Stream functions for divergence-free vector fields
In 1990, Von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector field u in a 3D bounded domain that is tangential to the boundary can be written as the curl of a vector field vanishing on the boundary of the domain. We extend this result to higher dimension and to Lipschitz boundaries in a form suitable for integration in flat space, showing that u can be written as the divergence of an antisymmetric matrix field. We also demonstrate how obtaining a kernel for such a matrix field is dual to obtaining a Biot-Savart kernel for the domain. Compiled on Friday 22 May 2020 at 12:37 1. Overview 1 2. Background material 3 3. Proof of main result 4 4. Higher regularity 7 5. 3D vector potentials 7 6. A Biot-Savart kernel? 8 Acknowledgments 9 References 9 1. Overview Let u be a divergence-free vector field on a bounded Lipschitz domain Ω ⊆ Rd, d ≥ 2, that is tangential to the boundary. For a simply connected domain, it is well known that in two dimensions, u = ∇⊥ψ := (−∂2ψ, ∂1ψ) for a stream function, ψ, vanishing on the boundary. It is also well known that in three dimensions, we can write u = curlψ, where now the vector potential ψ is a divergence-free vector field tangential to the boundary. Perhaps somewhat less well-known is that ψ can also be chosen (non-uniquely) to vanish on the boundary, though sacrificing the divergence-free condition. This 3D form of the vector potential was developed in [7, 20], where it is studied in Sobolev, Hölder spaces, for C1,1, C∞ boundaries, respectively. In higher dimension, we can no longer use a vector field as the potential; instead, we will use an antisymmetric matrix field A vanishing on the boundary, for which u = divA, the divergence applied to A row-by-row. This was the manner it was utilized in [15], without, however, the key antisymmetric condition. Our main result is Theorem 1.1. Theorem 1.1. Let H be the space of divergence-free vector fields on Ω that are tangential to the boundary and that have L2 coefficients. Let Hc be the closed subspace of curl-free vector fields (see (3.1)) in H, let H0 be its orthogonal complement in H, and let X0 := {A ∈ H 0 (Ω)d×d : A antisymmetric}. Then H0 = divX0, and there exists a bounded linear map S : H0 → X0 with divSu = u. Specializing to d = 2, 3, we can write H0 = { ∇⊥H1 0 (Ω), d = 2, curl3H 1 0 (Ω) 3, d = 3. Because the term matrix potential is commonly used in the literature for other purposes, we will adopt the 2D terminology for all dimensions, calling A the stream function for u. 1