scholarly journals On a System of Equations of a Non-Newtonian Micropolar Fluid

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
G. M. de Araújo ◽  
M. A. F. de Araújo ◽  
E. F. L. Lucena
Keyword(s):  
Weak Solutions ◽  
Micropolar Fluid ◽  
Dirichlet Boundary ◽  
Coupled System ◽  
Dirichlet Boundary Conditions ◽  
System Of Equations ◽  
Existence Of Weak Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Compactness Arguments

We investigate a problem for a model of a non-Newtonian micropolar fluid coupled system. The problem has been considered in a bounded, smooth domain ofR3with Dirichlet boundary conditions. The operator stress tensor is given byτ(e(u))=[(ν+ν0M(|e(u)|2))e(u)]. To prove the existence of weak solutions we use the method of Faedo-Galerkin and compactness arguments. Uniqueness and periodicity of solutions are also considered.

scholarly journals Weighted Steklov problem under nonresonance conditions

2018 ◽  
Vol 36 (4) ◽  
pp. 87-105
Author(s):  
Jonas Doumatè ◽  
Aboubacar Marcos
Keyword(s):  
Boundary Condition ◽  
Nonlinear Problem ◽  
Weak Solutions ◽  
Existence Results ◽  
Smooth Domain ◽  
Existence Of Weak Solutions ◽  
Steklov Problem ◽  
Bounded Smooth Domain ◽  
Carathéodory Function ◽  
Bounded Smooth

We deal with the existence of weak solutions of the nonlinear problem $-\Delta_{p}u+V|u|^{p-2}u$ in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$ which is subject to the boundary condition $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=f(x,u)$. Here $V\in L^{\infty}(\Omega)$ possibly exhibit both signs which leads to an extension  of particular cases in literature and $f$ is a Carathéodory function that satisfies some additional conditions. Finally we prove, under and between nonresonance condtions, existence results for the problem.

On steady flows of fluids with pressure– and shear–dependent viscosities

2005 ◽  
Vol 461 (2055) ◽  
pp. 651-670 ◽  
Author(s):  
M. Franta ◽  
J. Málek ◽  
K. R. Rajagopal
Keyword(s):  
Shear Rate ◽  
Weak Solutions ◽  
Velocity Gradient ◽  
Dirichlet Boundary ◽  
Incompressible Fluids ◽  
Dirichlet Boundary Conditions ◽  
Steady Flows ◽  
Existence Of Weak Solutions ◽  
Wide Range ◽  
Homogeneous Dirichlet Boundary Conditions

There are many technologically important problems such as elastohydrodynamics which involve the flows of a fluid over a wide range of pressures. While the density of the fluid remains essentially constant during these flows whereby the fluid can be approximated as being incompressible, the viscosity varies significantly by several orders of magnitude. It is also possible that the viscosity of such fluids depends on the shear rate. Here we consider the flows of a class of incompressible fluids with viscosity that depends on the pressure and shear rate. We establish the existence of weak solutions for the steady flows of such fluids subjected to homogeneous Dirichlet boundary conditions and to specific body forces that are not necessarily assumed to be small. A novel aspect of the study is the manner in which we treat the pressure that allows us to establish its compactness, as well as that of the velocity gradient. The method draws upon the physics of the problem, namely that the notion of incompressibility is an idealization that is attained by letting the compressibility of the fluid to tend to zero.

scholarly journals A GRADIENT-LIKE NONAUTONOMOUS EVOLUTION PROCESS

2010 ◽  
Vol 20 (09) ◽  
pp. 2751-2760 ◽  
Author(s):  
TOMÁS CARABALLO ◽  
JOSÉ A. LANGA ◽  
FELIPE RIVERO ◽  
ALEXANDRE N. CARVALHO
Keyword(s):  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Evolution Process ◽  
Pullback Attractor ◽  
Damped Wave Equation ◽  
Bounded Solutions ◽  
Pullback Attractors ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

In this paper we consider a dissipative damped wave equation with nonautonomous damping of the form [Formula: see text] in a bounded smooth domain Ω ⊂ ℝn with Dirichlet boundary conditions, where f is a dissipative smooth nonlinearity and the damping β : ℝ → (0, ∞) is a suitable function. We prove, if (1) has finitely many equilibria, that all global bounded solutions of (1) are backwards and forwards asymptotic to equilibria. Thus, we give a class of examples of nonautonomous evolution processes for which the structure of the pullback attractors is well understood. That complements the results of [Carvalho & Langa, 2009] on characterization of attractors, where it was shown that a small nonautonomous perturbation of an autonomous gradient-like evolution process is also gradient-like. Note that the evolution process associated to (1) is not a small nonautonomous perturbation of any autonomous gradient-like evolution processes. Moreover, we are also able to prove that the pullback attractor for (1) is also a forwards attractor and that the rate of attraction is exponential.

A Note on Symmetry of Solutions for a Class of Singular Semilinear Elliptic Problems

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Alessandro Trombetta
Keyword(s):  
Elliptic Equation ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Dirichlet Boundary Conditions ◽  
Moving Plane Method ◽  
Plane Method ◽  
Semilinear Elliptic ◽  
Moving Plane ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

AbstractWe prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equationin bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

Radiative effects for some bidimensional thermoelectric problems

2016 ◽  
Vol 5 (4) ◽  
Author(s):  
Luisa Consiglieri
Keyword(s):  
Steady State ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Divergence Form ◽  
Radiative Effects ◽  
Existence Of Weak Solutions ◽  
Nonlinear Radiation ◽  
Radiation Type

AbstractThere are two main objectives in this paper. One is to find sufficient conditions to ensure the existence of weak solutions for some bidimensional thermoelectric problems. At the steady-state, these problems consist of a coupled system of elliptic equations of the divergence form, commonly accomplished with nonlinear radiation-type conditions on at least a nonempty part of the boundary of a

scholarly journals Weak Solutions of a Coupled System of Urysohn-Stieltjes Functional (Delayed) Integral Equations

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
M. M. A. Al-Fadel
Keyword(s):  
Banach Space ◽  
Integral Equations ◽  
Weak Solutions ◽  
Reflexive Banach Space ◽  
Coupled System ◽  
Functional Integral ◽  
Existence Of Weak Solutions ◽  
Functional Integral Equations ◽  
Stieltjes Type

We study the existence of weak solutions for the coupled system of functional integral equations of Urysohn-Stieltjes type in the reflexive Banach spaceE. As an application, the coupled system of Hammerstien-Stieltjes functional integral equations is also studied.

scholarly journals Weak Solutions for ap-Laplacian Impulsive Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Wei Dong ◽  
Jiafa Xu ◽  
Xiaoyan Zhang
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Boundary Conditions ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Impulsive Differential Equation ◽  
Dirichlet Boundary ◽  
Point Theory ◽  
Existence Results ◽  
Dirichlet Boundary Conditions

By the virtue of variational method and critical point theory, we give some existence results of weak solutions for ap-Laplacian impulsive differential equation with Dirichlet boundary conditions.

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