What Slip Boundary Conditions Induce a Well–Posed Problem for the Navier–Stokes Equation?
نویسنده
چکیده
The system (1), (2) describes the motion of a viscous incompressible fluid with a constant density (we assume that it equals one). We denote by u the velocity, by p the pressure, by f the specific body force and by ν the coefficient of viscosity. The equation (1) expresses the balance of momentum and the equation (2) represents the condition of incompressibility. By a well–posed problem we mean a problem which possesses the existence of a weak solution and under an additional assumption on smoothness of the solution, also its uniqueness. In order to obtain such a problem, we must add an appropriate boundary condition. The system (1), (2) is usually considered with the homogeneous Dirichlet boundary condition
منابع مشابه
Divergence-free wavelet bases on the hypercube: Free-slip boundary conditions, and applications for solving the instationary Stokes equations
We construct wavelet Riesz bases for the usual Sobolev spaces of divergence free functions on (0, 1)n that have vanishing normals at the boundary. We give a simultaneous space-time variational formulation of the instationary Stokes equations that defines a boundedly invertible mapping between a Bochner space and the dual of another Bochner space. By equipping these Bochner spaces by tensor prod...
متن کاملError Estimates for Finite-Element Navier-Stokes Solvers without Standard Inf-Sup Conditions∗∗∗∗
The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C elements for velocity and C elements for pressure. A stabilit...
متن کاملA Study of the Navier-stokes Equations with the Kinematic and Navier Boundary Conditions
Abstract. We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in R with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential...
متن کاملDegeneracy in the Blasius Problem
The Navier-Stokes equations for the boundary layer are transformed, by a similarity transformation, into the ordinary Blasius differential equation which, together with appropriate boundary conditions constitutes the Blasius problem, f ′′′(η) + 1 2 f(η)f ′′(η) = 0, f(0) = 0, f ′(0) = 0, f ′(∞) = 1. The well-posedness of the Navier-Stokes equations is an open problem. We solve this problem, in t...
متن کاملWell-Posed Boundary Conditions for the Navier-Stokes Equations
In this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations. With those, well-posedness can be proven for a general initial-boundary value problem. The procedure is exemplified on the linearized Navier–Stokes equations in two and three space dimensions on a general domain.
متن کامل