Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain
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Mathematische Annalen
Optimal Sobolev regularity for the stokes equations on a 2D wedge domain Matthias Köhne1 · Jürgen Saal1 · Laura Westermann1 Received: 11 February 2019 / Revised: 20 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this note we prove that the solution of the stationary and the instationary Stokes equations subject to perfect slip boundary conditions on a 2D wedge domain admits optimal regularity in the L p -setting, in particular it is W 2, p in space. This improves known results in the literature to a large extend. For instance, in Maier and Saal (Discrete Contain Dyn Syst Ser S 7(5):1045–1063, 2014) [Theorem 1.1 and Corollary 3] it is proved that the Laplace and the Stokes operator in the underlying setting have maximal regularity. In that result the range of p admitting W 2, p regularity, however, is restricted to the interval 1 < p < 1 + δ for small δ > 0, depending on the opening angle of the wedge. This note gives a detailed answer to the question, whether the optimal Sobolev regularity extends to the full range 1 < p < ∞. We will show that for the Laplacian this does only hold on a suitable subspace, but, depending on the opening angle of the wedge domain, not for every p ∈ (1, ∞) on the entire L p -space. p On the other hand, for the Stokes operator in the space of solenoidal fields L σ we obtain optimal Sobolev regularity for the full range 1 < p < ∞ and for all opening angles less than π . Roughly speaking, this relies on the fact that an existing “bad” part of L p for the Laplacian is complemented to the space of solenoidal vector fields. Mathematics Subject Classification Primary: 35Q30 · 76D03 · 35K67; Secondary: 76D05 · 35K65
Communicated by Y. Giga.
B
Jürgen Saal [email protected] Matthias Köhne [email protected] Laura Westermann [email protected]
1
Mathematisches Institut, Angewandte Analysis, Heinrich-Heine-Universität Düsseldorf, 40204 Düsseldorf, Germany
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M. Köhne et al.
1 Introduction and main results It is well-known that regularity properties for PDE on non-smooth domains are important for many applications. The main objective of this note is to derive best possible regularity in the L p -setting for the instationary Stokes equations subject to perfect slip boundary conditions ∂t u − Δu + ∇π = div u = curl u = 0, u · ν = u(0) =
f 0 0 u0
in in on in
(0, ∞) × G, (0, ∞) × G, (0, ∞) × ∂G, G,
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(1)
on a two-dimensional wedge type domain given as G := (x1 , x2 ) ∈ R2 : 0 < x2 < x1 tan θ0 .
(2)
Here ν denotes the outer normal vector at ∂G, θ0 ∈ (0, π ) the opening angle of the wedge, and curl u = ∂1 u 2 − ∂2 u 1 . There exist approaches to an L p -theory for classical elliptic and parabolic problems on domains with conical boundary points, see, e.g., the classical monographs [10,22], or also [3,4] for the heat equation subject to Dirichlet boundary conditions. In contrast to that, corresponding results for the Stokes equations are very rare, in particular for the instationary case. For the s
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