell's Equations ◽  
Standing Waves ◽  
Bounded Domains ◽  
Maxwell System ◽  
Fibering Method

AbstractWe look for standing waves of nonlinear Schrödinger equationcoupled with Maxwell’s equations. We use the variational formulation introduced by Benci and Fortunato in 1992 for studying an eigenvalue problem for the Schrödinger-Maxwell system in bounded domains. We establish the existence of multiple standing waves both in the homogeneous and the non-homogeneous cases by means of the fibering method introduced by Pohozaev.

On the existence of a solution for an adsorption dynamic model with the Langmuir isotherm

2014 ◽  
Vol 25 (5) ◽  
pp. 629-653 ◽  
Author(s):  
J. R. FERNÁNDEZ ◽  
M. C. MUÑIZ ◽  
C. NÚÑEZ
Keyword(s):  
Langmuir Isotherm ◽  
A Priori ◽  
Time Discretization ◽  
Surface Concentration ◽  
Uniqueness Result ◽  
Discrete Problem ◽  
Adsorption Model ◽  
Adsorption Dynamic ◽  
Existence Of A Solution ◽  
Dynamical Boundary Condition

In this paper, we study an adsorption model arising in the dynamics of several surfactants at the air-water interface, where the Langmuir isotherm is employed for modelling the time-dependent surface concentration, providing a nonlinear dynamical boundary condition. Existence of a weak solution is proved by using the Rothe method for a semi-discrete problem in time. After obtaining some a priori estimates and passing to the limit in the time discretization parameter, we conclude that the original Langmuir problem has a bounded solution. An uniqueness result is also given.

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