2308/2308.01616.md

Title: Maximal regularity of Stokes problem with dynamic boundary condition — Hilbert setting

URL Source: https://arxiv.org/html/2308.01616

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Why HTML? Report Issue Back to Abstract Download PDF Abstract 1Introduction 2Notation and functional analytic setting 3Regularity theory for the elliptic problem 4Proof of the main theorem References License: arXiv.org perpetual non-exclusive license arXiv:2308.01616v2 [math.AP] 22 May 2025 Maximal regularity of Stokes problem with dynamic boundary condition — Hilbert setting Tomáš Bárta, Paige Davis, Petr Kaplický Abstract

For the evolutionary Stokes problem with dynamic boundary conditions, we show the maximal regularity of weak solutions in time. Due to the characterization of 𝑅 -sectorial operators on Hilbert spaces, the proof reduces to identifying the appropriate functional analytic setting and proving that the corresponding operator is sectorial, i.e., that it generates an analytic semigroup.

1Introduction

Certain materials, like polymer melts, can slip over solid surfaces. Such boundary behavior is described by slip velocity models; see [10, Section 6] for an overview. Moreover, it has been observed that the slip is often not constant but varies over time, depending on the fluid’s current state. Such fluids need to be represented using dynamic slip models. They were first proposed in [17] in a general form

𝑢 𝜏 + 𝜆 𝜏 ⁢ ∂ 𝑡 𝑢 𝜏

𝜑 ⁢ ( 𝜎 𝑤 ) ,

where 𝑢 𝜏 is the slip velocity, 𝑡 stands for the time, 𝜆 𝜏 is the slip relaxation time, 𝜎 𝑤 stands for the wall shear stress and 𝜑 should be determined based on the rheological properties of the fluid under consideration.

The mathematical studies of problems with dynamic boundary conditions in the context of fluid mechanics started by the thesis of Maringová, [15]. She studied the existence of solutions to systems of (Navier)-Stokes type under various constitutive relations for the extra stress tensor and the modified dynamic boundary condition 𝑠 ⁢ ( 𝑢 𝜏 ) + ∂ 𝑡 𝑢 𝜏

− 𝜎 𝑤 with a given–possibly nonlinear–function 𝑠 . These results were later published in [1].

We are interested in the optimal regularity of problems with dynamic boundary conditions in the context of Lebesgue spaces. Specifically, we focus on the linear Stokes problem. First, we find the result interesting. Second, it provides a basis for studying the regularity of more complex systems. Moreover, the linear theory can be considered as a tool for the reconstruction of pressure; see [20].

We study the problem

∂ 𝑡 𝑢 − Δ ⁢ 𝑢 + ∇ 𝑝

𝑓  in  ⁢ 𝐼 × Ω ,

(1)

div ⁡ 𝑢

0  in  ⁢ 𝐼 × Ω ,

(2)

𝛽 ⁢ ∂ 𝑡 𝑢 + ( 2 ⁢ 𝐷 ⁢ 𝑢 ⋅ 𝜈 ) 𝜏 + 𝛼 ⁢ 𝑢 𝜏

𝛽 ⁢ 𝑔  in  ⁢ 𝐼 × ∂ Ω ,

(3)

𝑢 𝜈

0  in  ⁢ 𝐼 × ∂ Ω ,

(4)

𝑢

𝑢 0  in  ⁢ { 0 } × Ω

(5)

𝑢

𝑣 0  in  ⁢ { 0 } × ∂ Ω

(6)

in a bounded domain Ω ⊂ ℝ 𝑑 , 𝑑 ≥ 2 with 𝐶 2 , 1 boundary and a time interval 𝐼

( 0 , 𝑇 ) , 𝑇

0 . The constants 𝛼 ∈ ℝ , 𝛽

0 , the functions 𝑓 : 𝐼 × Ω → ℝ 𝑑 , 𝑔 : 𝐼 × ∂ Ω → ℝ 𝑑 , 𝑢 0 : Ω → ℝ 𝑑 and 𝑣 0 : ∂ Ω → ℝ 𝑑 are given. Subscripts ( ⋅ ) 𝜏 and ( ⋅ ) 𝜈 denote the tangential and the normal part of the vectors. We look for unknown functions 𝑢 : 𝐼 × Ω → ℝ 𝑑 and 𝑝 : 𝐼 × Ω → ℝ . Let us mention that we permit 𝛼 < 0 , however, only 𝛼 ≥ 0 seems to be physically relevant.

The notion of the weak solution is adopted (with a small modification) from [15, Section 5]. We work in Banach spaces

𝒢

{ ( 𝑢 , 𝑢 𝑏 ) ∈ 𝐻 𝜎 1 ⁢ ( Ω ) × 𝐿 𝜈 2 ⁢ ( ∂ Ω ) : 𝑢 𝑏

𝛾 ⁢ ( 𝑢 ) } , ℋ

𝐿 𝜎 2 ⁢ ( Ω ) × 𝐿 𝜈 2 ⁢ ( ∂ Ω )

with norms

‖ ( 𝑢 , 𝑢 𝑏 ) ‖ 𝒢 2

2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 , ‖ ( 𝑢 , 𝑢 𝑏 ) ‖ ℋ 2

‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + 𝛽 ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 .

Definitions of all mentioned function spaces can be found in Subsection 2.1. Note, that 𝒢 is a dense subset of ℋ .

The duality pairing between 𝒢 and its dual space 𝒢 ∗ , denoted ⟨ ⋅ , ⋅ ⟩ 𝒢 , extends the scalar product in ℋ ; see [15, Section 3.1].

When dealing with a function from 𝒢 we write only the first component of the vector. The trace of the function is automatically considered as the second component.

Definition 1.

Let 0 < 𝑇 ≤ + ∞ , 𝛼 ∈ ℝ , 𝛽 > 0 , Ω ⊂ ℝ 𝑑 , Ω ∈ 𝐶 0 , 1 , 𝑓 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 1 ⁢ ( [ 0 , 𝑇 ) , 𝐻 𝜎 1 ⁢ ( Ω ) ∗ ) , 𝑔 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 1 ⁢ ( [ 0 , 𝑇 ) , 𝐿 𝜈 2 ⁢ ( ∂ Ω ) ) , 𝑢 0 ∈ 𝐿 𝜎 2 ⁢ ( Ω ) and 𝑣 0 ∈ 𝐿 𝜈 2 ⁢ ( ∂ Ω ) . We say that 𝑢 is a weak solution to the problem (1)–(6) if 𝑢 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 2 ⁢ ( [ 0 , 𝑇 ) , 𝒢 ) ∩ 𝐶 𝑙 ⁢ 𝑜 ⁢ 𝑐 ⁢ ( [ 0 , 𝑇 ) , ℋ ) ∩ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 ∞ ⁢ ( [ 0 , 𝑇 ) , ℋ ) , ∂ 𝑡 𝑢 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 1 ⁢ ( 0 , 𝑇 , 𝒢 ∗ ) , 𝑢 ⁢ ( 0 )

( 𝑢 0 , 𝑣 0 ) in ℋ , and the equations (1) and (3) are satisfied in the weak sense, i.e.,

⟨ ∂ 𝑡 𝑢 , 𝜑 ⟩ 𝒢 + 2 ⁢ ∫ Ω 𝐷 ⁢ 𝑢 : 𝐷 ⁢ 𝜑 + 𝛼 ⁢ ∫ ∂ Ω 𝑢 ⁢ 𝜑

⟨ ( 𝑓 , 𝑔 ) , 𝜑 ⟩ 𝒢

(7)

almost everywhere on ( 0 , 𝑇 ) and for all 𝜑 ∈ 𝒢 .

Note that 𝛽 is hidden in (7) in the definition of ⟨ ⋅ , ⋅ ⟩ 𝒢 . Under the regularity assumptions in Definition 1, equality (3) makes no sense when understood pointwise. Instead, this part of the boundary condition is hidden in the weak formulation (7). It can be derived pointwise only if the regularity of the data and solution is better; see Theorem 3.

We are interested in the maximal regularity of weak solutions with respect to the problem data, i.e., the right hand side functions 𝑓 and 𝑔 , and the initial values 𝑢 0 and 𝑣 0 . In order to state the precise conditions for the initial values we need to introduce spaces

𝒳 0

𝐿 𝜎 2 ⁢ ( Ω ) × 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) , 𝒳 1

{ ( 𝑢 , 𝑢 𝑏 ) ∈ 𝐻 𝜎 2 ⁢ ( Ω ) × 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) : 𝛾 ⁢ ( 𝑢 )

𝑢 𝑏 } ,

‖ ( 𝑓 , 𝑔 ) ‖ 𝒳 0

‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ 𝑔 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) , ‖ ( 𝑢 , 𝑣 ) ‖ 𝒳 1

‖ 𝑢 ‖ 𝐻 2 ⁢ ( Ω ) + ‖ 𝑣 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ,

𝒳 1 − 1 𝑞 , 𝑞

( 𝒳 0 , 𝒳 1 ) 1 − 1 𝑞 , 𝑞 .

The last space is the real interpolation space between 𝒳 0 and 𝒳 1 . It turns out that this is the optimal (largest possible) space for the initial data to guarantee 𝐿 𝑞 -maximal regularity of solutions to the non-homogeneous abstract Cauchy problem; see [14, Section 2.2.1] for details.

Before we formulate the main theorem, we need some preparation for nonaxisymmetric domains; this Lemma is a consequence of Lemma 4 below.

Lemma 2.

Let Ω be nonaxisymmetric. There exists 𝛼 0 < 0 such that for all 𝑢 ∈ 𝐻 1 ⁢ ( Ω ) with 𝑢 ⋅ 𝜈

0 on ∂ Ω

2 ⁢ 𝛼 0 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 + 4 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 ≥ 0 .

From this point forward, 𝛼 0 always refers to the fixed constant from Lemma 2.

Our main theorem follows.

Theorem 3.

Let one of the following conditions be met:

a.

𝑇 ∈ ( 0 , + ∞ ) ,

b.

𝑇

• ∞ , 𝛼
  0 ,

c.

𝑇

• ∞ , 𝛼 ∈ ( 𝛼 0 , 0 ] , Ω nonaxisymmetric.

For every 𝑞 ∈ ( 1 , + ∞ ) , there exists a constant 𝐶 > 0 such that, for every ℱ

( 𝑓 , 𝑔 ) ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) and ( 𝑢 0 , 𝑣 0 ) ∈ 𝒳 1 − 1 / 𝑞 , 𝑞 there exists a weak solution 𝑢 of (1)–(6), it is unique, and satisfies 𝑢 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐻 2 ⁢ ( Ω ) ) , ∂ 𝑡 𝑢 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐿 2 ⁢ ( Ω ) ) . Moreover, there exists a function 𝑝 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐻 1 ⁢ ( Ω ) ) such that (1)–(6) hold pointwise almost everywhere and

‖ ∂ 𝑡 𝑢 ⁢ ( 𝑡 ) ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐿 2 ⁢ ( Ω ) ) + ‖ 𝑢 ⁢ ( 𝑡 ) ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐻 2 ⁢ ( Ω ) ) + ‖ 𝑝 ⁢ ( 𝑡 ) ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐻 1 ⁢ ( Ω ) ) ≤ 𝐶 ⁢ ( ‖ ℱ ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ ( 𝑢 0 , 𝑣 0 ) ‖ 𝒳 1 − 1 / 𝑞 , 𝑞 ) .

(8)

Our approach to the problem is as follows. We rewrite the problem (1)–(4) as an abstract Cauchy problem

∂ 𝑡 𝒰

𝒜 ⁢ 𝒰 + ℱ ⁢ ( 𝑡 ) ,

(9)

on a Hilbert space 𝒳 0 . Since the problem combines evolutionary equations in the interior of Ω and on its boundary, the space 𝒳 0 must be a product of spaces in the interior and on the boundary of Ω , compare [8, 7]. We show below that 𝒜 is the generator of an analytic semigroup 𝒯 . Then the Variation-Of-Constants-Formula

𝒰 ⁢ ( 𝑡 )

𝒯 ⁢ ( 𝑡 ) ⁢ 𝒰 0 + ∫ 0 𝑡 𝒯 ⁢ ( 𝑡 − 𝑠 ) ⁢ ℱ ⁢ ( 𝑠 ) ⁢ 𝑑 𝑠

(10)

defines a mild solution to (9). This mild solution actually has better properties if 𝒰 0 and ℱ are sufficiently good. Namely, since 𝒳 0 is a Hilbert space we have maximal 𝐿 𝑝 regularity, i.e., for every ℱ ∈ 𝐿 𝑝 ⁢ ( 𝐼 , 𝒳 0 ) the solution 𝒰 given by (10) with 𝒰 0

0 satisfies 𝒜 ⁢ 𝒰 , 𝒰 ˙ ∈ 𝐿 𝑝 ⁢ ( 𝐼 , 𝒳 0 ) and

‖ 𝒰 ˙ ‖ 𝐿 𝑝 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ 𝒜 ⁢ 𝒰 ‖ 𝐿 𝑝 ⁢ ( 𝐼 , 𝒳 0 ) ≤ 𝐶 ⁢ ‖ ℱ ‖ 𝐿 𝑝 ⁢ ( 𝐼 , 𝒳 0 )

with a constant 𝐶

0 independent of ℱ , compare [6] or [13, Corollary 1.7]. Since the mild solution is very regular we show that it is actually a weak solution from Definition 1. Uniqueness of the weak solution then concludes the argumentation.

Apart from articles [15] and [1] we are aware only of the work [19], that appeared recently. In this article its authors study whether the solutions to (1)–(6) are given by an analytic semigroup in spaces 𝐿 𝜎 𝑝 ⁢ ( Ω ) × 𝐿 𝜈 𝑝 ⁢ ( ∂ Ω ) with 𝑝 > 1 . The result is rather involved but does not cover our result since we work in 𝒳 0

𝐿 𝜎 2 ⁢ ( Ω ) × 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) . A variant of dynamic boundary conditions appeared also in a different context. In [22] they appeared as general boundary conditions that turn a given elliptic differential operator to the generator of a semigroup of positive contraction operators. There are many works on dynamic boundary conditions (or Wentzell1 boundary conditions) in the context of parabolic and hyperbolic equations without the incompressibility constraint and without the pressure. Our main example are the results in [7] where the maximal 𝐿 𝑝 regularity is proved for a very general class of parabolic systems equipped with a general dynamic boundary condition. The presented article can be considered as the first step to a parallel theory for the Stokes problem.

In the following section we give the basic notation and define the operator 𝒜 . Elliptic theory is studied in Section 3. The proof of Theorem 3 is given in Section 4.

2Notation and functional analytic setting 2.1Notation and function spaces

The constant 𝛼 0 < 0 is a fixed constant from Lemma 2. If 𝑧 : ℝ 𝑑 → ℝ 𝑑 then ( ∇ 𝑧 ) 𝑖 ⁢ 𝑗

∂ 𝑗 𝑧 𝑖 and ( 𝐷 ⁢ 𝑧 ) 𝑖 ⁢ 𝑗

1 2 ⁢ ( ∂ 𝑗 𝑧 𝑖 + ∂ 𝑖 𝑧 𝑗 ) for 𝑖 , 𝑗 ∈ { 1 , … , 𝑑 } . If 𝐴 , 𝐵 are matrices, then 𝐴 ⁢ 𝐵 denotes the matrix product, e.g., ( [ ∇ 𝑧 ] ⁢ 𝑧 ) 𝑖

∂ 𝑘 𝑧 𝑖 ⁢ 𝑧 𝑘 for 𝑖 ∈ { 1 , … , 𝑑 } , while 𝐴 : 𝐵

𝑎 𝑖 ⁢ 𝑗 ⁢ 𝑏 𝑖 ⁢ 𝑗 . We use the summation convention over repeated indices. For two vectors 𝑎 , 𝑏 ∈ ℝ 𝑑 , 𝑎 ⋅ 𝑏 denotes the scalar product in ℝ 𝑑 .

We recall that Ω ⊂ ℝ 𝑑 is a bounded domain with 𝐶 2 , 1 boundary, 𝐼

( 0 , 𝑇 ) for some 𝑇

0 .

The standard Sobolev and Sobolev-Slobodeckii spaces over Ω with integrability 2 and differentiability 𝑠

0 are denoted by 𝐻 𝑠 . Further,

𝒟 𝜎

{ 𝑢 ∈ 𝐶 0 ∞ ⁢ ( Ω ) ; div ⁡ 𝑢

0 } , 𝐿 𝜎 2 ⁢ ( Ω )

closure of  𝒟 𝜎  in  𝐿 2 ⁢ ( Ω ) ,

𝐻 𝜎 1 ⁢ ( Ω )

𝐻 1 ∩ 𝐿 𝜎 2 ⁢ ( Ω ) , 𝐻 𝜎 2 ⁢ ( Ω )

𝐻 2 ∩ 𝐿 𝜎 2 ⁢ ( Ω ) .

We write 𝛾 for the trace operator. If 𝑤 is a function defined on Ω with trace 𝛾 ⁢ ( 𝑤 ) on ∂ Ω we denote 𝑤 𝜈 its normal part and 𝑤 𝜏 its tangential part on ∂ Ω . By 𝜈 ⁢ ( 𝑥 ) we denote the unit outer normal vector to ∂ Ω at point 𝑥 ∈ ∂ Ω . Equalities of functions are understood almost everywhere with respect to the corresponding Hausdorff measure.

We remark that if 𝑤 ∈ 𝐻 𝜎 1 ⁢ ( Ω ) then div ⁡ 𝑤

0 in Ω in the weak sense and 𝑤 𝜈

0 on ∂ Ω . Consequently, if we define

𝐿 𝜈 2 ⁢ ( ∂ Ω )

{ 𝑤 ∈ 𝐿 2 ⁢ ( ∂ Ω ) : 𝑤 𝜈

0 a.e. on  ∂ Ω } ,

𝐻 𝜈 1 2 ⁢ ( ∂ Ω )

{ 𝑤 ∈ 𝐻 1 2 ⁢ ( ∂ Ω ) : 𝑤 𝜈

0 ⁢  on  ∂ Ω } ,

𝐻 𝜈 3 2 ⁢ ( ∂ Ω )

{ 𝑤 ∈ 𝐻 3 2 ⁢ ( ∂ Ω ) ; 𝑤 𝜈

0 ⁢  on  ∂ Ω } ,

then 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω )

𝛾 ⁢ ( 𝐻 𝜎 1 ⁢ ( Ω ) ) and 𝐻 𝜈 3 / 2 ⁢ ( ∂ Ω )

𝛾 ⁢ ( 𝐻 𝜎 2 ⁢ ( Ω ) ) .

The Helmholtz-Weyl decomposition yields 𝐿 2 ⁢ ( Ω )

𝐺 2 ⁢ ( Ω ) ⊗ 𝐿 𝜎 2 ⁢ ( Ω ) where

𝐺 2 ⁢ ( Ω )

{ 𝑤 ∈ 𝐿 2 ⁢ ( Ω ) ; 𝑤

∇ 𝑝 , 𝑝 ∈ 𝐻 1 } ,

see, e.g., [9, Theorem III.1.1]. The continuous Leray projection of 𝐿 2 ⁢ ( Ω ) to 𝐿 𝜎 2 ⁢ ( Ω ) is denoted 𝑃 : 𝐿 2 ⁢ ( Ω ) → 𝐿 𝜎 2 ⁢ ( Ω ) .

2.2Definition of the operator 𝒜

The operator 𝒜 is considered on the space 𝒳 0 . The domain of 𝒜 is defined as 𝐷 ⁢ ( 𝒜 )

𝒳 1 . Finally, we set

𝒜 ⁢ ( 𝑢 𝑢 𝑏 )

( 𝑃 ⁢ Δ ⁢ 𝑢 − 𝛽 − 1 ⁢ [ ( 2 ⁢ 𝐷 ⁢ 𝑢 ⋅ 𝜈 ) 𝜏 + 𝛼 ⁢ 𝑢 𝑏 ] ) for  ⁢ ( 𝑢 𝑢 𝑏 ) ∈ 𝐷 ⁢ ( 𝒜 ) .

(11) 3Regularity theory for the elliptic problem

Before proving that ( 𝒜 , 𝐷 ⁢ ( 𝒜 ) ) generates an analytic semigroup in 𝒳 0 , we establish some preliminary results on the existence and regularity of solutions to the following system.

𝜆 ⁢ 𝑢 − Δ ⁢ 𝑢 + ∇ 𝜋

𝑓  in  ⁢ Ω ,

(12)

div ⁡ 𝑢

0  in  ⁢ Ω ,

(13)

𝜆 ⁢ 𝑢 𝜏 + 𝛽 − 1 ⁢ [ ( 2 ⁢ 𝐷 ⁢ 𝑢 ⋅ 𝜈 ) 𝜏 + 𝛼 ⁢ 𝑢 𝜏 ]

ℎ  in  ⁢ ∂ Ω ,

(14)

𝑢 𝜈

0  in  ⁢ ∂ Ω .

(15)

Since the operator 𝒜 is defined on a product space, we retain this structure also in this part of the presentation. However, this is not strictly necessary, because the second component of the space is the trace of the first.

In this part we work in the space 𝒢 . We recall its definition

𝒢

{ ( 𝑢 , 𝑢 𝑏 ) ∈ 𝐻 𝜎 1 ⁢ ( Ω ) × 𝐿 𝜈 2 ⁢ ( ∂ Ω ) : 𝑢 𝑏

𝛾 ⁢ ( 𝑢 ) }

with norm

‖ ( 𝑢 , 𝑢 𝑏 ) ‖ 𝒢 2

2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 .

The norm in 𝒢 is equivalent to the norm in 𝐻 1 ⁢ ( Ω ) due to Korn’s and Poincaré’s inequalities, originally established in [12]; see also [2, Proposition 3.13]. For reader’s convenience, we present it here. The essential part of Lemma 4 is taken from [2, Proposition 3.13]. The last equivalence of norms is the standard Korn’s inequality.

Lemma 4.

Let Ω be a bounded Lipschitz domain. Then, for all 𝑢 ∈ 𝐻 1 ⁢ ( Ω ) with 𝑢 𝜈

0 on ∂ Ω , we have

‖ 𝑢 ‖ 𝐻 1 ⁢ ( Ω ) ∼ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω )

if Ω is nonaxisymmetric, and

‖ 𝑢 ‖ 𝐻 1 ⁢ ( Ω ) ∼ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ 𝑢 𝜏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) ∼ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω )

if Ω is arbitrary. Here, “ ∼ ” denotes the equivalence of two norms.

The first part of this lemma proves Lemma 2. We continue with the definition of a weak solution to (12)–(15).

Definition 5.

Let ( 𝑓 , ℎ ) ∈ 𝐿 2 ⁢ ( Ω ) × 𝐿 2 ⁢ ( ∂ Ω ) (complex valued) and let 𝛼 ∈ ℝ , 𝜆 ∈ ℂ . We say that ( 𝑢 , 𝑢 𝑏 ) ∈ 𝒢 is a weak solution to (12)–(15) if

𝜆 ⁢ ∫ Ω 𝑢 ⁢ 𝜑 ¯ + ∫ Ω 2 ⁢ 𝐷 ⁢ 𝑢 : ∇ 𝜑 ¯ + ( 𝛽 ⁢ 𝜆 + 𝛼 ) ⁢ ∫ ∂ Ω 𝑢 𝑏 ⁢ 𝜑 ¯ 𝑏

∫ Ω 𝑓 ⁢ 𝜑 ¯ + ∫ ∂ Ω 𝛽 ⁢ ℎ ⁢ 𝜑 ¯ 𝑏

(16)

holds for every ( 𝜑 , 𝜑 𝑏 ) ∈ 𝒢 .

Note again that the boundary condition (14) is embedded within the weak formulation (16) and cannot be expressed pointwise under regularity assumptions of Definition 5. However, if 𝑢 is more regular, e.g., 𝑢 ∈ 𝐻 2 ⁢ ( Ω ) ∩ 𝐻 𝜎 1 ⁢ ( Ω ) , then one can show that (12)–(15) hold pointwise almost everywhere in Ω or ∂ Ω ; see Proposition 17.

We will use the standard definition of the sector

Definition 6.

For 𝜔 ∈ ℝ , 𝜃 ∈ ( 0 , 𝜋 ) we define

𝑆 𝜃 , 𝜔

{ 𝜆 ∈ ℂ ; 𝜆 ≠ 𝜔 , | arg ⁡ ( 𝜆 − 𝜔 ) | < 𝜃 } .

Throughout this article, arg denotes the continuous branch of the argument function, taking values in [ − 𝜋 , 𝜋 ) .

We aim to prove results on the existence, uniqueness and estimates of the weak solutions. Before formulating these results we need some preparatory lemmata. We start with a simple lemma on properties of complex numbers.

Lemma 7.

Let 𝜃 ∈ ( 0 , 𝜋 ) and Arg be a continuous branch of argument. Then

| 𝑎 ⁢ 𝜆 + 𝑏 ⁢ 𝜇 | ≥ cos ⁡ ( 𝜃 / 2 ) ⁢ ( 𝑎 ⁢ | 𝜆 | + 𝑏 ⁢ | 𝜇 | )

(17)

for all 𝑎 , 𝑏

0 and all 𝜆 , 𝜇 ∈ ℂ ∖ { 0 } such that | Arg ⁡ ( 𝜆 ) − Arg ⁡ ( 𝜇 ) | ≤ 𝜃 . In particular,

| 𝑎 ⁢ 𝜆 + 𝑏 | ≥ cos ⁡ ( 𝜃 / 2 ) ⁢ ( 𝑎 ⁢ | 𝜆 | + 𝑏 )

(18)

for all 𝑎 , 𝑏

0 and all 𝜆 ∈ 𝑆 𝜃 , 0 ¯ .

Proof.

To prove (18), we realize that due to the fact that 𝑏 > 0 we can separately treat the cases when Im ⁡ 𝜆 > 0 and Im ⁡ 𝜆 ≤ 0 . Then it is sufficient to set Arg

arg and apply (17) for 𝜆 ≠ 0 , whereas for 𝜆

0 is (18) obvious.

To prove (17), we set 𝜔

( Arg ⁡ 𝜆 + Arg ⁡ 𝜇 ) / 2 and 𝛾

𝜃 / 2 . Then we have

| arg ⁡ ( 𝑒 − 𝑖 ⁢ 𝜔 ⁢ 𝑎 ⁢ 𝜆 ) | , | arg ⁡ ( 𝑒 − 𝑖 ⁢ 𝜔 ⁢ 𝑏 ⁢ 𝜇 ) | ≤ 𝛾 ,

(19)

i.e., 𝑒 − 𝑖 ⁢ 𝜔 ⁢ 𝑎 ⁢ 𝜆 and 𝑒 − 𝑖 ⁢ 𝜔 ⁢ 𝑏 ⁢ 𝜇 belong to 𝑆 𝛾 , 0 ¯ . Obviously, for any 𝑧 ∈ 𝑆 𝛾 , 0 ¯ we have Re ⁡ 𝑧 ≥ | 𝑧 | ⁢ cos ⁡ 𝛾 . Now, we can estimate

| 𝑎 ⁢ 𝜆 + 𝑏 ⁢ 𝜇 |

| 𝑒 − 𝑖 ⁢ 𝜔 ⁢ ( 𝑎 ⁢ 𝜆 + 𝑏 ⁢ 𝜇 ) | ≥ Re ⁡ ( 𝑒 − 𝑖 ⁢ 𝜔 ⁢ ( 𝑎 ⁢ 𝜆 + 𝑏 ⁢ 𝜇 ) )

Re ⁡ ( 𝑒 − 𝑖 ⁢ 𝜔 ⁢ 𝑎 ⁢ 𝜆 ) + Re ⁡ ( 𝑒 − 𝑖 ⁢ 𝜔 ⁢ 𝑏 ⁢ 𝜇 ) ≥ cos ⁡ 𝛾 ⁢ ( 𝑎 ⁢ | 𝜆 | + 𝑏 ⁢ | 𝜇 | ) .

∎

The next lemma deals with a fundamental estimate needed for proof of existence of the weak solutions and also for spectral estimates.

Lemma 8.

Let 𝛼 ∈ ℝ , 𝛽

0 , and let 𝜔 ∈ ℝ be such that

∃ 𝐶

0 , ∀ 𝑢 ∈ 𝒢 : 2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ( 𝛼 + 𝛽 ⁢ 𝜔 ) ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 + 𝜔 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 ≥ 𝐶 ⁢ ‖ 𝑢 ‖ 𝒢 2 .

(20)

Then for every 𝜃 ∈ ( 0 , 𝜋 ) there exists 𝑐 > 0 such that for all 𝜆 ∈ 𝑆 𝜃 , 𝜔 ¯ and 𝒰

( 𝑢 , 𝑢 𝑏 ) ∈ 𝒢 the following inequality holds

| 𝜆 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + 2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ( 𝛼 + 𝛽 ⁢ 𝜆 ) ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 | ≥ 𝑐 ⁢ ‖ 𝒰 ‖ 𝒢 2 + 𝑐 ⁢ | 𝜆 − 𝜔 | ⁢ ‖ 𝒰 ‖ ℋ 2 .

(21) Proof.

The estimate (21) clearly holds for 𝑢

( 0 , 0 ) ∈ 𝒢 . Take 𝜆 ∈ 𝑆 𝜃 , 𝜔 ¯ and 𝑢 ∈ 𝒢 ∖ { ( 0 , 0 ) } arbitrary. Denote

𝑀 ⁢ ( 𝜆 )

𝜆 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + 2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ( 𝛼 + 𝛽 ⁢ 𝜆 ) ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 .

Then

𝑀 ⁢ ( 𝜆 )

( 𝜆 − 𝜔 ) ⁢ ( ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + 𝛽 ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 ) + 2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ( 𝛼 + 𝛽 ⁢ 𝜔 ) ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 + 𝜔 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 .

(22)

Let us observe that 2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ( 𝛼 + 𝛽 ⁢ 𝜔 ) ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 + 𝜔 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2

0 by (20) and 𝜆 − 𝜔 ∈ 𝑆 𝜃 , 0 ¯ . Equality (22) together with Lemma 7 and (20) imply

| 𝑀 ⁢ ( 𝜆 ) | ≥ 𝑐 ⁢ ‖ 𝒰 ‖ 𝒢 2 + 𝑐 ⁢ | 𝜆 − 𝜔 | ⁢ ‖ 𝒰 ‖ ℋ 2 .

∎

Remark 9.

The condition (20) is a version of the Poincaré-Korn inequality. Note that it is valid if any of the following conditions is met

a.

𝛼

0 and 𝜔 ≥ 0 ,

b.

Ω nonaxisymmetric , 𝛼 ∈ ( 𝛼 0 , 0 ] and 𝜔 ≥ 0 ,

c.

𝛼 ≤ 0 and 𝜔

− 𝛼 / 𝛽 .

Indeed, this follows for the cases a and c directly from Lemma 4. In the case b one also needs to exploit Lemma 2.

In the next proposition we prove the existence and uniqueness of weak solutions in the set  𝒢 .

Proposition 10.

Let 𝛼 ∈ ℝ , 𝛽 > 0 , 𝜃 ∈ ( 0 , 𝜋 ) , and 𝜔 ≥ 0 be such that one of the conditions a, b, c of Remark 9 be satisfied. Then there exists 𝐶 > 0 such that for all ( 𝑓 , ℎ ) ∈ 𝐿 2 ⁢ ( Ω ) × 𝐿 2 ⁢ ( ∂ Ω ) , 𝜆 ∈ 𝑆 𝜃 , 𝜔 ¯ there exists a unique weak (complex-valued) solution ( 𝑢 , 𝑢 𝑏 ) ∈ 𝒢 of (12)–(15) satisfying 𝑢 𝑏 ∈ 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) . Further, there exists a unique 𝜋 ∈ 𝐿 2 ⁢ ( Ω ) with ∫ Ω 𝜋

0 such that

∫ Ω 𝑓 ⁢ 𝜑 ¯ + ∫ ∂ Ω 𝛽 ⁢ ℎ ⁢ 𝜑 ¯ 𝑏

𝜆 ⁢ ∫ Ω 𝑢 ⁢ 𝜑 ¯ + ∫ Ω 2 ⁢ 𝐷 ⁢ 𝑢 : ∇ 𝜑 ¯ − ∫ Ω 𝜋 ⁢ div ⁡ 𝜑 ¯ + ( 𝛽 ⁢ 𝜆 + 𝛼 ) ⁢ ∫ ∂ Ω 𝑢 𝑏 ⁢ 𝜑 ¯ 𝑏

(23)

for all 𝜑 ∈ 𝐻 1 ⁢ ( Ω ) . The following estimate holds

‖ 𝑢 ‖ 𝒢 + | 𝜆 − 𝜔 | ⁢ ‖ 𝑢 ‖ ℋ + ‖ 𝜋 ‖ 𝐿 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐿 2 ⁢ ( ∂ Ω ) ) .

(24) Proof.

We define the sesquilinear form

𝐵 ⁢ ( 𝒰 , 𝒱 )

𝜆 ⁢ ∫ Ω 𝑢 ⁢ 𝑣 ¯ + ∫ Ω 2 ⁢ 𝐷 ⁢ 𝑢 : ∇ 𝑣 ¯ + ( 𝛼 + 𝛽 ⁢ 𝜆 ) ⁢ ∫ ∂ Ω 𝑢 𝑏 ⁢ 𝑣 ¯ 𝑏

on 𝒢 × 𝒢 where 𝒰

( 𝑢 , 𝑢 𝑏 ) , 𝒱

( 𝑣 , 𝑣 𝑏 ) . Lemma 8 and Remark 9 imply the existence of 𝐶

0 independent of 𝒰 and 𝜆 such that

| 𝐵 ⁢ ( 𝒰 , 𝒰 ) |

| 𝜆 ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + 2 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ( 𝛼 + 𝛽 ⁢ 𝜆 ) ⁢ ‖ 𝑢 𝑏 ‖ 𝐿 2 ⁢ ( ∂ Ω ) 2 | ≥ 𝐶 ⁢ ‖ 𝒰 ‖ 𝒢 2

Moreover, the form 𝐵 is bounded from above on 𝒢 .

By the Lax-Milgram theorem (see, e.g., [18]), for ℱ ∈ 𝒢 ∗ defined by ℱ ⁢ ( Φ )

∫ Ω 𝑓 ⁢ 𝜑 ¯ + ∫ ∂ Ω 𝛽 ⁢ ℎ ⁢ 𝜑 ¯ 𝑏 for Φ ∈ 𝒢 there exists a unique 𝒰

( 𝑢 , 𝑢 𝑏 ) ∈ 𝒢 such that 𝐵 ⁢ ( Φ , 𝒰 )

ℱ ⁢ ( Φ ) for every Φ

( 𝜑 , 𝜑 𝑏 ) ∈ 𝒢 , i.e., (16) holds. By the trace theorem, 𝑢 𝑏

𝛾 ⁢ ( 𝑢 ) ∈ 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) . Estimate (24) of 𝑢 follows from 𝐵 ⁢ ( 𝒰 , 𝒰 )

ℱ ⁢ ( 𝒰 ) , properties of 𝐵 and ℱ , Lemma 8 and Remark 9.

Let us now prove existence and uniqueness of 𝜋 . By [9, Theorem III.5.3] any weak solution 𝑢 defined in Definition 5 can be associated with a pressure 𝜋 ∈ 𝐿 2 ⁢ ( Ω ) satisfying

∫ Ω 𝑓 ⁢ 𝜑 ¯

𝜆 ⁢ ∫ Ω 𝑢 ⁢ 𝜑 ¯ + ∫ Ω 2 ⁢ 𝐷 ⁢ 𝑢 : ∇ 𝜑 ¯ − ∫ Ω 𝜋 ⁢ div ⁡ 𝜑 ¯

(25)

for any 𝜑 ∈ 𝐻 1 ⁢ ( Ω ) with 𝛾 ⁢ ( 𝜑 )

0 . The pressure is defined uniquely up to an additive constant. Let us further require the constant to be chosen in such a way that the pressure has zero mean over Ω . Then the validity of the estimate (24) for pressure follows from [9, Lemma IV.1.1] and the estimate (24) for 𝑢 .

For 𝜑 ∈ 𝐻 1 such that 𝜑 𝜈

0 on ∂ Ω we can find 𝑧 ∈ 𝐻 1 ⁢ ( Ω ) such that 𝛾 ⁢ ( 𝑧 )

0 on ∂ Ω and div ⁡ 𝑧

div ⁡ 𝜑 in Ω ; see [9, Theorem III.3.1]. Now, 𝑧 is an admissible test function in (25) and 𝜑 − 𝑧 ∈ 𝐻 𝜎 1 ⁢ ( Ω ) is an admissible test function in (16). Subtracting the so obtained equalities one gets (23) for all 𝜑 ∈ 𝐻 1 such that 𝜑 𝜈

0 on ∂ Ω

∎

Remark 11.

Note, that the mapping ( 𝑓 , ℎ ) ∈ 𝐿 𝜎 2 ⁢ ( Ω ) × 𝐿 𝜈 2 ⁢ ( ∂ Ω ) ↦ ( 𝑢 , 𝜋 ) ∈ 𝒢 × 𝐿 2 ⁢ ( Ω ) from the previous theorem is linear and bounded.

Since the parametres 𝛼 ∈ ℝ and 𝛽

0 are fixed, we do not track the dependence of the constant 𝐶 on these parameters in Proposition 10 and also in all further estimates.

Before we state our result on regularity of weak solutions we need to prove a lemma on existence of a special function satisfying boundary conditions.

Lemma 12.

There exists 𝐶 > 0 such that for every ℎ ∈ 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) there exists 𝑤 ∈ 𝐻 2 ⁢ ( Ω ) with properties 1) div ⁡ ( 𝑤 )

0 in Ω , 2) 𝛾 ⁢ ( 𝑤 )

0 on ∂ Ω , 3) ( 2 ⁢ 𝛾 ⁢ ( 𝐷 ⁢ 𝑤 ) ⋅ 𝜈 ) 𝜏

ℎ on ∂ Ω and 4) ‖ 𝑤 ‖ 𝐻 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) .

Remark 13.

Regularity of 𝑤 ∈ 𝐻 2 ⁢ ( Ω ) together with 1) and 2) imply 𝑤 ∈ 𝐻 𝜎 2 ⁢ ( Ω ) .

Proof of Lemma 12.

Step 1: We construct a function 𝑧 ∈ 𝐻 2 ⁢ ( Ω ) satisfying conditions 2)-4) and, additionally, 5) div ⁡ 𝑧

0 on ∂ Ω . By the inverse trace theorem (see, e.g., [16, Theorem 2.5.8]) there exists 𝑧 ∈ 𝐻 2 ⁢ ( Ω ) such that 𝛾 ⁢ ( 𝑧 )

0 and 𝛾 ⁢ ( ∇ 𝑧 ) ⁢ 𝜈

ℎ on ∂ Ω and ‖ 𝑧 ‖ 𝐻 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) . This function 𝑧 obviously satisfies 2) and 4). Since 𝛾 ⁢ ( 𝑧 )

0 on ∂ Ω , it follows that 𝛾 ⁢ ( ∇ 𝑧 ) ⁢ 𝜉

0 for any tangent vector 𝜉 to ∂ Ω . Consequently,

[ 𝛾 ⁢ ( ∇ 𝑧 ) 𝑇 ⁢ 𝜈 ] ⋅ 𝜉

𝜉 𝑇 ⁢ 𝛾 ⁢ ( ∇ 𝑧 ) 𝑇 ⁢ 𝜈

[ 𝛾 ⁢ ( ∇ 𝑧 ) ⁢ 𝜉 ] 𝑇 ⁢ 𝜈

0 ⁢ 𝜈

0 .

Thus, we obtain ( 2 ⁢ [ 𝛾 ⁢ ( 𝐷 ⁢ 𝑧 ) ] ⁢ 𝜈 ) 𝜏

( [ 𝛾 ⁢ ( ∇ 𝑧 ) ] ⁢ 𝜈 ) 𝜏 + ( [ 𝛾 ⁢ ( ∇ 𝑧 ) ] 𝑇 ⁢ 𝜈 ) 𝜏

( [ 𝛾 ⁢ ( ∇ 𝑧 ) ] ⁢ 𝜈 ) 𝜏

ℎ 𝜏

ℎ , which confirms 3). Further, Héron’s formula (see [11, Lemme 3.3] or [3, Lemma 3.5]) yields

div ⁡ 𝑧

div ∂ Ω ⁡ ( 𝑧 𝜏 ) + [ 𝛾 ⁢ ( ∇ 𝑧 ) ] ⁢ 𝜈 ⋅ 𝜈 − 2 ⁢ 𝐾 ⁢ 𝑧 𝜈 on  ∂ Ω .

(26)

In the formula, 𝐾 denotes the mean curvature of ∂ Ω and div ∂ Ω denotes the surface divergence. All three terms on the right-hand side of (26) are zero since 𝛾 ⁢ ( 𝑧 )

0 on ∂ Ω and [ 𝛾 ⁢ ( ∇ 𝑧 ) ] ⁢ 𝜈 ⋅ 𝜈

ℎ ⋅ 𝜈

0 . So, 5) holds.

Step 2: It remains to correct the solenoidality of 𝑧 without destroying the conditions 2 ) − 4 ) . To do this we apply [5, Theorem 2] to the problem div ⁡ 𝜁

div ⁡ 𝑧 in Ω . Since div ⁡ 𝑧 ∈ 𝐻 0 1 ⁢ ( Ω ) and ∫ Ω div ⁡ 𝑧

∫ ∂ Ω 𝑧 𝜈

0 there exists a solution 𝜁 ∈ 𝐻 0 2 ⁢ ( Ω ) of this problem such that ‖ 𝜁 ‖ 𝐻 2 ≤ 𝐶 ⁢ ‖ div ⁡ 𝑧 ‖ 𝐻 1 ≤ 𝐶 ⁢ ‖ 𝑧 ‖ 𝐻 2 ≤ 𝐶 ⁢ ‖ ℎ ‖ 𝐻 1 / 2 .

Finally, it remains to define 𝑤

𝑧 − 𝜁 . This function satisfies all conditions 1)–4). ∎

Theorem 14.

Under the assumptions of Proposition 10 the unique weak solution ( 𝑢 , 𝑢 𝑏 ) of (12)–(15) and the associated pressure 𝜋 satisfy ( 𝑢 , 𝑢 𝑏 ) ∈ 𝐷 ⁢ ( 𝒜 ) , 𝜋 ∈ 𝐻 1 ⁢ ( Ω ) for all 𝑓 ∈ 𝐿 2 ⁢ ( Ω ) , ℎ ∈ 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) . Moreover, there exists 𝐶

0 independent of 𝜆 , 𝑓 , ℎ such that

‖ 𝑢 𝑏 ‖ 𝐻 3 / 2 ⁢ ( ∂ Ω ) + ‖ 𝑢 ‖ 𝐻 2 ⁢ ( Ω ) + ‖ 𝜋 ‖ 𝐻 1 ⁢ ( Ω ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) .

(27) Proof.

According to the definition of 𝐷 ⁢ ( 𝒜 ) it suffices to show 𝑢 ∈ 𝐻 2 ⁢ ( Ω ) , 𝜋 ∈ 𝐻 1 ⁢ ( Ω ) together with the estimate (27). Note that the estimate of the boundary value 𝑢 𝑏 follows from the estimate of 𝑢 in 𝐻 2 ⁢ ( Ω ) by the trace theorem.

For 𝜆 ∈ ℝ we rewrite the system in the form

− Δ ⁢ 𝑢 + ∇ 𝜋

𝑓 − 𝜆 ⁢ 𝑢 , div ⁡ 𝑢

0 ,  in  ⁢ Ω ,

( 2 ⁢ 𝐷 ⁢ 𝑢 ⋅ 𝜈 ) 𝜏 + ( 𝛽 ⁢ 𝜆 + 𝜂 + 𝛼 ) ⁢ 𝑢 𝜏

𝛽 ⁢ ℎ + 𝜂 ⁢ 𝑢 𝜏 , 𝑢 𝜈

0  in  ⁢ ∂ Ω ,

where 𝜂

| 𝛼 | + 1 . Any of assumptions a–c of Remark 9 implies ( 𝛽 ⁢ 𝜆 + 𝜂 + 𝛼 ) ≥ 1 . We have ‖ 𝑓 − 𝜆 ⁢ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐿 2 ⁢ ( ∂ Ω ) ) and ‖ 𝛽 ⁢ ℎ + 𝜂 ⁢ 𝑢 𝜏 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) by Proposition 10. Therefore, we can apply [2, Theorem 4.5] to get the estimate (27).

If 𝜆 ∈ ℂ we still have a weak solution 𝑢 of (12)–(15) by Proposition 10. We would like to apply a complex valued analogue of [2, Theorem 4.5] to

− Δ ⁢ 𝑣 + ∇ 𝜎

𝑓 ~ , div ⁡ 𝑣

0 ,  in  ⁢ Ω ,

( 2 ⁢ 𝐷 ⁢ 𝑣 ⋅ 𝜈 ) 𝜏 + 𝛼 ~ ⁢ 𝑣 𝜏

ℎ ~ , 𝑣 𝜈

0  in  ⁢ ∂ Ω ,

(28)

where 𝑓 ~

𝑓 − 𝜆 ⁢ 𝑢 ∈ 𝐿 2 ⁢ ( Ω ) , 𝛼 ~

𝛽 ⁢ 𝜆 + 𝜂 + 𝛼 ∈ 𝑆 𝜃 , 1 ¯ and ℎ ~

𝛽 ⁢ ℎ + 𝜂 ⁢ 𝑢 𝜏 ∈ 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) with norms independent of 𝜆 . The proof presented in [2] works also in the complex valued situation with minor changes.

As in that article, we can again assume without loss of generality that ℎ ~

0 . In fact, if ℎ ~ ≠ 0 we consider a solenoidal function 𝑤 ∈ 𝐻 𝜎 2 ⁢ ( Ω ) satisfying the equation (28)2 on the boundary. Such a function exists and is independent of 𝛼 ~ due to Lemma 12 and satisfies ‖ 𝑤 ‖ 𝐻 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ‖ ℎ ~ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) . Then it suffices to study the solution to (28) with the right hand side 𝑓 ~ + Δ ⁢ 𝑤 ∈ 𝐿 2 ⁢ ( Ω ) and ℎ ~

0 .

To show the regularity of a weak solution to (28) with ℎ ~

0 and of the associated pressure we apply the method of difference quotients as in [2]. It can be followed almost line by line. The only difference is in obtaining regularity at the boundary in the tangent direction since our parameter 𝛼 ~ is complex. We present here the main idea of this estimate in the case that we deal with the flat portion of the boundary. Let 𝑥 0 ∈ ∂ Ω , 𝑟 > 0 , 𝑈 := 𝐵 ⁢ ( 𝑥 0 , 𝑟 ) , 2 ⁢ 𝑈 := 𝐵 ⁢ ( 𝑥 0 , 2 ⁢ 𝑟 ) be such that ∂ Ω ∩ 2 ⁢ 𝑈 is a subset of a hyperplane perpendicular to 𝑒 𝑑 . We test the weak formulation of the equation (28) by the complex conjugate 𝐷 𝑘 − ℎ ⁢ ( 𝜁 2 ⁢ 𝐷 𝑘 ℎ ⁢ 𝑣 ) ¯ , where 𝜁 ∈ 𝒟 ⁢ ( 2 ⁢ 𝑈 ) , 𝜁 ≥ 𝜒 𝑈 is a cut-off function that localizes our consideration to the neighborhood of the flat boundary and 𝐷 𝑘 ℎ is a difference quotient of size ℎ ≠ 0 taken in the direction 𝑒 𝑘 parallel to the boundary, i.e., 𝐷 𝑘 ℎ ⁢ 𝑣 ⁢ ( 𝑥 )

( 𝑣 ⁢ ( 𝑥 + ℎ ⁢ 𝑒 𝑘 ) − 𝑣 ⁢ ( 𝑥 ) ) / ℎ for 𝑥 ∈ ℝ 𝑑 . We can follow the computation in the section (i) of the proof of [2, Theorem 4.5] almost line by line to get

2 ⁢ ∫ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝐷 ⁢ 𝑣 | 2 + 𝛼 ~ ⁢ ∫ ∂ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝑣 𝜏 | 2 ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ‖ 𝜋 ‖ 𝐿 2 ⁢ ( Ω ) 2 + ‖ ∇ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) 2 ) .

Here comes the only difference in the argumentation, since to estimate the left hand side from below we need to employ Lemma 7 with 𝛼 ~ − 1 ∈ 𝑆 𝜃 , 0 ¯

| 2 ⁢ ∫ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝐷 ⁢ 𝑣 | 2 + 𝛼 ~ ⁢ ∫ ∂ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝑣 𝜏 | 2 |

| 2 ⁢ ∫ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝐷 ⁢ 𝑣 | 2 + ∫ ∂ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝑣 𝜏 | 2 + ( 𝛼 ~ − 1 ) ⁢ ∫ ∂ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝑣 𝜏 | 2 |

≥ 𝑐 ⁢ ( 2 ⁢ ∫ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝐷 ⁢ 𝑣 | 2 + ∫ ∂ Ω 𝜁 2 ⁢ | 𝐷 𝑘 ℎ ⁢ 𝑣 𝜏 | 2 ) .

Hence, one can continue as in [2] to conclude that solutions of (28) satisfy

‖ 𝑣 ‖ 𝐻 2 ⁢ ( 𝑈 ) + ‖ 𝜎 ‖ 𝐻 1 ⁢ ( 𝑈 ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ~ ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ~ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) )

which implies (27) by the flat portion of ∂ Ω . The full estimate (27) is obtained by localization and flattening the boundary. For details see [4].

∎

Remark 15.

It follows from Theorem 14, Proposition 10 and Remark 11 that the mapping associating the pressure with zero mean to the problem data, ( 𝑓 , ℎ ) ∈ 𝐿 𝜎 2 ⁢ ( Ω ) × 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) ↦ 𝜋 ∈ 𝐻 1 ⁢ ( Ω ) , is linear and bounded from 𝐿 𝜎 2 ⁢ ( Ω ) × 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) to 𝐻 1 ⁢ ( Ω ) .

Remark 16.

It seems to us that in [2] the result corresponding to the previous Theorem is announced for bounded domains Ω with 𝐶 1 , 1 boundary. As the main reference for the technique that allows to get the result in the neighborhood of the nonflat boundary is presented [4]. We are not able to reconstruct the proof for 𝐶 1 , 1 domains and we want to remark that also in [21] and [4] it is assumed that the boundary of Ω is 𝐶 3 and 𝐶 2 , 1 respectively.

The proof of regularity up to the boundary is done in the following steps. In order to avoid troubles with nonflat boundary, the problem is reformulated as a regularity problem with flat boundary and finally for this problem the technique of differences is used to show regularity of its solutions.

Let us discuss the flattening of the boundary in more detail. We assume that 𝑥 0 ∈ ∂ Ω , 𝑟 > 0 and Ω ∩ 𝐵 ⁢ ( 𝑥 0 , 𝑟 )

{ 𝑥 ∈ 𝐵 ⁢ ( 𝑥 0 , 𝑟 ) ; 𝑥 𝑑 > 𝐻 ⁢ ( 𝑥 1 , … , 𝑥 𝑑 − 1 ) } , where 𝐻 : ℝ 𝑑 − 1 → ℝ is a given function parametrizing the boundary of Ω . Moreover, the coordinate system corresponding to 𝑥 0 is chosen in such a way that 𝐻 ⁢ ( 0 , … , 0 )

0 , ∇ ′ 𝐻 ⁢ ( 0 , … , 0 )

0 . A modified solution is defined by the formula 𝑢 ~ ⁢ ( 𝑥 ′ , 𝑥 𝑑 − 𝐻 ⁢ ( 𝑥 ′ ) ) := ( 𝑢 ′ ⁢ ( 𝑥 ) , 𝑢 𝑑 ⁢ ( 𝑥 ) − ∇ ′ 𝐻 ⁢ ( 𝑥 ′ ) ⋅ 𝑢 ′ ⁢ ( 𝑥 ) ) for 𝑥 ∈ Ω ∩ 𝐵 ⁢ ( 𝑥 0 , 𝑟 ) , where 𝑥 ′

( 𝑥 1 , … , 𝑥 𝑑 − 1 ) , 𝑢 ′

( 𝑢 1 , … , 𝑢 𝑑 − 1 ) , ∇ ′

( ∂ 1 , … , ∂ 𝑑 − 1 ) . Note that it is defined on a subset of { 𝑥 ∈ ℝ 𝑑 ; 𝑥 𝑑 > 0 } . The term ∇ ′ 𝐻 ⁢ ( 𝑥 ′ ) ⋅ 𝑢 ′ ⁢ ( 𝑥 ) is subtracted from the last component of 𝑢 to enforce div ⁡ 𝑢 ~

0 in new coordinates.

It can be shown that the function 𝑢 ~ then again solves a variant of the Stokes problem and that this function is 𝐻 2 ⁢ ( 𝐵 ⁢ ( ( 𝑥 1 , … , 𝑥 𝑑 − 1 , 0 ) , 𝜌 ) ∩ { 𝑥 ∈ ℝ 𝑑 ; 𝑥 𝑑

0 } ) for some 𝜌

0 . One should reconstruct from this fact that also the original function 𝑢 is in 𝐻 2 . However, the term ∇ ′ 𝐻 ⁢ ( 𝑥 ′ ) ⋅ 𝑢 ′ ⁢ ( 𝑥 ) stands in the way. One needs to show that it also belongs to 𝐻 2 . Here one uses the choice of the coordinate system. It is clear that some information on third derivative of 𝐻 is needed, e.g., 𝐻 ∈ 𝐶 2 , 1 . In [4, Section 4, page 1096] a variant 𝐻 ∈ 𝑊 3 , 3 is also discussed.

Proposition 17.

Under the assumptions of Proposition 10 let ℱ

( 𝑓 , ℎ ) ∈ 𝐿 𝜎 2 ⁢ ( Ω ) × 𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω ) . The weak solution 𝒰 := ( 𝑢 , 𝑢 𝑏 ) ∈ 𝒢 of (12)–(15) belongs to 𝐷 ⁢ ( 𝒜 ) and satisfies 𝜆 ⁢ 𝒰 − 𝒜 ⁢ 𝒰

ℱ .

Proof.

Since ( 𝑢 , 𝑢 𝑏 ) ∈ 𝐷 ⁢ ( 𝒜 ) and the associated pressure 𝜋 ∈ 𝐻 1 by Theorem 14, we have 𝑢 ∈ 𝐻 2 and by (23)

∫ Ω 𝑓 ⁢ 𝜑 ¯ + ∫ ∂ Ω 𝛽 ⁢ ℎ ⁢ 𝜑 ¯ 𝑏

𝜆 ⁢ ∫ Ω 𝑢 ⁢ 𝜑 ¯ + 2 ⁢ ∫ Ω 𝐷 ⁢ 𝑢 : ∇ 𝜑 ¯ − ∫ Ω 𝜋 ⁢ div ⁡ 𝜑 ¯ + ( 𝛽 ⁢ 𝜆 + 𝛼 ) ⁢ ∫ ∂ Ω 𝑢 𝑏 ⁢ 𝜑 ¯ 𝑏

(29)

= 𝜆 ⁢ ∫ Ω 𝑢 ⁢ 𝜑 ¯ + 2 ⁢ ∫ ∂ Ω [ 𝐷 ⁢ 𝑢 ] ⁢ 𝜈 ⁢ 𝜑 ¯ 𝑏 − ∫ Ω Δ ⁢ 𝑢 ⁢ 𝜑 ¯ + ∫ Ω ∇ 𝜋 ⁢ 𝜑 ¯ + ( 𝛽 ⁢ 𝜆 + 𝛼 ) ⁢ ∫ ∂ Ω 𝑢 𝑏 ⁢ 𝜑 ¯ 𝑏

for any 𝜑 ∈ 𝐻 1 , 𝜑 𝑏

𝛾 ⁢ ( 𝜑 ) with ( 𝜑 𝑏 ) 𝜈

0 on ∂ Ω . It follows 𝜆 ⁢ 𝑢 − Δ ⁢ 𝑢 + ∇ 𝜋

𝑓 a.e. in Ω which gives 𝜆 ⁢ 𝑢 − 𝑃 ⁢ ( Δ ⁢ 𝑢 )

𝑓 . Inserting the pointwise equality 𝜆 ⁢ 𝑢 − Δ ⁢ 𝑢 + ∇ 𝜋

𝑓 into (29) for a general 𝜑 ∈ 𝐻 1 , 𝜑 𝑏

𝛾 ⁢ ( 𝜑 ) with ( 𝜑 𝑏 ) 𝜈

0 on ∂ Ω we obtain

∫ ∂ Ω 𝛽 ⁢ ℎ ⁢ 𝜑 ¯ 𝑏

2 ⁢ ∫ ∂ Ω [ 𝐷 ⁢ 𝑢 ] ⁢ 𝜈 ⁢ 𝜑 ¯ 𝑏 + ( 𝛽 ⁢ 𝜆 + 𝛼 ) ⁢ ∫ ∂ Ω 𝑢 𝑏 ⁢ 𝜑 ¯ 𝑏 .

Due to the regularity of ( 𝑢 , 𝑢 𝑏 ) we get 𝛽 ⁢ ℎ

2 ⁢ ( [ 𝐷 ⁢ 𝑢 ] ⁢ 𝜈 ) 𝜏 + ( 𝛽 ⁢ 𝜆 + 𝛼 ) ⁢ 𝑢 𝑏 a.e. on ∂ Ω . ∎

4Proof of the main theorem 4.1Uniqueness of the weak solutions

Our Definition 1 of the weak solution differs from the one in [15, Definition 5.1] in the assumption on regularity of the right-hand side function ( 𝑓 , 𝑔 ) and of the solution ∂ 𝑡 𝑢 . That is why we present here a simple proof of the uniqueness of the weak solutions.

Lemma 18.

In the situation of Definition 1, let 𝑢 , 𝑣 be two weak solutions corresponding to the same data 𝑓 , 𝑔 , 𝑢 0 , 𝑣 0 . Then 𝑢

𝑣 .

Proof.

We define 𝑤

𝑢 − 𝑣 . Then 𝑤 is a weak solution corresponding to the trivial data. In particular, it solves (7) with zero right hand side. From this equation we read that actually ∂ 𝑡 𝑤 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 2 ⁢ ( [ 0 , 𝑇 ) , 𝒢 ∗ ) and consequently 𝑤 is the unique weak solution on any ( 0 , 𝑇 ∗ ) with 𝑇 ∗ ∈ ( 0 , 𝑇 ) in the spirit of [15, Definition 5.1], for uniqueness see [15, Theorem 5.1]. It follows that 𝑤

0 and 𝑢

𝑣 . ∎

4.2The operator 𝒜 generates an analytic semigroup

We show that ( 𝒜 , 𝐷 ⁢ ( 𝒜 ) ) is densely defined, closed, its resolvent set contains a sector and resolvent estimates are satisfied there; see (30). We start with

Proposition 19.

Let 𝛼 ∈ ℝ , 𝛽

0 . 𝐷 ⁢ ( 𝒜 ) is dense in 𝒳 0 and ( 𝒜 , 𝐷 ⁢ ( 𝒜 ) ) is a closed operator.

Proof.

We first prove density of 𝐷 ⁢ ( 𝒜 ) in 𝒳 0 . Let ( 𝑓 , ℎ ) ∈ 𝒳 0 and 𝜀 > 0 . Due to the density of 𝐻 3 / 2 ⁢ ( ∂ Ω ) in 𝐻 1 / 2 ⁢ ( ∂ Ω ) there exists ℎ ~ 1 ∈ 𝐻 3 / 2 ⁢ ( ∂ Ω ) such that ‖ ℎ − ℎ ~ 1 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) < 𝜀 . Then we orthonormally project ℎ ~ 1 to the tangent bundle of ∂ Ω and denote the resulting function ℎ 1 . Since Ω has 𝐶 2 , 1 boundary, the ortonormal projection does not spoil the regularity of ℎ . Indeed, ℎ 1 can be written as ℎ 1 ⁢ ( 𝑥 )

ℎ ~ 1 ⁢ ( 𝑥 ) − ⟨ ℎ ~ 1 ⁢ ( 𝑥 ) , 𝜈 ⁢ ( 𝑥 ) ⟩ ⁢ 𝜈 ⁢ ( 𝑥 ) . Consequently, ℎ 1 ∈ 𝐻 𝜈 3 / 2 ⁢ ( ∂ Ω ) . Moreover, since ℎ 𝜈

0 we have ‖ ℎ − ℎ 1 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ≤ ‖ ℎ − ℎ ~ 1 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) < 𝜀 . We now find 𝑓 1 ∈ 𝐻 𝜎 2 ⁢ ( Ω ) such that 𝛾 ⁢ ( 𝑓 1 )

ℎ 1 . Its existence follows from [3, Corollary 3.8]. Finally, by definition of 𝐿 𝜎 2 ⁢ ( Ω ) there exists 𝑓 2 ∈ 𝒟 𝜎 such that ‖ ( 𝑓 − 𝑓 1 ) − 𝑓 2 ‖ 𝐿 𝜎 2 ⁢ ( Ω ) < 𝜀 . Then ( 𝑓 1 + 𝑓 2 , ℎ 1 ) ∈ 𝐷 ⁢ ( 𝒜 ) is the desired approximation of ( 𝑓 , ℎ ) ∈ 𝒳 0 .

To show closedness of 𝒜 let 𝒰

( 𝑢 , 𝑏 ) , 𝒰 𝑛

( 𝑢 𝑛 , 𝑏 𝑛 ) ∈ 𝐷 ⁢ ( 𝒜 ) be such that 𝒰 𝑛 → 𝒰 in 𝒳 0 and ℱ 𝑛 := 𝒜 ⁢ 𝒰 𝑛 → ℱ in 𝒳 0 as 𝑛 → + ∞ . In particular,

𝛾 ⁢ ( 𝑢 𝑛 )

𝑏 𝑛 , 𝑢 𝑛 → 𝑢  in  𝐿 𝜎 2 ⁢ ( Ω ) , 𝑏 𝑛 → 𝑏  in  𝐻 𝜈 1 / 2 ⁢ ( ∂ Ω )  as  𝑛 → + ∞ .

Since { 𝐺 𝑛 }

{ 𝜆 ⁢ 𝒰 𝑛 − ℱ 𝑛 } is a bounded sequence in 𝒳 0 we can apply Theorem 14 to the equation 𝜆 ⁢ 𝒰 𝑛 − 𝒜 ⁢ 𝒰 𝑛

𝐺 𝑛 with 𝜆

max ⁡ ( 0 , − 𝛼 / 𝛽 ) + 1 . We get that the sequence { ‖ 𝑢 𝑛 ‖ 𝐻 2 } is bounded, so a subsequence { 𝑣 𝑛 } of { 𝑢 𝑛 } converges weakly in 𝐻 2 ⁢ ( Ω ) to some 𝑣 . By the convergence 𝑢 𝑛 → 𝑢 in 𝐿 𝜎 2 ⁢ ( Ω ) we have 𝑣

𝑢 and necessarily 𝑢 ∈ 𝐻 2 ⁢ ( Ω ) . Due to the continuity of the trace mapping, the embeddings and the Leray projection 𝑃 we also get 𝛾 ⁢ ( 𝑣 𝑛 ) ⇀ 𝛾 ⁢ ( 𝑢 ) in 𝐻 3 / 2 ⁢ ( ∂ Ω ) , div ⁡ 𝑣 𝑛 ⇀ div ⁡ 𝑢 in 𝐻 1 ⁢ ( Ω ) , and 𝑃 ⁢ Δ ⁢ 𝑣 𝑛 ⇀ 𝑃 ⁢ Δ ⁢ 𝑢 in 𝐿 𝜎 2 ⁢ ( Ω ) . Thus, we conclude that 𝑏

𝛾 ⁢ ( 𝑢 ) , 𝑢 ∈ 𝐻 𝜎 2 ⁢ ( Ω ) , ( 𝑢 , 𝑏 ) ∈ 𝐷 ⁢ ( 𝒜 ) , and 𝒜 ⁢ 𝒰

ℱ . ∎

Theorem 20.

Let 𝛼 ∈ ℝ , 𝛽

0 . The operator ( 𝒜 , 𝐷 ⁢ ( 𝒜 ) ) is sectorial. More precisely, if 𝜔 ∈ ℝ is such that 𝛼 , 𝛽 and 𝜔 satisfy one of the conditions in Remark 9 then for any 𝜃 ∈ ( 0 , 𝜋 ) there exists 𝐶

0 such that

𝑆 𝜃 , 𝜔 ¯ ⊂ 𝜌 ⁢ ( 𝒜 ) , ∀ 𝜆 ∈ 𝑆 𝜃 , 𝜔 ¯ : | 𝜆 − 𝜔 | ⁢ ‖ ( 𝜆 − 𝒜 ) − 1 ‖ ≤ 𝐶 .

(30)

If 1) 𝛼

0 or 2) 𝛼

𝛼 0 and Ω nonaxisymmetric, then 𝜔 can be chosen negative.

Proof.

Let 𝜃 ∈ ( 0 , 𝜋 ) and 𝜆 ∈ 𝑆 𝜃 , 𝜔 ¯ . By Proposition 17, for every ℱ

( 𝑓 , ℎ ) ∈ 𝒳 0 there exists a solution 𝒰

( 𝑢 , 𝑢 𝑏 ) ∈ 𝐷 ⁢ ( 𝒜 ) satisfying ( 𝜆 − 𝒜 ) ⁢ 𝒰

ℱ , i.e., the operator 𝜆 − 𝒜 : 𝐷 ⁢ ( 𝒜 ) → 𝒳 0 is surjective. Since any solution to ( 𝜆 − 𝒜 ) ⁢ 𝒰

ℱ corresponds to a unique weak solution of (12)–(15) (by Proposition 10), 𝜆 − 𝒜 is also injective. The operator 𝜆 − 𝒜 is closed by Proposition 19 and we obtain 𝜆 ∈ 𝜌 ⁢ ( 𝒜 ) . In particular, ( 𝜆 − 𝒜 ) − 1 is bounded.

Next, we establish the resolvent estimate, i.e., the inequality | 𝜆 − 𝜔 | ⁢ ‖ 𝒰 ‖ 𝒳 0 ≤ 𝐶 ⁢ ‖ ℱ ‖ 𝒳 0 . This can be reformulated for 𝒰

( 𝑢 , 𝑢 𝑏 ) and ℱ

( 𝑓 , ℎ ) as

| 𝜆 − 𝜔 | ⁢ ( ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ 𝑢 𝑏 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) .

In Proposition 10 we have already proved

| 𝜆 − 𝜔 | ⁢ ‖ 𝑢 ‖ 𝐿 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) .

(31)

From the second component of the equation ( 𝜆 − 𝒜 ) ⁢ 𝒰

ℱ (see (14)), we have

| 𝜆 | ⁢ ‖ 𝑢 𝑏 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ≤ | 𝛼 | ⁢ ‖ 𝑢 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) + 𝐶 ⁢ ‖ 𝐷 ⁢ 𝑢 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) + ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ,

and therefore

| 𝜆 − 𝜔 | ∥ 𝑢 𝑏 ∥ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ≤ ( | 𝜆 | + | 𝜔 | ) ∥ 𝑢 𝑏 ∥ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ≤ ( | 𝛼 | + | 𝜔 | ) ∥ 𝑢 ∥ 𝐻 1 / 2 ⁢ ( ∂ Ω ) + 𝐶 ∥ 𝐷 𝑢 ∥ 𝐻 1 / 2 ⁢ ( ∂ Ω ) + ∥ ℎ ∥ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) .

(32)

We estimate the terms containing 𝑢 on the right-hand side by the trace theorem and by Theorem 14 as

‖ 𝑢 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) + ‖ 𝐷 ⁢ 𝑢 ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ≤ 𝐶 ⁢ ‖ 𝑢 ‖ 𝐻 2 ⁢ ( Ω ) ≤ 𝐶 ⁢ ( ‖ 𝑓 ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) )

to get (30) combining (31) and (32).

If moreover 1) or 2) holds, then the statement is already established with 𝜔

0 . Moreover, 0 ∈ 𝜌 ⁢ ( 𝐴 ) , implying that a neighborhood of zero belongs to 𝜌 ⁢ ( 𝐴 ) . Fix 𝜔 < 0 within this neighborhood. Then, for appropriate 𝜃 ′ < 𝜃 we have 𝑆 𝜃 ′ , 𝜔 ⊂ 𝜌 ⁢ ( 𝐴 ) , and the resolvent estimates hold on this sector (by standard arguments). As 𝜃 ′ → 𝜋 when 𝜃 → 𝜋 , it follows that the resolvent estimates hold on 𝑆 𝜃 , 𝜔 for each 𝜃 < 𝜋 , with 𝐶 depending on 𝜃 . ∎

Corollary 21.

The operator ( 𝒜 , 𝐷 ⁢ ( 𝒜 ) ) generates an analytic semigroup { 𝒯 ⁢ ( 𝑡 ) } 𝑡

0 ⊂ ℒ ⁢ ( 𝒳 0 ) . There exist constants 𝜔 ∈ ℝ and 𝐶

0 such that the semigroup satisfies for any 𝑡

0

‖ 𝒯 ⁢ ( 𝑡 ) ‖ ℒ ⁢ ( 𝒳 0 ) ≤ 𝐶 ⁢ 𝑒 𝜔 ⁢ 𝑡 .

(33)

If moreover 1) 𝛼

0 or 2) 𝛼

𝛼 0 and Ω nonaxisymmetric, 𝜔 can be chosen negative.

Proof.

The statement follows directly from [14, Proposition 2.1.1] and Theorem 20. ∎

Now we are ready to prove the main result, Theorem 3.

4.3Proof of Theorem 3

We begin by proving the regularity of the weak solution and the estimate (8) under assumptions b or c. Let one of them hold, in particular 𝐼

( 0 , + ∞ ) . Then 𝒜 is densely defined, closed and generates a bounded analytic semigroup 𝒯 on the Hilbert space 𝒳 0 by Proposition 19 and Corollary 21. We moreover have the estimate (33) with 𝜔 < 0 at our disposal. From [13, Theorem 1.1, Corollary 1.7 and (1.9)] the operator 𝒜 has maximal 𝐿 𝑞 regularity, i.e., the mild solution 𝒰 0 of (9) (see (10)) with 𝒰 ⁢ ( 0 )

0 satisfies 𝒰 ˙ 0 , 𝒜 ⁢ 𝒰 0 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) and

‖ 𝒰 ˙ 0 ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ 𝒜 ⁢ 𝒰 0 ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) ≤ 𝐶 ⁢ ‖ ℱ ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) .

Let us denote 𝒰 1 ⁢ ( 𝑡 ) := 𝒯 ⁢ ( 𝑡 ) ⁢ 𝒰 0 for 𝑡 > 0 the mild solution of (9) with ℱ

0 and 𝒰 ⁢ ( 0 )

𝒰 0 . Since 𝒰 0 ∈ 𝒳 1 − 1 / 𝑞 , 𝑞 , the function 𝑡 ↦ 𝒜 ⁢ 𝒰 1 ⁢ ( 𝑡 ) (and therefore also 𝑡 ↦ 𝒰 ˙ 1 ⁢ ( 𝑡 ) ) belongs to 𝐿 𝑞 ⁢ ( ( 0 , 1 ) , 𝒳 0 ) and the inequality

‖ 𝒰 ˙ 1 ‖ 𝐿 𝑞 ⁢ ( ( 0 , 1 ) , 𝒳 0 ) + ‖ 𝒜 ⁢ 𝒰 1 ‖ 𝐿 𝑞 ⁢ ( ( 0 , 1 ) , 𝒳 0 ) ≤ 𝐶 ⁢ ‖ 𝒰 0 ‖ 𝒳 1 − 1 / 𝑞 , 𝑞

holds (see [14, Proposition 2.2.2 and formula (2.2.3)]). From the properties of analytic semigroups (see [14, Proposition 2.1.1]), we get for 𝑡

0

‖ 𝒜 ⁢ 𝒰 1 ⁢ ( 𝑡 ) ‖ 𝒳 0

1 𝑡 ⁢ ‖ 𝑡 ⁢ 𝒜 ⁢ 𝒯 ⁢ ( 𝑡 ) ⁢ 𝒰 0 ‖ 𝒳 0 ≤ 𝐶 ⁢ 𝑒 𝜔 ⁢ 𝑡 𝑡 ⁢ ‖ 𝒰 0 ‖ 𝒳 0 ≤ 𝐶 ⁢ 𝑒 𝜔 ⁢ 𝑡 𝑡 ⁢ ‖ 𝒰 0 ‖ 𝒳 1 − 1 / 𝑞 , 𝑞 .

Since 𝜅 ⁢ ( 𝑡 ) := 𝑒 𝜔 ⁢ 𝑡 / 𝑡 satisfies 𝜅 ∈ 𝐿 𝑞 ⁢ ( 1 , + ∞ ) it follows that

‖ 𝒰 ˙ 1 ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ 𝒜 ⁢ 𝒰 1 ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) ≤ 𝐶 ⁢ ‖ 𝒰 0 ‖ 𝒳 1 − 1 / 𝑞 , 𝑞 .

Hence, the solution 𝒰

( 𝑢 , 𝑢 𝑏 )

𝒰 0 + 𝒰 1 of (9) satisfies 𝒰 ˙ , 𝒜 ⁢ 𝒰 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) , and

‖ 𝒰 ˙ ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ 𝒜 ⁢ 𝒰 ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) ≤ 𝐶 ⁢ ( ‖ ℱ ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ 𝒰 0 ‖ 𝒳 1 − 1 / 𝑞 , 𝑞 ) .

For a.e. 𝑡

0

− 𝒜 ⁢ 𝒰 ⁢ ( 𝑡 )

ℱ ⁢ ( 𝑡 ) − 𝒰 ˙ ⁢ ( 𝑡 ) in  𝒳 0

and 𝒰 ⁢ ( 𝑡 ) is also the unique weak solution to (12)–(15) with 𝜆

0 and the right hand side ( 𝑓 ~ , ℎ ~ )

( 𝑓 ⁢ ( 𝑡 ) − ∂ 𝑡 𝑢 ⁢ ( 𝑡 ) , ℎ ⁢ ( 𝑡 ) − ∂ 𝑡 𝑢 𝑏 ) ∈ 𝒳 0 . It follows from Theorem 14, with assumptions a or b of Remark 9, that the functions 𝑢 ⁢ ( 𝑡 ) , 𝑢 𝑏 ⁢ ( 𝑡 ) and the associated pressure 𝜋 ⁢ ( 𝑡 ) satisfy the estimate

‖ 𝑢 𝑏 ⁢ ( 𝑡 ) ‖ 𝐻 3 2 ⁢ ( ∂ Ω ) + ‖ 𝑢 ⁢ ( 𝑡 ) ‖ 𝐻 2 ⁢ ( Ω ) + ‖ 𝜋 ⁢ ( 𝑡 ) ‖ 𝐻 1 ⁢ ( Ω )

≤ 𝐶 ⁢ ( ‖ 𝑓 ⁢ ( 𝑡 ) ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ∂ 𝑡 𝑢 ⁢ ( 𝑡 ) ‖ 𝐿 2 ⁢ ( Ω ) + ‖ ℎ ⁢ ( 𝑡 ) ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) + ‖ ∂ 𝑡 𝑢 𝑏 ⁢ ( 𝑡 ) ‖ 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) .

(34)

Since ( 𝑓 ~ , ℎ ~ ) ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐿 2 ⁢ ( Ω ) ) × 𝐿 𝑞 ⁢ ( 𝐼 , 𝐻 1 / 2 ⁢ ( ∂ Ω ) ) measurability of the mapping 𝑡

0 ↦ 𝜋 ⁢ ( 𝑡 ) ∈ 𝐻 1 ⁢ ( Ω ) follows from Remark 15. Integrating (34) and using regularity of 𝒰 ˙ we obtain (8). It remains to show that 𝒰 is actually the unique weak solution of (1)–(6). The function 𝒰 satisfies 𝒰 ∈ 𝐶 ⁢ ( [ 0 , + ∞ ) , ℋ ) ∩ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 ∞ ⁢ ( [ 0 , + ∞ ) , ℋ ) , 𝒰 ˙ ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 1 ⁢ ( [ 0 , + ∞ ) , 𝒢 ∗ ) and the equation (7) holds almost everywhere in ( 0 , + ∞ ) . As 𝒰 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 ∞ ⁢ ( [ 0 , + ∞ ) , ℋ ) we have 𝑢 𝑏 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 2 ⁢ ( [ 0 , ∞ ) , 𝐿 2 ⁢ ( Ω ) ) . The initial values are attained by [14, Proposition 2.1.1 and Proposition 2.1.4 (i)] and Proposition 19. To show that 𝑢 is a weak solution of (1)–(6) it remains to prove 𝑢 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 2 ⁢ ( [ 0 , + ∞ ) , 𝐻 1 ⁢ ( Ω ) ) . We know 𝒰 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐷 ⁢ ( 𝒜 ) ) , 𝒰 ∈ 𝐿 ∞ ⁢ ( 𝐼 , 𝒳 0 ) , consequently 𝑢 ∈ 𝐿 𝑞 ⁢ ( 𝐼 , 𝐻 2 ⁢ ( Ω ) ) , 𝑢 ∈ 𝐿 ∞ ⁢ ( 𝐼 , 𝐿 2 ⁢ ( Ω ) ) and an interpolation theorem gives

‖ 𝑢 ⁢ ( 𝑡 ) ‖ 𝐻 1 ⁢ ( Ω ) 2 ≤ 𝐶 ⁢ ‖ 𝑢 ⁢ ( 𝑡 ) ‖ 𝐻 2 ⁢ ( Ω ) ⁢ ‖ 𝑢 ⁢ ( 𝑡 ) ‖ 𝐿 2 ⁢ ( Ω ) .

It is enough to integrate this inequality over the time interval to get 𝑢 ∈ 𝐿 𝑙 ⁢ 𝑜 ⁢ 𝑐 2 ⁢ ( [ 0 , + ∞ ) , 𝐻 1 ⁢ ( Ω ) ) . This concludes the proof of the regularity properties in the case of assumptions b or c.

If the assumption a holds we can proceed similarly. Recall 𝐼

( 0 , 𝑇 ) with 𝑇 ∈ ( 0 , + ∞ ) . First we note that from analyticity of 𝒜 we get that the mild solution 𝒰 defined in (10) satisfies 𝑈 ∈ 𝐿 ∞ ⁢ ( 𝐼 , 𝒳 0 ) . Then we rewrite the equation (9) as

∂ 𝑡 𝒰

𝒜 ~ ⁢ 𝒰 + ℱ ~ ⁢ ( 𝑡 ) ,

with 𝒜 ~ := 𝒜 − 𝜆 0 , ℱ ~ ⁢ ( 𝑡 ) := ℱ ⁢ ( 𝑡 ) + 𝜆 0 ⁢ 𝒰 and 𝜆 0

max ⁡ ( 0 , − 𝛼 / 𝛽 ) + 1 . The constant 𝜆 0 is chosen such that 𝑆 𝜃 , 0 ¯ is a subset of the resolvent set of 𝒜 ~ ; see Proposition 10 and Theorem 14. The function ℱ ~ and the semigroup 𝒯 ~ generated by the operator 𝒜 ~ can be estimated

‖ ℱ ~ ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) ≤ 𝐶 ⁢ ( ‖ ℱ ‖ 𝐿 𝑞 ⁢ ( 𝐼 , 𝒳 0 ) + ‖ 𝒰 0 ‖ 𝒳 0 ) and ∃ 𝜔 < 0 , ∀ 𝑡

0 : ‖ 𝒯 ~ ⁢ ( 𝑡 ) ‖ ℒ ⁢ ( 𝒳 0 ) ≤ 𝐶 ⁢ 𝑒 𝜔 ⁢ 𝑡 .

The rest of the proof of regularity can be done as in the cases b or c.

It remains to show that the equations (1)–(6) hold almost everywhere. It is clear for (2) a.e. in 𝐼 × Ω and for (4) a.e. in 𝐼 × ∂ Ω . We have already identified, under assumptions of the theorem, the mild and the weak solutions, and we reconstructed the pressure 𝜋 so that for 𝜑 ∈ 𝐻 1 ⁢ ( Ω ) with 𝜑 𝜈

0 at ∂ Ω at almost every 𝑡 ∈ 𝐼

∫ Ω ∂ 𝑡 𝑢 𝜑 + 𝛽 ∫ ∂ Ω ∂ 𝑡 𝑢 𝜑 + ∫ Ω ( 2 𝐷 𝑢 : 𝐷 𝜑 − 𝜋 div 𝜑 ) + 𝛼 ∫ ∂ Ω 𝑢 𝜑

∫ Ω 𝑓 𝜑 + 𝛽 ∫ ∂ Ω 𝑔 𝜑 .

Regularity of 𝑢 and 𝜋 allows us to use the Divergence theorem in the third integral to get for 𝜑 ∈ 𝐻 1 ⁢ ( Ω ) with 𝜑 𝜈

0 at ∂ Ω and 𝜂 ∈ 𝒟 ⁢ ( 𝐼 )

∫ 𝐼 ∫ Ω ( ∂ 𝑡 𝑢 + Δ ⁢ 𝑢 + ∇ 𝜋 − 𝑓 ) ⁢ 𝜑 ⁢ 𝜂 + ∫ 𝐼 ∫ ∂ Ω ( 𝛽 ⁢ ∂ 𝑡 𝑢 + 2 ⁢ 𝐷 ⁢ 𝑢 ⁢ 𝜈 + 𝛼 ⁢ 𝑢 − 𝑔 ) ⁢ 𝜑 ⁢ 𝜂

0 .

It follows that (1) must hold a.e. in 𝐼 × Ω and (3) must hold a.e. in 𝐼 × ∂ Ω .

We have already discussed that the initial values are attained in ℋ , which also means that (5) holds a.e. in { 0 } × Ω and (6) holds a.e. in { 0 } × ∂ Ω . ∎

Acknowledgements

This study was funded by the Czech Science Foundation project 20-11027X. We thank the anonymous referees for their valuable remarks and suggestions.

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