scholarly journals MODELING OF FLOWS THROUGH A CHANNEL BY THE NAVIER–STOKES VARIATIONAL INEQUALITIES

2021 ◽  
Vol 61 (SI) ◽  
pp. 89-98
Author(s):  
Stanislav Kračmar ◽  
Jiří Neustupa
Keyword(s):  
Boundary Condition ◽  
Variational Inequalities ◽  
A Priori Estimates ◽  
A Priori ◽  
Energy Inequality ◽  
Point Of View ◽  
Navier Stokes ◽  
Artificial Boundary ◽  
Physical Point ◽  
Artificial Boundary Conditions

We deal with a mathematical model of a flow of an incompressible Newtonian fluid through a channel with an artificial boundary condition on the outflow. We explain how several artificial boundary conditions formally follow from appropriate variational formulations and the wayone expresses the dynamic stress tensor. As the boundary condition of the “do nothing”–type, that is predominantly considered to be the most appropriate from the physical point of view, does not enable one to derive an energy inequality, we explain how this problem can be overcome by using variational inequalities. We derive a priori estimates, which are the core of the proofs, and present theorems on the existence of solutions in the unsteady and steady cases.

scholarly journals An Energy Inequality and Its Applications of Nonlocal Boundary Conditions of Mixed Problem for Singular Parabolic Equations in Nonclassical Function Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Moussa Zakari Djibibe ◽  
Kokou Tcharie ◽  
N. Iossifovich Yurchuk
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
A Priori Estimates ◽  
A Priori ◽  
Energy Inequality ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Function Spaces ◽  
Priori Estimates ◽  
Singular Parabolic Equations

The aim of this paper is to establish a priori estimates of the following nonlocal boundary conditions mixed problem for parabolic equation: ∂v/∂t-(a(t)/x2)(∂/∂x)(x2∂v/∂x)+b(x,t)v=g(x,t), v(x, 0)=ψ(x), 0≤x≤ℓ, v(ℓ, t)=E(t), 0≤t≤T, ∫0ℓx3v(x,t)dx=G(t), 0≤t≤ℓ. It is important to know that a priori estimates established in nonclassical function spaces is a necessary tool to prove the uniqueness of a strong solution of the studied problems.

The non-Lipschitz stochastic Cahn–Hilliard–Navier–Stokes equations in two space dimensions

2021 ◽  
pp. 2250003
Author(s):  
Chengfeng Sun ◽  
Qianqian Huang ◽  
Hui Liu
Keyword(s):  
A Priori Estimates ◽  
Stokes Equations ◽  
A Priori ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Priori Estimates ◽  
Convergence Method ◽  
Two Space Dimensions ◽  
Lipschitz Conditions ◽  
Weak Convergence Method

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained.

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