Existence,uniqueness,and asymptotic stability results for the 3-D steady and unsteady Navier–Stokes equations on multi-connected domains with inhomogeneous boundary conditions

Abstract

We consider both stationary and time-dependent solutions of the 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain $Ω \subset R^{3}$ with inhomogeneous boundary values on $\partial Ω = Γ$ ; here Γ is a union of disjoint surfaces $Γ_{0}, Γ_{1}, \dots, Γ_{l}$ . Our starting point is Leray’s classic problem, which is to find a weak solution $u \in H^{1} (Ω)$ of the stationary problem assuming that on the boundary $u = β \in H^{1 / 2} (Γ)$ . The general flux condition $\sum_{j = 0}^{l} \int_{Γ_{j}} β \cdot n d S = 0$ must be satisfied due to compatibility considerations. Early results on this problem including the initial results in (J. Math. Pures Appl. 12 (1933) 1–82) assumed the more restrictive flux condition $\int_{Γ_{j}} β \cdot n d S = 0$ for each $j = 1, \dots, l$ . More recent results, of which those in (An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II 1994 Springer–Verlag) and (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are particularly representative, assume only the general flux condition in exchange for size restrictions on the data. In this paper we also assume only the general flux condition throughout, and for virtually the same size restrictions on the data as in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) we obtain the existence of a weak solution that matches that found in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) when the assumptions imposed here and those assumed in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are both met; additionally we demonstrate that this solution is unique. For slightly stronger size restrictions we obtain the existence and uniqueness of solutions of both Leray’s problem and global mild solutions of the corresponding time-dependent problem, while showing that both the stationary and time-dependent solutions we construct are a bit stronger than weak solutions. The settings in which we establish our results allow us to culminate our discussion by showing that our time-dependent solutions converge to each other exponentially in time, so that in particular our stationary solutions are asymptotically stable. We also discuss additional features which allow for data of increased size on certain domains, including those which are thin in a generalized sense.

Keywords

Navier Stokes equations inhomogenious boundary data multi-connected domains general flux condition mild solutions asymptotic stability

1. Introduction

We consider the 3-D Navier–Stokes equations for viscous incompressible homogeneous flow with inhomogeneous boundary conditions $\begin{array}{l} (1.1a) & v_{t} - ν Δ v + (v \cdot \nabla) v + \nabla p = g, \\ (1.1b) & \nabla \cdot v = 0, \\ (1.1c) & v = β, x \in \partial Ω . \end{array}$ Here Ω is a bounded Lipschitz spatial domain in $R^{3}$ whose boundary $\partial Ω = Γ$ is a union of disjoint surfaces $Γ_{0}, Γ_{1}, \dots, Γ_{l}$ . We have that $v = (v_{1}, v_{2}, v_{3})$ with $v_{i} = v_{i} (x, t)$ , $x \in Ω$ , $t \in [0, \infty)$ , $i \in {1, 2, 3}$ . The external force is $g = (g_{1}, g_{2}, g_{3})$ with $g_{i} = g_{i} (x, t)$ , $p = p (x, t)$ is the pressure, and ν is the viscosity coefficient. Since the solution v must satisfy $\nabla \cdot v = 0$ , an integration by parts shows that the boundary data β must satisfy the compatibility condition $\begin{matrix} (1.2) & \sum_{j = 0}^{l} \int_{Γ_{j}} β \cdot n d S = 0 \end{matrix}$ known as the general flux condition. The archetypal special case of the domains Ω under consideration are annular regions $Ω_{R} = {x \in R^{3} | R_{1} ⩽ | x | ⩽ R_{0}}$ where $0 < R_{1} < R_{0}$ , in which case $l = 1$ and $Γ_{i} = {x \in R^{3} ∣ | x | = R_{i}}$ , $i = 0, 1$ .

The problem of finding a weak stationary solution $v = v (x)$ of (1.1a)–(1.1c), (1.2) with $v \in H^{1} (Ω)$ under the assumption $β \in H^{1 / 2} (Γ)$ is commonly known as Leray’s problem. Since its original treatment in [19] this problem has received the attention of several authors (see e.g. [1,5,6,9], [10, VIII.4] [12–14,18–20,23,24,27]), in which the standard approach to handling the inhomogeneous boundary conditions is to assume that $v = u + b$ where b is a divergence-free vector satisfying $b = β$ on Γ. Substituting accordingly into the stationary version of (1.1a)–(1.1c) results in the following equations for u: $\begin{array}{l} (1.3a) & - ν Δ u + (b \cdot \nabla) u + (u \cdot \nabla) b + (u \cdot \nabla) u + \nabla p = g + ν Δ b - (b \cdot \nabla) b, \\ (1.3b) & \nabla \cdot u = 0, \\ (1.3c) & u = 0, x \in \partial Ω . \end{array}$ The equations (1.3a)–(1.3c) constitute a modified Navier–Stokes system with zero (no-slip) boundary conditions prescribed for u; accordingly in this regard we assume relative to u that $D (- Δ) = {v \in H^{2} (Ω) | v |_{Γ} = 0}$ . The function b can be chosen so that there exists a constant $C_{Γ}$ such that $\begin{matrix} (1.4) & ‖ \nabla b ‖_{L^{2} (Ω)} ⩽ ‖ b ‖_{H^{1} (Ω)} ⩽ C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)} \end{matrix}$ (see e.g. [10] for further discussion regarding (1.4)).

Here we will treat both the stationary and time-dependent versions of (1.3a)–(1.3c). Let P be the Leray projection onto the divergence-free vectors, let $H \equiv P L^{2} (Ω)$ , let $f = P g$ , and let A denote the Stokes operator $A = - P Δ$ with domain $D (A) = H \cap D (- Δ)$ . In this way A is equipped in standard fashion with zero boundary conditions relative to u (see e.g. [11] for further discussion and references regarding P and the operator A). It is well-known that A is a positive definite operator on H with eigenvalues $0 < λ_{1} < λ_{2} < \dots$ over corresponding eigenspaces $E_{1}, E_{2}, \dots$ . Moreover (see e.g. [11]) P is a bounded operator on $L^{q} (Ω)$ for $1 < q < \infty$ , while $e^{- t A}$ is an analytic semigroup and $A^{- 1}$ is a well-defined compact operator on $X_{q} \equiv P L^{q} (Ω)$ ; for ease of notation we set $‖ v ‖_{q} = ‖ v ‖_{X_{q}}$ . With these considerations we apply P to both sides of (1.3a) to rewrite the system (1.3a)–(1.3c) as $\begin{matrix} (1.5) & ν A u + P (b \cdot \nabla) u + P (u \cdot \nabla) b + P (u \cdot \nabla) u = f - P (b \cdot \nabla) b + ν P Δ b . \end{matrix}$ Note that the operator $P Δ$ in the last term of (1.5) is distinct from $- A$ since A is equipped with zero boundary conditions.

The corresponding time-dependent version of (1.1a)–(1.1c) in this formulation is $\begin{array}{l} (1.6a) & \frac{d}{d t} u + ν A u + P (b \cdot \nabla) u + P (u \cdot \nabla) b + P (u \cdot \nabla) u = f - P (b \cdot \nabla) b + ν P Δ b, \\ (1.6b) & u (x, 0) = u_{0} (x) \end{array}$ which is in the form of an initial-value problem for $u = u (x, t)$ for $t \in [0, \infty)$ . We note that at this stage the term $ν P Δ b$ should be interpreted in a weak sense to be described more rigorously in the working reformulations of (1.5) and (1.6a)–(1.6b) we use below.

The first results for (1.3a)–(1.3c) (see e.g. [9,18,19]) were weak-solution existence results that depended on finding for each $ϵ > 0$ a solenoidal vector $b = b_{ϵ}$ satifying $\begin{matrix} (1.7) & | \int_{Ω} v \cdot \nabla b_{ϵ} d x | ⩽ ϵ \int_{Ω} | \nabla v |^{2} d x . \end{matrix}$ The relation (1.7) is sometimes referred to as Leray’s inequality. In particular it is known that (1.7) holds if the restricted flux condition $\begin{matrix} (1.8) & \int_{Γ_{j}} β \cdot n d S = 0 \end{matrix}$ holds for all $j = 0, 1, \dots, l$ , as was assumed in [9,18,19]. In fact in a number of geometries the restrictive condition (1.8) must be satisfied in order for (1.7) to hold. This was shown in annular domains in [23] and for a wider class of geometrical settings in [14].

Existence results for (1.3a)–(1.3c) not requiring (1.7) or (1.8) have been recently established by assuming smallness assumptions on β or on a portion of b. In [10, VIII, Theorem 4.1], the existence of a weak solution to (1.3a)–(1.3c) was established under the condition $Σ_{j = 1}^{l} c_{j} | \int_{Γ_{j}} β \cdot n d S | < ν$ , where the $c_{j}$ are computable constants depending on Ω. The exposition in [10] uses a decomposition of b into a rotational component rot ω and a component consisting of a superposition of fundamental solutions of the Laplacian Δ, each such solution corresponding to one of the $Γ_{i}$ . As in [14,24] the rotational component is shown to be negligible in the calculations so that it does not figure into the derivation of the smallness condition.

The expostion in [14] employs a similar decompostion $b = h +$ rot ω but in this case $h \in V_{har} (Ω)$ where $V_{har} (Ω)$ is a finite-dimensional subspace of harmonic vectors with an orthogonal basis ${φ_{j}}_{j = 1}^{l}$ . It is shown in [14] that there exists a weak solution $u \in H_{0}^{1} (Ω)$ of (1.3a)–(1.3c) with $f = 0$ if h satisfies $\begin{matrix} (1.9) & ‖ h ‖_{3} ⩽ ν C_{l}^{- 1} \end{matrix}$ where $C_{l}$ is the best constant of the Sobolev embedding $H_{0}^{1} (Ω) \subset L^{6} (Ω)$ . The smallness condition (1.9) again does not depend on the rotational part rot ω, and it is shown that there are coefficients ${α_{j k}}_{j . k = 1}^{l}$ such that $h = \sum_{j, k = 1}^{l} α_{j k} (\int_{Γ_{k}} β \cdot n d S) φ_{j}$ so that $‖ h ‖_{3} ⩽ \sum_{j, k = 1}^{l} α_{j k} | \int_{Γ_{k}} β \cdot n d S | ‖ φ_{j} ‖_{3}$ . Thus (1.9) can also be expressed in the form $Σ_{j = 1}^{l} c_{j} | \int_{Γ_{j}} β \cdot n d S | < ν$ where now $c_{j} = C_{l} ‖ φ_{j} ‖_{3} (\sum_{k = 1}^{l} α_{j k})$ .

In both [10] and [14] the elimination of the dependence of the rotational component is accomplished as in [24] by replacing rot ω with rot $θ_{ϵ} ω$ where $θ_{ϵ}$ is a suitable cutoff function with support in $Ω_{ϵ} = {x \in Ω | dist (x, Γ) \equiv d (x) < 2 e^{- 1 / ϵ}}$ . The resultant $b_{ϵ}$ still satisfies $div b_{ϵ} = 0$ and $b_{ϵ} = β$ on Γ, meanwhile $ω \in H^{2} (Ω) \subset W^{1, 6} (Ω) \subset L^{\infty} (Ω)$ , so that $| b_{ϵ} (x) | ⩽ \frac{ϵ}{d (x)} | w (x) | + | \nabla ω (x) |$ . Hence for all vectors $v \in H_{0}^{1} (Ω)$ it follows that $‖ b_{ϵ} v ‖_{2} ⩽ ϵ ‖ ω ‖_{\infty} ‖ \frac{v}{d} ‖_{2} + ‖ (v \cdot \nabla) ω ‖_{L^{2} (Ω_{ϵ})} ⩽ (ϵ ‖ ω ‖_{\infty} + M_{0} ‖ \nabla ω ‖_{L^{3} (Ω_{ϵ})}) ‖ \nabla v ‖_{2}$ for a generic constant $M_{0}$ by respectively applying Young’s inequality and an appropriate Sobolev inequality to the first and second terms on the right-hand side. It then follows that for all vectors $v \in H_{0}^{1} (Ω)$ we have for $ψ \equiv$ rot ω that $‖ ψ v ‖_{2} ⩽ c_{ω} (ϵ + μ (ϵ)) ‖ v ‖_{H_{0}^{1} (Ω)}$ where $μ (ϵ) \to 0$ as $ϵ \to 0$ and $c_{ω}$ depends on appropriate norms of ω, i.e. there is a constant $κ (ϵ)$ such that $\begin{matrix} (1.10) & ‖ ψ v ‖_{2} ⩽ κ (ϵ) ‖ \nabla v ‖_{2} \end{matrix}$ and such that $κ (ϵ) \to 0$ as $ϵ \to 0$ . For the results in this paper we will have need to extend (1.10) to the case $v = ψ$ via an approximation argument.

In the case of an annular region $Ω = Ω_{R} = {x \in R^{3} | R_{1} ⩽ | x | ⩽ R_{0}}$ with $l = 1$ the general flux condition (1.2) becomes $\int_{Γ_{1}} β \cdot n d S = - \int_{Γ_{0}} β \cdot n d S$ . In this case $V_{har} (Ω)$ is one-dimensional and h has the explicit form $(\int_{Γ_{1}} β \cdot n d S) φ$ where φ is proportional to $\nabla (\frac{1}{| x |})$ . This was used in [14] to show that (1.9) is satisfied in this case provided that $\begin{matrix} (1.11) & | \int_{Γ_{1}} β \cdot n d S | ⩽ ν C_{s}^{'} \frac{R_{0} R_{1}}{{(R_{0}^{3} - R_{1}^{3})}^{1 / 3}} \end{matrix}$ where $C_{s}^{'} = 3^{5 / 6} 2^{2 / 3} π^{4 / 3}$ . The estimate (1.11) can allow $| \int_{Γ_{1}} β \cdot n d S |$ to be of increased size depending on the sizes of $R_{0}$ and $R_{1}$ as well as how close they are to each other, as we will discuss in greater detail in Section 5 below. Results similar to (1.11) can be found in [5] on generalized annular regions using different techniques.

We begin the discussion of the results of this paper with (1.5). For appropriately smooth solenoidal vector fields v and w we have that $(v \cdot \nabla) w = div (v \otimes w)$ for the appropriate tensor product ⊗, where for a vector field w we set $div (w) \equiv \nabla \cdot w$ ; applying $ν^{- 1} A^{- 1}$ to both sides of (1.5) we obtain the following working reformulation of (1.5): $\begin{array}{l} u + ν^{- 1} A^{- 1} [P div (b \otimes u) + P div (u \otimes b) + P div (u \otimes u)] \\ (1.12) & = ν^{- 1} A^{- 1} (f - P div (b \otimes b)) + A^{- 1} P Δ b . \end{array}$ For simplicity of notation we set $‖ f ‖_{H} = L$ . In Section 2 we will obtain for $q \in [3, 4]$ an explicit bound $M_{s}$ on the size of $‖ ν^{- 1} A^{- 1} (f - P (b \cdot \nabla) b) + A^{- 1} P Δ b ‖_{q}$ , which we will use in proving the next result via a contraction-mapping argument.

Theorem 1.
For $3 ⩽ q ⩽ 4$ , $p = q / 2$ , $r = (q - 3) / (2 q)$ , for a given $η \in (1, 2)$ , and for h as above there are generic constants $M_{p}$ , $M_{q}$ , $B_{p}$ , and $τ_{q}$ such that if $\begin{matrix} (1.13) & λ_{1}^{- 3 / 4} M_{q} L + η^{2} {(\frac{1}{λ_{1}})}^{2 r} M_{p} ‖ h ‖_{q}^{2} + ν {(\frac{1}{λ_{1}})}^{r} η (1 + η τ_{q}) ‖ h ‖_{q} ⩽ ν^{2} / (4 M_{p} B_{p}) \end{matrix}$ then ( 1.12 ) has a unique solution $u_{s} \in X_{q}$ . When $q = 3$ and the conditions ( 1.9 ) and ( 1.13 ) are both met (with $f = 0$ as in [ 14 ]), then the solution $u_{s}$ coincides with the solution established in [ 14 ], so that $u_{s} \in X_{3} \cap H_{0}^{1} (Ω)$ and $u_{s}$ is the unique weak solution of Leray’s problem. When $3 < q ⩽ 4$ the unique solutions of ( 1.12 ) solve Leray’s problem with $H^{s}$ -regularity for $s > 1$ .

From the second term and third terms on the left-hand side of (1.13) we have in particular the respective upper-bound estimates $‖ h ‖_{q} ⩽ ν / (2 η M_{p} B_{p}^{1 / 2} λ_{1}^{- r})$ and $‖ h ‖_{q} ⩽ ν / (4 η (1 + η τ_{q}) M_{p} B_{p} λ_{1}^{- r})$ . These estimates are of the form (1.9) in having right-hand sides proportional to ν, and in this sense closely match (1.9), in particular when $q = 3$ . Note that these estimates can also be expressed in the form $Σ_{j = 1}^{l} c_{j} | \int_{Γ_{j}} β \cdot n d S | < ν$ , where e.g. in the first estimate we now have that $c_{j} = 2 η M_{p} B_{p}^{1 / 2} λ_{1}^{- r} ‖ φ_{j} ‖_{q} (\sum_{k = 1}^{l} α_{j k})$ ; here the $φ_{j}$ and $α_{j k}$ are as in the discussion following (1.9). In the case $Ω = Ω_{R}$ , our upper bounds via the above considerations look like (1.11) with $C_{s}^{'}$ replaced by a suitable constant in the case $q = 3$ , and for $3 < q ⩽ 4$ we replace $(R_{0} R_{1}) {(R_{0}^{3} - R_{1}^{3})}^{- 1 / 3}$ by ${(R_{0} R_{1})}^{(2 q - 3) / q} {(R_{0}^{2 q - 3} - R_{1}^{2 q - 3})}^{- 1 / q}$ which e.g. in the case $q = 4$ is ${(R_{0} R_{1})}^{5 / 4} {(R_{0}^{5} - R_{1}^{5})}^{- 1 / 4}$ . In the same way as in (1.11) these estimates can allow $| \int_{Γ_{1}} β \cdot n d S |$ to be of increased size as will be discussed in Section 5. Meanwhile the term $λ_{1}^{- r}$ adds an additional capacity for $q > 3$ to allow for data of increased size on a general class of “thin” domains; this feature will also be discussed in Section 5, along with a further discussion of results on thin domains, including those in [2,3] and the particularly representative results in [15–17,22,25,26].

When $q = 3$ the similarity of the conditions (1.9) and (1.13) in giving size restrictions on $‖ h ‖_{3}$ proportional to ν means that the two conditions will be both met for a significant portion of the allowable data, lending $H^{1}$ -regularity to Theorem 1 when $q = 3$ and adding the fundamental property of uniqueness to the existence results in [14]. That the solutions found in [14] meet the conditions of Theorem 1 when $q = 3$ and in particular satisfy (1.12) when (1.9) and (1.13) both hold will be shown toward the end of Section 2. Otherwise the results overlap, with the presence of the term $λ_{1}^{- r}$ in particular giving further generality to (1.13) for $q \in (3, 4]$ as will be made clear in Section 5. Meanwhile, for $3 < q ⩽ 4$ the solutions of (1.12) will be shown to also be suitable weak solutions toward the end of Section 2. That these solutions in fact have enhanced regularity will be stated in detail in Theorem 5 below and shown in Section 4.

We now discuss the time-dependent case (1.6a)–(1.6b). Viewing solutions as trajectories in a Banach space we let $u (t)$ denote $u (x, t)$ , and with this convention we consider the following integral-equation version (and working reformulation) of (1.6a)–(1.6b): $\begin{array}{l} u (t) = & e^{- ν t A} u_{0} + \int_{0}^{t} e^{- ν (t - s) A} [f (s) - P div (b \otimes b)] d s + ν \int_{0}^{t} A e^{- ν (t - s) A} A^{- 1} P Δ b d s \\ (1.14) & - \int_{0}^{t} e^{- ν (t - s) A} [P div (b \otimes u (s)) + P div (u (s) \otimes b) + P div (u (s) \otimes b)] d s . \end{array}$ As is standard we refer to solutions of (1.14) as mild solutions.

For $u_{0} \in X_{q}$ but now for $q \in (3, 4]$ we set $G (t) = e^{- t A} u_{0} + \int_{0}^{t} e^{- (t - s) A} [f (s) - P div (b \otimes b)] d s + ν \int_{0}^{t} A e^{- (t - s) A} A^{- 1} P Δ b d s$ and make the standard assumption that there is a constant L such that ${sup}_{0 ⩽ t < \infty} ‖ f (t) ‖_{H} ⩽ L$ . We will obtain in Section 2 below an explicit bound M on ${sup}_{0 ⩽ t < \infty} ‖ G (t) ‖_{q}$ which we will use to establish the following result.
Theorem 2.
For h as above, for $3 < q ⩽ 4$ , for any $η \in (1, 2)$ , and for $r = (q - 3) / (2 q)$ and $s = (3 q - 6) / (4 q)$ there are generic constants $K_{i}$ , $i = 0, 1, 2, 3, 4$ such that if $\begin{matrix} (1.15) & K_{0} {(\frac{2}{ν λ_{1}})}^{r} (K_{1} η ‖ h ‖_{q} + K_{2} ‖ u_{0} ‖_{q}) + K_{3} {(\frac{2}{ν λ_{1}})}^{2 r} ‖ h ‖_{q}^{2} + K_{4} {(\frac{2}{ν λ_{1}})}^{s} L ⩽ 1 \end{matrix}$ then ( 1.14 ) has a unique global solution $u \in C_{B} ([0, \infty); X_{q})$ . The constants $K_{i}$ involve the same generic constants as in ( 1.13 ) together with some additional constants arising from the properties of $e^{- ν t A}$ .

From the middle term on the left-hand side of (1.15) we have the upper-bound estimate $‖ h ‖_{q} ⩽ ν^{r} [K_{3}^{- 1 / 2} {(λ_{1} / 2)}^{r}]$ with a similar estimate coming from the first term in (1.15). These estimates are comparable to both (1.11) and (1.13), but only require dependence on a fractional power on ν, which in fact goes to zero as $q ↓ 3$ . Meanwhile, similarly to the stationary case we again have the capacity to allow for data of increased size on certain annular domains $Ω = Ω_{R}$ , and additionally on a general class of “thin” domains as will be discussed in Section 5. At the end of Section 2 we will show that the solutions of (1.14) constructed in Theorem 2 are suitable weak solutions of (1.6a)–(1.6b).

For the solutions constructed under the conditions of Theorem 2 and satisfying some additional conditions of the same type as those given by (1.15) (to be detailed in Section 3 below), the next theorem shows that as $t \to \infty$ such solutions converge to each other at an exponentioal rate.
Theorem 3.
Under the conditions of Theorem 2 and additional similar conditions we have that there exists a generic constant K such that any two solutions u and v of ( 1.14 ) satisfy $‖ u (t) - v (t) ‖_{q} ⩽ K e^{- ν (λ_{1} / 4) t}$ for all $t ⩾ 0$ .

Meanwhile, it is fairly clear that solutions of (1.12) are also solutions of (1.14); for completeness we will show this in Section 2 below. Moreover, since the powers on $ν^{- 1}$ in (1.15) are fractional powers significantly lower than those appearing in (1.13), by satisfying (1.13) the solutions established by Theorem 1 will for typically small ν also satisfy (1.15) (with $u_{s} = u_{0}$ ) and are thus among the solutions established by Theorem 2. Thus from Theorems 2 and 3 we have the following result, which shows that the solutions $u_{s}$ of (1.12) given by Theorem 1 are exponentially asymptotically stable.
Theorem 4.
Under the conditions of Theorems 1 , 2 , and 3 , all solutions u of ( 1.14 ) satisfy $‖ u (t) - u_{s} ‖_{4} ⩽ K_{2} e^{- ν (λ_{1} / 4) t}$ for all $t ⩾ 0$ .

The next theorem obtains for $q \in (3, 4]$ enhanced regularity for the solutions established in Theorem 1 and Theorem 2.
Theorem 5.
For $q \in (3, 4]$ suppose that $ς \in (0, δ)$ for $δ = (q - 3) / (2 q)$ . Then under the conditions of Theorems 1 and 2 , along with some additional similar conditions, we have that the solutions $u_{s}$ of ( 1.12 ) established in Theorem 1 satisfy $u_{s} \in D (A^{1 / 2 + ς})$ , and if $u_{0} \in D (A^{1 / 2 + ς})$ the solutions u of ( 1.14 ) established in Theorem 2 satisfy $u \in C_{B} ([0, \infty); D (A^{1 / 2 + ς})$ .

With this slightly increased regularity we see that our solutions for $q \in (3, 4]$ are in fact a bit stronger than typical weak solutions, and that this increased regularity persists for all time in the case of the solutions u of (1.14). We could establish higher degrees of regularity by imposing additional regularity asumptions on the data; we do not choose to pursue this here so as to remain in the context of Leray’s problem.

Theorems 1 through 4 will be proven in Section 3 below. In the next section we will provide background material and results needed for the later sections, including explicit estimates for M and $M_{s}$ on which the proofs of Theorems 1 and 2 rely. Theorem 5 will be established in Section 4. The way in which certain domains allow for data of increased size will be discussed in Section 5. In Section 6 we will conclude with some contextual remarks.
2. Preliminaries

We first express the Sobolev inequalities on Ω in terms of the operator $B = - Δ$ ; we have that $\begin{matrix} (2.1) & ‖ υ ‖_{q} ⩽ M_{1} {‖ B^{m_{1} / 2} υ ‖}_{p}^{1 - θ} {‖ B^{m_{2} / 2} υ ‖}_{p}^{θ} \end{matrix}$ for all $v \in D (B^{m / 2})$ , $m = max {m_{1}, m_{2}}$ , where $q ⩽ 6 / (3 - p [(1 - θ) m_{1} + θ m_{2}])$ and $M_{1} = M_{1} (θ, p, q, m_{1}, m_{2}, Ω)$ . Here B is equipped with zero boundary conditions as in the introduction. By [11, Proposition 1.4] $D (A^{γ})$ as a subset of $X_{q}$ is continuously embedded into $X_{q} \cap W^{2 γ, q} (Ω)$ for any $γ ⩾ 0$ , and thus we have for a constant $M_{0} = M_{0} (θ, p, q, Ω)$ and for q as above that for all $v \in D (A^{m / 2})$ $\begin{matrix} (2.2) & ‖ υ ‖_{q} ⩽ M_{1} {‖ B^{m_{1} / 2} υ ‖}_{p}^{1 - θ} {‖ B^{m_{2} / 2} υ ‖}_{p}^{θ} ⩽ M_{0} {‖ A^{m_{1} / 2} υ ‖}_{p}^{1 - θ} {‖ A^{m_{2} / 2} υ ‖}_{p}^{θ} . \end{matrix}$ For most of our calculations below we will focus on the following special case of (2.l) and (2.2): $\begin{matrix} (2.3) & ‖ υ ‖_{q} ⩽ M_{q, p} {‖ A^{θ / 2} υ ‖}_{p} \end{matrix}$ where $q ⩽ 6 / (3 - θ p)$ and $M_{q, p} = M_{1} (θ, p, q, Ω)$ . Since A is a positive definite operator on H with eigenvalues $0 < λ_{1} < λ_{2} < \dots$ the Poincaré estimate $\begin{matrix} (2.4) & {‖ A^{γ} g ‖}_{2} ⩾ λ_{1}^{γ} ‖ g ‖_{2} \end{matrix}$ follows from the functional calculus for any $γ > 0$ , and for the semigroup $exp (- t A)$ we have for a constant $d_{p}$ the decay estimate $\begin{matrix} (2.5) & {‖ e^{- ν t A} υ ‖}_{p} ⩽ d_{p} ‖ υ ‖_{p} e^{- ν λ_{1} t} . \end{matrix}$ Since A generates an analytic semigroup, for each $p \in (1, \infty)$ there is a constant $c_{p}$ such that $\begin{matrix} (2.6) & {‖ A^{β} e^{- t A} υ ‖}_{p} ⩽ c_{p} t^{- β} ‖ υ ‖_{p} \end{matrix}$ for any $β ⩾ 0$ . (For convenience we include the case $β = 0$ in (2.6); the operator $e^{- t A}$ is not necessarily a contraction if $p \neq 2$ , but it is still a bounded operator.) See e.g [11] and the references contained therein for further background on (2.4)–(2.6). As noted in the introduction, in addition to being a projection operator on a Hilbert space, in fact P is a bounded operator on all the spaces $L^{p} (Ω)$ for each $p \in (1, \infty)$ , i.e. (see e.g. [11]) there exists a constant $C_{p}$ such that $\begin{matrix} (2.7) & ‖ P υ ‖_{p} ⩽ C_{p} ‖ υ ‖_{p} \end{matrix}$ for all $v \in L^{P} (Ω)$ and each $p \in (1, \infty)$ . Meanwhile we set $A^{- 1 / 2} P (w \cdot \nabla) v = T (w \otimes v)$ where $T \equiv A^{- 1 / 2} P div$ ; a duality argument (see e.g. [11, Lemma 2.1]) shows that the operator T extends to a bounded operator from $L^{p} (Ω)$ to $X_{p}$ , $p \in (1, \infty)$ , and accordingly we set $B_{p} = ‖ T ‖_{p}$ .

For the bound $M_{s}$ on $‖ ν^{- 1} A^{- 1} [f - P div (b \otimes b)] + A^{- 1} P Δ b ‖_{q}$ for $3 ⩽ q ⩽ 4$ , we first note for any $θ \in [0, 1)$ that $‖ A^{- 1} P div (b \otimes b) ‖_{q} = ‖ A^{- (1 / 2 - θ)} [A^{- θ} A^{- 1 / 2} P div (b \otimes b)] ‖_{q}$ ; we thus have from the triangle inequality and (2.4) that $\begin{array}{l} {‖ ν^{- 1} A^{- 1} [f - P div (b \otimes b)] + A^{- 1} P Δ b ‖}_{q} \\ ⩽ ν^{- 1} λ_{1}^{- (1 - θ_{1})} {‖ A^{- θ_{1}} f ‖}_{q} \\ (2.8) & + ν^{- 1} λ_{1}^{- (1 / 2 - θ_{2})} {‖ A^{- θ_{2}} A^{- 1 / 2} P div (b \otimes b) ‖}_{q} + {‖ A^{- 1} P Δ b ‖}_{q} . \end{array}$ From (2.3) with $p = 2$ and $θ_{1} = θ = (3 q - 6) / (4 q)$ we have for $M_{q} \equiv M_{q, 2}$ that $‖ A^{- θ} f ‖_{q} ⩽ M_{q} ‖ f ‖_{2}$ , and from (2.3) with $p = q / 2$ , $θ_{2} = θ = 3 / (2 q)$ , and $M_{p} \equiv M_{2 p, p}$ we have that $‖ A^{- θ_{2}} A^{- 1 / 2} P div (b \otimes b) ‖_{q} ⩽ M_{p} ‖ A^{- 1 / 2} P div (b \otimes b) ‖_{q / 2}$ . From the remarks following (2.7) we have that $‖ A^{- 1 / 2} P div (b \otimes b) ‖_{q / 2} = ‖ T (b \otimes b) ‖_{q / 2} ⩽ B_{p} ‖ b \otimes b ‖_{q / 2}$ , while another duality argument shows that the operator $A^{- 1} P Δ$ also extends to a bounded operator from $L^{q} (Ω)$ to $X_{q}$ , $q \in (1, \infty)$ ; to see this, we follow the proof of [11, Lemma 2.1]: we have from the embedding $D (A^{γ}) \subset X_{r} \cap W^{2 γ, r} (Ω)$ that the operator $(\partial^{2} / (\partial x^{2})) I A^{- 1} : X_{r} \to L^{r} (Ω)$ is continuous where $r^{- 1} + q^{- 1} = 1$ and I denotes the injection $X_{r} \subset L^{r} (Ω)$ . Since $I^{*}$ is P acting on $L^{q} (Ω)$ , we have by duality that ${[(\partial^{2} / (\partial x^{2})) I A^{- 1}]}^{*} = A^{- 1} P (\partial^{2} / (\partial x^{2}))$ extends to a bounded operator on $L^{q} (Ω)$ . This shows the boundedness result for the operator $A^{- 1} P Δ$ and accordingly we set $τ_{q} = ‖ A^{- 1} P Δ ‖_{q}$ . Meanwhile we have that $1 - θ_{1} = (q + 6) / (4 q)$ and $1 / 2 - θ_{2} = (q - 3) / (2 q)$ ; combining these observations with (2.8) we now set $r = (q - 3) / (2 q)$ , then for $s = (q + 6) / (4 q)$ we have that $\begin{array}{l} {‖ ν^{- 1} A^{- 1} (f + P div (b \otimes b)) + A^{- 1} P Δ b ‖}_{q} \\ ⩽ ν^{- 1} λ_{1}^{- s} M_{q} ‖ f ‖_{2} \\ + ν^{- 1} λ_{1}^{- r} M_{p} B_{p} ‖ b \otimes b ‖_{q / 2} + τ_{q} ‖ b ‖_{q} \\ (2.9) & ⩽ ν^{- 1} λ_{1}^{- s} M_{q} L + ν^{- 1} λ_{1}^{- r} M_{p} B_{p} ‖ b ‖_{q}^{2} + τ_{q} ‖ b ‖_{q} . \end{array}$ We have that $‖ b ‖_{q} ⩽ ‖ h ‖_{q} + ‖ ψ ‖_{q}$ and for $q = 4$ we have that $| ψ |^{4} = | ψ |^{2} | ψ |^{2}$ so that $‖ ψ ‖_{4} = ‖ ψ v ‖_{2}^{1 / 2}$ with $v = ψ$ . Now by standard results there exist $C_{0}^{\infty} (Ω)$ functions $ψ_{δ}$ such that $D^{α} ψ_{δ} \to D^{α} ψ$ in $L^{2} (Ω^{'})$ as $δ \to 0$ for all suitable multi-indices α on any $Ω^{'} \subset \subset Ω$ . For example, let the function ρ be defined by $ρ (x) = c exp (1 / (| x | - 1)$ for $| x | ⩽ 1$ and $ρ (x) = 0$ for $| x | ⩽ 1$ where c is such that $\int ρ d x = 1$ , then we can set $ψ_{δ} (x) \equiv δ^{- 3} \int_{Ω} ρ (\frac{x - y}{δ}) ψ (y) d y$ for $δ < dist (x, Γ)$ . We have that $ψ_{δ} \in H_{0}^{1} (Ω)$ , that $D_{i} ψ_{δ} = {(D_{i} ψ)}_{δ}$ by integration by parts and hence $\nabla \cdot ψ_{δ} = {(\nabla \cdot ψ)}_{δ} = 0$ , and we have that (1.10) holds with $v = ψ_{δ}$ . Now choose a sequence of subdomains $Ω_{m}^{'}$ such that $| Ω ∖ Ω_{m}^{'} | \to 0$ where $| Ω |$ denotes the measure of Ω. Since $‖ \nabla b ‖_{2}^{2} = ‖ \nabla h ‖_{2}^{2} + ‖ \nabla ψ ‖_{2}^{2}$ we have that $‖ \nabla ψ ‖_{2}$ is bounded by the right-hand side of (1.4), so we can choose m large enough so that for a given $ϵ > 0$ we have that $| ‖ ψ ‖_{4, Ω} - ‖ ψ ‖_{4, Ω_{m}^{'}} | < ϵ$ and that $| ‖ \nabla ψ ‖_{2, Ω} - ‖ \nabla ψ ‖_{2, Ω_{m}^{'}} | < ϵ$ . Meanwhile $‖ ψ ψ_{δ} ‖_{2, Ω_{m}^{'}}^{1 / 2} \to ‖ ψ ‖_{4, Ω_{m}^{'}}$ and $‖ \nabla ψ_{δ} ‖_{2, Ω_{m}^{'}} \to ‖ \nabla ψ ‖_{2, Ω_{m}^{'}}$ as $δ \to 0$ . With these observations we see that (1.10) holds with v replaced by ψ, and combining with (1.4) we have according to the above remarks that $‖ ψ ‖_{4} ⩽ κ (ϵ) ‖ \nabla ψ ‖_{2} ⩽ κ (ϵ) C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)}$ . Hence for any given choice of b we can by adjusting ψ as above choose $κ (ϵ)$ small enough so that in particular $‖ ψ ‖_{4} ⩽ σ ‖ h ‖_{4}$ for a given $σ \in (0, 1)$ . Setting $η = 1 + σ$ we thus have that $‖ b ‖_{4} ⩽ η ‖ h ‖_{4}$ . For $q \in [3, 4)$ we set $| ψ |^{q} = | ψ |^{q - 2} | ψ |^{2}$ and observe that $\frac{1}{d^{(q - 2) / 2}} ⩽ \frac{1}{d}$ and hence $‖ \frac{v}{d^{(q - 2) / 2}} ‖_{2} ⩽ ‖ \frac{v}{d} ‖_{2}$ . The calculations leading to the extension of (1.10) to $v = ψ$ then follow similarly as in the case $q = 4$ using the regularity properties of ω. Combining these observations with (2.9) we have that $\begin{array}{l} {‖ ν^{- 1} A^{- 1} (f + P (b \cdot \nabla) b) + A^{- 1} P Δ b ‖}_{q} \\ (2.10) & ⩽ ν^{- 1} λ_{1}^{- s} M_{q} L + η^{2} ν^{- 1} λ_{1}^{- r} M_{p} B_{p} ‖ h ‖_{q}^{2} + η τ_{q} ‖ h ‖_{q} \end{array}$ for some $η \in (1, 2)$ . The constant $M_{s}$ can now be taken to be the right-hand side of (2.10).

Note that in the course of deriving (2.10) we have established the following result:

Theorem 6.
For any $p \in [3, 4]$ we can choose a divergence-free vector $b = h + ψ$ satisfying $b = β$ on Γ and satisfying ( 1.4 ) such that $\begin{matrix} (2.11) & ‖ b ‖_{q} ⩽ η ‖ h ‖_{q} \end{matrix}$ for any given $η \in (1, 2)$ .

Turning to (1.14), we first show that solutions $u_{s}$ of (1.12) are also solutions to (1.14). Setting $W_{s} = b \otimes u_{s} + u_{s} \otimes b + u_{s} \otimes u_{s}$ and $B = f (s) - P div (b \otimes b)$ we substitute $u_{s}$ into the right-hand side of (1.14), and noting that in this case $u_{0} = u_{s}$ we obtain $e^{- t A} u_{s} + \int_{0}^{t} e^{- ν (t - s) A} B d s + \int_{0}^{t} A e^{- ν (t - s) A} A^{- 1} P Δ b d s - \int_{0}^{t} e^{- ν (t - s) A} P div W_{s} d s$ . Changing variables using the time-independence of $u_{s}$ , b, and f, this becomes $e^{- t A} u_{s} + \int_{0}^{t} e^{- ν s A} B d s + \int_{0}^{t} A e^{- ν s A} A^{- 1} P Δ b d s + \int_{0}^{t} e^{- ν s A} P div W_{s} d s$ . Using the semigroup properties, including $\frac{d}{d t} e^{- ν s A} v = A e^{- ν s A} v$ , we obtain $e^{- t A} u_{s} + ν^{- 1} A^{- 1} (I - e^{- ν t A}) B + (I - e^{- ν t A}) A^{- 1} P Δ b - ν^{- 1} A^{- 1} (e^{- ν t A} - I) P div W_{s}$ ; collecting terms and using (1.12) this becomes $e^{- ν t A} u_{s} + (I - e^{- ν t A}) (ν^{- 1} A^{- 1} B + A^{- 1} P Δ b - ν^{- 1} A^{- 1} P div W_{s}) = e^{- ν t A} u_{s} + (I - e^{- ν t A}) = u_{s}$ . Thus (1.14) is satisfied by the solution of (1.12).

Now for $u_{0} \in X_{q}$ with $q \in (3, 4]$ we set $G (t) = e^{- t A} u_{0} + \int_{0}^{t} e^{- (t - s) A} [f (s) - P div (b \otimes b)] d s + ν \int_{0}^{t} A e^{- (t - s) A} A^{- 1} P Δ b d s$ as in the introduction, and we now show that $G (t)$ remains in $X_{q}$ for all $t ⩾ 0$ . We first note from (2.1) that for smooth enough v there exists a constant $M_{q}$ such that $‖ v ‖_{q} ⩽ M_{q} ‖ v ‖_{H^{1} (Ω)}$ for each $q \in (3, 4]$ , and together with (1.4) we then have that $‖ b ‖_{q} ⩽ M_{q} C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)}$ . The following result will also be useful in estimating ${sup}_{0 ⩽ t < \infty} ‖ G (t) ‖_{q}$ , as well as in providing the basis for the proofs of Theorems 2 and 3.
Theorem 7.
Suppose $q > 3$ and set $θ = 3 / q$ . Set $δ = 1 - (1 + θ) / 2$ , then there exists a generic constant $C_{0}$ and a constant $a_{δ} = a_{δ} (δ)$ such that for each $v, w \in C_{B} ([0, \infty); X_{q})$ we have that $\begin{array}{l} \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (v (s) \otimes w (s)) ‖}_{q} d s \\ ⩽ C_{0} \int_{0}^{t} \frac{1}{{[ν (t - s)]}^{(1 + θ) / 2}} {‖ v (s) ‖}_{q} {‖ w (s) ‖}_{q} e^{- ν (λ_{1} / 2) (t - s)} d s \\ (2.12) & ⩽ C_{0} a_{δ} {(\frac{2}{ν λ_{1}})}^{δ} sup_{0 ⩽ t < \infty} ({‖ v (t) ‖}_{q} {‖ w (t) ‖}_{q}) \end{array}$ for all $t ⩾ 0$ .

To prove this result we use (2.3) with $θ = 3 / q$ and $p = q / 2$ along with the semigroup properties of $e^{- t A}$ to see that there is a constant $M_{q}$ such that $\begin{array}{l} {‖ e^{- ν (t - s) A} P div (v (s) \otimes w (s)) ‖}_{q} \\ = ‖ A^{1 / 2} e^{- ν (t - s) A / 2} (e^{- ν (t - s) A / 2} A^{- 1 / 2} P div (v (s) \otimes w (s))) ‖_{q} \\ (2.13) & ⩽ M_{q} {‖ A^{(1 + θ) / 2} e^{- ν (t - s) A / 2} [e^{- ν (t - s) A / 2} A^{- 1 / 2} P div (v (s) \otimes w (s))] ‖}_{p} \end{array}$ where we have noted that $A^{1 / 2} A^{θ / 2} = A^{(1 + θ) / 2}$ and that powers of A commute with $e^{- τ A}$ for any $τ ⩾ 0$ . Using the factorization $e^{- ν (t - s) A} = e^{- ν (t - s) A / 2} e^{- ν (t - s) A / 2}$ while applying (2.6) and the definition of the operator T following (2.7) we have that $\begin{array}{l} \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (v (s) \otimes w (s)) ‖}_{q} d s \\ (2.14) & ⩽ M_{q} \int_{0}^{t} \frac{2 c_{p}}{{[ν (t - s)]}^{(1 + θ) / 2}} {‖ e^{- ν (t - s) A / 2} T (v (s) \otimes w (s)) ‖}_{p} d s \end{array}$ so that applying (2.5) to (2.14) we have that $\begin{array}{l} \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (v (s) \otimes w (s)) ‖}_{q} d s \\ (2.15) & ⩽ M_{q} \int_{0}^{t} \frac{2 d_{p} c_{p}}{{[ν (t - s)]}^{(1 + θ) / 2}} {‖ T (v (s) \otimes w (s)) ‖}_{p} e^{- ν (t - s) λ_{1} / 2} d s . \end{array}$ Meanwhile $‖ T (v (s) \otimes w (s)) ‖_{p} ⩽ B_{p} ‖ v (s) \otimes w (s) ‖_{p} ⩽ B_{p} ‖ v (s) ‖_{2 p} ‖ w (s) ‖_{2 p} = B_{p} ‖ v (s) ‖_{q} ‖ w (s) ‖_{q}$ where as above $B_{p} = ‖ T ‖_{p}$ . Applying this observation to (2.15) we have that $\begin{array}{l} \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (v (s) \otimes w (s)) ‖}_{q} d s \\ (2.16) & ⩽ M_{q} B_{p} \int_{0}^{t} \frac{2 d_{p} c_{p}}{{[ν (t - s)]}^{(1 + θ) / 2}} {‖ v (s) ‖}_{q} {‖ w (s) ‖}_{q} e^{- ν (t - s) λ_{1} / 2} d s . \end{array}$ This establishes the first inequality in (2.12) with $C_{0} = 2 M_{q} B_{p} d_{p} c_{p}$ . The second inequality follows from the estimate $\begin{matrix} (2.17) & \int_{0}^{t} \frac{e^{- ν (t - s) λ_{1} / 2}}{{[ν (t - s)]}^{(1 + θ) / 2}} d s ⩽ a_{δ} {(\frac{2}{ν λ_{1}})}^{δ}, \end{matrix}$ proofs of which can be found e.g. in [2,8].

To estimate ${sup}_{0 ⩽ t < \infty} ‖ G (t) ‖_{q}$ we first apply Theorem 7 with $v = w = b$ to obtain that $\begin{array}{l} \int_{0}^{t} {‖ e^{- ν (t - s) A} (b \cdot \nabla) b ‖}_{q} d s & ⩽ C_{0} a_{δ} {(\frac{2}{ν λ_{1}})}^{δ} ‖ b ‖_{q} ‖ b ‖_{q} \\ (2.18) & ⩽ 2 η^{2} M_{q} B_{p} d_{p} c_{p} a_{r} {(\frac{2}{ν λ_{1}})}^{r} ‖ h ‖_{q}^{2} \end{array}$ where for $θ = 3 / q$ we set $r = δ = 1 - (1 + 3 / q) / 2 = (q - 3) / (2 q)$ and in the second line we have applied (2.11).

Since b is time-independent we have that $\int_{0}^{t} A e^{- ν (t - s) A} A^{- 1} P Δ b d s = \int_{0}^{t} A e^{- ν s A} A^{- 1} P Δ b d s = - \int_{0}^{t} \frac{d}{d s} e^{- ν s A} A^{- 1} P Δ b d s = - (e^{- ν t A} - I) A^{- 1} P Δ b$ , so that from (2.5) and (2.11) and the remarks regarding the operator $A^{- 1} P Δ$ following (2.8) we have that $\begin{matrix} (2.19) & {‖ ν \int_{0}^{t} A e^{- ν (t - s) A} A^{- 1} P Δ b d s ‖}_{q} ⩽ ν (d_{q} e^{- ν λ_{1} t} + 1) {‖ A^{- 1} P Δ b ‖}_{q} ⩽ ν (d_{q} + 1) η τ_{q} ‖ h ‖_{q} . \end{matrix}$ Meanwhile, we have from (2.3) that there exists a constant $M_{2}$ such that $‖ \int_{0}^{t} e^{- ν (t - s) A} f d s ‖_{q} ⩽ \int_{0}^{t} ‖ e^{- ν (t - s) A} f ‖_{q} d s ⩽ \int_{0}^{t} M_{2} ‖ A^{θ / 2} e^{- ν (t - s) A} f ‖_{2} d s$ where now $θ = (3 q - 6) / (2 q)$ ; since $q \in (3, 4]$ we have that $θ / 2 \in (1 / 4, 3 / 8]$ . From developments similar to those that led to the proof of Theorem 7 we then have for $s = δ = 1 - (1 + θ) / 2 = (6 - q) / (4 q)$ that $\begin{matrix} (2.20) & {‖ \int_{0}^{t} e^{- ν (t - s) A} f d s ‖}_{q} ⩽ \int_{0}^{t} M_{2} \frac{e^{- ν (t - s) λ_{1} / 2}}{{[ν (t - s)]}^{(1 + θ) / 2}} ‖ f ‖_{2} d s ⩽ 2 M_{2} c_{2} a_{s} L {(\frac{2}{ν λ_{1}})}^{s}, \end{matrix}$ and we note that $d_{2} = 1$ where $d_{2}$ is defined by (2.5).

Combining (2.18)–(2.20) and applying (2.5) to $‖ e^{- ν t A} u_{0} ‖_{q}$ we thus have, substituting r, s as above, that $\begin{array}{l} sup_{0 ⩽ t < \infty} {‖ G (t) ‖}_{q} ⩽ & d_{q} ‖ u_{0} ‖_{q} + 2 η^{2} M_{q} B_{p} d_{p} c_{p} a_{r} {(\frac{2}{ν λ_{1}})}^{r} ‖ h ‖_{q}^{2} \\ (2.21) & + ν (d_{q} + 1) η τ_{q} ‖ h ‖_{q} + 2 M_{2} c_{2} a_{s} L {(\frac{2}{ν λ_{1}})}^{s} . \end{array}$ The estimate M on ${sup}_{0 ⩽ t < \infty} ‖ G (t) ‖_{q}$ discussed in the remarks preceding Theorem 2 is thus established by the right-hand side of (2.21), with explicit dependence exhibited on $‖ u_{0} ‖_{q}$ , L, and $‖ h ‖_{q}$ .

We now discuss in what sense the solution of (1.12) is a weak solution of (1.3a)–(1.3c). For a smooth solenoidal vector function φ we take the inner product of both sides of (1.12) with $ν A φ$ . We will show in Section 4 below that $u \in D (A^{s / 2}) \subset H^{s} (Ω)$ for some $s > 1$ when $3 < q ⩽ 4$ . Accordingly we use the self-adjointness of A and P to obtain $\begin{array}{l} ν (A^{1 / 2} u, A^{1 / 2} φ) + ([div (b \otimes u) + div (u \otimes b) + div (u \otimes u)], φ) \\ (2.22) & = (f - P div (b \otimes b), φ) + ν (A^{- 1 / 2} P Δ b, A^{1 / 2} φ) . \end{array}$ We have in particular that $u \in D (A^{1 / 2})$ ; from this we can confirm the validity of each term in (2.22): the first term on the left-hand side clearly makes sense, and for the second term we have that $\begin{matrix} (2.23) & div (b \otimes u) + div (u \otimes b) + div (u \otimes u) = (b \cdot \nabla) u + (u \cdot \nabla) b + (u \cdot \nabla) u . \end{matrix}$ Each of the terms on the right-hand side of (2.23) is a product of square-integrable functions, so each of these terms can be integrated against the smooth test function φ. Similarly the first term on the right-hand side of (2.22) makes sense, while $A^{- 1 / 2} P Δ b = A^{- 1 / 2} P div (\nabla b) = T (\nabla b)$ which is square-integrable as of course is $A^{1 / 2} φ$ . Thus the equation (2.22) makes sense and can easily be transformed into more recognizable forms including the weak equation used in [14] (see (2.24) below) by integration by parts and by suitably manipulating P and the powers of A. This shows that solutions of (1.12) guaranteed by Theorem 1 are suitably weak solutions of (1.5) for the cases in which $3 < q ⩽ 4$ .

For the case $q = 3$ we first note that the solutions established in [14] solve the following weak formulation for (1.3a)–(1.3c) for smooth solenoidal vector functions ψ: $\begin{array}{l} ν (\nabla u, \nabla ψ) + ([(b \cdot \nabla) u + (u \cdot \nabla) b + (u \cdot \nabla) u], ψ) \\ (2.24) & = - ((b \cdot \nabla) b, ψ) - ν (\nabla b, \nabla ψ) . \end{array}$ By standard calculation $(\nabla u, \nabla ψ) = (A^{1 / 2} u, A^{1 / 2} ψ) = (u, A ψ)$ and by the properties of P we have that $((b \cdot \nabla) b, ψ) = ((b \cdot \nabla) b, P ψ) = (P (b \cdot \nabla) b, ψ)$ and similarly for the other term on the left-hand side of (2.23). Replacing ψ by $A^{- 1} φ = P A^{- 1} φ$ we have that $- ν (\nabla b, \nabla ψ) = - ν (\nabla b, \nabla P A^{- 1} φ)$ . By standard calculation and by duality we then have that $- ν (\nabla b, \nabla ψ) = ν (b, Δ P A^{- 1} φ) = ν (A^{- 1} P Δ b, φ)$ . Using the self-adjointness of $A^{- 1}$ and applying (2.23) we then have that u is a solution of $\begin{array}{l} ν (u, φ) + (A^{- 1} P [div (b \otimes u) + div (u \otimes b) + div (u \otimes u)], φ) \\ (2.25) & = (- A^{- 1} P div (b \otimes b), φ) + ν (A^{- 1} P Δ b, φ) . \end{array}$ Since φ is arbitrary we see that the solution u established in [14] is also a solution of (1.12). By the statement of Theorem 1, this means that if conditions (1.9) and (1.13) are both met, then the $H^{1}$ -solution found in [14] in fact is the unique solution of (1.12), which establishes the assertion pertaining to this in Theorem 1 in the case $q = 3$ .

We now show that the solution of (1.14) found in Theorem 2 is a suitable weak solution of (1.6a)–(1.6b). We rewrite (1.14) as $\begin{matrix} (2.26) & u (t) = e^{- t ν A} u_{0} + \int_{0}^{t} e^{- (t - s) ν A} F (s) d s + ν \int_{0}^{t} A^{1 / 2} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s \end{matrix}$ where $F (s) \equiv f (s) - P div (b \otimes b) - (P div (b \otimes u (s)) + P div (u (s) \otimes b) + P div (u (s) \otimes u (s))$ and assume for simplicity that $u_{0} \in D (A)$ . Then for a smooth test vector function v we have that $\begin{array}{l} (u_{t}, v) = & (\frac{d}{d t} e^{- t ν A} u_{0}, v) + (\frac{d}{d t} \int_{0}^{t} e^{- (t - s) ν A} F (s) d s, v) \\ (2.27) & + ν (\frac{d}{d t} \int_{0}^{t} A^{1 / 2} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s, v) \end{array}$ and by the properties of $e^{- t ν A}$ , the self-adjointness of powers of A, and using Leibniz’s rule we have that $\begin{array}{l} (\frac{d}{d t} e^{- t ν A} u_{0}, v) + (\frac{d}{d t} \int_{0}^{t} e^{- (t - s) ν A} F (s) d s, v) \\ = (- ν A e^{- t ν A} u_{0}, v) + (\int_{0}^{t} A^{- 1 / 2} \frac{d}{d t} e^{- (t - s) ν A} F (s) d s, A^{1 / 2} v) + (F (t), v) \\ = (- ν A^{1 / 2} e^{- t ν A} u_{0}, A^{1 / 2} v) - (\int_{0}^{t} A^{- 1 / 2} (ν A) e^{- (t - s) ν A} F (s) d s, A^{1 / 2} v) + (F (t), v) \\ = - ([ν A^{1 / 2} e^{- t ν A} u_{0} + ν A^{1 / 2} \int_{0}^{t} e^{- (t - s) ν A} F (s) d s], A^{1 / 2} v) + (F (t), v) \\ (2.28) & = - ν (A^{1 / 2} [e^{- t ν A} u_{0} + \int_{0}^{t} e^{- (t - s) ν A} F (s) d s], A^{1 / 2} v) + (F (t), v) \end{array}$ and similarly we have that $\begin{array}{l} ν (\frac{d}{d t} \int_{0}^{t} A^{1 / 2} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s, v) \\ = ν (\frac{d}{d t} \int_{0}^{t} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s, A^{1 / 2} v) \\ = ν (\int_{0}^{t} \frac{d}{d t} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s, A^{1 / 2} v) + ν (A^{- 1 / 2} P Δ b, A^{1 / 2} v) \\ = ν (A^{1 / 2} \int_{0}^{t} A^{- 1 / 2} \frac{d}{d t} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s, A^{1 / 2} v) + ν (A^{- 1 / 2} P Δ b, A^{1 / 2} v) \\ = ν (A^{1 / 2} \int_{0}^{t} A^{- 1 / 2} (- A) e^{- (t - s) A} A^{- 1 / 2} P Δ b d s, A^{1 / 2} v) + ν (A^{- 1 / 2} P Δ b, A^{1 / 2} v) \\ (2.29) & = - (A^{1 / 2} [ν \int_{0}^{t} A^{1 / 2} e^{- (t - s) A} A^{- 1 / 2} P Δ b d s], A^{1 / 2} v) + ν (A^{- 1 / 2} P Δ b, A^{1 / 2} v) . \end{array}$ Set $F_{b} (t) \equiv P div (b \otimes u (t)) + P div (u (t) \otimes b) + P div (u (t) \otimes u (t))$ and $f_{b} (s) \equiv f - P div (b \otimes b)) + A^{- 1 / 2} P Δ b$ , then by adding the last lines of (2.28), (2.29) and substituting into (2.26), we have that $(u_{t}, v) + ν (A^{1 / 2} u, A^{1 / 2} v) + (F_{b} (t), v) = (f_{b} (s), v)$ , i.e. u is a solution of the time-dependent version of (2.25). This concludes this section of preliminary observations, and we are now in a position to prove our main results.
3. Proof of the main theorems

For the proof of Theorem 1 we first use T to rewrite (1.12) as $\begin{array}{l} u = & - ν^{- 1} A^{- 1 / 2} T [(b \otimes u) + (u \otimes b) + (u \otimes u)] \\ (3.1) & + ν^{- 1} A^{- 1} [f + P div (b \otimes b)] + A^{- 1} P Δ b . \end{array}$ As above we set $G_{s} = ν^{- 1} A^{- 1} [f + P div (b \otimes b)] + A^{- 1} P Δ b$ , and setting $E_{s} = {v \in X_{q} | ‖ v - G_{s} ‖_{q} ⩽ M_{s}}$ we have that $E_{s}$ is a subspace of the metric space $X_{q}$ under the metric $ρ_{s} (v, w) = ‖ v - w ‖_{q}$ ; $E_{s}$ is nonempty since $G_{s} \in E_{s}$ by virtue of (2.10). We define a map $S_{s}$ on $E_{s}$ by setting $\begin{array}{l} S_{s} v = & - ν^{- 1} A^{- 1 / 2} T [(b \otimes v + (v \otimes b) + (v \otimes v)] \\ (3.2) & + ν^{- 1} A^{- 1} [f + P div (b \otimes b)] + A^{- 1} P Δ b . \end{array}$ For any $v \in E_{s}$ we have as in the development of (2.7) with $p = q / 2$ that $\begin{array}{l} ‖ S_{s} v - G_{s} ‖_{q} & = ν^{- 1} λ_{1}^{- (q - 3) / (2 q)} ‖ A^{- 3 / (2 q)} T [(b \otimes v + (v \otimes b) + (v \otimes v)] ‖_{q} \\ ⩽ ν^{- 1} M_{p} λ_{1}^{- (q - 3) / (2 q)} {‖ T [(b \otimes v + (v \otimes b) + (v \otimes v)]) ‖}_{p} \\ ⩽ ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} {‖ (b \otimes v) + (v \otimes b) + (v \otimes v)] ‖}_{p} \\ (3.3) & ⩽ ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} [‖ b ‖_{q} ‖ v ‖_{q} + ‖ v ‖_{q} ‖ b ‖_{q} + ‖ v ‖_{q} ‖ v ‖_{q}] . \end{array}$ By the definition of $E_{s}$ we have that $‖ v ‖_{q} ⩽ ‖ v - G_{s} ‖_{q} + ‖ G_{s} ‖_{q} ⩽ 2 M_{s}$ . Combining this with (2.11) and (3.3) we have that $\begin{matrix} (3.4) & ‖ S_{s} v - G_{s} ‖_{q} ⩽ ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} (2 η ‖ h ‖_{q} + 2 M_{s}) 2 M_{s} . \end{matrix}$ Thus $S_{s} : E \to E$ provided that $\begin{matrix} (3.5) & ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} (2 η ‖ h ‖_{q} + 2 M_{s}) 2 M_{s} ⩽ M_{s} \end{matrix}$ which holds if $\begin{matrix} (3.6) & 4 ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} (η ‖ h ‖_{q} + M_{s}) ⩽ 1 . \end{matrix}$ Let $v, w \in E_{s}$ and set $W \equiv v - w$ then by calculations similar to those used in the development of (3.3) we have for any $v, w \in E_{s}$ that $\begin{array}{l} ‖ S_{s} v - S_{s} w ‖_{q} & ⩽ ν^{- 1} λ_{1}^{- (q - 3) / (2 q)} ‖ A^{- 3 / (2 q)} T [(b \otimes W + W \otimes b + W \otimes v + w \otimes W] ‖_{q} \\ (3.7) & ⩽ ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} [2 ‖ b ‖_{q} ‖ W ‖_{q} + ‖ W ‖_{q} ‖ v ‖_{q} + ‖ w ‖_{q} ‖ W ‖_{q}] . \end{array}$ Applying (2.11) and using the definition of $ρ_{s}$ in the remarks following (3.1) we have that $\begin{matrix} (3.8) & ρ_{s} (S v, S w) ⩽ ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} (2 η ‖ h ‖_{q} + ‖ v ‖_{q} + ‖ w ‖_{q}) ρ_{s} (v, w) \end{matrix}$ and hence $S_{s}$ is a contraction on $E_{s}$ if $\begin{matrix} (3.9) & 2 ν^{- 1} M_{p} B_{p} λ_{1}^{- (q - 3) / (2 q)} (η ‖ h ‖_{q} + 2 M_{s}) < 1 . \end{matrix}$ Note that (3.6) and (3.9) both hold if (3.6) holds, from which we obtain the condition (1.13) guaranteeing that $S_{s} : E \to E$ is a contraction, and we observe that its fixed point is the unique solution of (1.12) in the metric space E. This completes the proof of the existence and uniqueness of solutions of (1.12) in $X_{q}$ for $q \in [3, 4]$ ; we will establish the $H^{1}$ -regularity of these solutions for $q \in (3, 4]$ in Section 4 below ( $H^{1}$ -regularity of the solution for $q = 3$ was addressed above in the discussion following (2.25)).

For the proof of Theorem 2, using the definition of $G (t)$ from the remarks following (1.14) we rewrite (1.14) as $\begin{matrix} (3.10) & u (t) = G (t) - \int_{0}^{t} e^{- ν (t - s) A} P div [(b \otimes u (s)) + (u (s) \otimes b) + (u (s) \otimes u (s))] d s . \end{matrix}$ Setting $E = {v \in X_{q} | {sup}_{0 ⩽ t < \infty} ‖ v (t) - G (t) ‖_{q} ⩽ M}$ we have that E is a subspace of the metric space $C_{B} ([0, \infty); X_{q})$ under the metric $ρ (v, w) = {sup}_{0 ⩽ t < \infty} ‖ v (t) - w (t) ‖_{q}$ ; E is nonempty since $G \in E$ by virtue of (2.21). We define a map S on E for each $v \in E$ by $\begin{matrix} (3.11) & (S v) (t) = G (t) - \int_{0}^{t} e^{- ν (t - s) A} P div [(b \otimes v (s)) + (v (s) \otimes b) + (v (s) \otimes v (s))] d s . \end{matrix}$ We have that $\begin{array}{l} {‖ (S v) (t) - G (t) ‖}_{q} \\ ⩽ \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (b \otimes v (s)) ‖}_{q} d s \\ (3.12) & + \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (v (s) \otimes b) ‖}_{q} d s + \int_{0}^{t} {‖ e^{- ν (t - s) A} P div (v (s) \otimes v (s)) ‖}_{q} d s . \end{array}$ Applying Theorem 7 to each term on the right-hand side of (3.12) for appropriate choices of v and w in each case and collecting terms involving b and v we have for $C_{0} = 2 M_{q} B_{p} d_{p} c_{p}$ and $r = δ = (q - 3) / (2 q)$ as before that $\begin{matrix} (3.13) & {‖ (S v) (t) - G (t) ‖}_{q} ⩽ C_{0} a_{r} {(\frac{2}{ν λ_{1}})}^{r} [2 ‖ b ‖_{q} sup_{0 ⩽ t < \infty} {‖ v (t) ‖}_{q} + sup_{0 ⩽ t < \infty} ({‖ v (t) ‖}_{q} {‖ v (t) ‖}_{q})] . \end{matrix}$ By the definition of E we have for all $t ⩾ 0$ that $‖ v (t) ‖_{q} ⩽ ‖ v - G ‖_{q} + ‖ G ‖_{q} ⩽ 2 M$ . Applying this and (2.11) to (3.13) we have that $\begin{matrix} (3.14) & {‖ (S v) (t) - G (t) ‖}_{q} ⩽ C_{0} a_{r} {(\frac{2}{ν λ_{1}})}^{r} 4 [η ‖ h ‖_{q} M + M^{2}] . \end{matrix}$ We have that $S : E \to E$ if the right-hand side of (3.14) is bounded by M, i.e. if $\begin{matrix} (3.15) & 8 M_{q} B_{p} d_{p} c_{p} a_{r} [η ‖ h ‖_{q} + M] {(\frac{2}{ν λ_{1}})}^{r} ⩽ 1 . \end{matrix}$

Now let $v, w \in E$ and set $V (t) = v (t) - w (t)$ and $V_{0} = v_{0} - w_{0}$ then $\begin{array}{l} (S v) (t) - (S w) (t) \\ = e^{- t A} V_{0} \\ - \int_{0}^{t} e^{- (t - s) A} P div [(b \otimes V (s)) + (V (s) \otimes b) \\ (3.16) & + (v (s) \otimes V (s)) + (V (s) \otimes w (s))] d s . \end{array}$ With a similar application of Theorem 7 to (3.16) while collecting terms and using the bound $‖ v (t) ‖_{q} ⩽ 2 M$ for any $v \in E$ we have for $C_{0}$ , r, and $θ = 3 / q$ as above that $\begin{array}{l} {‖ (S v) (t) - (S w) (t) ‖}_{q} \\ ⩽ C_{0} \int_{0}^{t} \frac{1}{{[ν (t - s)]}^{(1 + θ) / 2}} [2 ‖ b ‖_{q} {‖ V (s) ‖}_{q} + 4 M {‖ V (s) ‖}_{q}] e^{- ν (t - s) λ_{1} / 2} d s \\ (3.17) & ⩽ C_{0} a_{r} {(\frac{2}{ν λ_{1}})}^{r} [2 ‖ b ‖_{q} sup_{0 ⩽ t < \infty} {‖ V (t) ‖}_{q} + 4 M sup_{0 ⩽ t < \infty} {‖ V (t) ‖}_{q}] . \end{array}$ Using (2.11) and the definition of ρ we have from (3.17) that $\begin{matrix} (3.18) & ρ (S v, S w) ⩽ 2 C_{0} a_{r} {(\frac{2}{ν λ_{1}})}^{r} [η ‖ h ‖_{q} + 2 M] ρ (v, w) \end{matrix}$ from which we see with $δ = r$ and $C_{0}$ as above that S is a contraction on E if $\begin{matrix} (3.19) & 4 M_{q} B_{p} d_{p} c_{p} [η ‖ h ‖_{q} + 2 M] a_{r} {(\frac{2}{ν λ_{1}})}^{r} < 1 . \end{matrix}$ Note that (3.15) and (3.19) both hold if (3.15) holds, from which we obtain (1.15) with $K_{0} = 8 M_{q} B_{p} d_{p} c_{p} a_{r}$ , $K_{1} = [1 + ν (d_{q} + 1) τ_{q}]$ , $K_{2} = d_{q}$ , $K_{3} = 16 η^{2} {(M_{q} B_{p} d_{p} c_{p} a_{r})}^{2}$ , and $K_{4} = 16 M_{2} d_{p} c_{p} c_{r} c_{2} c_{s}$ . This completes the proof of Theorem 2.

For the proof of Theorem 3, we have similar to the development of the first line of (3.17) that $w = u - v$ satisfies $\begin{array}{l} {‖ w (t) ‖}_{q} ⩽ & d_{q} ‖ w_{0} ‖_{q} e^{- ν λ_{1} t} \\ (3.20) & + C_{0} \int_{0}^{t} \frac{1}{{[ν (t - s)]}^{(1 + θ) / 2}} (2 ‖ b ‖_{q} + 4 M) {‖ w (s) ‖}_{q} e^{- ν (t - s) λ_{1} / 2} d s \end{array}$ where $w_{0} = u_{0} - v_{0}$ . Using (2.11) and using the factorization $e^{- ν (t - s) λ_{1} / 2} = {(e^{- ν (t - s) λ_{1} / 4})}^{2}$ we have from (3.20) that $\begin{array}{l} {‖ w (t) ‖}_{q} ⩽ & d_{q} ‖ w_{0} ‖_{q} e^{- ν λ_{1} t} \\ (3.21) & + C_{0} \int_{0}^{t} \frac{e^{- ν (λ_{1} / 4) (t - s)}}{{[ν (t - s)]}^{(1 + θ) / 2}} (2 η ‖ h ‖_{q} + 4 M) {‖ w (s) ‖}_{q} e^{- ν (λ_{1} / 4) (t - s)} d s . \end{array}$ With further exponential-term factorization, noting that $e^{- ν λ_{1} t} ⩽ e^{- ν (λ_{1} / 4) t}$ , and by setting $C_{1} \equiv C_{0} (2 η ‖ h ‖_{q} + 4 M)$ we have from (3.21) that $\begin{matrix} (3.22) & {‖ w (t) ‖}_{q} ⩽ d_{q} ‖ w_{0} ‖_{q} e^{- ν (λ_{1} / 4) t} + C_{1} e^{- ν (λ_{1} / 4) t} \int_{0}^{t} \frac{e^{- ν (λ_{1} / 4) (t - s)}}{{[ν (t - s)]}^{(1 + θ) / 2}} {‖ w (s) ‖}_{q} e^{ν (λ_{1} / 4) s} d s . \end{matrix}$ Setting $ϕ (t) = ‖ w (t) ‖_{4} e^{ν (λ_{1} / 4) t}$ and multiplying both sides of (3.22) by $e^{ν (λ_{1} / 4) t}$ we obtain $\begin{matrix} (3.23) & ϕ (t) ⩽ d_{q} ‖ w_{0} ‖_{q} + C_{1} \int_{0}^{t} \frac{e^{- ν (λ_{1} / 4) (t - s)}}{{[ν (t - s)]}^{(1 + θ) / 2}} ϕ (s) d s . \end{matrix}$ For any $T > 0$ we clearly have that $ϕ \in C ([0, T])$ so there exists a $t_{*} \in [0, T]$ such that $ϕ (t_{*}) = {sup}_{0 ⩽ t < T} ϕ (t)$ . Then from (3.23) we have that $\begin{array}{l} ϕ (t) & ⩽ d_{q} ‖ w_{0} ‖_{q} + C_{1} ϕ (t_{*}) \int_{0}^{t} \frac{e^{- ν (λ_{1} / 4) (t - s)}}{{[ν (t - s)]}^{(1 + θ) / 2}} d s \\ (3.24) & ⩽ d_{q} ‖ w_{0} ‖_{q} + C_{1} ϕ (t_{*}) a_{r} {(\frac{4}{ν λ_{1}})}^{r} \end{array}$ where in the last line of (3.24) we have used calculations similar to those used in the proof of Theorem 7. Using (2.16) and the definition of $C_{1}$ as above, we have that if $C_{1} a_{r} {(\frac{4}{ν λ_{1}})}^{r} ⩽ 1 / 2$ , i.e. if $\begin{matrix} (3.25) & 8 M_{q} B_{p} d_{p} c_{p} (η ‖ h ‖_{q} + 2 M) a_{r} {(\frac{4}{ν λ_{1}})}^{r} ⩽ 1 \end{matrix}$ then from (3.24) we have for all $t \in [0, T]$ that $\begin{matrix} (3.26) & ϕ (t) ⩽ ϕ (t_{*}) ⩽ 2 d_{q} ‖ w_{0} ‖_{q} . \end{matrix}$ Since T is arbitrary we have by multiplying both sides of (3.26) by $e^{- ν (λ_{1} / 4) t}$ that $\begin{matrix} (3.27) & {‖ w (t) ‖}_{q} ⩽ 2 d_{q} ‖ w_{0} ‖_{q} e^{- ν (λ_{1} / 4) t} \end{matrix}$ for all $t > 0$ , which is the conclusion of Theorem 3 with $K = 2 d_{q} ‖ w_{0} ‖_{q}$ . Note that the condition (3.25) is similar to the condition (3.15), which is the basis for (1.15). This completes the proof of Theorem 3.

4. Regularity

For $G (t) = e^{- t A} u_{0} + \int_{0}^{t} e^{- (t - s) A} [f (s) - P div (b \otimes b)] d s - \int_{0}^{t} A e^{- (t - s) A} A^{- 1} P Δ b d s$ , we first show that $G \in C_{B} ([0, \infty); D (A^{1 / 2})$ . We first note that similarly to the proof of Theorem 7 we have for $v \in X_{q}$ , $q \in (3, 4]$ and for $w \in H^{1} (Ω)$ that $\begin{array}{l} \int_{0}^{t} {‖ A^{1 / 2} e^{- ν (t - s) A} P (v \cdot \nabla) w ‖}_{2} d s \\ ⩽ \int_{0}^{t} {‖ A^{θ / 2} A^{1 / 2} e^{- ν (t - s) A} (v \cdot \nabla) w ‖}_{p} d s \\ (4.1) & ⩽ 2 M_{p} B_{p} d_{p} c_{p} \int_{0}^{t} \frac{1}{{[ν (t - s)]}^{(1 + θ) / 2}} ‖ v ‖_{q} ‖ \nabla w ‖_{2} e^{- ν (λ_{1} / 2) (t - s)} d s \end{array}$ for $p = 2 q / (2 + q)$ and $θ = 3 / q$ so that $(1 + θ) / 2 = (q + 3) / (2 q)$ . Similarly to the development of the second line of (2.16) and using (1.4) and (2.11) we then have for $δ = 1 - (1 + θ) / 2 = (q - 3) / (2 q)$ that $\begin{matrix} (4.2) & \int_{0}^{t} {‖ A^{1 / 2} e^{- ν (t - s) A} (v \cdot \nabla) w ‖}_{2} d s ⩽ 2 M_{p} B_{p} d_{p} c_{p} a_{δ} {(\frac{2}{ν λ_{1}})}^{δ} ‖ v ‖_{q} ‖ w ‖_{H^{1} (Ω)} . \end{matrix}$ Rewriting G as $G (t) = e^{- t A} u_{0} + \int_{0}^{t} e^{- (t - s) A} [f (s) - P (b \cdot \nabla) b] d s + ν \int_{0}^{t} A e^{- (t - s) A} A^{- 1} P Δ b d s$ , we combine (4.2) with $v = w = b$ together with (1.4) and (2.11) to obtain that $\begin{matrix} (4.3) & \int_{0}^{t} {‖ A^{1 / 2} e^{- ν (t - s) A} (b \cdot \nabla) b ‖}_{2} d s ⩽ 2 η M_{p} B_{p} d_{p} c_{p} a_{δ} {(\frac{2}{ν λ_{1}})}^{δ} ‖ h ‖_{q} C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)} . \end{matrix}$ Meanwhile we have that $ν \int_{0}^{t} A e^{- ν (t - s) A} A^{- 1} P Δ b d s = ν \int_{0}^{t} A^{1 / 2} e^{- ν (t - s) A} A^{- 1 / 2} P div (\nabla b) d s = ν \int_{0}^{t} A^{1 / 2} e^{- ν (t - s) A} T (\nabla b) d s$ , so that $ν \int_{0}^{t} A^{1 / 2} (A e^{- ν (t - s) A} A^{- 1} P Δ b) d s = \int_{0}^{t} ν A e^{- ν s A} T (\nabla b) d s = - \int_{0}^{t} \frac{d}{d s} e^{- ν s A} T (\nabla b) b d s = - (e^{- ν t A} - I) T (\nabla b)$ and hence from (1.4) and similarly to the development of (2.19) we have that $\begin{matrix} (4.4) & {‖ ν \int_{0}^{t} A^{1 / 2} (A e^{- ν (t - s) A} A^{- 1} P Δ b) d s ‖}_{2} ⩽ (e^{- ν λ_{1} t} + 1) ‖ T (\nabla b ‖_{2} ⩽ 2 ν B_{2} C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)} . \end{matrix}$ Similarly to the proof of Theorem 7 we have that $\begin{matrix} (4.5) & {‖ \int_{0}^{t} A^{1 / 2} e^{- ν (t - s) A} f d s ‖}_{2} ⩽ \int_{0}^{t} M_{2} \frac{e^{- ν (t - s) λ_{1} / 2}}{{[ν (t - s)]}^{1 / 2}} ‖ f ‖_{2} d s ⩽ 2 M_{2} c_{2} a_{2} L {(\frac{2}{ν λ_{1}})}^{1 / 2} . \end{matrix}$

We now assume that $u_{0} \in D (A^{1 / 2})$ so that $‖ A^{1 / 2} e^{- ν t A} u_{0} ‖_{2} = ‖ e^{- ν t A} (A^{1 / 2} u_{0}) ‖_{2} ⩽ ‖ A^{1 / 2} u_{0} ‖_{2}$ . Combining this with (4.3)–(4.5) we thus have that $\begin{array}{l} sup_{0 ⩽ t < \infty} {‖ A^{1 / 2} G (t) ‖}_{2} ⩽ & {‖ A^{1 / 2} u_{0} ‖}_{2} + η C_{Γ} K_{1, 0} {(\frac{2}{ν λ_{1}})}^{δ} ‖ h ‖_{q} ‖ β ‖_{H^{1 / 2} (Γ)} s \\ (4.6) & + K_{1, 1} ‖ β ‖_{H^{1 / 2} (Γ)} + K_{1, 2} L {(\frac{2}{ν λ_{1}})}^{1 / 2} \end{array}$ where $K_{1, 0} = 2 M_{p} B_{p} d_{p} c_{p} a_{δ}$ , $K_{1, 1} = 2 ν B_{2} C_{Γ}$ , and $K_{1, 2} = 2 M_{2} c_{2} a_{2}$ . We set $M_{1}$ to be the right-hand side of (4.6) and set $E_{1} = {v \in C_{B} ([0, T_{1}]; D (A^{1 / 2})) | {sup}_{0 ⩽ t < \infty} ‖ A^{1 / 2} (v (t) - G (t)) ‖_{2} ⩽ M_{1}}$ then $E_{1}$ is a closed subset of the metric space $C ([0, T_{1}]; D (A^{1 / 2}))$ for each $T_{1} > 0$ under the metric $ρ_{1} (v, w) = {sup}_{0 ⩽ t < \infty} ‖ A^{1 / 2} (v (t) - w (t)) ‖_{2}$ . $E_{1}$ is nonempty since we have just shown via (4.6) that G is in $E_{1}$ for any $T_{1} > 0$ . We now consider the following integral-equation version of (1.6a)–(1.6b): $\begin{array}{l} u (t) = & e^{- t A} u_{0} + \int_{0}^{t} e^{- (t - s) A} [f (s) - P (b \cdot \nabla) b] d s \\ + ν \int_{0}^{t} A e^{- (t - s) A} A^{- 1} P Δ b d s \\ (4.7) & - \int_{0}^{t} e^{- (t - s) A} [P (b \cdot \nabla) u + P (u \cdot \nabla) b + P (u \cdot \nabla) u] d s . \end{array}$ The following result follows from standard methods:

Theorem 8.
There exists a $T_{1} > 0$ such that ( 4.7 ) has a unique solution $u_{1} \in E_{1}$ .

The proof uses a standard contraction-mapping approach using (4.2) and can otherwise safely be omitted. We now let $[0, T_{0})$ be a maximal interval of existence for the solutions $u_{1} \in C ([0, T_{1}]; D (A^{1 / 2}))$ and assume that $‖ h ‖_{q}$ , $‖ u_{0} ‖_{q}$ , and L satisfy the combined conditions of Theorems 2 and 3. We have from (2.3) with $θ = 1$ and $p = 2$ that $u_{1} (t) \in X_{q}$ for any $q ⩽ 6$ and in particular for $q \in (3, 4]$ . Since solutions of (4.7) are solutions of (1.14), we therefore have for any such q that $u_{1} (t) = u (t)$ for all $t \in [0, T_{0})$ where u is the unique solution of (1.14) consructed in Theorem 2. By calculations similar to those used in the development of (4.2) we have that $\begin{array}{l} {‖ A^{1 / 2} u (t) ‖}_{2} ⩽ & K_{1, 0} {(\frac{2}{ν λ_{1}})}^{δ} [(‖ b ‖_{q} + ‖ u ‖_{q}) ‖ \nabla u ‖_{2} + ‖ u ‖_{q} ‖ \nabla b ‖_{2}] + {‖ A^{1 / 2} G (t) ‖}_{2} \\ ⩽ & K_{1, 0} {(\frac{2}{ν λ_{1}})}^{δ} (η ‖ h ‖_{q} + M) {‖ A^{1 / 2} u ‖}_{2} \\ (4.8) & + 2 K_{1, 0} {(\frac{2}{ν λ_{1}})}^{δ} M C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)} + M_{1} . \end{array}$ Choosing $‖ h ‖_{q}$ and M small enough so that $K_{1, 0} {(\frac{2}{ν λ_{1}})}^{δ} (η ‖ h ‖_{q} + M) ⩽ 1 / 2$ and collecting terms in (4.7) we have that $\begin{matrix} (4.9) & {‖ A^{1 / 2} u (t) ‖}_{2} ⩽ 2 K_{1, 0} {(\frac{2}{ν λ_{1}})}^{δ} M C_{Γ} ‖ β ‖_{H^{1 / 2} (Γ)} + 2 M_{1} . \end{matrix}$ From (4.8) we see that the solution constructed in Theorem 8 stays bounded on $[0, T_{0})$ and so by standard methods can be extended to be a solution on $[0, \infty)$ . Since $u_{1} (t) = u (t)$ as noted above, we have that the solution u constructed in Theorem 2 in fact satisfies $u \in C_{B} ([0, \infty); D (A^{1 / 2}))$ . Since as noted above in the discussion following Theorem 3, by satisfying (1.13) the solutions established by Theorem 1 with $q \in (3, 4]$ also satisfy (1.15) (with $u_{s} = u_{0}$ ) and are solutions of (1.14); we therefore also have that $u_{s} \in D (A^{1 / 2})$ .

We now bootstrap these results by reworking (4.1) slightly to obtain that $\begin{array}{l} \int_{0}^{t} {‖ A^{1 / 2 + ς} e^{- ν (t - s) A} P (v \cdot \nabla) w ‖}_{2} d s \\ ⩽ \int_{0}^{t} {‖ A^{ς} A^{θ / 2} A^{1 / 2} e^{- ν (t - s) A} (v \cdot \nabla) w ‖}_{p} d s \\ (4.10) & ⩽ 2 M_{p} B_{p} d_{p} c_{p} \int_{0}^{t} \frac{1}{{[ν (t - s)]}^{ς + (1 + θ) / 2}} ‖ v ‖_{q} ‖ \nabla w ‖_{2} e^{- ν (λ_{1} / 2) (t - s)} d s \end{array}$ where ς satisfies $0 < ς < δ$ for $δ = 1 - (1 + θ) / 2 = (q - 3) / (2 q)$ ; we have that $ς + (1 + θ) / 2 < 1$ so that the integrand in the second line of (4.10) is well-defined. Proceeding along these lines we can show that in fact $G \in C_{B} ([0, \infty); D (A^{1 / 2 + ς}))$ so that using arguments similar to those above we can conclude that $u \in C_{B} ([0, \infty); D (A^{1 / 2 + ς}))$ and that $u_{s} \in D (A^{1 / 2 + ς})$ for $q \in (3, 4]$ . With these observations we have established Theorem 5.
5. Domains allowing data of increased size

If on the right-hand side of (1.11) in the case $Ω = Ω_{R}$ we set $R_{1} = ζ R_{0}$ for $0 < ζ < 1$ then $R_{0} R_{1} / {(R_{0}^{3} - R_{1}^{3})}^{1 / 3} = ζ R_{0} / {[(1 - ζ^{3})]}^{1 / 3}$ . If ζ is close to one, i.e. $R_{0}$ and $R_{1}$ are close (i.e. the domain is thin), then the right-hand side of (1.11) can be of increased size if $R_{0}$ is of moderate size or larger. Similarly the right-hand side of (1.11) can be of increased size if $R_{0}$ is large and ζ is not too small. Similar considerations apply to allow for data of increased size for $3 < q ⩽ 4$ when we replace $(R_{0} R_{1}) {(R_{0}^{3} - R_{1}^{3})}^{- 1 / 3}$ by the qualitatively similar expression ${(R_{0} R_{1})}^{(2 q - 3) / q} {(R_{0}^{2 q - 3} - R_{1}^{2 q - 3})}^{- 1 / q}$ as was discussed in the remarks following Theorem 1.

Meanwhile, as was noted in [2,3], the eigenvalue $λ_{1}$ can be of increased size for certain types of domains, thus giving a further capacity to allow for data of increased size in (1.15) (and in (1.13) for $3 < q ⩽ 4$ ) due to the presence of $λ_{1}^{- r}$ on the left-hand side of both estimates. The archetypal example of such domains in the simply-connected case are of the form $Ω = Ω^{'} \times [0, ϵ]$ where $Ω^{'}$ is an arbitrary 2-d domain and $0 < ϵ ≪ 1$ . The eigenvalues of the Laplace operator are $γ_{n} = γ_{n}^{'} + {(\frac{n π}{ϵ})}^{2}$ where the $γ_{n}^{'}$ are the corresponding eigenvalues of the 2-d Laplacian on $Ω^{'}$ , and in particular $γ_{1} = γ_{1}^{'} + {(\frac{π}{ϵ})}^{2}$ is of increased size if ϵ is small. These eigenvalues are of similar size to those of the Stokes operator (see e.g. [7] and the references contained therein). Note that the domain can be of arbitrary volume since there is no size restriction on the area of $Ω^{'}$ .

On annular domains $Ω_{R}$ the radial parts of the eigenfunctions of $- Δ$ use spherical Bessel functions, and the first of these are $j_{0} (x)$ and $n_{0} (x)$ . These have the explicit expressions $j_{0} (x) = \frac{sin x}{x}$ and $n_{0} (x) = - \frac{cos x}{x}$ (see e.g. [21]). For the radial part $ψ (r)$ of the first eigenfunction we set $ψ (r) = n_{0} (k_{0}) j_{0} (\frac{k_{0} r}{a}) - j_{0} (k_{0}) n_{0} (\frac{k_{0} r}{a})$ where $a = R_{1}$ , then $ψ (a) = 0$ , and for $b = R_{0}$ we have after using some algebra and the explicit expressions for $j_{0} (x)$ and $n_{0} (x)$ that $ψ (b) = 0$ provided that $k_{0}$ is a root of $tan \frac{k_{0} r}{a} = tan k_{0}$ . Using the periodicity of $tan x$ we see that this equation is solved if $\frac{k_{0} r}{a} = k_{0} + π$ from which we obtain that $k_{0} = \frac{π a}{b - a}$ . Since $γ_{1} = {(\frac{k_{0}}{a})}^{2}$ we thus have that $γ_{1} = {(\frac{π}{b - a})}^{2}$ .

This is the same expression as the first eigenvalue of $- Δ$ on $Ω = Ω^{'} \times [0, ϵ]$ if the thinness parameter ϵ is replaced by the corresponding thinness parameter $b - a = R_{0} - R_{1}$ . Again the size of $λ_{1}$ will be similar; in fact if v is the first eigenfunction of A we have that $λ_{1} = \frac{(A v, v)}{(v, v)} = \frac{(\nabla v, \nabla v)}{(v, v)} = \frac{(- Δ v, v)}{(v, v)} ⩾ γ_{1}$ so that ${(\frac{π}{b - a})}^{2} ⩽ λ_{1}$ . These examples show that with the terms $λ_{1}^{- r}$ present in (1.13) and (1.15) we have introduced an additional factor that allows for data of increased size. Here also in similar fashion the formula for $γ_{1}$ is independent of the volume of $Ω_{R}$ . Moreover, our results are very general as in [2,3] since they are stated simply in terms of $λ_{1}^{- 1}$ and not in terms of any other specifics of a particular domain. It is to be noted, however, that we are careful to use the phrase “data of increased size” rather than “large data” given that the estimates (1.9), (1.13), and (1.15) are nonetheless smallness conditions imposed on the data. Indeed, on the domains discussed in this section the configurations of the domains allow for the increased effects of viscosity to more greatly influence the dynamics; for example on thin domains the boundaries of the domain and their strong “no-slip” effects are always in close proximity to the flow.

The papers [2,3] were, as noted in the introduction, part of a series of works by a number of authors devoted to the study of the Navier–Stokes equations on thin domains. One motivation for these works as first laid out in [22] was the idea that thinner domains could support data of increased size while still obtaining global existence and regularity. The initial work in [22], as in many of the results to follow, focused on the case of periodic boundary conditions in thin rectangular solids, but one result in [22] concerned the zero Dirichlet case, and the work in [2] was designed to improve this result. Robust results for annular domains with applications to geophysical flow can be found in [25,26]. Similar considerations apply on arbitrary domains if the size of the data is concentrated in the high frequencies as explored in [4].

It should be noted that, unlike the cases considered here, thin periodic domains can support data that can be legitimately described as large. Indeed, in this case there are no dissipative influences coming from the boundary. As noted, the results in [22] were followed by a series of results in this direction, for which the results in [15–17] are particularly representative, and provide a comprehensive discussion and references.

6. Conclusion

The stability result given by Theorem 4 has physical significance in that in a number of natural settings corresponding to constant flow on the boundary we expect to observe either steady flow or flow that quickly settles into steady flow. Theorem 4 verifies this for the cases at hand from a mathematical standpoint.

Since the early papers of Leray, the existence of unique solutions stronger than weak solutions has only been established on domains with certain geometrical features or for small data. Our results and observations in Sections 4 and 5 fit into this context while making assumptions only slightly stronger than those assumed for the weak-solution results in [10] and [14].

It would be of interest to consider the case of time-dependent boundary conditions $β = β (t)$ , and the continuous dependence on such data would be a potential topic for future study, in particular in handling the case that $β (t) \to β$ in a suitable space as $t \to \infty$ .

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