Werner Varnhorn

Boundary value problems for the Stokes equations with jumps in open sets

A boundary value problem for the Stokes system is studied in a cracked domain in ℝ n , n > 2, where the Dirichlet condition is specified on the boundary of the domain. The jump of the velocity and the jump of the stress tensor in the normal direction are prescribed on the crack. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence, a maximum modulus estimate for the Stokes system is proved.

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

Abstract In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods. The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nystrom’s method with the trapezoidal rule. For this, a convergence analysis is carried out.

The poisson equation with weights in exterior domains of Rn

We consider Poissons equation in G with Dirichlet boundary condition u=Φ on ∂G. Here is an exterior domain with a smooth compact boundary [∂G]. In the present paper we prove existence and uniqueness of the solution u in the Soboley space parovided . Here the aditional condition f e Lρ(G) is necessary and corresponds to a certain decay behaviour of f. We show that u tends to zero as |x| → ∞ if r > n. This result can be applied to weighted versions of Poissons equation and leads to statements about the rate of decay of the solution

Necessary and sufficient conditions on local strong solvability of the Navier-Stokes system

Consider in a smooth bounded domain Ω ⊆ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, the Navier–Stokes system with the initial value and the external force f = div F, F ∈ L 2(0, T; L 2(Ω)). As is well-known, there exists at least one weak solution on [0, T) × Ω in the sense of Leray–Hopf; then it is an important problem to develop conditions on the data u 0, f as weak as possible to guarantee the existence of a unique strong solution u ∈ L s (0, T; L q (Ω)) satisfying Serrins condition with 2 < s < ∞, 3 < q < ∞ at least if T > 0 is sufficiently small. Up to now there are known several sufficient conditions, yielding a larger class of corresponding local strong solutions, step by step, during the past years. Our following result is optimal in a certain sense yielding the largest possible class of such local strong solutions: let E be the weak solution of the linearized system (Stokes system) in [0, T) × Ω with the same data u 0, f. Then we show that the smallness condition ‖E‖ L s (0,T; L q (Ω)) ≤ ϵ* with some constant ϵ* = ϵ*(Ω, q) > 0 is sufficient for the existence of such a strong solution u in [0, T). This leads to the following sufficient and necessary condition: Given F ∈ L 2(0, ∞; L 2(Ω)), there exists a strong solution u ∈ L s (0, T; L q (Ω)) in some interval [0, T), 0 < T ≤ ∞, if and only if E ∈ L s (0, T′; L q (Ω)) with some 0 < T′ ≤ ∞.

On strong solutions of the Stokes equations in exterior domains

We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -δu + ▽p=f inG, ▽ ·u=g inG, u=Φ on γ in an exterior domainG ⊂ℝn (n⩾3) with boundary γ of class C2. Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ▽ ·Ν=g inG, Ν = 0 on γ, the exterior domain problem is reduced to the entire space problem and an interior problem.

THE POISSON EQUATION IN HOMOGENEOUS SOBOLEV SPACES

We consider Poisson’s equation in an n-dimensional exterior domain G( n≥ 2) with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certain L q (G)-spaces there exists a solution in the homogeneous Sobolev space S 2,q (G), containing functions being local in L q (G) and having second-order derivatives in L q (G). Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimension n + 1, independent of q.

On Eulerian and Lagrangian representation of steady incompressible fluid flow

FOR THE description of fluid flow there are, in principle, two approaches, the Eulerian approach and the Lagrangian approach. The first one describes the flow of its velocity u = (u,(x), v,(x), uJx)) = u(x) in every point x = (xi, x2, x3) of the domain G containing the fluid. The second one uses the trajectory x = (xi(t), x2(t), x3(t)) = x(t) = X(t, x0) of a single particle of fluid, which at initial time t = 0 is located at some point x0 E G. This second approach is of great importance for the numerical analysis and computation of fluid flow [l-4], while the first one has often also been used in connection with theoretical questions [5-81. In the present paper we consider the stationary motion of a viscous incompressible fluid in a bounded domain G c R3 with a sufficiently smooth boundary S. Because for steady flow the streamlines and the trajectories of the fluid particles coincide, both approaches mentioned above are correlated by the autonomous system of characteristic ordinary differential equations

Time delay and Lagrangian approximation for Navier–Stokes flow

Abstract We consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for 0<t≤T

Boundary Integral Equations for the Stokes Problem in Exterior Domains of ℝ n

{0 < t \le T}

Finite differences and boundary element methods for non-stationary viscous incompressible flow

in a bounded domain Ω⊆ℝ 3