On the $$\mathrm {L}^p$$ L p -theory of the Navier–Stokes equations on three-dimensional bounded Lipschitz domains

Tolksdorf, Patrick

doi:10.1007/s00208-018-1653-4

Cited by 17 publications

(32 citation statements)

References 22 publications

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“…There is, of course, the Lipschitz approach to even more general non-smooth domains. Existence and analyticity of the Stokes semigroup on L p σ on Lipschitz domains is proved, for instance, in [23,28,30]. Note that the Lipschitz approach does not provide full W 2,p Sobolev regularity which, however, might be crucial for the treatment of related quasilinear problems.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain

2019

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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 [21, 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. On the other hand, for the Stokes operator in the space of solenoidal fields L p 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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain

2019

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“…To circumvent this problem, we shall write (u(s) · ∇)u(s) as div(u(s) ⊗ u(s)) and then consider e −(t−s)A Pdiv as one composite operator on L p . That this is well-defined for 3/2−ε < p < 3+ε and a bounded Lipschitz domain Ω, was proven by the third author in [34]. To prove convergence of the iteration scheme, it will be important that u(s)⊗u(s) and [∇d(s)] ⊤ ∇d(s) exhibit the same decay rate in the time variable, because in this case, both integrands in the mild formulation of u behave similarly with respect to the time variable.…”

Section: Introductionmentioning

confidence: 85%

“…The following result characterises the domains of the square roots of A and B defined above. Proposition 2.3 (see [34] and [20]). Let Ω ⊂ R 3 be a bounded Lipschitz domain.…”

Section: Preliminariesmentioning

confidence: 98%

Nematic Liquid Crystals in Lipschitz Domains

Choudhury¹,

Hussein²,

Tolksdorf³

2018

SIAM J. Math. Anal.

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We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in L 3 σ for the fluid and W 1,3 for the director field. The analysis of such models, so far, has been restricted to domains with smooth boundaries.

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“…This is a fatal fact as in three dimensions this condition exhibits only the interval 3/2 < p < 3 while it is crucial for the existence theory of the Navier-Stokes equations in the critical space L ∞ (0, ∞; L 3 (Ω)) to have information for p in the interval [3, 3 + ε), cf., [20,12,11,30,31] for the cases of the whole space and bounded smooth/Lipschitz domains. In the following, we review the existence proof of Lang and Méndez and point out a slight modification in order to recover the interval (2.3) for exterior Lipschitz domains Ω with connected boundary.…”

Section: 1mentioning

confidence: 99%

“…Recently, the study of the Stokes operator on bounded Lipschitz domains was continued by Kunstmann and Weis [22] and the first author of this article [30,31]. In [22], the property of maximal regularity and the boundedness of the H ∞ -calculus were established yielding a short proof to reveal the domain of the square root of the Stokes operator as W 1,p 0,σ (Ω), see [30]. The purpose of this paper is to continue the study of the Stokes operator in the case of exterior Lipschitz domains Ω.…”

Section: Introductionmentioning

confidence: 99%