On the L^p-theory of the Navier--Stokes equations on three-dimensional bounded Lipschitz domains

2017/03/03 by Tolksdorf, Patrick · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1703.01091

Abstract

On a bounded Lipschitz domain Ω⊂ ℝ^d, d ≥ 3, we continue the study of Shen and of Kunstmann and Weis of the Stokes operator on L^p_σ (Ω). We employ their results in order to determine the domain of the square root of the Stokes operator as the space W^{1 , p}_{0 , σ} (Ω) for | (1)/(p) - (1)/(2) | < (1)/(d) + ε and some ε > 0. This characterization provides gradient estimates as well as L^p-L^q-mapping properties of the corresponding semigroup. In the three-dimensional case this provides a means to show the existence of solutions to the Navier--Stokes equations in the critical space L^∞ (0 , ∞ ; L³_σ (Ω)) whenever the initial velocity is small in the L³-norm. Finally, we present a different approach to the L^p-theory of the Navier--Stokes equations by employing the maximal regularity proven by Kunstmann and Weis.

Cited by

• On well-posedness theory of very weak solutions to Navier-Stokes equations on irregular domains with nonhomogeneous Dirichlet boundary data
• H^∞-calculus for Stokes operators on rough and on unbounded domains
• The magnetohydrodynamical system in bounded non smooth domains of dimension n≥3

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