scholarly journals Pullback attractors for 2D MHD equations on time-varying domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Daomin Cao ◽  
Xiaoya Song ◽  
Chunyou Sun
Keyword(s):  
Weak Solutions ◽  
Dirichlet Boundary Conditions ◽  
Mhd Equations ◽  
Pullback Attractor ◽  
Time Varying ◽  
Cylindrical Domains ◽  
Spatial Domains ◽  
Asymptotic Dynamics ◽  
Definition Of ◽  
Homogeneous Dirichlet Boundary Conditions

<p style='text-indent:20px;'>In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula> and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.</p>

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 Large Time Existence of Special Strong Solutions to MHD Equations in Cylindrical Domains

2017 ◽  
Vol 20 (3) ◽  
pp. 1013-1034
Author(s):  
Bernard Nowakowski ◽  
Gerhard Ströhmer ◽  
Wojciech M. Zaja̧czkowski
Keyword(s):  
Large Time ◽  
Strong Solutions ◽  
Mhd Equations ◽  
Cylindrical Domains ◽  
Time Existence

scholarly journals An r-Order Finite-Time State Observer for Reaction-Diffusion Genetic Regulatory Networks with Time-Varying Delays

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaofei Fan ◽  
Yantao Wang ◽  
Ligang Wu ◽  
Xian Zhang
Keyword(s):  
Finite Time ◽  
Regulatory Networks ◽  
Reaction Diffusion ◽  
State Observer ◽  
Genetic Regulatory Networks ◽  
Dirichlet Boundary Conditions ◽  
Time Varying ◽  
Error System ◽  
Delay Dependent ◽  
T State

It will be settled out for the open problem of designing an r-order finite-time (F-T) state observer for reaction-diffusion genetic regulatory networks (RDGRNs) with time-varying delays. By assuming the Dirichlet boundary conditions, aiming to estimate the mRNA and protein concentrations via available network measurements. Firstly, sufficient F-T stability conditions for the filtering error system have been investigated via constructing an appropriate Lyapunov–Krasovskii functional (LKF) and using several integral inequalities and (reciprocally) convex technique simultaneously. These conditions are delay-dependent and reaction-diffusion-dependent and can be checked by MATLAB toolbox. Furthermore, a method is proposed to design an r-order F-T state observer, and the explicit expressions of observer gains are given. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.

Some Elements of Functional Analysis

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Functional Analysis ◽  
Weak Solutions ◽  
Vector Fields ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dual Spaces ◽  
Type Spaces ◽  
Definition Of

Before introducing the concept of Leray’s weak solutions to the incompressible Navier–Stokes equations, classical definitions of Sobolev spaces are required. In particular, when it comes to the analysis of the Stokes operator, suitable functional spaces of incompressible vector fields have to be defined. Several issues regarding the associated dual spaces, embedding properties, and the mathematical way of considering the pressure field are also discussed. Let us first recall the definition of some functional spaces that we shall use throughout this book. In the framework of weak solutions of the Navier– Stokes equations, incompressible vector fields with finite viscous dissipation and the no-slip property on the boundary are considered. Such H1-type spaces of incompressible vector fields, and the corresponding dual spaces, are important ingredients in the analysis of the Stokes operator.

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