Global wellposedness of a free boundary problem for the Navier-stokes equations in an exterior domain

doi:10.15406/fmrij.2017.01.00008

eISSN: 2577-8242

Fluid Mechanics Research International Journal

Review Article Volume 1 Issue 2

Global wellposedness of a free boundary problem for the Navier-stokes equations in an exterior domain

Yoshihiro Shibata

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Department of Mathematics and Research Institute of Science and Engineering, Waseda University, Japan

Correspondence: Yoshihiro Shibata, Department of Mathematics and Research Institute of Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Received: May 29, 2017 | Published: October 18, 2017

Citation: Shibata Y. Global wellposedness of a free boundary problem for the navier-stokes equations in an exterior domain. Fluid Mech Res Int. 2017;1(2):56-72. DOI: 10.15406/fmrij.2017.01.00008

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Abstract

In this paper, we prove a unique existence theorem of globally in time strong solutions to free boundary problem for the Navier-Stokes equations in an exterior domain in the case that initial data are small enough. The key step is to prove decay properties of locally in time solutions, which is derived by combination of maximal L_p-L_q regularity with L_p-L_q decay estimates for solutions of slightly perturbed Stokes equations with free boundary condition in an exterior domain.

Keywords: navier-stokes equations, free boundary problem, global well-posedness, exterior domain, polynomial decay, maximal l_p-l_q regularity

Introduction

This paper deals with global well-posedness of the following free boundary problem for the Navier-Stokes equations. Let $Ω$ be an exterior domain in the N dimensional Eucledian space R^N occupied by an incompressible viscous fluid. Let $Γ$ be the boundary of $Ω$ that is a C² compact hyper surface with the unit outer normal n. Let $Ω_{t}$ be the evolution of $Ω$ at time t. Let $Γ_{t}$ be the boundary of $Ω_{t}$ with the unit outer normal n_t. Let $ρ$ and $µ$ be positive numbers denoting the mass density and the viscosity coeﬃcient, respectively. Let $u =^{⊤} (u_{1}, . . ., u_{N})$ be an N-vector of functions describing the velocity field, where $^{⊤} M$ denotes the the transposed M, and let p be a scalar function describing the pressure field. We consider the initial boundary value problem for the Navier-Stokes equations in $Ω_{t}$ given by

$\begin{array}{l} ρ (\partial t u + u \cdot \nabla u) - D i v (µ D (u) - Ρ I) = 0, d i v u = 0 i n \cup_{0 < t < T} Ω_{t} \times {t}, \\ (µ D (u) - Ρ I) n_{t} = 0, V_{Γ t} = n_{t} . u o n \cup_{0 < t < T} Γ_{t} \times {t}, \\ u |_{t}_{= 0} = u_{0}, Ω_{t} |_{t}_{= 0} = Ω \end{array}$ (1)

Here, $D (u) = \nabla u +^{⊤} \nabla u$ denotes the doubled deformation tensor, I the N × N identity matrix, and $V_{Γ t}$ the evolution speed of the surface $Γ_{t}$ in the n_t direction. Moreover, for any matrix field K with (i, j) componentK_ij, the quantity Div K is an N vector of functions whose i^th component is $\sum_{j = 1}^{N} \partial_{j} K_{i j}, \partial_{j} = \partial / \partial x_{j}$ and for any N vector of function $w = (w_{1}, ..., w_{N}), d i v w = \sum_{j = 1}^{N} \partial_{j} w_{i}$ and the quantity $w \cdot \nabla w$ is an N -vector of functions whose i^th component is $\sum_{j = 1}^{N} \partial_{j} w_{i}$

One phase problem for the Navier-Stokes equations formulated in (1) with $(µ D (u) - Ρ I) n t = c_{σ} H n_{t}$ in place of $(µ D (u) - Ρ I) n t = 0$ has been received wide attention for many years, where H is the doubledmean curvature of $Γ_{t}$ and $c_{σ}$ is a non-negative constant describing the coeﬃcient of surface tension. Inparticular, the following two cases have been studied by many mathematicians: (1) the motion of an isolated liquid mass and (2) the motion of a viscous incompressible ﬂuid contained in an ocean of inﬁnite extent. In case (1), the initial domain $Ω_{0}$ is bounded and local well posedness in the case that $c σ > 0$ was proved by Solonnikov^1-4 in the L₂ Sobolev-Slobodetskii space, by Schweizer⁵ in the semi group setting, and by Moglievskii and Solonnikov⁶ in the Holder spaces. And, in the case that $c σ = 0$ , local wellposedness was proved by Solonnikov,⁷ Mucha and Zaja¸czkowski^8,9 in the L_pSobolev-Slobodetskii space, and by Shibata and Shimizu^10,11 in the L_pin time and L_qin space setting. Global wellposedness in the case that $c σ = 0$ for small initial data by Solonnikov^4,7 in the L_pSobolev-Slobodetskii space and by Shibata¹² in the L_pin time and L_qin space setting. Global wellposedness in the case that $c σ > 0$ was proved under the assumption that the initial domain $Ω_{0}$ is suﬃciently close to a ball and initial data are very small by Solonnikov¹³ in the L2 Sobolev-Slobodetskii space, by Padula and Solonnikov¹⁴ in the Holder spaces, and by Shibata¹⁵ in the L_pin time and L_qin space setting.

In case (2), the initial domain $Ω_{0}$ is a perturbed layer like:
$Ω_{0} = {x \in ℝ^{3} | - b < x 3 < η (x'), x' = (x 1, x 2) \in ℝ^{2}}$ and local wellposed was proved by Allain,¹⁶ Beale¹⁷ and Tani¹⁸ in the L2 Sobolev-Slobodetskii space when $c σ > 0$ and by Abels¹⁹ in the L_pSobolev-Slobodetskii space when $c σ = 0$ .

Global wellposedness for small initial velocity was proved in the L₂ Sobolev-Slobodetskii space by Beale²⁰ and Tani and Tanaka²¹ in the case that $c σ > 0$ and by Sylvester²² in the case that $c σ = 0$ . The
decay rate was studied by Beale and Nishida,²³ Sylvestre,²⁴ Hataya²⁵ and Hataya and Kawashima.²⁶ In the case of the Ocean problem without bottom, $Ω_{0} = {x \in ℝ^{3} | x_{3} < η (x'), x' = (x_{1}, x_{2}) \in ℝ^{2}}$ .

In this case, global well posedness for small initial data and the decay properties of solutions have been studied by Saito and Shibata.^28,29 Recently, local well-posedness for the one phase problem of the Navie-Stokes equations was proved in the general unbounded domain case by Shibata¹² in the $c σ = 0$ case and by Shibata^29,30 in the case $c σ > 0$ .

We remark that two phase problem of the Navier-Stokes equations has been studied by many math- ematicians,^31-46 and references therein. Although many papers dealt with global well-posend, as far as the author knows, global well-posedness of free boundary problem for the Navier-Stokes equations in an exterior domain has never be treated, and the purpose of this paper is to prove global well-posedness of problem (1) in the L_pin time and L_qin space setting. Since only polynomial decays are obtained for solutions of Stokes equations with free boundary conditions in the exterior domain case;^47,48 it is necessary to choose a large exponent p to guarantees L_pintegrability of solutions, so that the maximal L_p- L_qregularity for the Stokes equations with free boundary consition proved in Shibata^30,49,50 and also in Pruess and Simonett⁴⁰ in the dfferent p and q case are one of essential tools.

Now we consider the transformation that transforms $Ω_{t}$ to a ﬁxed domain. If $Ω$ is a bounded domain, then we have the exponential stability of the corresponding Stokes equations with free boundary conditions in some quotient space, so that we can use the Lagrange transformation to transform $Ω_{t} t o Ω$ .^7,12 But, is now an exterior domain, so that solutions of the Stokes equations with free boundary conditions decay polynomially as mentioned above. Thus, the Lagrange transformation is not available, because the polynomial decay does not seem to be enough to control the term $\int_{0}^{t} \nabla u d s$ . Another known transformation is the Hanzawa one. But, this transformation requires at least the $W_{q}^{3 - 1 / q} (N < q)$ regularity of the height function representing $Γ$ , and such regularity is usually derived from surface tension. In our case, surface tension is not taken into account, so that such regularity is unable to be obtained. To overcome such diﬃculty, our idea is to use the Lagrange transformation only near $Γ$ .

Let R be a positive number such that $Ο = ℝ^{N} \ Ω \subset B_{R / 2}$ , where $B_{L} = {x \in ℝ^{N} | | x | < L}$ , and let $κ b e a C^{\infty}$ function such that $κ (ξ) = 1 f o r | ξ | \leq R a n d κ (ξ) = 0 f o r | ξ | \geq 2 R$ . Let $v (ξ, t) a n d q (ξ, t)$ be the velocity ﬁeld and the pressure ﬁeld in Lagrange coordinates ${ξ}$ . Let us consider a transformation,

$x = L (ξ, t) : = ξ + \int_{0}^{t} κ (ξ) v (ξ, s) d s$ (2)

Let $δ$ be a positive number such that the transformation: $x = L (ξ, t)$ is bijective from $Ω$ on to $Ω_{t} = {x = L (ξ, t) | ξ \in Ω}$ for each $t \in (0, T)$ provided that

$\int_{0}^{T} ‖ \nabla (κ (\cdot) v (\cdot, s)) ∥ L_{\infty} (Ω) d s \leq δ$ (3)

Since $δ$ will be chosen as a small positive number eventually, we may assume that $0 < δ \leq 1$ . Let

$l_{i j} (t) = δ_{i j} + \int_{0}^{t} \frac{\partial}{\partial ξ_{j}} (κ (ξ) v_{i} (ξ, s)) d s (v =^{⊤} (v_{1} =, ..., v_{N})), A (t) {(l_{i j} (t))}^{- 1} = (a_{i j} (t))$ (4)

where $δ_{i j}$ are the Kronecker delta symbols, that is $δ_{i i} = 1$ and $δ_{i j} = 0$ for $i \neq j$ . Here and hereafter, a function $a = a (ξ, t)$ is written simply by a(t) and (a_ij) denotes the $N \times N$ matrix whose (i, j) component is a_ij, unless confusion may occur. For a while, we assume that the $N \times N$ matrix $(l_{i j} (t))$ is invertible.
Let

$Ω_{t} = {x = L (ξ, t) | ξ \in Ω}, Γ_{t} = {x = L (ξ, t) | ξ \in Γ},$

$u (x, t) = v (L^{- 1} (x, t), t) a n d p (x, t) = q (L^{- 1} (x, t), t)$ in Eq. (1), and then, v and q satisfy the following equations:

${\begin{cases} \partial_{t} v - D i v (µ D (v) - q I) = f (v), i n Ω \times (0, T), \\ d i v v = g (v) = d i v g (v) i n Ω \times (0, T), \\ (µ D (v) - q I) n = h (v) o n Γ \times (0, T), \\ v |_{t = 0} = u_{0} i n Ω . \end{cases}$ (5)

Here, f (v) is consisting of some linear combinations of nonlinear functions of the forms

$\begin{array}{l} V_{0} (\int_{0}^{t} \nabla (κ v) d s) \nabla^{2} v, V_{1} (\int_{0}^{t} \nabla (κ v) d s) \partial_{t} v, \\ V_{2} (\int_{0}^{t} \nabla (κ v) d s) (\int_{0}^{t} \nabla^{2} (κ v) d s) \nabla v, (1 - κ) {(^{⊤}}^{A) - 1} {v \cdot (^{⊤} A \nabla v)}; \end{array}$ (6)

and g(v) and g(v) and h(v) are nonlinear functions of the forms:

$g (v) = V_{3} (\int_{o}^{t} \nabla (κ v) d s) \nabla v; g (v) = V_{4} (\int_{0}^{t} \nabla (κ v) d s) v; h (v) = V_{5} (\int_{0}^{t} \nabla (κ v) d s) \nabla v$ (7)

with some nonlinear functions V_isuch that V_i(0) = 0 except for i = 2.

The main result of this paper is the following theorem that shows the unique existence theorem of global in time solutions of Eq. (5) and asymptotics as $t \to \infty$ .

Theorem 1

Let $N \geq 3$ and let q₁ and q₂ be exponents such that $\max (N, \frac{2 N}{N - 2}) < q_{2} < \infty a n d 1 / q_{1} = 1 / q_{2} + 1 / N$ . Let b and p be numbers defined by

$b = \frac{3 N}{2 q_{2}} + \frac{1}{2}, p = \frac{2 q_{2} (1 + σ)}{q_{2} - N}$ (8)

with some very small positive number $σ$ . Then, there exists an $\in > 0$ such that if initial data $u_{0} \in B_{q_{2, p}}^{2 (1 - 1 / p)} {(Ω)}^{N} \cap B_{q_{1 / 2}, p}^{2 (1 - 1 / p)} {(Ω)}^{N}$ satisfies the compatibility condition:

${div u}_{0} = 0 in Ω, D (u_{0}) n - < D (u_{0}) n, n > n = 0 o n Γ$ (9)

and the smallness condition: with $I = ∥ u_{0} ∥ B_{q_{2}, p}^{2 (1 - 1 / p)} + ∥ u_{0} ∥ B_{q_{1 / 2}, p}^{2 (1 - 1 / p)}$ , then Eq. (5) admits unique solutions v and q with

$v \in L_{p} ((0, \infty), H_{q_{2}}^{2} (Ω)^{N}) \cap H_{p}^{1} ((0, \infty), L_{q_{2}} (Ω)^{N}), q \in L_{p} ((0, \infty), H_{q_{2}}^{1} (Ω) + {\hat{H}}_{q_{2}, 0}^{1} (Ω))$ ,

possessing the estimate ${[v]}_{\infty} \leq C ϵ$ with

$\begin{array}{l} {[v]}_{T} = {\int_{0}^{T} {(< s >^{b} ∥ v (\cdot, s) ∥ H_{\infty}^{1} (Ω))}^{p} d s \\ + {\int_{0}^{T} (< s >^{(b - \frac{N}{2 q_{1}})} ∥ v (\cdot, s) ∥ H_{q_{_{1}}}^{1} (Ω))^{p} d s + (\underset{0 < s < T}{s u p} < s >^{\frac{N}{2_{q_{_{1}}}}} ∥ v (\cdot, s) ∥ L_{q_{1}} (Ω))}^{p} \\ + {\int_{0}^{T} (< s >^{(b - \frac{N}{2 q_{2}})} ∥ v (\cdot, s) ∥ H_{q_{2}}^{2} (Ω) + ∥ \partial_{t} v (\cdot, s) ∥ L_{q 2} (Ω))}^{p} d s}^{\frac{1}{p}} . \end{array}$

Here, $< s > = {(1 + s^{2})}^{1 / 2}$ and C is a constant that is independent of $\in$ .

Remark 2

Let $p' = p / (p - 1),$ that is $1 / p' = 1 - 1 / p .$ And then,

$\frac{1}{p'} = \frac{(1 + 2 σ) q_{2} + N}{2 q_{2} (1 + σ)}$

We choose $σ > 0$ small enough in such a way that the following relations hold,

$\begin{array}{l} 1 < q_{1} < 2, \frac{N}{q_{1}} > b > \frac{1}{p^{'}}, (\frac{N}{q_{1}} - b) p > 1, (b - \frac{N}{2 q_{2}}) p > 1, b \geq \frac{N}{2 q_{1}}, \\ b \geq \frac{N}{q_{2}}, (\frac{N}{2 q_{2}} + \frac{1}{2}) p^{'} < 1, (b - \frac{N}{2 q_{2}}) p^{'} > 1, \frac{N}{q_{2}} + \frac{2}{p} < 1. \end{array}$ (11)

Remark 3

The exponent q₂ is used to control the nonlinear terms, so that q₂ is chosen as $N < q_{2} < \infty .$

Let

$\frac{1}{q_{1}} = \frac{1}{N} + \frac{1}{q_{2}}, \frac{1}{q_{3}} = \frac{1}{q_{1}} + \frac{1}{q_{2}} .$ (12)

And the condition: $q_{2} > \frac{2 N}{N - 2}$ implies that q₁ > 2 and q₃ > 1 which is necessary to prove Theorem 1.

Thus, we assume that

$\max (N, \frac{2 N}{N - 2}) < q_{2} < \infty$ .

Remark 4

We can choose δ > 0 so small that x = L(ξ, t) is a diffeomorphism with suitable regularity from $Ω$ onto $Ω t$ , so that the original problem (1) is globally well-posed.

Finally, we explain several symbols used in this paper. We use bold small letters to denote N -vectors of functions and bold capital letters to denote N × N matrix of functions. For a scalor function h = h(x), $\nabla h$ is an N vector whose i^th component is $\partial_{i} h = \partial h / \partial x_{i}$ . For v(x) = (v₁(x), . . . , v_N (x)), $\nabla v$ is an N × N matrix of functions whose (i, j) component is $\partial_{i} v_{j}$ . Given exponent $1 < q < \infty$ , let q′ = q/(q − 1).

Let $L_{q} (Ω), H_{q}^{m} (Ω) a n d B_{q, p}^{s} (Ω)$ be the standard Lebesgue, Sobolev, and Besov spaces on $Ω$ , while $∥ • ∥ L_{q} (Ω),$ $∥ • ∥ H_{q}^{m} (Ω),$ $∥ • ∥ B_{q, p}^{s} (Ω)$ denote their norms, respectively. For a Banach space X with norm $∥ • ∥ X$ ,

Let ${(f_{1}, . . ., f_{d}) | f_{i} \in X (i = 1, . . ., d)}$ , while the norm of Xd is written simply by $∥ • ∥ X$ , which is deﬁned by $∥ f ∥ X = \sum_{j = 1}^{d} ∥ f_{j} ∥ X$ for $f = (f_{1}, . . ., f_{d}) \in X_{d}$ . Let ${\hat{H}}_{q_{, 0}}^{1} (Ω) = {θ \in L_{q, l o c} (Ω) | \nabla θ \in L_{q} {(Ω)}^{N}, θ | Γ = 0}$

${\hat{H}}_{q}^{1} (Ω) = {\hat{H}}_{q_{, 0}}^{1} (Ω) + {p = p_{1} + p_{2} | p_{1} \in H_{q}^{1} (Ω), p_{2} \in {\hat{H}}_{q_{, 0}}^{1} (Ω)}$

For $1 \leq p \leq \infty, L_{p} ((a, b), X)$ and $H_{m}^{p} ((a, b), X)$ denote the standard Lebesgue and Sobolev spaces of X-valued functions deﬁned on an interval (a, b), while $∥ • ∥ L_{p} ((a, b), X)$ and $∥ • ∥ H_{m}^{p} ((a, b), X)$ denote their norms, respectively. The letter C denotes generic constants and C_a,b,c,… means that the constant C_a,b,c,… depends on a, b, c... The values of C and C_a,b,c… may change from line to line.

After Introduction (Sect. 1), the paper is organized as follows: In Sect. 2, we reformulate problem (5) by using the formula:

$\int_{0}^{t} \nabla (κ (ξ) v (ξ, s) d s = \int_{0}^{T} \nabla (κ (ξ) v (ξ, s) d s - \int_{0}^{T} \nabla (κ (ξ) v (ξ, s) d s$ .

In Sect. 3, we give estimations of the nonlinear terms. In Sect. 4, we explain how to prolong local in time solutions to the infinite time interval $(0, \infty)$ . Finally, in Sect. 5, we prove Theorem 1 by using maximal L_p-L_q regularity and L_p-L_q decay estimate for solutions of the perturbed Stokes equations with free boundary condition in an exterior domain, which was proved in Shibata.^30,47

Another formulation of Eq. (5)

Let T > 0 and let $v \in H_{p}^{1} ((0, T), L_{q} {(Ω)}^{N}) \cap L_{p} ((0, T), H_{q}^{2} (Ω)), q \in L_{p} ((0, T), H_{q}^{1} (Ω) + {\hat{H}}_{q, 0}^{1} (Ω))$ (13)

be solutions of Eq. (5) satisfying the condition (3). In what follows, we rewrite Eq. (5) in order that the nonlinear terms have suitable decay properties.

Let A(t) = (a_ij(t)) be the N×N matrix given in (4), and let $n_{t} = {}^{⊤}{(n_{t 1}, . . ., n_{t N})} a n d n = {}^{⊤}{(n_{1}, . . ., n_{N})},$ and then by the transformation (2), we have

$a_{ji} (t) \frac{\partial}{\partial ξ_{j}}, \frac{\partial}{\partial x_{j}} = \sum_{j = 1}^{N} n_{ti} = d (t) \sum_{j = 1}^{N} a_{ji} (t) n_{j}$ (14)

where $d (t) = | T A (t) n | .$

$a_{j i} (t)= δ_{i j} + {\tilde{a}}_{i j} (t), J (t) = 1 + \tilde{J} (t), l_{i j} (t) = δ_{i j} + {\tilde{l}}_{i j} (t)$ , (15)

Where $l_{i j}$ are given in (4) and J is the Jacobian of the transformation (2), that is J = $J = d e t (l_{i j}),$ and then

${\tilde{a}}_{i j} (t) = b_{i j} (\int_{0}^{t} \nabla (κ (ξ) v (ξ, s)) d s), \tilde{J} (t) = K (\int_{0}^{t} \nabla (κ (ξ) v (ξ, s)) d s),$

${\tilde{l}}_{i j} (t) = m_{i j} (\int_{0}^{t} \nabla (κ (ξ) v (ξ, s)) d s) : = \int_{0}^{t} \frac{\partial}{\partial ξ_{j}} (κ (ξ) v_{i} (ξ, t)) d s)$ (16)

With some smooth functions b_ijand K deﬁned on {w | |w| ≤ 1} such that b_ij(0) = K(0) = 0, where w is the corresponding variable to $\int_{0}^{t} \nabla (κ (ξ) v (ξ, s)) d s$ Let $u (x, t) = v (ξ, t) a n d p (x, t) = q (ξ, t)$ and then u and p satisfy Eq. (1). By (14),

$\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} = D_{i j, t} (v) : = D_{i j} (v) + {\tilde{D}}_{i j} (t) Δ v$

With

$D_{i j} (v) = \frac{\partial v_{i}}{\partial ξ_{j}} + \frac{\partial v_{j}}{\partial ξ_{j}}, {\tilde{D}}_{i j} (t) Δ v = \sum_{K = 1}^{N} {\tilde{a}}_{k j} (t) \frac{\partial v_{i}}{\partial ξ_{k}} + {\tilde{a}}_{k i} (t) \frac{\partial v_{j}}{\partial ξ_{k}}$ (17)

We also have an important formula:

$d i v u = \sum_{j = 1}^{N} \frac{\partial u_{j}}{\partial x_{j}} = \sum_{j, k = 1}^{N} J (t) a_{k j} (t) \frac{\partial v_{j}}{\partial ξ_{k}} = \sum_{j, k = 1}^{N} \frac{\partial}{\partial ξ_{k}} (J (t) a_{k j} (t) v_{j})$ (18)

which implies that

$\sum_{j, k = 1}^{N} {\tilde{a}}_{k j} (t) = \tilde{J} (t) a_{k j} (t) \frac{\partial v_{j}}{\partial ξ_{k}} = \sum_{j, k = 1}^{N} \frac{\partial}{\partial ξ_{k}} {({\tilde{a}}_{k j} (t) + \tilde{J} (t) a_{k j} (t)) v_{j}}$ (19)

And then, Eq. (5) is written as follows:

${\begin{cases} \sum_{i = 1}^{N} l_{i s} (t) (\partial_{t} v_{i} + (1 - κ) \sum_{j, k = 1}^{N} v_{j} a_{k j} (t) \frac{\partial v_{i}}{\partial ξ_{k}}) \\ - μ \sum_{i, j, k = 1}^{N} l_{i s} (t) a_{k j} (t) \frac{\partial}{\partial ξ_{k}} D_{i, j, t} (v) - \frac{\partial q}{\partial ξ_{s}} = 0 i n Ω \times (0, T), \\ \sum_{j, k = 1}^{N} J (t) a_{k j} (t) \frac{\partial v_{j}}{\partial ξ_{k}} = \sum_{j, k = 1}^{N} \frac{\partial}{\partial ξ_{k}} (J (t) a_{k j} (t) v_{j}) = 0 i n Ω \times (0, T) \\ μ \sum_{i, j, k = 1}^{N} l_{i s} (t) a_{k j} (t) D_{i, j, t} (v) n_{k} - q n_{s} = 0 o n Γ \times (0, T), \\ v |_{t = 0} = u_{0} i n Ω \end{cases}$ (20)

Where s runs from 1 through N. Here, we have used the fact that $(l_{i j}) = A^{- 1}$ which follows from (4).

In order to get some decay properties of the nonlinear terms, we write

$\int_{0}^{t} \nabla (κ (ξ) v (ξ, s) d s = \int_{0}^{T} \nabla (κ (ξ) v (ξ, s) d s - \int_{0}^{T} \nabla (κ (ξ) v (ξ, s) d s$ .

In (16), by the Taylor formula we write

$\begin{array}{l} a_{i j} (t) = a_{i j} (T) + A_{i j} (t), l_{_{i j}} (t) = l_{_{i j}} (T) + L_{_{i j}} (t), \\ D_{i j, t} = D_{i j, T} (v) + D_{i j} (t) Δ v J (t) = J (T) + J (t) \end{array}$ (21)

With

$\begin{array}{l} A_{i j} (t) = \int_{0}^{1} - {b^{'}}_{i j} \int_{0}^{T} \nabla (κ (ξ) v (ξ, s) d s - θ \int_{t}^{T} \nabla (κ (ξ) v (ξ, s) d s) d θ \int_{t}^{T} \nabla (κ (ξ) v (ξ, s) d s, \\ L_{i j} (t) = - \int_{t}^{T} \frac{\partial}{\partial ξ_{j}} κ (ξ) v_{i} (ξ, s) d s, D_{i j} (t) Δ v = \sum_{κ = 1}^{N} (A_{k j} (t) \frac{\partial v_{i}}{\partial_{ξ_{k}}} + A_{k i} \frac{\partial v_{j}}{\partial_{ξ_{k}}}), \\ J (t) = - \int_{0}^{1} K^{'} (\int_{0}^{T} \nabla (κ (ξ) v (ξ, s) d s - θ \int_{t}^{T} \nabla (κ (ξ) v (ξ, s) d s) d θ \int_{t}^{T} \nabla (κ (ξ) v (ξ, s) d s \end{array}$

Where ${b^{'}}_{i j}$ and $K^{'}$ are derivatives of $b_{i j}$ and $K$ with respect to w. By the relation:

$\sum_{s = 1}^{N} l_{i s} (T) a_{s m} (T) = δ_{i m},$ (22)

the first equation in (20) is rewritten as follows:

$\partial_{t} v_{m} - μ \sum_{j, k = 1}^{N} a_{j k} (T) \frac{\partial}{\partial ξ_{k}} (μ D_{m j, T} (v) - δ_{m j} q = f_{m} (v)$

With

$\begin{array}{l} f_{m} (v) = - \sum_{s = 1}^{N} a_{s m} (T) {\sum_{i = 1}^{N} L_{i s} (t) \partial_{t} v_{i} + \sum_{i, j, k = 1}^{N} (1 - κ) l_{i s} (t) a_{k j} (t) v_{i} \frac{\partial v_{i}}{\partial ξ_{k}}} \\ + μ \sum_{s = 1}^{N} a_{s m} (T) {\sum_{i, j, k = 1}^{N} l_{i s} (T) a_{k j} (T) \frac{\partial}{\partial ξ_{k}} (D_{i j} (t) \nabla v) + \sum_{i, j, k = 1}^{N} l_{i s} (T) A_{k j} (t) \frac{\partial}{\partial ξ_{k}} D_{i j, t} (v)} \\ + \sum_{i, j, k = 1}^{N} L_{i s} (t) a_{k j} (t) \frac{\partial}{\partial ξ_{k}} D_{i j, t} (v) \end{array}$ (23)

Next, by (18)

$\tilde{d i v v} = g (v) = d i v g (v)$

With

$\begin{array}{l} \tilde{d i v v} = \sum_{j, k = 1}^{N} J (T) a_{k j} (T) \frac{\partial v_{i}}{\partial ξ_{k}} = \sum_{j, k = 1}^{N} \frac{\partial}{\partial ξ_{k}} J (T) a_{k j} (T) v_{j} \\ g (v) = \sum_{j, k = 1}^{N} (J (T) A_{k j} (t) + J (t) a_{k j} (t)) \frac{\partial v_{j}}{\partial ξ_{k}} \\ g_{k} (v) = \sum_{j = 1}^{N} (J (T) A_{k j} (t) + J (t) a_{k j} (t) v_{j}, g (v) = (g_{1} (v) .... g_{N} (v)) \end{array}$ (24)

Finally, we consider the boundary condition. Let $\tilde{n}$ be an N -vector deﬁned on $ℝ^{N}$ such that $\tilde{n} = n$ on $Γ and ∥ \tilde{n} ∥ H_{\infty}^{2} (ℝ^{N}) \leq C$ . In what follows, $\tilde{n}$ is simply written by $n = {}^{⊤}{(n_{1}, . . ., n_{N})}$ . By (14) and (22)

$\sum_{j, k = 1}^{N} a_{j k} (T) (μ D_{m j, T} (v) - δ_{m j} q) n_{k} = h_{m} (v)$

with

$h_{m} (v) = - μ \sum_{j, k = 1}^{N} (a_{k j} (T) D_{m j} (t) \nabla v + A_{k j} (t) D_{m j, t} (v)) n_{k} - μ \sum_{i, j, k, s = 1}^{N} a_{s m} (T) L_{i s} (t) a_{k j} (t) D_{i j, t} (v) n_{k}$ (25)

By (18),

$μ \sum_{j, k = 1}^{N} a_{k j} (T) \frac{\partial}{\partial ξ_{k}} (μ D_{m j, T} (v) - δ_{m j} q) = J {(T)}^{- 1} \sum_{j, k = 1}^{N} \frac{\partial}{\partial ξ_{k}} {J (T) a_{j k} (T) (D_{m j, T} (v) δ_{m j} q)}$

And

$\sum_{j, k = 1}^{N} a_{k j} (T) D_{m j, T} (v) n_{k} - (\sum_{s = 1}^{N} a_{s m} (T) n_{s}) q = \sum_{k = 1}^{N} [\sum_{j = 1}^{N} {J (T) a_{k j} (T) (D_{m j, T} (v) - δ_{m j} q}] n_{k}$

Thus, letting

$S_{m k} (v, q) = \sum_{j = 1}^{N} J (T) a_{k j} (T) (µ D_{m j, T} (v) - δ_{m j} q), \tilde{S} (v, q) = (S_{i j} (v, q)),$

$f (v) = {}^{⊤}{(f_{1} (v), ..., f_{N} (v))}, h (v) = {}^{⊤}{(h_{1} (v), ..., h_{N} (v))},$

and using (18), we see that v and q satisfy the following equations:

${\begin{cases} \partial_{t} v - J (^{T) - 1} D i v \tilde{S} (v, q) = f (v) i n Ω \times (0, T), \\ \tilde{d i v v} = g (v) = d i v g (v) i n Ω \times (0, T), \\ \tilde{S} (v, q) n = h (v) o n Γ \times (0, T), \\ v |_{t = 0} = u_{0} in Ω . \end{cases}$ (26)

Estimates for the nonlinear terms

Let f (v), g(v), and h(v) are functions deﬁned in Sect. 2. In this section, we estimate these functions. In what follows we write

$\begin{array}{l} ∥ < t >^{α} w ∥ L_{p} ((0, T), X) = {\int_{0}^{T} {(< t >^{α} ∥ w (\cdot, t) ∥ X)}^{p} d t}^{\frac{1}{p}} 1 \leq p < \infty, \\ ∥ < t >^{α} w ∥ L_{\infty} ((0, T), X) = \underset{0 < t < T}{esssup} < t >^{α} ∥ w (\cdot, t) ∥ X p = \infty . \end{array}$

First, we prove that

$∥ < t >^{b} f ∥ L_{p} ((0, T), L_{q 1 / 2} (Ω) + ∥ < t > f ∥ L_{p} ((0, T), L_{q_{2}} (Ω)) \leq C (I + {[v]}_{T}^{2}) .$ (27)

with $I = {‖ u_{0} ‖}_{B_{q 1_{/ 2}, p}^{2 (1 - 1 / p)} (Ω)} + {‖ u_{0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)}$ . Here and in what follows, C denotes generic constants independent of $I, [v] T, δ, a n d T$ . The value of C may change from line to line. Since we choose I small enough eventually, we may assume that $0 < I \leq 1$ . Especially, we use the estimates:

$I^{2} \leq I, I {[v]}_{T} \leq \frac{1}{2} (I^{2} + {[v]}_{T}^{2}) \leq I + {[v]}_{T}^{2}$

Since

$\begin{array}{l} \int_{α}^{β} ∥ \nabla (κ v (\cdot, s)) ∥ L_{\infty} (Ω) \leq C {(1 + α)}^{- b + \frac{1}{p^{'}}} {(\int_{α}^{β} {(< s >^{b} ∥ v (\cdot, s) ∥ H_{\infty}^{1} (Ω))}^{p} d s)}^{\frac{1}{p}} \\ \int_{α}^{β} ∥ \nabla^{2} (κ v (\cdot, s)) ∥ L_{q} (Ω) \leq C {(1 + α)}^{- b + \frac{N}{2 q_{2}} + \frac{1}{p^{'}}} {(\int_{α}^{β} {(< s >^{b - \frac{N}{2 q_{2}}} ∥ v (\cdot, s) ∥ H_{q_{2}}^{2} (Ω))}^{p} d s)}^{\frac{1}{p}} \end{array}$

for any $0 \leq α < β \leq T$ , where $q \in [1, q_{2}]$ , we have

$\begin{array}{l} \int_{α}^{β} ∥ \nabla (κ v (\cdot, s)) ∥ L_{\infty} (Ω) d s \leq C {[v]}_{T} {(1 + α)}^{- b + \frac{1}{p^{'}}} \\ \int_{α}^{β} ∥ \nabla^{2} (κ v (\cdot, s)) ∥ L_{q} (Ω) \leq C {[v]}_{T} \end{array}$ (28)

for any $0 \leq α < β \leq T$ , where $q \in [1, q]$ , because $^{b > \frac{N}{2 q_{2}} + \frac{1}{p^{'}}}$ as follows from (11). By real interpolation theorem, we have

$\begin{array}{l} \underset{t \in (0, T)}{s u p} < t >^{b - \frac{N}{2 q_{2}}} {‖ v (\cdot, t) ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq C {‖ u_{0} ‖}_{_{_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)}}} \\ + ‖ < t >^{b - \frac{N}{2 q_{2}}} {v ‖}_{L_{p} ((0, T), H_{q_{2}}^{2} (Ω))} + ‖ < t >^{b - \frac{N}{2 q_{2}}} \partial t v ‖_{_{L_{p} ((0, T) L_{q_{2}} (Ω))}} \end{array}$ (29)

To prove (29), we introduce an operator $T (t)$ acting of $g \in B_{q_{2}, p}^{2 (1 - 1 / p)} ℝ^{N}$ defined by

$T (t) g = F^{- 1} [e^{- (| ξ | + 1) t} F [g] (ξ)],$ (30)

where $F a n d F^{- 1}$ denote the Fourier transform in $ℝ^{N}$ and its inverse transform. We have

$e^{γ t} T (t) g ∥_{L p ((0, \infty), H_{q}^{2} (ℝ^{N}))} + e^{γ t} \partial t T (t) g ∥_{L p ((0, \infty), L_{q (ℝ^{N}))}} \leq C | | g | |_{_{_{_{B_{q_{2}, p}^{2 (1 - 1 / p)}} (ℝ^{N})}}}$ (31)

Given f(t) defned on (0, T) with $f |_{t = 0} = 0$ , let

$[e_{T} f] (t) = {\begin{cases} 0 (t < 0), \\ f (t) (0 < t < T), \\ f (2 T - t) (T < t < 2 T), \\ 0 (2 T < t), \end{cases}$ (32)

and then $[e_{T} f] (t) = f (t)$ for $t \in (0, T)$ and

$\partial t [e_{T} f] (t) = {\begin{cases} 0 (t < 0), \\ \partial t f (t) (0 < t < T), \\ - (\partial t f) (2 T - t) (T < t < 2 T), \\ 0 (2 T < t), \end{cases}$ (33)

Let ${\tilde{u}}_{0}$ be an N-vector of functions in $B_{q_{2}, p}^{2 (1 - 1 / p)} {(ℝ^{N})}^{N}$ such that ${\tilde{u}}_{0} = u_{0}$ in $Ω$ and ${‖ {\tilde{u}}_{0} ‖}_{B_{q 1_{/ 2}, p}^{2 (1 - 1 / p)} ℝ^{N}} < C {‖ u_{0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)}$ .

Let $z = e_{T} [< t >^{b - \frac{N}{2 q_{2}}} v - T (| t |) {\tilde{u}}_{0}] + T (| t |) {\tilde{u}}_{0} f o r t \in ℝ$ . Since $< t >^{b - \frac{N}{2 q_{2}}} v - T (| t |) {\tilde{u}}_{0} |_{t = 0} = 0 i n Ω$ by (31), (32) and (33),

$\begin{array}{l} ∥ z ∥_{L_{p} ((0, \infty), H_{q_{2}}^{2} (Ω))} + ∥ \partial_{t} z ∥_{L_{p} ((0, \infty), L_{q_{2}} (Ω))} \\ < C (| | < t >^{b - \frac{N}{2 q_{2}}} v ‖_{L_{p} ((0, T), H_{q_{2}}^{2} (Ω))} + < t >^{b - \frac{N}{2 q_{2}}} \partial_{t} v ‖_{L_{p} ((0, T) L_{q_{2}} (Ω))} + {‖ u_{0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \end{array}$ (34)

It is known (Tanabe⁵¹) that $L_{p} ((0, \infty), E_{1}) \cap H_{p}^{1} ((0, \infty), E_{0})$ is continuously imbedded into $B U C ((0, \infty), {(E_{0}, E_{1})}_{1 - 1 / p, p})$ , where E₀ and E₁ are two Banach spaces such that E₁ is a dense subset of E₀, and BUC denotes the set of all uniformly bounded continuous functions . Noting that
$z (t) = < t >^{b - \frac{N}{2 q_{2}}} v (t) f o r t \in (0, T)$ , we have

$\underset{0 < t < T)}{s u p} < t >^{b - \frac{N}{2 q_{2}}} {‖ v (t) ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq \underset{t \in (0, \infty)}{s u p} {‖ z (t) ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq C {‖ z ‖}_{L p ((0, \infty), H_{q_{2}}^{2} (Ω))} + {‖ \partial_{t} z ‖}_{L p ((0, \infty), L_{q_{2}} (Ω))}$

which, combined with (34), furnishes (29).

Since $\frac{2}{p} + \frac{N}{q_{2}} < 1, B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)$ is continuously imbedded into $H_{\infty}^{1} (Ω)$ so that by (29)

$< t >^{b - \frac{N}{2 q_{2}}} {‖ v (t) ‖}_{_{L \infty ((0, T), H_{\infty}^{1} (Ω))}} \leq C (I + {[v]}_{T})$ (35)

Applying (3), (28) and (29) to the formulas in (15) and (16) and using the fact that $- b + \frac{1}{p^{'}} \frac{N}{2 q_{2}} \leq - \frac{N}{2 q_{2}}$ and $- b + \frac{N}{2 q_{2}} \leq - \frac{N}{2 q_{2}}$ ,which follows from (11), give

${‖ (a_{i j} (t), J (t), l_{_{i j}} (t), A_{i j} (t), J (t), L_{i j} (t) ‖}_{L_{\infty} (Ω)} \leq C$ ,

${‖ (A_{i j} (t), J (t), L_{i j} (t) ‖}_{L_{\infty} (Ω)} \leq C \int_{t}^{T} \nabla (κ v (\cdot, s)) ∥_{L_{\infty}} (Ω) d s \leq C {[v]}_{T} + < t >^{- b + \frac{1}{p^{'}}} \leq C {[v]}_{T} < t >^{- \frac{N}{2 q_{2}}},$

${‖ \nabla (a_{i j} (t), J (t), l_{_{i j}} (t), A_{i j} (t), J (t), L_{i j} (t) ‖}_{L_{q} (Ω)} \leq C \int_{0}^{T} \nabla^{2} (κ v (\cdot, s)) ∥_{L_{q}} \leq C {[v]}_{T},$ (36)

$\begin{array}{l} {‖ \partial_{t} (a_{i j} (t), J (t), l_{_{i j}} (t), A_{i j} (t), J (t), L_{i j} (t) ‖}_{L_{\infty} (Ω)} \\ \leq C ‖ \nabla (κ v (\cdot, t)) ‖,_{L_{\infty} (Ω)} \leq C (I + [_{v] T}) < t >^{- b + \frac{N}{2 q_{2}}} \leq C (I + [_{v] T}) < t >^{\frac{N}{2 q_{2}}} \end{array}$

for any $t \in (0, T]$ , where $q \in [1, q_{2}]$ . Moreover, we have

$({\tilde{a}}_{i j}, \tilde{J}, {\tilde{l}}_{_{i j}}, {\tilde{A}}_{i j}, J, L_{i j}) (x, t) = 0 f o r x \notin B_{2 R} a n d t \in [0, T]$ (37)

By (36) and (37),

${‖ (a_{s m} (T) L_{i s} (t) \partial_{t} v_{i}) ‖}_{L_{q} (Ω)} C {[v]}_{T} + < t >^{- b + \frac{1}{p^{'}}} {‖ \partial_{t} v_{i} (t) ‖}_{L_{q_{2}} (Ω)}$

for any $q \in [1, q_{2}]$ . Since $\frac{1}{p^{'}} < b - \frac{N}{2 q_{2}}$ as follows from (11), we have

${‖ < t >^{b} a_{s m} (T) L_{i s} (t) \partial_{t} v_{i}) ‖}_{L_{p} ((0, T), L_{q} (Ω))} \leq C (I + {[v]}_{T}^{2})$

for any $q \in [1, q_{2}]$

Next, by Holder’s inequality,

$< t >^{b} {‖ v \cdot \nabla v (\cdot, t) ‖}_{L_{q_{1}} / 2} (Ω) \leq < t >^{\frac{N}{2 q_{1}}} {‖ \nabla v (\cdot, t) ‖}_{L_{q_{1}} (Ω)}$

so that by (36), we have

${‖ < t >^{b} a_{s m} (T) l_{i s} a_{k j} v_{j} \frac{\partial v_{i}}{\partial ξ_{k}} ‖}_{L_{p} ((0, T), L_{q_{1}} / 2 (Ω))} \leq C {[v]}_{T}^{2} .$

Since

$< t >^{b} {‖ v \cdot \nabla v (\cdot, t) ‖}_{L_{q_{2}} (Ω)} \leq < t >^{\frac{N}{2 q_{2}}} {‖ \nabla v (\cdot, t) ‖}_{_{L_{\infty} (Ω)}} < t >^{b - \frac{N}{2 q_{2}}} {‖ \nabla v (\cdot, t) ‖}_{L_{q_{2}} (Ω)},$

by (36)

${‖ < t >^{b} a_{s m} (T) l_{i s} a_{k j} v_{j} \frac{\partial v_{i}}{\partial ξ_{k}} ‖}_{L_{p} ((0, T), L_{q_{2}} (Ω))} \leq C (I + {[v]}_{T}) [v] T \leq C (I + {[v]}_{T}^{2})$

Since

$\frac{\partial}{\partial ξ_{k}} (D_{i j} (t) \nabla v) = \sum_{m = 1}^{N} (A_{m j} (t) \frac{\partial^{2} v_{i}}{\partial ξ_{k} \partial ξ_{m}} + A_{m i} (t) \frac{\partial^{2} v_{j}}{\partial ξ_{k} \partial ξ_{m}}) + \sum_{m = 1}^{N} ((\frac{\partial}{\partial ξ_{m}} A_{m j} (t) \frac{\partial v_{i}}{\partial ξ_{m}} + \frac{\partial}{\partial ξ_{k}} A_{m i} (t)) \frac{\partial v_{j}}{\partial ξ_{m}})$

by (36)

$< t >^{b} {‖ \frac{\partial}{\partial ξ_{k}} (D_{i j} (.) \nabla v) ‖}_{L_{q} (Ω)} \leq C {[v]}_{T} {< t >^{b - \frac{N}{2 q_{2}}} {‖ \nabla^{2} v (. t) ‖}_{L_{q} (Ω)} + < t >^{b} {‖ \nabla v (\cdot, t) ‖}_{_{L_{\infty} (Ω)}}}$ (38)

for any $q \in [1, q_{2}]$ and therefore

${‖ < t >^{b} a_{s m} (T) l_{i s} (T) a_{k j} (T) \frac{\partial}{\partial ξ_{k}} (D_{i j} (.) \nabla v) ‖}_{L_{p} ((0, T), L_{q} (Ω))} \leq C {[v]}_{T}^{2}$

for any $q \in [1, q_{2}]$ . Since

$\frac{\partial}{\partial ξ_{k}} (D_{i j, T} (v)) = \sum_{m = 1}^{N} (a_{m j} (T) \frac{\partial^{2} v_{i}}{\partial ξ_{k} \partial ξ_{m}} + a_{m i} (T) \frac{\partial^{2} v_{j}}{\partial ξ_{k} \partial ξ_{m}}) + \sum_{m = 1}^{N} ((\frac{\partial}{\partial ξ_{m}} a_{m j} (T) \frac{\partial v_{i}}{\partial ξ_{m}} + \frac{\partial}{\partial ξ_{k}} a_{m i} (T)) \frac{\partial v_{j}}{\partial ξ_{m}}),$

by (36)

$< t >^{b} {‖ a_{s m} (T) l_{i s} (T) A_{k j} (t) \frac{\partial}{\partial ξ_{k}} (D_{i j, T} (v) ‖}_{L_{q} (Ω)} \leq C {[v]}_{T} {< t >^{b - \frac{N}{2 q_{2}}} {‖ \nabla^{2} v (. t) ‖}_{L_{q} (Ω)} + < t >^{b} {‖ \nabla v (\cdot, t) ‖}_{_{L_{\infty} (Ω)}}}$ ,

so that

$< t >^{b} {‖ a_{s m} (T) l_{i s} (T) A_{k j} (T) \frac{\partial}{\partial ξ_{k}} (D_{i j, T} (v) ‖}_{L_{p} ((0, T), L_{q} (Ω))} \leq C {[v]}_{T}^{2}$

for any $q \in [1, q_{2}]$ . Analogously, we have

${‖ < t >^{b} a_{s m} (T) L_{i s} a_{k j} \frac{\partial}{\partial ξ_{k}} (D_{i j, T} (v) ‖}_{L_{p} ((0, T), L_{q} (Ω))} \leq C {[v]}_{T}^{2}$

for any $q \in [1, q_{2}]$ . Summing up, we have obtained (27).

Next, we consider 𝔤 and g. To estimate the $H_{p}^{\frac{1}{2}}$ norm, we use the following lemma.

Lemma 5.

Let $f \in H_{\infty}^{1} (ℝ, L_{\infty} (Ω))$ and $g \in H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))$ . Assume that $f (x, t) = 0 f o r (x, t) \notin B_{R} \times ℝ$ .

Then,

${‖ f g ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))} \leq C {‖ f ‖}_{H_{\infty}^{1} (ℝ, L_{\infty} (Ω))} {‖ g ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q_{2}} (Ω))}$ (39)

Proof: To prove the lemma, we use the fact that

$H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω)) = {(L_{p} (ℝ, L_{q} (Ω)), H_{p}^{1} (ℝ, L_{q} (Ω)))}_{[\frac{1}{2}]},$ (40)

where ${(\cdot, \cdot)}_{[σ]}$ denotes a complex interpolation functor. Let $q \in [1, q_{2}]$ . Noting that $f (x, t) = 0$ for $(x, t) \notin B_{R} \times ℝ$ , we have

${‖ \partial_{t} (f g) ‖}_{L_{q} (Ω)} \leq {‖ \partial_{t} f ‖}_{L_{\infty} (Ω)} {‖ g ‖}_{L_{q_{2}} (Ω)} + {‖ f ‖}_{L_{\infty} (Ω)} + {‖ \partial_{t} g ‖}_{L_{q_{2}} (Ω)},$

and therefore

${‖ \partial_{t} (f g) ‖}_{L_{p} (ℝ, L_{q} (Ω))} \leq C {‖ f ‖}_{H_{\infty}^{1} (ℝ, L_{\infty} (Ω))} {‖ g ‖}_{H_{p}^{1} (ℝ, L_{q_{2}} (Ω))}$ .

for any $q \in [1, q_{2}]$ . Moreover, we easily see that

${‖ (f g) ‖}_{L_{p} (ℝ, L_{q} (Ω))} \leq C {‖ f ‖}_{L_{\infty} (ℝ, L_{\infty} (Ω))} {‖ g ‖}_{L_{p} (ℝ, L_{q_{2}} (Ω))} .$

Thus, by (40), we have (39), which completes the proof of Lemma 5.

To use the maximal L_p-L_q estimate, we have to extend 𝔤, g and h to R. For this purpose, we introduce an extension operator ${\tilde{e}}_{T}$ . Let f be a function deﬁned on (0, T ) such that $f |_{t = T} = 0$ , and then ${\tilde{e}}_{T}$ is an operator acting on f deﬁned by

$[{\tilde{e}}_{T} f] (t) = {\begin{cases} 0 (t > T), \\ f (t) (0 < t < T), \\ f (- t) (- T < t < 0), \\ 0 (t < - T) . \end{cases}$ (41)

Lemma 6

Let $1 < p < \infty, 1 \leq q \leq q_{2}$ and $0 \leq a \leq b$ . Let $f \in H_{\infty}^{1} (0, T, L_{\infty} (Ω))$ and $g \in H_{p}^{1} ((0, T), L_{q_{2}} (Ω)) \cap L_{p} ((0, T) H_{q}^{2} (Ω))$ . Assume $f |_{t = T} = 0$ and $f = 0 f o r (x, t) \notin B_{R} \times ℝ$ . Let $< t > = {(1 + t^{2})}^{1 / 2}$ . Then we have

$\begin{array}{l} {‖ {\tilde{e}}_{T} (< t >^{a} f \nabla g) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))} \leq C < t >^{\frac{N}{2 q_{2}}} f ‖_{H_{\infty}^{1} ((0, \infty), L_{\infty} (Ω))} \\ \times (‖ < {g ‖}_{L_{p} ((0, T) H_{q}^{2} (Ω))} + ‖ < t >^{b - \frac{N}{2 q_{2}}} {\partial_{t} g ‖}_{L_{p} ((0, T), L_{q_{_{2}}} (Ω))} + {‖ {g |}_{t = 0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)}) \end{array}$ (42)

Proof: Let $f_{0} (t) = < t >^{a - b + \frac{N}{2 q_{2}}} f (t)$ and $g_{0} (t) = < t >^{b + \frac{N}{2 q_{2}}} g (t)$ , and then $< t >^{a} f \nabla g = f_{0} \nabla g_{0}$ . Let $T (t)$ be the operator given in (30) and let h be a function in $B_{q_{2}, p}^{2 (1 - 1 / p)} (ℝ^{N})$ such that $h = {g |}_{t = 0} i n Ω$ and $h = {g |}_{t = 0} {‖ h ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq {‖ {g |}_{t = 0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)}$ . Recall the operator eT defined in (32) and note that ${g_{0} |}_{t = 0} = {g |}_{t = 0} = T (t) h |_{t = 0 i n Ω}$ . Let $\tilde{g} (t) = e_{T} [g_{0} - T (\cdot) h] (t) + T (t) h$

for t > 0 and let

$[ι g] (t) = {\begin{cases} \tilde{g} (t) (t > 0), \\ \tilde{g} (- t) (t < 0), \end{cases} [ι f] (t) = {\begin{cases} 0 (t > T), \\ f_{0} (t) (0 < t < T), \\ f_{0} (- t) (- T < t < 0), \\ 0 (t < - T) . \end{cases}$

Since $\tilde{g} (t) = g_{0} (t) f o r 0 < t < T$ , we have

$\begin{array}{l} {\tilde{e}}_{T} [< t >^{a} f \nabla g] (t) = {\begin{cases} 0 (t > T), \\ f_{0} (t) \nabla g_{0} (t) (0 < t < T), \\ f_{0} (- t) \nabla g_{0} (- t) (- T < t < 0), \\ 0 (t < - T) . \end{cases} = {\begin{cases} 0 (t > T), \\ f_{0} (t) \nabla \tilde{g} (t) (0 < t < T), \\ f_{0} (- t) \nabla \tilde{g} (- t) (- T < t < 0), \\ 0 (t < - T) . \end{cases} \\ = [ι f] (t) \nabla [ι g] (t) . \end{array}$

By Lemma 5,

${‖ {\tilde{e}}_{T} [< t >^{a} f \nabla g] ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))} = {‖ [ι f] \nabla [ι g] ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))} \leq C {‖ [ι f] ‖}_{H_{\infty}^{1} (ℝ, L_{q} (Ω))} {‖ \nabla (ι g) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))}$

Since, $f_{0} |_{t = T} = 0$ we have

${‖ [ι f] ‖}_{H_{\infty}^{1} (ℝ, L_{q} (Ω))} = 2 {‖ [f_{0}] ‖}_{H_{\infty}^{1} ((0, T), L_{\infty} (Ω))} \leq ‖ < t >^{\frac{N}{2 q_{2}}} {f ‖}_{_{H_{\infty}^{1} ((0, T), L_{\infty} (Ω))}},$

Because $a - b \leq 0$ .

To estimate ${‖ \nabla (ι g) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))}$ , we use the fact that $H_{p}^{1} (ℝ, L_{q_{2}} (Ω)) \cap L_{p} (ℝ, H_{q_{2}}^{2} (Ω))$ is continuously imbedded into $H_{p}^{\frac{1}{2}} (ℝ, H_{q_{_{2}}}^{1} (Ω))$ which was proved by Meyries and Schnaubelt⁵² in case of p = q2 and by Shibata³⁰ for any $1 < p, q_{2} < \infty$ . Using this fact and (31), we have

${‖ \nabla (ι g) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q_{2}} (Ω))} \leq C {‖ (ι g) ‖}_{H_{p}^{1} (ℝ, L_{q_{2}} (Ω))} + {‖ (ι g) ‖}_{L_{p} (ℝ, H_{q_{2}}^{2} (Ω))}$

$\leq C ({‖ \tilde{g} ‖}_{H_{p}^{1} ((0, \infty), L_{q_{2}} (Ω))} = {‖ \tilde{g} ‖}_{L_{p} ((0, \infty), H_{q_{2}}^{2} (Ω))})$

$\begin{array}{l} \leq C ({‖ g_{0} - T (\cdot) h ‖}_{H_{p}^{1} ((0, T), L_{q_{2}} (Ω))} + {‖ g_{0} - T (\cdot) h ‖}_{L_{p} ((0, T), H_{q_{2}}^{2} (Ω))} \\ + {‖ T (\cdot) h ‖}_{H_{p}^{1} ((0, T), L_{q_{2}} (Ω))} + {‖ T (\cdot) h ‖}_{L_{p} ((0, \infty), H_{q_{2}}^{2} (Ω))}) \end{array}$

$\leq C (‖ < t >^{b - \frac{N}{2 q_{2}}} {\partial_{t} g ‖}_{L_{p} ((0, T), L_{q_{_{2}}} (Ω))}) + ‖ < t >^{b - \frac{N}{2 q_{2}}} {\partial_{t} g ‖}_{L_{p} ((0, T) H_{q}^{2} (Ω))} + {‖ {g |}_{t = 0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)}$

This completes the proof of Lemma 6.

Recall the definitions of g(v) and h_m(v) given in (24) and (25). By Lemma 6 and (36)

${‖ {\tilde{e}}_{T} (< t >^{a} g (v) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))}$

$\begin{array}{l} \leq \sum_{j, k = 1}^{N} ‖ < t >^{\frac{N}{2 q_{2}}} (J (T) A_{k j} (\cdot) + T (\cdot) a_{k j} (\cdot)) ∥ H_{\infty}^{1} ((0, T), L_{\infty} (Ω)) \\ \times (‖ < t >^{b - \frac{N}{2 q_{2}}} {v ‖}_{L_{p} ((0, T) H_{q_{2}}^{2} (Ω))} + ‖ < t >^{b - \frac{N}{2 q_{2}}} {\partial_{t} v ‖}_{L_{p} ((0, T), L_{q_{_{2}}} (Ω))} + {‖ u_{0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \\ \leq C (I + [_{v}^{]}) \end{array}$ (43)

for any $a \in [0, b]$ and $q \in [1, q_{2}]$ . Analogously, we have

${‖ {\tilde{e}}_{T} (< t >^{a} h (v) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))} \leq C (I + {[v]}_{T}^{2})$ (44)

for any $a \in [0, b]$ and $q \in [1, q_{2}]$ . Analogously, we have

Next, by (36), (37) and (41),

$\begin{array}{l} {‖ {\tilde{e}}_{T} [< t >^{a} g (v)] ‖}_{L_{p} (ℝ, H_{q}^{1} (Ω))} \\ \leq \sum_{j, k = 1}^{N} ‖ < t >^{\frac{N}{2 q_{2}}} (J (T) A_{k j} (\cdot) + J (\cdot) a_{k j} (\cdot)) ‖_{L_{\infty} ((0, T), L_{\infty} (Ω))} ‖ < t >^{b - \frac{N}{2 q_{2}}} {v ‖}_{L_{p} ((0, T) H_{q_{2}}^{2} (Ω))} \\ \leq C [_{v}^{]} \end{array}$ (45)

for any $a \in [0, b]$ and $q \in [1, q_{2}]$ . Analogously, we have

${‖ {\tilde{e}}_{T} (< t >^{b} h(v) ‖}_{L_{p} ((0, T) H_{q}^{1} (Ω))} \leq C {[v]}_{T}^{2}$ (46)

for any $a \in [0, b]$ and $q \in [1, q_{2}]$ . Since

$\partial_{t} g_{k} (v) = \sum_{j = 1}^{N} ‖ (J (T) \partial_{t} A_{k j} (t) + (\partial_{t} J (t) a_{k j} (t) + J (t) \partial_{t} a_{k j} (t)) v_{j} + \sum_{j = 1}^{N} ‖ (J (T) A_{k j} (t) + J (t) a_{k j} (t)) v_{j}$

and since ${‖ J (T) a_{k j} (t) J (t) ‖}_{L_{\infty} (Ω)} \leq C$ as follows from (36), by (37) we have

$\begin{array}{l} {‖ {\tilde{e}}_{T} [< t >^{a} \partial_{t} g_{k} (v)] ‖}_{L_{p} ((0, T), L_{q} (Ω))} \\ \leq \sum_{j = 1}^{N} (< t >^{\frac{N}{2 q_{2}}} \partial_{t} (A_{k j}, J, a_{k j}) ‖_{L_{\infty} ((0, T), L_{\infty} (Ω))} ‖ < t >^{b - \frac{N}{2 q_{2}}} {v ‖}_{L_{p} ((0, T) L_{q_{2}} (Ω)} \\ + (< t >^{\frac{N}{2 q_{2}}} (A_{k j}, J) ‖_{L_{\infty} ((0, T), L_{\infty} (Ω))} ‖ < t >^{b - \frac{N}{2 q_{2}}} {\partial_{t} v ‖}_{L_{p} ((0, T) L_{q_{2}} (Ω)}, \end{array}$

which, combined with (36), furnishes that

${‖ {\tilde{e}}_{T} [< t >^{a} g (v)] ‖}_{L_{p} (ℝ, L_{q} (Ω))} \leq C (I + {[v]}_{T}^{2})$ (47)

for any $a \in [0, b]$ and $q \in [1, q_{2}]$ .

Prolongation of local in time solutions

Before proving Theorem 1, we state a unique existence theorem of locally in time solutions to Eq. (5), which can be proved by a standard argumentation based on maximal L_p-L_q regularity theorem for the Stokes equations with free boundary condition.^29,40

Theorem 7

Let $N < q_{2} < \infty$ and $2 < p < \infty$ . Assume that 2/p + N/q₂ < 1. Then, given T > 0, there exists an $\in_{0} > 0$ depending on T such that if initial data $u_{0} \in B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)$ satisfies the condition:

${‖ u_{0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq \in_{0}$ (48)

and the compatibility condition (9), then Eq. (5) admits unique solutions v and q with

$v \in H_{p}^{1} ((0, T), L_{q_{2}} {(Ω)}^{N}) \cap L_{p} ((0, T), H_{q_{2}}^{2} {(Ω)}^{N}), q \in L_{p} ((0, T), H_{q_{2}}^{1} (Ω) + {\hat{H}}_{q_{2}, 0}^{1})$

Let T be a positive number > 2 and let v and q be solutions of Eq. (5) satisfying (13) and (3). In
view of Theorem 7, such solutions v and q exist uniquely provided that

${‖ u_{0} ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq \in_{0}$ (49)

Thus, we assume that $0 < \in \leq \in_{0}$ in Theorem 1. Let [v]_T and $I$ be the quantities defined in Theorem 1 in Sect. 1 Introduction. And then, if we prove that there exists a constant M > 0 independent of ϵ and T such that

${[v]}_{T} \leq M (I + {[v]}_{T}^{2})$ (50)

then we can prolong v and q beyond T. Namely, there exist v₁ and q₁ with

$v_{1} \in H_{p}^{1} ((T, T + 1), L_{q_{2}} {(Ω)}^{N}) \cap L_{p} ((T, T + 1) H_{q_{2}}^{2} {(Ω)}^{N}), q_{1} \in L_{p} ((T, T + 1), H_{q_{2}}^{1} (Ω) {\hat{H}}_{q_{2}, 0}^{1} (Ω))$

such that v₁ and q₁ are solutions to the equations:

${\begin{cases} \partial_{t} v_{1} - Div (µ D (v_{1}) - q I) = f (v_{1}), in Ω \times (T, T + 1), \\ div v_{1} = g (v_{1}) = div g (v_{1}) in Ω \times (T, T + 1), \\ (µ D (v_{1}) - q I) n = h (v_{1}) on Γ \times (0, T), \\ {v_{1} |}_{t = T} = v(\cdot, T) in Ω . \end{cases}$ (51)

Here, f (v₁) is consisting of some linear combinations of nonlinear terms of the forms

$V_{0} (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s)) \nabla^{2} v_{1}, V_{1} (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s) \nabla^{2} v_{1})) \partial_{t} v_{1}$

$\begin{array}{l} V_{1} (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s)) (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s) \nabla^{2} v_{1})) \nabla v_{1}, \\ (1 - κ) ({}^{⊤}A) - 1 {v_{1} ({}^{⊤}A \nabla v_{1})}; \end{array}$

and $g (v_{1})$ , g(v₁) and h(v₁) have the following forms:

$\begin{array}{l} g (v_{1}) = V_{3} (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s)) \nabla v_{1}; g (v_{1}) = V_{4} (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s)) v_{1}; \\ h (v_{1}) = V_{5} (\int_{0}^{T} (\nabla (κ v) d s + \int_{T}^{t} (\nabla (κ v_{1}) d s)) \nabla v_{1}, \end{array}$

where V_i are the same nonlinear functions as in (6) and (7).

In fact, the inequality (50) yields that there exists a small constant $ε \in (0, ε_{0}]$ such that if $I \leq ε$ , then

${[v]}_{T} \leq {(2 M)}^{- 1} - \sqrt{{(2 M)}^{- 2} - I} = 2 M I + O (I^{2}) .$

Thus, we may assume that

${[v]}_{T} \leq 3 M I$ (52)

By (29) and (52) we have

${‖ v (\cdot, T) ∥ ‖}_{B_{q_{2}, p}^{2 (1 - 1 / p)} (Ω)} \leq M_{1} ε$

with some positive constant M₁ independent of T . Thus, noting that

$\int_{0}^{T} ∥ \nabla v (\cdot, t) ∥ L_{\infty} (Ω) d t \leq C {(\int_{0}^{T} < t >^{- b p^{'}} d s)}^{1 / p^{'}} {[v]}_{T} \leq M_{2} ϵ$

with some positive constant M₂ independent of T, choosing $ϵ > 0$ smaller if necessary, we can show the existence of v₁ and q₁. Thus, setting

$v_{2} (, t) = {\begin{cases} v (, t) 0 < t < T, \\ v_{1} (\cdot, t) T < t < T + 1, \end{cases} q_{2} = {\begin{cases} q (, t) 0 < t < T, \\ q_{2} (\cdot, t) T < t < T + 1, \end{cases}$

we see that v₂ and q₂ satisfy the regularity condition:

$v_{2} \in H_{p}^{1} ((0, T + 1), L_{q_{2}} {(Ω)}^{N}) \cap L_{p} ((0, T + 1), H_{q_{2}}^{2} (Ω)), q_{2} \in L_{p} ((0, T + 1), H_{q_{2}}^{1} (Ω) + {\hat{H}}_{q_{2, 0}}^{1} (Ω))$

and Eq. (5) replacing T by T + 1. Repeating this argument, we can prolong v to time interval $(0, \infty)$ .

This completes the proof of Theorem 1. Therefore, we prove (50).

A Proof of Theorem 1

Let v and q be the same N-vector of functions and the function as in Sect. 4. We prove that v satisfies (50). And, we recall that T > 2. As was seen in Sect. 2, v and q satisfy Eq. (26). To estimate v, we write v by, $v = w + u$ where w is a solution to the equations:

${\begin{cases} \partial_{t} w + λ_{0} w - J (^{T) - 1} Div \tilde{s} (w, r) = f (v) in Ω \times (0, T), \\ div w = {\tilde{e}}_{T} [g(v)]=div {\tilde{e}}_{T} [g(v)] in Ω \times (0, T), \\ \tilde{s} (w, r) = {\tilde{e}}_{T} [h] on Γ \times (0, T), \\ w |_{t = 0} = u_{0} in Ω \end{cases}$ (53)

with some pressure term r, and u is a solution to the equations:

${\begin{cases} \partial_{t} u - J (^{T) - 1} Div \tilde{S} (u, p) = - λ_{0} w in Ω \times (0, T), \\ div u = 0 in Ω \times (0, T), \\ \tilde{S} (u, p) = 0 on Γ \times (0, T), \\ u |_{t = 0} = 0 in Ω \end{cases}$ (54)

To estimate w, we quote the maximal L_p-L_q regularity theorem due to Shibata.³³ Let us consider the equations:

${\begin{cases} \partial_{t} w + λ_{0} w - J (^{T) - 1} D i v \tilde{S} (w, r) = f in Ω \times (0, T), \\ \tilde{div} w = g = d i v g in Ω \times (0, T), \\ \tilde{S} (w, r) = h on Γ \times (0, T), \\ w |_{t = 0} = w_{0} in Ω \end{cases}$ (55)

And then, we have

Theorem 8

Let $Ω$ be an exterior domain in $ℝ^{N}$ whose boundary $Γ$ is a C₂ hyper surface. Let $1 < p, q < \infty$ and T > 0. Assume that

$\begin{array}{l} w_{0} \in B_{q, p}^{2 (1 - 1 / p)} (Ω), f \in L_{p} ((0, T), L_{q} (^{Ω) N}), g \in H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω)) \cap L_{p} (ℝ, H_{q}^{1} (Ω)), \\ g \in H_{p}^{1} (ℝ, L_{q} (^{Ω) N}), h \in H_{p}^{\frac{1}{2}} (ℝ, L_{q} (^{Ω) N}) \cap L_{p} (ℝ, H_{q}^{1} (^{Ω) N}) \end{array}$

and that w₀ satisfies the compatibility condition:

$w_{0} - g |_{t = 0} \in B_{q, p}^{2 (1 - 1 / p)} (Ω)$

and in addition

${(µ D (w_{0}) - h |_{t = 0})}_{τ} = 0 o n Γ$

If $2 / p + 1 / q < 1$ , where $d_{τ} = d - < d, n > n$ . Then, there exists a positive number $λ_{0}$ such that Eq. (55) admits unique solutions w and r with

$w \in L_{p} ((0, T), H_{q}^{2} {(Ω)}^{N}) \cap H_{p}^{1} ((0, T), L_{q} {(Ω)}^{N}), r \in L_{p} ((0, T), H_{q}^{1} (Ω) + {\hat{H}}_{q, 0}^{2} (Ω))$

Possessing the estimate:

$\begin{array}{l} {‖ w ‖}_{L_{p} ((0, T), H_{q}^{2} (Ω))} + ‖ \partial_{t} w ‖_{L_{p} ((0, T), L_{q} (Ω))} \leq C_{q} {{‖ w_{0} ‖}_{B_{q, p}^{2 (1 - 1 / p)} (Ω)} + ∥ f ∥ L_{p} ((0, T), L_{q} (Ω)) \\ + {‖ (g, h) ‖}_{H_{p}^{\frac{1}{2}} (ℝ, L_{q} (Ω))} ‖ (g, h) ‖ L_{p} (ℝ, L_{q} (Ω)) + {‖ \partial_{t} g ‖}_{L_{p} (ℝ, L_{q} (Ω))}}, \end{array}$

where C_q is a constant that depends on q but is independent of T. Applying Theorem 8 yields that there exists a large $λ_{0} > 0$ such that Eq. (53) admits unique solutions w and r with

$w \in H_{p}^{1} ((0, T), L_{q} {(Ω)}^{N}) \cap L_{p} ((0, T), H_{q}^{2} {(Ω)}^{N}) (q \in \frac{q_{1}}{2}, q_{2})$

Possessing the estimate:

$(‖ < t >^{b} {\partial_{t} w ‖}_{L_{p} ((0, T) L_{q_{1} / 2} (Ω) \cap L_{q_{_{2}}} (Ω))} + ‖ < t >^{b} {\partial_{t} w ‖}_{L_{p} ((0, T), H_{q_{1} / 2}^{2} (Ω) \cap H_{q 2}^{2} (Ω)} \leq C (I + {[v]}_{T}^{2})$ . (56)

In fact, $f (v), {\tilde{e}}_{T} [g (v)], {\tilde{e}}_{T} [g (v)] a n d {\tilde{e}}_{T} [h (v)]$ satisfy (27), (43), (44), (45), (46), and (47), so that we know the existence of w possessing the estimate:

${\partial_{t} w ‖}_{L_{p} ((0, T) L_{q_{1} / 2} (Ω) \cap L_{q_{_{2}}} (Ω))} + ‖ < t >^{b} {\partial_{t} w ‖}_{L_{p} ((0, T), H_{q_{1} / 2}^{2} (Ω) \cap H_{q_{2}}^{2} (Ω)} \leq C (I + {[v]}_{T}^{2})$

with some constant C depending on q>sub>1

/2 and q₂. Let a = min(1, b), and then

$\tilde{w} : = {<t>}^{a} w$ satisfies the equations:

${\begin{cases} \partial_{t} \tilde{w} + λ_{0} \tilde{w} - J (^{T) - 1} Div \tilde{S} (\tilde{w}, < t >^{a} r) = < t >^{a} f + λ_{0} a t < t >^{a - 2} w in Ω \times (0, T), \\ \tilde{div} \tilde{w} = {\tilde{e}}_{T} [< t >^{a} g (v)] = d i v {\tilde{e}}_{T} [< t >^{a} g (v)] in Ω \times (0, T), \\ \tilde{S} (\tilde{w}, < t >^{a} r) = {\tilde{e}}_{T} [< t >^{a} h] on Γ \times (0, T), \\ \tilde{w} |_{t = 0} = u_{0} in Ω \end{cases}$ (57)

Since

${‖ t < t >^{a - 2} w ‖}_{L_{p} ((0, T), L_{q} (Ω))} \leq {‖ w ‖}_{L_{p} ((0, T), L_{q} (Ω))} \leq C (I + {[v]}_{T}^{2})$

as follows from the fact that $a - 1 \leq 0$ , we have

$(‖ < t >^{a} {\partial_{t} w ‖}_{L_{p} ((0, T) L_{q_{1} / 2} (Ω) \cap L_{q_{_{2}}} (Ω))} + ‖ < t >^{a} {w ‖}_{L_{p} ((0, T), H_{q_{1} / 2}^{2} (Ω) \cap H_{q 2}^{2} (Ω)} \leq C (I + {[v]}_{T}^{2})$ .

Repeating this argument finite times yields (56). In particular, by (56) we have

${[w]}_{T} \leq C (I + {[v]}_{T}^{2})$ (58)

Next, we consider g. Let ${T (t)}_{t \geq 0}$ be a C₀ analytic semigroup associated with problem (54). Shibata³³ proved the existence of ${T (t)}_{t \geq 0}$ satisfying the estimates:

${‖ T (t) f ‖}_{L_{p} (Ω)} \leq C t^{- \frac{N}{2} (\frac{1}{q} - \frac{1}{p})} {‖ f ‖}_{L_{q} (Ω)}, {‖ Δ T (t) f ‖}_{L_{p} (Ω)} \leq C t^{- \frac{1}{2} - \frac{N}{2} (\frac{1}{q} - \frac{1}{p})} {‖ f ‖}_{L_{q} (Ω)}$ (59)

for any t > 0 and $f \in L_{q} {(Ω)}^{N}$ provided that $1 < q \leq p \leq \infty$ and $q \leq q_{2}$ . To represent u by using ${T (t)}_{t \geq 0}$ , we introduce the solenoidal space $J_{q} (Ω)$ defined by

$J_{q} (Ω) = {f \in L_{q} {(Ω)}^{N} | {(f, J {}^{T}A \nabla φ)}_{Ω} = 0 f o r a n y φ \in {\hat{H}}_{q^{'}, 0}^{1} (Ω)}$ (60)

Here, A is the matrix defined in (4) and J the function given in (15), and

${\hat{H}}_{q^{'}, 0}^{1} (Ω) = {φ \in L_{q^{'}, l o c} (Ω) | \nabla φ L_{q^{'}} {(Ω)}^{N}, φ |_{Γ} = 0}$ .

As was proved by Shibata,³⁰ we know that for any $f \in L_{q} {(Ω)}^{N}$ there exists a unique solution $ψ \in {\hat{H}}_{q^{'}, 0}^{1} (Ω)$ of the variational equation

${(A \nabla ψ, J {}^{T}A φ)}_{Ω} = {(f, J {}^{T}A \nabla φ)}_{Ω} = 0 f o r a n y φ \in {\hat{H}}_{q^{'}, 0}^{1} (Ω)$ . (61)

which possesses the estimate $∥ \nabla ψ ∥_{L_{q}} (Ω) \leq C_{q} ∥ f ∥_{L_{q}} (Ω)$ . Here C_q is a constant that is independent of v and T in view of (36). Given $f \in L_{q} {(Ω)}^{N}$ let $ψ \in {\hat{H}}_{q^{'}, 0}^{1} (Ω)$ be a unique solution of Eq.(61), and let P_qbe an operator acting on f defined $P_{q} f = f - \nabla ψ$ . And then, $P_{q} f \in J_{q} (Ω)$ and

$∥ P_{q} f ∥_{L_{q} (Ω)} \leq C q ∥ f ∥_{L q (Ω)}$ (62)

with some constant C_q that is independent of v and T. By Proposition 21 in Shibaata,³³ we have

$u (\cdot, t) = - λ_{0} \int_{0}^{t} T (t - s) (P w) (\cdot, s) d s$ . (63)

Combining (59) and (62) yields that

$\begin{array}{l} _{{‖ \nabla^{j} u (\cdot, t) ‖}_{L_{r} (Ω)}} \leq C_{r, {\tilde{q}}_{1}} \int_{0}^{t - 1} (^{t - s)^{- \frac{j}{2} - \frac{N}{2} (\frac{1}{{\tilde{q}}_{1}} - \frac{1}{r})}} ∥ w (\cdot, s) ∥_{L_{{\tilde{q}}_{1}} (Ω)} d s \\ + C_{r, {\tilde{q}}_{2}} \int_{0}^{t - 1} (^{t - s)^{- \frac{j}{2} - \frac{N}{2} (\frac{1}{{\tilde{q}}_{2}} - \frac{1}{r})}} ∥ w (\cdot, s) ∥_{L_{{\tilde{q}}_{2}} (Ω)} d s \end{array}$ (64)

for j=0, 1, for any t > 1 and for any indices $r, {\tilde{q}}_{1} a n d {\tilde{q}}_{2}$ such that $1 < {\tilde{q}}_{1}, {\tilde{q}}_{2} \leq r \leq \infty$ and ${\tilde{q}}_{1}, {\tilde{q}}_{2} \leq q_{2},$ where $\nabla^{0} u = u a n d \nabla^{1} u = \nabla u$ .

Recall that T > 2. In what follows, we prove that

${(\int_{2}^{T} (< t >^{b} ‖ u (\cdot, t) ‖ H_{\infty}^{1} (Ω))^{p} d t)}^{\frac{1}{p}} \leq C (I + {[v]}_{T}^{2})$ (65)

$\underset{2 \leq t \leq T}{S u p} (< t >^{\frac{N}{2 q_{1}}} {‖ u (\cdot, t) ‖}_{L_{q_{1}}} \leq C (I + {[v]}_{T}^{2})$ (66)

${(\int_{2}^{T} (< t >^{b - \frac{N}{2 q_{1}}} ‖ u (\cdot, t) ‖ H_{q_{1}}^{1} (Ω))^{p} d t)}^{\frac{1}{p}} \leq C (I + {[v]}_{T}^{2})$ (67)

${(\int_{2}^{T} (< t >^{b - \frac{N}{2 q_{2}}} ‖ u (\cdot, t) ‖ H_{q_{2}}^{1} (Ω))^{p} d t)}^{\frac{1}{p}} \leq C (I + {[v]}_{T}^{2})$ (68)

By (64) with $r = \infty$ , ${\tilde{q}}_{1} = q_{1} / 2$ and ${\tilde{q}}_{2} = q_{2}$ ,

${‖ u (\cdot, t) ‖}_{H_{\infty}^{1} (Ω)} \leq C \int_{0}^{t} {‖ T (t - s) w (\cdot, s) ‖}_{H_{\infty}^{1} (Ω)} d s = C (I_{\infty} (t) + I I_{\infty} (t) + I I I_{\infty} (t))$

With

$\begin{array}{l} I_{\infty} (t) = \int_{0}^{t / 2} (^{t - s) - \frac{N}{q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I_{\infty} (t) = \int_{t / 2}^{t - 1} (^{t - s) - \frac{N}{q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I I_{\infty} (t) = \int_{t - 1}^{t} (^{t - s) - \frac{N}{2 q_{2}} - \frac{1}{2}} {‖ w (\cdot, s) ‖}_{L_{q_{2} / 2} (Ω)} d s . \end{array}$

Since

$\begin{array}{l} I_{\infty} (t) \leq (^{t / 2) - \frac{N}{q_{1}}} {(\int_{0}^{t / 2} < s >^{- b p'} d s)}^{1 / p^{'}} {(\int_{0}^{t / 2} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}^{p} d s)}^{1 / p} \\ \leq C (^{b p' - 1) - 1 / p^{'}} (I + {[v]}_{T}^{2}) t^{^{- \frac{N}{q_{1}}}} \end{array}$

as follows from the condition: $b p' > 1$ in (11), by the condition: $(\frac{N}{q_{1}} - b) p > 1$ in (11), we have

$\int_{2}^{T} {(< t >^{b} I_{\infty} (t))}^{p} d t \leq C \int_{2}^{T} < t >^{- (\frac{N}{q_{1}} - b) p} d t {(I + {[v]}_{T}^{2})}^{p} \leq C {((\frac{N}{q_{1}} - b) p - 1)}^{- 1} {(I + {[v]}_{T}^{2})}^{p}$ .

By Holder’s inequality

$\begin{array}{l} < t >^{b} I I_{\infty} (t) \leq C {\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{q_{1}}} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s \\ \leq C (^{{\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{q_{1}}} d s) 1 / p^{'}} (^{\int_{t / 2}^{t - 1} (^{t - s) - \frac{N}{q_{1}}} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s) 1 / p} \\ \leq C {(\frac{N}{q_{1}} - 1)}^{- 1 / p^{'}} (^{{\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{q_{1}}} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s)^{1 / p}} \end{array}$

Because $N / q_{1} = N / q_{2} + 1 > 1$ . By the change of integration order and (56),

$\begin{array}{l} \int_{2}^{T} (< t >^{b} I I_{\infty} (t))^{p} \leq C {(\frac{N}{q_{1}} - 1)}^{- \frac{p}{p^{'}}} \int_{2}^{T} d t \int_{t / 2}^{t - 1} (^{t - s) - \frac{N}{q_{1}}} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s \\ \leq C {(\frac{N}{q_{1}} - 1)}^{- \frac{p}{p^{'}}} \int_{1}^{T - 1} (< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)})^{p} d s {\int_{s + 1}^{2 s} (t - s)}^{- \frac{N}{q_{1}}} d t \\ \leq C {(\frac{N}{q_{1}} - 1)}^{- p} {(I + {[v]}_{T}^{2})}^{p} . \end{array}$

Since $\frac{N}{2 q_{2}} + \frac{1}{2} < 1$ as follows from q₂ > N, by Holder’s inequality,

$\begin{array}{l} < t >^{b} I I I_{\infty} (t) \leq C {\int_{t - 1}^{t} (t - s)}^{- \frac{N}{2 q_{2}} - \frac{1}{2}} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)} d s \\ \leq C (^{{\int_{t - 1}^{t} (t - s)}^{- \frac{N}{2 q_{2}} - \frac{1}{2}} d s) 1 / p^{'}} (^{{\int_{t - 1}^{t} (t - s)}^{- \frac{N}{2 q_{2}} - \frac{1}{2}} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s) 1 / p} \\ \leq C {(- \frac{N}{2 q_{2}} - \frac{1}{2})}^{^{- 1 / p^{'}}} (^{{\int_{t - 1}^{t} (t - s)}^{- \frac{N}{2 q_{2}} - \frac{1}{2}} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s) 1 / p} . \end{array}$

By the change of integration order, we have

$\begin{array}{l} \int_{2}^{T} (^{< t >^{b} I I I_{\infty} (t)) p} d t \leq C {(1 - \frac{N}{2 q_{2}})}^{- \frac{p}{p^{'}}} \int_{2}^{T} d t \int_{t - 1}^{t} (^{t - s) - \frac{N}{2 q_{2}} - \frac{1}{2}} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s \\ \leq C {(1 - \frac{N}{2 q_{2}})}^{- \frac{p}{p^{'}}} \int_{1}^{T} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s \int_{s}^{s + 1} (^{t - s) - \frac{N}{2 q_{2}} - \frac{1}{2}} d t \\ \leq C {(1 - \frac{N}{2 q_{2}})}^{- p} {(I + {[v]}_{T}^{2})}^{p} . \end{array}$

Summing up, we have obtained (65). Next, we prove (66).

By (64) with $r = q_{1}, {\tilde{q}}_{1} = q_{1} / 2 a n d {\tilde{q}}_{2} = q_{1}$ ,

${‖ u (\cdot, t) ‖}_{L_{q_{1}} (Ω)} \leq C (I_{q_{1}, \infty} (t) + I I_{q_{1}, 1} (t) + I I I_{q_{1}, 1} (t))$

With

$\begin{array}{l} I_{q_{1}, 1} (t) = {\int_{0}^{t / 2} (t - s)}^{- \frac{N}{2 q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I_{q_{1}, 1} (t) = {\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{2 q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I I_{q_{1}, 1} (t) = \int_{t - 1}^{t} {‖ w (\cdot, s) ‖}_{L_{q_{1}} (Ω)} d s . \end{array}$

By (56)

$\begin{array}{l} I_{q_{1}, 1} (t) \leq (^{t / 2)^{- \frac{N}{2 q_{1}}}} (\int_{0}^{t / 2} < s >^{- b p^{'}} d s)^{1 / p^{'}} (\int_{0}^{t / 2} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)})^{p} d s)^{1 / p} \\ \leq C t^{^{^{- \frac{N}{2 q_{1}}}}} {(I + {[v]}_{T}^{2})}^{p} . \end{array}$

Analogously, by Holder’s inequality and (56),

$\begin{array}{l} I I_{q_{1}, 1} (t) = \int_{t / 2}^{t - 1} (^{t - s) - \frac{N}{2 q_{1}}} < s >^{- b} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ \leq C < t >^{- b} (^{\int_{t / 2}^{t - 1} (^{t - s) - \frac{N p^{'}}{2 q_{1}}} d s) 1 / p^{'}} (^{\int_{0}^{T} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s) 1 / p} \\ = C (^{1 - - \frac{N p^{'}}{2 q_{1}}) 1 / p^{'}} < t >^{- b - \frac{N}{2 q_{1}} + \frac{1}{p^{'}}} (I + {[v]}_{T}^{2}) \\ \leq C (^{1 - - \frac{N p^{'}}{2 q_{1}}) 1 / p^{'}} < t >^{\frac{N}{2 q_{1}}} (I + {[v]}_{T}^{2}) \end{array}$

because $b > \frac{1}{p^{'}}$ . Finally, by (56),

$\begin{array}{l} I I I_{q_{1}, 1} (t) \leq C t^{- b} \int_{t - 1}^{t} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s \\ \leq C t^{- b} (^{\int_{t - 1}^{t} d s)^{1 / p^{'}}}) (^{\int_{0}^{T} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s)^{1 / p} \\ \leq C t^{- b} (I + {[v]}_{T}^{2}) . \end{array}$

Summing up, we have obtained (66). Next, we prove (67). By (64),

${‖ u (\cdot, t) ‖}_{H_{q_{1}}^{1} (Ω)} \leq C (I_{q_{1}, 2} (t) + I I_{q_{1}, 2} (t) + I I I_{q_{1}, 2} (t))$

with

$\begin{array}{l} I_{q_{1}, 2} (t) = {\int_{0}^{t / 2} (t - s)}^{- \frac{N}{2 q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I_{q_{1}, 2} (t) = {\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{2 q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I I_{q_{1}, 2} (t) = \int_{t - 1}^{t} {(t - s)}^{- \frac{1}{2}} {‖ w (\cdot, s) ‖}_{L_{q_{1}} (Ω)} d s . \end{array}$

By (56),

$\begin{array}{l} I_{q_{1}, 2} (t) \leq (^{t / 2)^{- \frac{N}{2 q_{1}}}} (\int_{0}^{t / 2} < s >^{- b p^{'}} d s)^{1 / p^{'}} (\int_{0}^{t / 2} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)})^{p} d s)^{1 / p} \\ \leq C t^{^{^{- \frac{N}{2 q_{1}}}}} (I + {[v]}_{T}^{2}) . \end{array}$

so that by the condition: $(\frac{N}{q_{1}} - b) p > 1$ in (11)

${(\int_{2}^{T} (< t >^{b - \frac{N}{2 q_{1}}} I_{q_{1}, 2} (t))}^{p} d t)^{1 / p} \leq C {((\frac{N}{q_{1}} - b) p - 1)}^{- 1 / p} (I + {[v]}_{T}^{2}) .$

By Holder’s inequality,

$\begin{array}{l} < t >^{b - \frac{N}{2 q_{1}}} I I_{q_{1}, 2} (t) \leq C < t >^{- \frac{N}{2 q_{1}}} {\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{2 q_{1}}} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s \\ \leq C < t >^{- \frac{N}{2 q_{1}}} (^{{\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N p^{'}}{2 q_{1}}} d s) 1 / p^{'}} (^{\int_{o}^{T} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s) 1 / p} \\ \leq C {(1 + t)}^{- (\frac{N}{q 1} - \frac{1}{p^{'}})} (I + {[v]}_{T}^{2}) . \end{array}$

Since $(\frac{N}{q_{1}} - \frac{1}{p^{'}}) p > 1$ as follows from $\frac{N}{q_{1}} = 1 + \frac{N}{q_{2}} > 1 = \frac{1}{p} + \frac{1}{p^{'}}$ , we have

${(\int_{2}^{T} {(< t >^{b - \frac{N}{q_{1}}} I I_{q_{1}, 2} (t))}^{p} d t)}^{1 / p} \leq C {((\frac{N}{q_{1}} - b) p - 1)}^{- 1 / p} (I + {[v]}_{T}^{2})$

Since $q_{1} / 2 < q_{1} < q_{2}$ , we have

${‖ w (\cdot, t) ‖}_{L_{q_{1}} (Ω)} \leq {‖ w (\cdot, t) ‖}_{_{L_{q_{1} / 2} (Ω)}^{\frac{q 2}{N + 2 q_{2}}}} {‖ w (\cdot, t) ‖}_{_{L_{q_{2}} (Ω)}^{\frac{N + q 2}{N + 2 q_{2}}}} .$

Let $α = \frac{q_{2}}{N + 2 q_{2}}$ and $β = \frac{N + q_{2}}{N + 2 q_{2}}$ then $α + β = 1$ , so that by (56) and Holder’s inequality

$\begin{array}{l} {‖ < t >^{b} w ‖}_{L_{p} ((0, T), L_{q_{1}} (Ω))} \\ \leq {(\int_{0}^{T} {(< t >^{b} {‖ w (\cdot, t) ‖}_{L_{q_{1} / 2} (Ω)})}^{p α} {(< t >^{b} {‖ w (\cdot, t) ‖}_{L_{q_{2}} (Ω)})}^{p β} d t)}^{1 / p} \\ \leq {(\int_{0}^{T} {(< t >^{b} {‖ w (\cdot, t) ‖}_{L_{q_{1} / 2} (Ω)})}^{p} d t)}^{α / p} {(\int_{0}^{T} {(< t >^{b} {‖ w (\cdot, t) ‖}_{L_{q_{2}} (Ω)})}^{p} d t)}^{β / p} \\ \leq C (I + {[v]}_{T}^{2}) \end{array}$ (69)

Since

$\begin{array}{l} < t >^{b - \frac{N}{2 q_{1}}} I I I_{q_{1}, 2} (t) \leq {\int_{t - 1}^{t} (t - s)}^{- \frac{1}{2}} < s >^{b - \frac{N}{2 q_{1}}} {‖ w (\cdot, s) ‖}_{L_{q_{1}} (Ω)} d s \\ \leq (^{{\int_{t - 1}^{t} (t - s)}^{- \frac{1}{2}} d s) 1 / p^{'}} (^{\int_{t - 1}^{t} (^{t - s) - \frac{1}{2}} d s (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1}} (Ω)}) p} d s) 1 / p}, \end{array}$

by the change of integration order, we have

$\begin{array}{l} \int_{2}^{T} (^{< t >^{b - \frac{N}{2 q_{1}}} I I I_{q_{1}, 2} (t)) p} d t \leq 2^{\frac{p}{p^{'}}} \int_{2}^{T} d t {\int_{t - 1}^{t} (t - s)}^{- \frac{1}{2}} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1}} (Ω)}) p} d s \\ \leq 2^{\frac{p}{p^{'}}} \int_{0}^{T} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1}} (Ω)}) p} d s {\int_{s}^{s + 1} (t - s)}^{- \frac{1}{2}} d t = 2^{p} {‖ < t >^{b} w ‖}_{L_{p} ((0, T), L_{q_{1}} (Ω))}, \end{array}$

which, combined with (69), furnishes that

${(\int_{2}^{T} (< t >^{b - \frac{N}{2 q_{1}}} I I I_{q_{1}, 2} (t))}^{p} d t)^{1 / p} \leq C (I + {[v]}_{T}^{2})$

Summing up, we have obtained (67).

Finally, we prove (68). By (64) with $r = q_{2}, {\tilde{q}}_{1} = q_{1} / 2$ and ${\tilde{q}}_{2} = q_{2}$ ,

${‖ u (\cdot, t) ‖}_{L_{q_{2}} (Ω)} \leq C (I_{q_{2}} (t) + I I_{q_{2}} (t) + I I I_{q_{2}} (t))$

with

$\begin{array}{l} I_{q_{2}} (t) = \int_{0}^{t / 2} {(t - s)}^{- \frac{N}{2} (\frac{2}{q_{1}} + \frac{1}{q_{2}})} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I_{q_{2}} (t) = {\int_{t / 2}^{t - 1} (t - s)}^{- \frac{N}{2} (\frac{2}{q_{1}} + \frac{1}{q_{2}})} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s, \\ I I I_{q_{2}} (t) = \int_{t - 1}^{t} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)} d s . \end{array}$

By Holder’s inequality,

$\begin{array}{l} I_{q_{2}} (t) \leq (^{t / 2) - \frac{N}{2} (\frac{2}{q_{1}} + \frac{1}{q_{2}})} {(\int_{0}^{t / 2} < s >^{b p^{'}} d s)}^{1 / p^{'}} {(\int_{0}^{t / 2} < s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{_{1}} / 2} (Ω)})^{p} d s)}^{1 / p} \\ \leq C < t >^{- \frac{N}{2} (\frac{2}{q_{1}} + \frac{1}{q_{2}})} (I + {[v]}_{T}^{2}) . \end{array}$

for $t \geq 2$ . Since $\frac{N}{2} (\frac{2}{q_{1}} + \frac{1}{q_{2}}) - (b - \frac{N}{2 q_{2}}) = \frac{N}{q_{1}} - b$

by the condition: $(\frac{N}{q_{1}} - b) p > 1$ in (11),

$\begin{array}{l} {(\int_{2}^{T} (< t >^{b - \frac{N}{2 q_{2}}} I_{q_{2}} (t))^{p} d t)}^{1 / p} \leq C {(\int_{2}^{T} t^{- (\frac{N}{q_{1}} - b) p} d t)}^{1 / p} (I + {[v]}_{T}^{2}) \\ \leq C {((\frac{N}{q_{1}} - b) p - 1)}^{- 1 / p} (I + {[v]}_{T}^{2}) . \end{array}$

Since

$\frac{N}{2} (\frac{2}{q_{1}} - \frac{1}{q_{2}}) = \frac{N}{2} (\frac{1}{q_{2}} + \frac{2}{N}) = \frac{N}{2 q_{2}} + 1 > 1,$

by Holder’s inequality

$\begin{array}{l} < t >^{b - \frac{N}{2 q_{2}}} I I_{q_{2}} (t) \leq C {\int_{t / 2}^{t - 1} (t - s)}^{- (\frac{N}{2 q_{2}} + 1)} < s >^{b - \frac{N}{2 q_{2}}} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)} d s \\ \leq C (^{{\int_{t / 2}^{t - 1} (t - s)}^{- (\frac{N}{2 q_{2}} + 1)} d s) 1 / p^{'}} (^{\int_{t / 2}^{t - 1} {(t - s)}^{- (\frac{N}{2 q_{2}} + 1)} (< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s)^{1 / p} \\ \leq C {(\frac{N}{2 q_{2}})}^{- 1 / p^{'}} (^{\int_{t / 2}^{t - 1} {(t - s)}^{- (\frac{N}{2 q_{2}} + 1)} (< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s)^{1 / p} . \end{array}$

so that by the change of integration order and (56)

$\begin{array}{l} \int_{2}^{T} (^{< t >^{b - \frac{N}{2 q_{1}}} I I_{q_{2}} (t)) p} d t \leq C (\frac{N}{2 q_{2}})^{- \frac{p}{p^{'}}} \int_{2}^{T} d t {\int_{t / 2}^{t - 1} (t - s)}^{- (\frac{N}{2 q_{2}} + 1)} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1} / 2} (Ω)}) p} d s \\ \leq C (\frac{N}{2 q_{2}})^{- \frac{p}{p^{'}}} \int_{0}^{T} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{1}_{/ 2}} (Ω)}) p} d s {\int_{s + 1}^{2 s} (t - s)}^{- (\frac{N}{2 q_{2}} + 1)} d t \leq C (\frac{N}{2 q_{2}})^{- p} (I + {[v]}_{T}^{2}) . \end{array}$

Analogously, by Holder’s inequality

$\begin{array}{l} < t >^{b - \frac{N}{2 q_{2}}} I I I_{q_{2}} (t) \leq C {\int_{t - 1}^{t} < s >}^{b - \frac{N}{2 q_{2}}} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)} d s, \\ \leq C (^{\int_{t - 1}^{t} d s) 1 / p^{'}} (^{\int_{t - 1}^{t} (< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s)^{1 / p} \\ \leq (^{\int_{t - 1}^{t} (< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s)^{1 / p}, \end{array}$

so that by the change of integration order and (56)

$\begin{array}{l} \int_{2}^{T} (^{< t >^{b - \frac{N}{2 q_{2}}} I I I_{q_{2}} (t)) p} d t \leq C \int_{2}^{T} d t \int_{t - 1}^{t} (< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)})^{p} d s \\ \leq C \int_{0}^{T} (^{< s >^{b} {‖ w (\cdot, s) ‖}_{L_{q_{2}} (Ω)}) p} d s \int_{s}^{s + 1} d t \leq C (I + {[v]}_{T}^{2}) . \end{array}$

Summing up, we have obtained (68).

Recalling that $T \geq 2$ , applying the maximal L_p-L_q regularity theorem due to Shibata³³ to Eq. (54) and using (56) give that

${‖ u ‖}_{L_{p} ((0, 2), H_{q}^{2} (Ω))} + {‖ \partial t u ‖}_{L_{p} ((0, 2), L_{q} (Ω))} \leq C q {‖ λ_{0} w ‖}_{L_{p} ((0, 2), L_{q} (Ω))} \leq C (I + {[v]}_{T}^{2})$ (70)

For any $q \in [q_{1} / 2, q_{2}]$ . Employing the same argumentation as that in proving (29), by real interpolation,we have

$\sup_{0 < t < 2} {‖ u (\cdot, t) ‖}_{B_{q, p}^{2 (1 - 1 / p)} (Ω)} \leq C (I + {[v]}_{T}^{2})$ (71)

for any $q \in [q_{1} / 2, q_{2}]$ . Combining (65), (66), (67), (68), (70), (71) and the Sobolev imbedding theorem, we have

$\begin{array}{l} ‖ < t >^{b} {u ‖}_{L_{p} ((0, T) H_{\infty}^{1} (Ω))} + ‖ < t >^{\frac{N}{2 q_{1}}} {u ‖}_{L_{\infty} ((0, T), L_{q_{_{1}}} (Ω))} \\ + ‖ < t >^{b - \frac{N}{2 q_{1}}} {u ‖}_{L_{p} ((0, T) H_{q_{1}}^{1} (Ω))} + ‖ < t >^{b - \frac{N}{2 q_{2}}} {u ‖}_{L_{p} ((0, T), L_{q_{_{2}}} (Ω))} \leq C (I + [_{v}^{]}) . \end{array}$ (72)

From (54), u satisfies the equations:

${\begin{cases} \partial_{t} u + λ_{0} v- J (^{T) - 1} Div \tilde{S} (u, p) = - λ_{0} w+ λ_{0} u in Ω \times (0, T), \\ \tilde{div u} = 0 in Ω \times (0, T), \\ \tilde{S} (u, p) = 0 on Γ \times (0, T), \\ u |_{t = 0} = 0 in Ω \end{cases}$

so that by Theorem 8,

$\begin{array}{l} ‖ < t >^{b - \frac{N}{2 q_{2}}} {u ‖}_{L_{p} ((0, T) H_{q_{2}}^{2} (Ω))} + ‖ < t >^{b - \frac{N}{2 q_{2}}} {\partial_{t} u ‖}_{L_{p} ((0, T), L_{q_{_{2}}} (Ω))} \\ + C ‖ < t >^{b - \frac{N}{2 q_{2}}} {(u, w) ‖}_{L_{p} ((0, T) L_{q_{_{2}}} (Ω))}, \end{array}$