Solenoidal Subspaces Research Articles

The Scalar Products of the Regular Analytic Vectors of the Laplace Operator in the Solenoidal Subspace

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing together with their first derivatives at selected points xn, n = 1, . . .,N, is a symmetric operator with deficiency indices (3N, 3N). The calculation of the scalar products of its regular analytic vectors is the key step in the construction of the resolvents of its self-adjoint extensions via Krein’s formula.

Analyticity of the Stokes semigroup in $BMO$-type spaces

We consider the Stokes semigroup in a large class of domains including bounded domains, the half-space and exterior domains. We will prove that the Stokes semigroup is analytic in a certain type of solenoidal subspaces of $BMO$.

The Stokes equations in layer domains on spaces of bounded and integrable functions

The Stokes equations in a layer domain are investigated. For the case of a two‐dimensional layer it is shown by resolvent estimates that there exists a semigroup of angle on solenoidal subspaces of and L1. Furthermore, the stationary problem for dimension two is solvable in these spaces. It is also shown that both results cannot be extended to three and higher dimensions in the same setting.

Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains

Consider the Stokes resolvent system in general unbounded domains $\Omega \subset {\mathbb{R}^n}$, $n\geq 2$, with boundary of uniform class $C^{3}$, and Navier slip boundary condition. The main result is the resolvent estimate in function spaces of the type ${\tilde{L}^q}$ defined as $L^q\cap L^2$ when $q\geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain. As a consequence, we get that the Stokes operator generates an analytic semigroup on a solenoidal subspace ${\tilde{L}^q}_\sigma(\Omega)$ of ${\tilde{L}^q}(\Omega)$.

On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow

In this work we study the efficiency of a spectral Petrov–Galerkin method for the linear and nonlinear stability analysis of the pipe or Hagen–Poiseuille flow. We formulate the problem in solenoidal primitive variables for the velocity field and the pressure term is eliminated from the scheme suitably projecting the equations on another solenoidal subspace. The method is unusual in being based on Chebyshev polynomials of selected parity for the radial variable, avoiding clustering of the quadrature points near the origin, satisfying appropriate regularity conditions at the pole and allowing the use of a fast cosine transform if required. Besides, this procedure provides good conditions for the time marching schemes. For the time evolution, we use semi-implicit time integration schemes. Special attention is given to the explicit treatment and efficient evaluation of the nonlinear terms via pseudospectral partial summations. The method provides spectral accuracy and the linear and nonlinear results obtained are in very good agreement with previous works. The scheme presented can be applied to other flows in unbounded cylindrical geometries.

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n, with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L p(Ω))nis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L p(Ω))n. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S 1) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.