scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

Attractors for the modifications of the three-dimensional Navier-Stokes equations

1994 ◽  
Vol 346 (1679) ◽  
pp. 173-190 ◽  
Keyword(s):  
Viscous Fluid ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Initial Boundary Value ◽  
Dimensional Domain

The modifications of the three-dimensional Navier-Stokes equations, which I suggested earlier for the description of viscous fluid flows with large gradients of velocities, are considered. It is proved that the first initial-boundary value problem for these equations in any bounded three-dimensional domain has a compact minimal global B-attractor. Some properties of the attractor are established.

scholarly journals Existence and long-time behavior of solutions to the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory

2021 ◽  
Author(s):  
Nguyen Toan
Keyword(s):  
Stokes Equations ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Time Behavior ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Voigt Model ◽  
Initial Boundary Value

In this paper, we study the long-time dynamical behavior of the non-autonomous velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory. We first investigate the existence and uniqueness of weak solutions to the initial boundary value problem for above-mentioned model. Next, we prove the existence of uniform attractor of this problem, where the time-dependent forcing term $f \in L^2_b(\mathbb{R}; H^{-1}(\Omega))$ is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in Yue, Wang (Comput. Math. Appl., 2020) in the case of non-autonomous and contain memory kernels which have not been studied before.

scholarly journals Global Well-posedness and Asymptotics of Full Compressible Non-resistive MHD System with Large External Potential Forces

2021 ◽  
Author(s):  
Wanrong Yang ◽  
Xiaokui Zhao
Keyword(s):  
Stationary Solution ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Unique Existence ◽  
Background Magnetic Field ◽  
External Forces ◽  
Mhd System ◽  
Initial Boundary Value

We consider the global well-posedness and asymptotic behavior of compressible viscous, heat-conductive, and non-resistive magnetohydrodynamics (MHD) fluid in a field of external forces over three-dimensional periodic thin domain $\Omega=\mathbb{T}^2\times(0,\delta)$. The unique existence of the stationary solution is shown under the adhesion and the adiabatic boundary conditions. Then, it is shown that a solution to the initial boundary value problem with the same boundary and periodic conditions uniquely exists globally in time and converges to the stationary solution as time tends to infinity. Moreover, if the external forces are small or disappeared in an appropriate Sobolev space, then $\delta$ can be a general constant. Our proof relies on the two-tier energy method for the reformulated system in Lagrangian coordinates and the background magnetic field which is perpendicular to the flat layer. Compared to the work of Tan and Wang (SIAM J. Math. Anal. 50:1432–1470, 2018), we not only overcome the difficulties caused by temperature, but also consider the big external forces.

Unsteady three-dimensional sources in deep water with an elastic cover and their applications

2013 ◽  
Vol 730 ◽  
pp. 392-418 ◽  
Author(s):  
Izolda V. Sturova
Keyword(s):  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniform Density ◽  
Forward Speed ◽  
Submerged Sphere ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Long Time ◽  
Initial Boundary Value

AbstractThe velocity potential is derived for a transient source of arbitrary strength undergoing arbitrary three-dimensional motion. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogeneous. The upper surface of the fluid is covered by a thin layer of elastic material of uniform density with lateral stress. The linearized initial boundary-value problem is formulated within the framework of the potential-flow theory, and the Laplace transform technique is employed to obtain the solution. The potential of a time-harmonic source with forward speed is obtained as a particular case. The far-field wave motion at long time is determined via the method of stationary phase. The problems of radiation (surge, sway and heave) of the flexural–gravity waves by a submerged sphere advancing at constant forward speed are investigated. The method of multipole expansions is used. Numerical results are obtained for the wave-making resistance and lift, added-mass and damping coefficients. The effects of an ice sheet and broken ice on the hydrodynamic loads are discussed in detail.

scholarly journals Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jian Liu ◽  
Ruxu Lian
Keyword(s):  
Strong Solution ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cylindrically Symmetric ◽  
Initial Boundary Value ◽  
Dependent Viscosity

This paper is concerned with the initial boundary value problem for the three-dimensional Navier-Stokes equations with density-dependent viscosity. The cylindrically symmetric strong solution is shown to exist globally in time and tend to the equilibrium state exponentially as time grows up.

scholarly journals Solvability of a problem for the equations of dynamics of one-temperature mixtures of heat-conducting viscous compressible fluids

2019 ◽  
Vol 486 (2) ◽  
pp. 159-162
Author(s):  
A. E. Mamontov ◽  
D. A. Prokudin
Keyword(s):  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Natural Generalization ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Component Mixture ◽  
Initial Boundary Value ◽  
Fourier Model

A system of partial differential equations governing the three-dimensional unsteady flow of a homogeneous two-component mixture of heat-conducting viscous compressible fluids (gases) is considered within the multivelocity approach. The model is complete in the sense that it retains all terms in the equations, which are a natural generalization of the Navier-Stokes-Fourier model for the motion of a single-component medium. The existence of weak solutions to the initial-boundary value problem describing the flow in a bounded domain is proved globally in time and the input data.

scholarly journals Axisymmetric flows in the exterior of a cylinder

2019 ◽  
Vol 150 (4) ◽  
pp. 1671-1698 ◽  
Author(s):  
K. Abe ◽  
G. Seregin
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Initial Boundary Value

AbstractWe study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$.

Pull-back attractors for three-dimensional Navier—Stokes—Voigt equations in some unbounded domains

2013 ◽  
Vol 143 (2) ◽  
pp. 223-251 ◽  
Author(s):  
Cung The Anh ◽  
Pham Thi Trang
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Three Dimensional ◽  
Unbounded Domains ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Uniform Attractor ◽  
Finite Dimensional ◽  
Initial Boundary Value

We study the first initial–boundary-value problem for the three-dimensional non-autonomous Navier–Stokes–Voigt equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Faedo–Galerkin method. We then show the existence of a unique minimal finite-dimensional pull-back $\smash{\mathcal D_\sigma}$-attractor for the process associated with the problem, with respect to a large class of non-autonomous forcing terms. We also discuss relationships between the pull-back attractor, the uniform attractor and the global attractor.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

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