Settings for the Stokes Problem As a first step to considering the Navier-Stokes equations, we will consider various settings for the Stokes problem of finding u⃗,p which satisfy −ν∇2u⃗ + ∇p = f⃗ in U ⊂ Rn 1 div u⃗ = 0. We could consider this system of equations on: 1. a. U = Rn : in this case we would have to impose some ”behavior at ∞" conditions such as u⃗ → ψ⃗ as | x⃗| → ∞ b. U ⊂ Rn, open bounde...

Abstract: In this paper, we present an algorithm for stability analysis of systems described by coupled linear Partial Differential Equations (PDEs) with constant coefficients and mixed boundary conditions. Our approach uses positive matrices to parameterize functionals which are positive or negative on certain function spaces. Applying this parameterization to construct Lyapunov functionals wi...

We study stationary measures for the two-dimensional Navier-Stokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of t...

We analyse how small irregularities of the solid surface effects the steady flow of a general viscous fluid at larger scales. In particular we consider generalised Stokes system for incompressible non-Newtonian fluids of power-law type with zero Dirichlet boundary conditions when the surface of boundary is rough. Namely, it contains microscopic surface irregularities and an amplitude, and a wav...

In this paper, we study the asymptotic behavior of the solution of a semilinear problem and of a Stokes problem, with periodic data, when the size of the domain increases. In particular, we prove exponential convergence to the solution of the corresponding problem with periodic boundary conditions.

We present a new ODE–type method of passing to the limit with the dimension of Galerkin projection for dissipative PDEs. We apply this method to trapping regions derived by Mattingly and Sinai to give a new proof of the existence and uniqueness of solutions to Navier–Stokes equations with periodic boundary conditions on the plane.

This paper gives another version of results due to Raugel and Sell, and similar results due to Moise, Temam and Ziane, that state the following: the solution of the Navier-Stokes equation on a thin three dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and the forcing term, where the control is larger th...

In this paper, we study the global well-posedness of the 2D compressible NavierStokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ) = ρβ with β > 3, then the 2D compressible Navier-Stokes equations with the periodic boundary conditions on the torus T2 ad...

in this paper we study the boundary layer problems in which boundary conditions are non-local. here we try to find the necessary conditions by the help of fundamental solution to the given adjoint equation. by getting help from these conditions, at first the boundary condition is changed from non-local to local. the main aim of this paper is to identify the location of the boundary layer. in ot...

In this study, free vibration of functionally graded rectangular plates for various types of boundary conditions has been presented . The properties of the plate are assumed as power- law form along the thickness direction , while poisson's ratio is kept constant. the linear vibration equations of functionally graded rectangular plates are derived based on first order shear deformation theory b...