Stokes-Dirichlet/Neuman problems and complex analysis
de J Jan Graaf · TU/e Research Portal · 2011
On a bounded and simply connected open set G ⊂ R 2 ∼= C, with a sufficiently smooth boundary ∂G, the following boundary value problem for a pair {φ, χ} of analytic functions is studied: φ , χ : G→ C , both analytic, [ zφ′ ± φ+ χ′ ]∣∣∣ ∂G = G ∈ L2(∂G), (0.1) Multiplication by i transforms the +version into the −version. Necessary and sufficient conditions on G for solvability and also results on the behaviour of the solution near ∂G are found. The original motivation for this study is to provide a sound mathematical link between 2D Stokes boundary value problems and 2D free boundary evolution equations of Hopper type, cf. [H], with ’arbitrary Hamiltonian’ , cf. [G]. During this, the interesting (and for the author unexpected) fact came up that both the Dirichlet and the Neumann Problem for the 2D-Stokes equations can be reduced to the problem (0.1). Full details of all this are in the underlying note. A brief overview now follows. On G ⊂ R 2 ∼= C, the stationary behaviour of a pressure-velocity flow pair {p, v}, where p : G → R and v : G → R , can often be modelled by Stokes’ equations { ∇ · T = 0 ∇ · v = 0 , with stress matrix T = −p I + [dv dx ] + [dv dx ]> . (0.2) Only Cartesian coordinates will be employed! It is classical folklore, scattered in the litterature, that there exists a bi-harmonic potential pair ψ, φ : G → R , (the stream function and Airy function, respectively), such that, cf. (1.3), v = ∇× (ψ e3), , T = 2 [ (Dφ)− (∆φ)I ] . (0.3) Consistency in T requires that φ and ψ are related: For z = x + iy ∈ G one necessarily has, cf. Appendix B, φ(x) + iψ(x) = zφ(z) + χ(z) , with analytic φ, χ : G→ C . (0.4) Also this is classical folklore. For a strongly related approach in the field of ’elasticity’ cf. [E] and [M] Ch 4. In the Appendices to this note full details are presented on ψ, φ, φ, χ and on the kinematic expressions derived from them. For a full set of the latter see (1.5). By means of the analytic potentials φ, χ we investigate boundary value problems for Stokes’ equations with respective boundary conditions: Stokes-Dirichlet: v ∣∣∣ ∂G ∈ L2(∂G) , Stokes-Neumann: T n ∣∣∣ ∂G ∈ H−1(∂G) . (0.5) As it turns out both problems can be reduced to (0.1). By means of a conformal mapping the problem (0.1) is then transformed to an integral operator equation on the unit circle.