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

This paper deals with global well-posedness of the following free boundary problem for the Navier-Stokes equations. Let $\Omega$ be an exterior domain in the N dimensional Eucledian space R^N occupied by an incompressible viscous fluid. Let $\Gamma$ be the boundary of $\Omega$ that is a C² compact hyper surface with the unit outer normal n. Let ${\Omega }_t$ be the evolution of $\Omega$ at time t. Let ${\Gamma }_t$ be the boundary of ${\Omega }_t$ with the unit outer normal n_t. Let $\rho$ and $\mu$ be positive numbers denoting the mass density and the viscosity coefficient, respectively. Let $u{=}^{\top }\left(u_1,...,u_N\right)$ be an N-vector of functions describing the velocity field, where ${}^{\top }M$ 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 ${\Omega }_t$ given by

$\begin{array}{l}\rho (\partial tu+u·\nabla u)-Div(\mu D(u)-{\rm P}I)=0, divu=0in {\displaystyle {\cup }_{0<t<T}{\Omega }_t}\times \left\{t\right\},\\ (\mu D(u)-{\rm P}I)n_t =0, V_{\Gamma t} =n_t.uon{\displaystyle {\cup }_{0<t<T}{\Gamma }_t}\times \left\{t\right\},\\ u{|}_t{}_{=0} =u_0, {\Omega }_t{|}_t{}_{=0} =\Omega \end{array}$ (1)

Here, $D\left(u\right)=\nabla u{+}^{\top }\nabla u$ denotes the doubled deformation tensor, I the N × N identity matrix, and $V_{\Gamma t}$ the evolution speed of the surface ${\Gamma }_t$ 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 ${\displaystyle \sum _{j=1}^N{\partial }_j{K}_{ij}} ,{\partial }_j=\partial /\partial x_j$ and for any N vector of function $w=(w_1,\mathrm{...},w_N),divw={\displaystyle \sum _{j=1}^N{\partial }_j{w}_i}$ and the quantity $w·\nabla w$ is an N -vector of functions whose i^th component is ${\displaystyle \sum _{j=1}^N{\partial }_j{w}_i}$

One phase problem for the Navier-Stokes equations formulated in (1) with $\left(\mu D\left(u\right)-{\rm P}I\right)nt=c_{\sigma }H{n}_t$ in place of $\left(\mu D\left(u\right)-{\rm P}I\right)nt=0$ has been received wide attention for many years, where H is the doubledmean curvature of ${\Gamma }_t$ and $c_{\sigma }$ 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 ${\Omega }_0$ is bounded and local well posedness in the case that $c\sigma >0$ 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 $c\sigma =0$ , 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 $c\sigma =0$ 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 $c\sigma >0$ was proved under the assumption that the initial domain ${\Omega }_0$ 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 ${\Omega }_0$ is a perturbed layer like:
${\Omega }_0 =\{x\in {ℝ}^3 |-b<x3 <\eta \left(x\prime \right),x\prime = (x1,x2)\in {ℝ}^2\}$ and local wellposed was proved by Allain,¹⁶ Beale¹⁷ and Tani¹⁸ in the L2 Sobolev-Slobodetskii space when $c\sigma >0$ and by Abels¹⁹ in the L_pSobolev-Slobodetskii space when $c\sigma =0$ .

Global wellposedness for small initial velocity was proved in the L₂ Sobolev-Slobodetskii space by Beale²⁰ and Tani and Tanaka²¹ in the case that $c\sigma >0$ and by Sylvester²² in the case that $c\sigma =0$ . 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, ${\Omega }_0 = \{x \in {ℝ}^3 |x_{3 }<\eta \left(x\prime \right),x\prime =(x_1,x_2) \in {ℝ}^2\}$ .

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 $c\sigma =0$ case and by Shibata^29,30 in the case $c\sigma >0$ .

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 ${\Omega }_t$ to a fixed domain. If $\Omega$ 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 ${\Omega }_{t}to\Omega$.^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 ${\displaystyle {\int }_0^t\nabla uds}$ . Another known transformation is the Hanzawa one. But, this transformation requires at least the $W_q^{3-1/q}\left(N<q\right)$ regularity of the height function representing $\Gamma$ , 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 $\Gamma$ .

Let R be a positive number such that ${\rm O}={ℝ}^N\backslash \Omega \subset B_{R/2}$ , where $B_L =\{x\in {ℝ}^N||x|<L\}$ , and let $\kappa bea{C}^{\infty }$ function such that $\kappa \left(\xi \right)=1for\left|\xi \right|\le Rand\kappa \left(\xi \right)=0for\left|\xi \right|\ge 2R$ . Let $v(\xi ,t)andq\left(\xi ,t\right)$ be the velocity field and the pressure field in Lagrange coordinates $\left\{\xi \right\}$ . Let us consider a transformation,

$x=L(\xi ,t):=\xi +{\displaystyle {\int }_0^t}\kappa (\xi )v(\xi ,s)ds$ (2)

Let $\delta$ be a positive number such that the transformation: $x=L\left(\xi ,t\right)$ is bijective from $\Omega$ on to ${\Omega }_t=\{x=L\left(\xi ,t\right)|\xi \in \Omega \}$ for each $t\in \left(0,T\right)$ provided that

${\displaystyle {\int }_0^T}\Vert \nabla (\kappa (·)v(·,s))\parallel L_{\infty }\left(\Omega \right) ds\le \delta$ (3)

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

${\mathcal{l}}_{ij} (t)={\delta }_{ij} +{\displaystyle {\int }_0^t}\frac{\partial }{\partial {\xi }_j}(\kappa \left(\xi \right)v_i(\xi ,s))ds(v{=}^{\top }(v_1=,\mathrm{...},v_N)), A\left(t\right) {({\mathcal{l}}_{ij}\left(t\right))}^{-1}=(a_{ij}\left(t\right))$ (4)

where ${\delta }_{ij}$ are the Kronecker delta symbols, that is ${\delta }_{ii} =1$ and ${\delta }_{ij} =0$ for $i \ne j$ . Here and hereafter, a function $a=a\left(\xi ,t\right)$ 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 $({\mathcal{l}}_{ij}\left(t\right))$ is invertible.
Let

${\Omega }_t=\{x=L\left(\xi ,t\right)|\xi \in \Omega \},{\Gamma }_t=\{x=L\left(\xi ,t\right)|\xi \in \Gamma \},$

$u\left(x,t\right)=v\left(L^{-1}\left(x,t\right),t\right)andp\left(x,t\right)=q\left(L^{-1}\left(x,t\right),t\right)$ in Eq. (1), and then, v and q satisfy the following equations:

$\{\begin{array}{l}{\partial }_{t}v-Div(\mu D(v)-qI)=f(v),in\Omega \times (0,T),\\ divv=g\left(v\right)=divg\left(v\right)in\Omega \times \left(0,T\right),\\ \left(\mu D\left(v\right)-qI\right)n=h\left(v\right)on\Gamma \times \left(0,T\right),\\ v{|}_{t=0} =u_0 in\Omega .\end{array}$ (5)

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

$\begin{array}{l}{V}_0({\displaystyle {\int }_0^t\nabla \left(\kappa v\right)ds){\nabla }^{2}v,V_1({\displaystyle {\int }_0^t\nabla \left(\kappa v\right)ds){\partial }_{t}v}},\\ V_2({\displaystyle {\int }_0^t\nabla \left(\kappa v\right)ds)}({\displaystyle {\int }_0^t{\nabla }^2(\kappa v)ds})\nabla v,\left(1-\kappa \right){{(}^{\top }}^A\{v·{(}^{\top }A\nabla v)\};\end{array}$ (6)

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

$g\left(v\right)=V_3({\displaystyle {\int }_o^t}\nabla \left(\kappa v\right)ds)\nabla v; g\left(v\right) =V_4({\displaystyle {\int }_0^t}\nabla \left(\kappa v\right)ds)v; h\left(v\right)=V_5({\displaystyle {\int }_0^t}\nabla \left(\kappa v\right)ds)\nabla v$ (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 $t\to \infty$ .

Let $N \ge 3$ and let q₁ and q₂ be exponents such that $\max \left(N, \frac{2N}{N-2}\right) <q_2<\infty and1/q_1=1/q_2+1/N$ . Let b and p be numbers defined by

$b=\frac{3N}{2q_2}+\frac{1}{2},p=\frac{2q_2(1+\sigma )}{q_2-N}$ (8)

with some very small positive number $\sigma$ . Then, there exists an $\in > 0$ such that if initial data $u_0\in B_{q_{2,p}}^{2(1-1/p)}{\left(\Omega \right)}^N \cap B_{q_{1/2},p}^{2(1-1/p)}{\left(\Omega \right)}^N$ satisfies the compatibility condition:

${\text{div u}}_{\text{0}}\text{ = 0}\text{in }\Omega ,D(u_0)n-<D(u_0)n,n>n=0 on\Gamma$ (9)

and the smallness condition: with $I = \parallel u_0\parallel B_{q_2,p}^{2(1-1/p)} + \parallel u_0\parallel B_{q_{1/2},p}^{2(1-1/p)}$ , then Eq. (5) admits unique solutions v and q with

$v\in L_p\left((0,\infty ),H_{q_2}^2 \left(\Omega \right){ }^N\right)\cap H_p^1 \left((0,\infty ),L_{q_2} \left(\Omega \right){ }^N\right), q\in L_p\left((0,\infty ),H_{q_2}^1\left(\Omega \right)+{\widehat{H}}_{q_2,0}^1(\Omega )\right)$ ,

possessing the estimate ${\left[v\right]}_{\infty }\le Cϵ$ with

$\begin{array}{l}{\left[v\right]}_T=\{{\displaystyle {\int }_0^T{\left(<s{>}^b\parallel v(·,s)\parallel H_{\infty }^1 (\Omega )\right)}^{p}ds}\\ +{{\displaystyle {\int }_0^T\left(<s{>}^{(b-\frac{N}{2q_1})}\parallel v(·,s)\parallel H_{q_{{}_1}}^1(\Omega ){)}^{p}ds+(\underset{0<s<T}{sup}<s{>}^{\frac{N}{2_{q_{{}_1}}}} \parallel v(·,s)\parallel L_{q_1}(\Omega )\right)}}^p\\ +{{{\displaystyle {\int }_0^T\left(<s{>}^{(b-\frac{N}{2q_2})}\parallel v(·,s)\parallel H_{q_2}^2(\Omega )+\parallel {\partial }_{t}v(·,s)\parallel L_{q2}(\Omega )\right)}}^{p}ds\}}^{\frac{1}{p}}.\end{array}$

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

Remark 2

Let $p\prime =p/\left(p-1\right),$ that is $1/p\prime =1-1/p.$ And then,

$\frac{1}{p\prime }=\frac{\left(1+2\sigma \right)q_2+N}{2q_2\left(1+\sigma \right)}$

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

$\begin{array}{l}1<q_1<2,\frac{N}{q_1}>b>\frac{1}{p^{\prime }},\left(\frac{N}{q_1}-b\right)p>1,\left(b-\frac{N}{2q_2}\right)p>1,b\ge \frac{N}{2q_1},\\ b\ge \frac{N}{q_2},\left(\frac{N}{2q_2}+\frac{1}{2}\right)p^{\prime }<1,\left(b-\frac{N}{2q_2}\right)p^{\prime }>1,\frac{N}{q_2}+\frac{2}{p}<1.\end{array}$ (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: $q_2>\frac{2N}{N-2}$ implies that q₁ > 2 and q₃ > 1 which is necessary to prove Theorem 1.

Thus, we assume that

$\max \left(N,\frac{2N}{N-2}\right)<q_2<\infty$ .

Remark 4

We can choose δ > 0 so small that x = L(ξ, t) is a diffeomorphism with suitable regularity from $\Omega$ onto $\Omega t$ , 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 $L_q \left(\Omega \right),H_q^m\left(\Omega \right)and{B}_{q,p}^s (\Omega )$ be the standard Lebesgue, Sobolev, and Besov spaces on $\Omega$ , while $\parallel •\parallel L_q (\Omega ),$ $\parallel •\parallel H_q^m\left(\Omega \right),$ $\parallel •\parallel B_{q,p}^s (\Omega )$ denote their norms, respectively. For a Banach space X with norm $\parallel •\parallel X$,

Let $\left\{\left(f_1,...,f_d\right)|f_i \in X \left(i=1,...,d\right)\right\}$ , while the norm of Xd is written simply by $\parallel •\parallel X$ , which is defined by $\parallel f\parallel X ={\displaystyle {\sum }_{j=1}^d\parallel f_j\parallel X }$ for $f =\left(f_1,...,f_d\right)\in X_d$ . Let ${\widehat{H}}_{q_{,0}}^1(\Omega )=\left\{\theta \in L_{q,loc}(\Omega )|\nabla \theta \in L_q{\left(\Omega \right)}^N ,\theta |\Gamma =0\right\}$

${\widehat{H}}_q^1(\Omega )={\widehat{H}}_{q_{,0}}^1(\Omega )+\left\{\mathfrak{p}={\mathfrak{p}}_1+{\mathfrak{p}}_2|{\mathfrak{p}}_1\in H_q^1(\Omega ),{\mathfrak{p}}_2\in {\widehat{H}}_{q_{,0}}^1(\Omega )\right\}$

For $1\le p\le \infty ,L_p\left(\left(a,b\right),X\right)$ and $H_m^p\left(\left(a,b\right),X\right)$ denote the standard Lebesgue and Sobolev spaces of X-valued functions defined on an interval (a, b), while $\parallel •\parallel L_p\left(\left(a,b\right),X\right)$ and $\parallel •\parallel H_m^p\left(\left(a,b\right),X\right)$ 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:

${\displaystyle {\int }_0^t\nabla (\kappa (\xi )v(\xi ,s)ds}={\displaystyle {\int }_0^T\nabla (\kappa (\xi )v(\xi ,s)ds}-{\displaystyle {\int }_0^T\nabla (\kappa (\xi )v(\xi ,s)ds}$ .

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 $v\in H_p^1 \left((0,T),L_q {(\Omega )}^N \right)\cap L_p\left((0,T),H_q^2 (\Omega )\right) ,q\in L_p\left((0,T),H_q^1 (\Omega )+{\widehat{H}}_{q,0}^1 (\Omega )\right)$ (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 $n_t = {}^{\top }\left(n_{t1},...,n_{tN}\right)andn={}^{\top }\left(n_1,...,n_N\right),$ and then by the transformation (2), we have

$a_{\text{ji}}\text{(t)}\frac{\partial }{\partial {\xi }_j},\frac{\partial }{\partial x_j}={\displaystyle \sum _{j=1}^N{n}_{\text{ti}}}=d(t){\displaystyle \sum _{j=1}^N{a}_{\text{ji}}}(t)n_j$ (14)

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

$a_{ji}\text{(t)=}{\delta }_{ij}+{\tilde{a}}_{ij} (t), J(t)=1+\tilde{J}(t),{\mathcal{l}}_{ij}(t)={\delta }_{ij}+{\tilde{\mathcal{l}}}_{ij}(t)$ ,     (15)

Where ${\mathcal{l}}_{ij}$ are given in (4) and J is the Jacobian of the transformation (2), that is J = $J= det\left({\mathcal{l}}_{ij}\right),$ and then

${\tilde{a}}_{ij} (t)=b_{ij }\left({\displaystyle {\int }_0^t\nabla (\kappa (\xi )v(\xi ,s))ds}\right),\tilde{J}(t)=K\left({\displaystyle {\int }_0^t\nabla (\kappa (\xi )v(\xi ,s))ds}\right),$

${\tilde{\mathcal{l}}}_{ij} (t)=m_{ij }\left({\displaystyle {\int }_0^t\nabla (\kappa (\xi )v(\xi ,s))ds}\right):={\displaystyle {\int }_0^t\frac{\partial }{\partial {\xi }_j}(\kappa (\xi )v_i(\xi ,t))ds)}$ (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 ${\displaystyle {\int }_0^t\nabla (\kappa (\xi )v(\xi ,s))ds}$ Let $u\left(x,t\right)=v\left(\xi ,t\right)andp\left(x,t\right)=q\left(\xi ,t\right)$ and then u and p satisfy Eq. (1). By (14),

$\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}=D_{ij,t}(v):=D_{ij}(v)+{\tilde{D}}_{ij}(t)\Delta v$

With

$D_{ij}(v)=\frac{\partial v_i}{\partial {\xi }_j}+\frac{\partial v_j}{\partial {\xi }_j},{\tilde{D}}_{ij}(t)\Delta v={\displaystyle \sum _{K=1}^N{\tilde{a}}_{kj}(t)\frac{\partial v_i}{\partial {\xi }_k}+{\tilde{a}}_{ki}(t)\frac{\partial v_j}{\partial {\xi }_k}}$ (17)

We also have an important formula:

$divu={\displaystyle \sum _{j=1}^N\frac{\partial u_j}{\partial x_j}}={\displaystyle \sum _{j,k=1}^{N}J(t)a_{kj}(t)\frac{\partial v_j}{\partial {\xi }_k}}={\displaystyle \sum _{j,k=1}^N\frac{\partial }{\partial {\xi }_k}(J(t)a_{kj}(t)v_j)}$ (18)

which implies that

${\displaystyle \sum _{j,k=1}^N{\tilde{a}}_{kj}(t)}=\tilde{J}(t)a_{kj}(t)\frac{\partial v_j}{\partial {\xi }_k}={\displaystyle \sum _{j,k=1}^N\frac{\partial }{\partial {\xi }_k}\left\{\left({\tilde{a}}_{kj}(t)+\tilde{J}(t)a_{kj}(t)\right)v_j\right\}}$ (19)

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

$\{\begin{array}{l}{\displaystyle \sum _{i=1}^N{\mathcal{l}}_{is}(t)({\partial }_t{v}_i+(1-\kappa )}{\displaystyle \sum _{j,k=1}^N{v}_j{a}_{kj}(t)\frac{\partial v_i}{\partial {\xi }_k})}\\ -\mu {\displaystyle \sum _{i,j,k=1}^N{\mathcal{l}}_{is}(t)a_{kj}(t)\frac{\partial }{\partial {\xi }_k}{D}_{i,j,t}(v)-\frac{\partial \mathfrak{q}}{\partial {\xi }_s}=0}in\Omega \times (0,T),\\ {\displaystyle \sum _{j,k=1}^{N}J(t)a_{kj}(t)\frac{\partial v_j}{\partial {\xi }_k}}={\displaystyle \sum _{j,k=1}^N\frac{\partial }{\partial {\xi }_k}(J(t)a_{kj}(t)v_j)=0}in\Omega \times (0,T)\\ \mu {\displaystyle \sum _{i,j,k=1}^N{\mathcal{l}}_{is}(t)a_{kj}(t)D_{i,j,t}(v)n_k-\mathfrak{q}{\mathfrak{n}}_s=0}on\Gamma \times (0,T),\\ \text{v}|{}_{t=0}=u_{0}in\Omega \end{array}$ (20)

Where s runs from 1 through N. Here, we have used the fact that $({\mathcal{l}}_{ij})=A^{-1}$ which follows from (4).

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

${\displaystyle {\int }_0^t\nabla (\kappa (\xi )v(\xi ,s)ds}={\displaystyle {\int }_0^T\nabla (\kappa (\xi )v(\xi ,s)ds}-{\displaystyle {\int }_0^T\nabla (\kappa (\xi )v(\xi ,s)ds}$ .

In (16), by the Taylor formula we write

$\begin{array}{l}{a}_{ij}(t)=a_{ij}(T)+A_{ij}(t),{\mathcal{l}}_{{}_{ij}}(t)={\mathcal{l}}_{{}_{ij}}(T)+{\mathcal{L}}_{{}_{ij}}(t),\\ D_{ij,t}=D_{ij,T}(\text{v})+D_{ij}(t)\Delta \text{v}J(t)=J(T)+\mathcal{J}(t)\end{array}$ (21)

With

$\begin{array}{l}{A}_{ij}(t)={\displaystyle {\int }_0^1-{b^{\prime }}_{ij}{\displaystyle {\int }_0^T\nabla (\kappa (\xi )v(\xi ,s)ds}-\theta {\displaystyle {\int }_t^T\nabla (\kappa (\xi )v(\xi ,s)ds})d\theta {\displaystyle {\int }_t^T\nabla (\kappa (\xi )v(\xi ,s)ds}},\\ {\mathcal{L}}_{ij}(t)=-{\displaystyle {\int }_t^T\frac{\partial }{\partial {\xi }_j}\kappa (\xi )v_i(\xi ,s)ds,D_{ij}(t)\Delta v={\displaystyle \sum _{\kappa =1}^N(A_{kj}(t)\frac{\partial v_i}{{\partial }_{{\xi }_k}}+A_{ki}\frac{\partial v_j}{{\partial }_{{\xi }_k}})}},\\ \mathcal{J}(t)=-{\displaystyle {\int }_0^1{K}^{\prime }}({\displaystyle {\int }_0^T\nabla (\kappa (\xi )v(\xi ,s)ds}-\theta {\displaystyle {\int }_t^T\nabla (\kappa (\xi )v(\xi ,s)ds})d\theta {\displaystyle {\int }_t^T\nabla (\kappa (\xi )v(\xi ,s)ds}\end{array}$

Where ${b^{\prime }}_{ij}$ and $K^{\prime }$ are derivatives of $b_{ij}$ and $K$ with respect to w. By the relation:

${\displaystyle \sum _{s=1}^N{\mathcal{l}}_{is}(T)a_{sm}(T)={\delta }_{im},}$ (22)

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

${\partial }_t{v}_m-\mu {\displaystyle \sum _{j,k=1}^N}{a}_{jk}(T)\frac{\partial }{\partial {\xi }_k}(\mu D_{mj,T}(v)-{\delta }_{mj}\mathfrak{q}=f_m(v)$

With

$\begin{array}{l}{f}_m\left(\text{v}\right)=-{\displaystyle \sum _{s=1}^N{a}_{sm}(T)\left\{{\displaystyle \sum _{i=1}^N{\mathcal{L}}_{is}(t){\partial }_t{v}_i+{\displaystyle \sum _{i,j,k=1}^N(1-\kappa ){\mathcal{l}}_{is}(t)a_{kj}(t)v_i\frac{\partial v_i}{\partial {\xi }_k}}}\right\}}\\ +\mu {\displaystyle \sum _{s=1}^N{a}_{sm}(T)\left\{{\displaystyle \sum _{i,j,k=1}^N{\mathcal{l}}_{is}}(T)a_{kj}(T)\frac{\partial }{\partial {\xi }_k}(D_{ij}(t)\nabla v)+{\displaystyle \sum _{i,j,k=1}^N{\mathcal{l}}_{is}}(T)A_{kj}(t)\frac{\partial }{\partial {\xi }_k}{D}_{ij,t}(v)\right\}}\\ +{\displaystyle \sum _{i,j,k=1}^N{\mathcal{L}}_{is}}(t)a_{kj}(t)\frac{\partial }{\partial {\xi }_k}{D}_{ij,t}(v)\end{array}$ (23)

Next, by (18)

$\widetilde{divv}=g\left(v\right)=divg\left(v\right)$

With

$\begin{array}{l}\widetilde{divv}={\displaystyle \sum _{j,k=1}^{N}J(T)a_{kj}(T)\frac{\partial v_i}{\partial {\xi }_k}={\displaystyle \sum _{j,k=1}^N}\frac{\partial }{\partial {\xi }_k}J(T)a_{kj}(T)v_j}\\ g(v)={\displaystyle \sum _{j,k=1}^N(J(T)A_{kj}(t)+\mathcal{J}(t)a_{kj}(t))\frac{\partial v_j}{\partial {\xi }_k}}\\ g_k(v)={\displaystyle \sum _{j=1}^N(J(T)A_{kj}(t)+\mathcal{J}(t)a_{kj}(t)v_j},g(v)=(g_1(v)\mathrm{....}{g}_N(v))\end{array}$ (24)

Finally, we consider the boundary condition. Let $\tilde{n}$ be an N -vector defined on ${ℝ}^N$ such that $\tilde{n}=n$ on $\Gamma \text{ and}\parallel \tilde{n}\parallel H_{\infty }^2 \left({ℝ}^N\right) \le C$ . In what follows, $\tilde{n}$ is simply written by $n={}^{\top }\left(n_1,...,n_N\right)$ . By (14) and (22)

${\displaystyle \sum _{j,k=1}^N{a}_{jk}(T)(\mu D_{mj,T}(v)-{\delta }_{mj}\mathfrak{q}\text{)}{n}_k=h_m(v)}$

with

$h_m(v)=-\mu {\displaystyle \sum _{j,k=1}^N\left(a_{kj}(T){\mathcal{D}}_{mj}(t)\nabla v+{\mathcal{A}}_{kj}(t)D_{mj,t}(v)\right)n_k}-\mu {\displaystyle \sum _{i,j,k,s=1}^N{a}_{sm}(T){\mathcal{L}}_{is}(t)a_{kj}(t)D_{ij,t}(v)n_k}$ (25)

By (18),

$\mu {\displaystyle \sum _{j,k=1}^N{a}_{kj}(T)\frac{\partial }{\partial {\xi }_k}(\mu D_{mj,T}\text{(v)}-{\delta }_{mj}\mathfrak{q}\text{)}}\text{=}J{(T)}^{-1}{\displaystyle \sum _{j,k=1}^N\frac{\partial }{\partial {\xi }_k}\left\{J(T)a_{jk}(T)(D_{mj,T}(v){\delta }_{mj}\mathfrak{q})\right\}}$

And

${\displaystyle \sum _{j,k=1}^N{a}_{kj}(T)D_{mj,T}(\text{v})n_k}-\left({\displaystyle \sum _{s=1}^N{a}_{sm}(T)n_s}\right)\mathfrak{q}={\displaystyle \sum _{k=1}^N\left[{\displaystyle \sum _{j=1}^N\left\{J(T)a_{kj}(T)(D_{mj,T}(v)-{\delta }_{mj}\mathfrak{q}\right\}}\right]}{n}_k$

Thus, letting

$S_{mk}(v,\mathfrak{q})={\displaystyle \sum _{j=1}^N}J(T)a_{kj} (T)(\mu D_{mj,T} (v)-{\delta }_{mj}\mathfrak{q}),\tilde{S}(v,\mathfrak{q})=\left(S_{ij} (v,\mathfrak{q})\right),$

$\text{f}\left(\text{v}\right)={}^{\top }(f_1\left(\text{v}\right),\mathrm{...},f_N\left(\text{v}\right)), h\left(\text{v}\right)={}^{\top }(h_1\left(\text{v}\right),\mathrm{...},h_N\left(\text{v}\right)),$

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

$\{\begin{array}{l}{\partial }_t\text{v}-J{(}^{T}Div\tilde{S}(\text{v},\mathfrak{q})=f(\text{v})in\Omega \times (0,T),\\ \widetilde{divv}=g\left(v\right)=divg\left(v\right)in\Omega \times (0,T),\\ \tilde{S}(v,\mathfrak{q})n=h(v)on\Gamma \times (0,T),\\ v|{}_{t=0}={\text{u}}_0\text{in}\Omega .\end{array}$ (26)

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}\parallel <t>{ }^{\alpha }w\parallel L_p((0,T),X) ={\left\{{\displaystyle {\int }_0^T{(<t{>}^{\alpha }\parallel w(·,t)\parallel X )}^p}dt\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}1\le p<\infty ,\\ \parallel <t>{ }^{\alpha }w\parallel L_{\infty }((0,T),X)=\underset{0<t<T}{\text{esssup}}<t{>}^{\alpha }\parallel w(·,t)\parallel Xp=\infty .\end{array}$

First, we prove that

$\parallel <t{>}^{b}f \parallel L_p((0,T),L_{q1/2}\left(\Omega \right) + \parallel < t >f \parallel L_p((0,T ),L_{q_2} \left(\Omega \right)) \le C(\mathcal{I} + {[v]}_T^2 ).$ (27)

with $\mathcal{I}={\Vert {\text{u}}_0\Vert }_{B_{q{1}_{/2},p}^{2(1-1/p)}\left(\Omega \right) }+{\Vert {\text{u}}_0\Vert }_{B_{q_2,p}^{2(1-1/p)}\left(\Omega \right) }$ . Here and in what follows, C denotes generic constants independent of $I,\left[v\right]T,\delta ,andT$ . The value of C may change from line to line. Since we choose I small enough eventually, we may assume that $0<I\le 1$ . Especially, we use the estimates:

${\mathcal{I}}^2 \le \mathcal{I}, \mathcal{I}{\left[v\right]}_T\le \frac{1}{2}({\mathcal{I}}^2+{\left[v\right]}_T^2)\le \mathcal{I}+{\left[v\right]}_T^2$

Since

$\begin{array}{l}{\displaystyle {\int }_{\alpha }^{\beta }\parallel \nabla (\kappa v(·,s))\parallel L_{\infty }\left(\Omega \right)\le C{(1+\alpha )}^{-b+\frac{1}{p^{\prime }}}}{\left({\displaystyle {\int }_{\alpha }^{\beta }{(<s{>}^b\parallel v(·,s)\parallel H_{\infty }^1 (\Omega ))}^{p}ds}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\\ {\displaystyle {\int }_{\alpha }^{\beta }\parallel {\nabla }^2(\kappa v(·,s))\parallel L_q\left(\Omega \right)\le C{(1+\alpha )}^{-b+\frac{N}{2q_2}+\frac{1}{p^{\prime }}}}{\left({\displaystyle {\int }_{\alpha }^{\beta }{(<s{>}^{b-\frac{N}{2q_2}}\parallel v(·,s)\parallel H_{q_2}^2(\Omega ))}^{p}ds}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\end{array}$

for any $0\le \alpha <\beta \le T$ , where $q\in \left[1,q_2\right]$ , we have

$\begin{array}{l}{\displaystyle {\int }_{\alpha }^{\beta }\parallel \nabla (\kappa v(·,s))\parallel L_{\infty }\left(\Omega \right)ds\le C{[v]}_T{(1+\alpha )}^{-b+\frac{1}{p^{\prime }}}}\\ {\displaystyle {\int }_{\alpha }^{\beta }\parallel {\nabla }^2(\kappa v(·,s))\parallel L_q\left(\Omega \right)\le C{[v]}_T}\end{array}$ (28)

for any $0\le \alpha <\beta \le T$ , where $q\in \left[1,q\right]$ , because ${}^{b>\frac{N}{2q_2}+\frac{1}{p^{\prime }}}$ as follows from (11). By real interpolation theorem, we have

$\begin{array}{l}\underset{ t\in (0,T)}{sup}<t{>}^{b- \frac{N}{2q_2}}{\Vert v(·,t)\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}\le C{\Vert {\text{u}}_0\Vert }_{{}_{{}_{B_{q_2,p}^{2(1-1/p)}(\Omega )}}}\\ + \Vert < t {>}^{b- \frac{N}{2q_2}}{\text{v}\Vert }_{L_p((0,T),H_{q_2}^2 \left(\Omega \right)) } +\Vert <t {>}^{b- \frac{N}{2q_2}}\partial t\text{v}\Vert {}_{{}_{L_p((0,T)L_{q_2}\left(\Omega \right))}}\end{array}$ (29)

To prove (29), we introduce an operator $\mathcal{T}(t)$ acting of $g\in B_{q_2,p}^{2(1-1/p)}{ℝ}^N$ defined by

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

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

$e^{\gamma t}\mathcal{T}(t)g{\parallel }_{Lp((0,\infty ),H_q^2( {ℝ}^N)) }+e^{\gamma t}\partial t\mathcal{T}(t)g{\parallel }_{Lp((0,\infty ),L_{q( {ℝ}^N))}}\le C\left|\right|g|{|}_{{}_{{}_{{}_{B_{q_2,p}^{2(1-1/p)}}({ℝ}^N)}}}$ (31)

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

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

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

$\partial t[e_{T}f](t)=\{\begin{array}{l}0(t<0),\\ \partial tf(t) (0<t<T),\\ -(\partial tf)(2T-t) (T<t<2T),\\ 0 (2T <t),\end{array}$ (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 $\Omega$ and ${\Vert {\tilde{\text{u}}}_0\Vert }_{B_{q{1}_{/2},p}^{2(1-1/p)} {ℝ}^N }<C{\Vert {\text{u}}_0\Vert }_{B_{q_2,p}^{2(1-1/p)}\left(\Omega \right) }$ .

Let $z=e_T[<t{>}^{b- \frac{N}{2q_2}}v-\mathcal{T}\left(\left|t\right|\right){\tilde{\text{u}}}_0]+\mathcal{T}\left(\left|t\right|\right){\tilde{\text{u}}}_{0}fort\in ℝ$ . Since $<t{>}^{b- \frac{N}{2q_2}}v-\mathcal{T}\left(\left|t\right|\right){\tilde{\text{u}}}_0{|}_{t=0}=0in\Omega$ by (31), (32) and (33),

$\begin{array}{l}\parallel z{\parallel }_{L_p((0,\infty ),H_{q_2}^2(\Omega ))}+ \parallel {\partial }_{t}z{\parallel }_{L_p((0,\infty ),L_{q_2} \left(\Omega \right))}\\ <C(||<t{>}^{b- \frac{N}{2q_2}}\text{v}\Vert {}_{L_p((0,T),H_{q_2}^2 \left(\Omega \right)) }+< t {>}^{b- \frac{N}{2q_2}}{\partial }_t\text{v}\Vert {}_{L_p((0,T)L_{q_2}\left(\Omega \right))}+{\Vert {\text{u}}_0\Vert }_{B_{q_2,p}^{2(1-1/p)}\left(\Omega \right) }\end{array}$ (34)

It is known (Tanabe⁵¹) that $L_p\left(\left(0,\infty \right),E_1\right)\cap H_p^1\left(\left(0,\infty \right),E_0\right)$ is continuously imbedded into $BUC\left((0,\infty ),{(E_0,E_1)}_{1-1/p,p}\right)$ , 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
$z\left(t\right)=< t {>}^{b- \frac{N}{2q_2}}v\left(t\right)fort\in \left(0,T\right)$ , we have

$\underset{ 0<t<T)}{sup}<t{>}^{b- \frac{N}{2q_2}}{\Vert v(t)\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}\le \underset{ t\in (0,\infty )}{sup}{\Vert z(t)\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}\le C{\Vert z\Vert }_{Lp((0,\infty ),H_{q_2}^2 \left(\Omega \right)) }+{\Vert {\partial }_{t}z\Vert }_{Lp((0,\infty ),L_{q_2}\left(\Omega \right)) }$

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

Since $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$p$}\right.+\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$q_2$}\right.<1,B_{q_2,p}^{2(1-1/p)}(\Omega )$ is continuously imbedded into $H_{\infty }^1(\Omega )$ so that by (29)

$<t{>}^{b- \frac{N}{2q_2}}{\Vert v(t)\Vert }_{{}_{L\infty ((0,T),H_{\infty }^1 \left(\Omega \right)) }}\le C(\mathcal{I}+{[\text{v}]}_T)$ (35)

Applying (3), (28) and (29) to the formulas in (15) and (16) and using the fact that $-b+\frac{1}{p^{\prime }}\frac{N}{2q_2}\le -\frac{N}{2q_2}$ and $-b+\frac{N}{2q_2}\le -\frac{N}{2q_2}$ ,which follows from (11), give

${\Vert (a_{ij}(t),J(t),{\mathcal{l}}_{{}_{ij}}(t),A_{ij}(t),\mathcal{J}(t),{\mathcal{L}}_{ij}(t)\Vert }_{L_{\infty } \left(\Omega \right)}\le C$ ,

${\Vert (A_{ij}(t),\mathcal{J}(t),{\mathcal{L}}_{ij}(t)\Vert }_{L_{\infty } \left(\Omega \right)}\le C{\displaystyle {\int }_t^T\nabla (\kappa v(·,s)){\parallel }_{L_{\infty }}\left(\Omega \right)ds\le C{[v]}_T}+<t{>}^{-b+\frac{1}{p^{\prime }}}\le C{[v]}_T<t{>}^{-\frac{N}{2q_2}},$

${\Vert \nabla (a_{ij}(t),J(t),{\mathcal{l}}_{{}_{ij}}(t),A_{ij}(t),\mathcal{J}(t),{\mathcal{L}}_{ij}(t)\Vert }_{L_q \left(\Omega \right)}\le C{\displaystyle {\int }_0^T{\nabla }^2(\kappa v(·,s)){\parallel }_{L_q}\le C{[v]}_T},$ (36)

$\begin{array}{l}{\Vert {\partial }_t(a_{ij}(t),J(t),{\mathcal{l}}_{{}_{ij}}(t),A_{ij}(t),\mathcal{J}(t),{\mathcal{L}}_{ij}(t)\Vert }_{L_{\infty } \left(\Omega \right)}\\ \le C\Vert \nabla (\kappa v(·,t))\Vert {,}_{L_{\infty }(\Omega )}\le C(\mathcal{I}+{[}_{\text{v}})<t{>}^{-b+ \frac{N}{2q_2}}\le C(\mathcal{I}+{[}_{\text{v}})<t{>}^{\frac{N}{2q_2}}\end{array}$

for any $t\in \left(0,T\right]$ , where $q\in \left[1,q_2\right]$ . Moreover, we have

$({\tilde{a}}_{ij},\tilde{J},{\tilde{\mathcal{l}}}_{{}_{ij}},{\tilde{A}}_{ij},\mathcal{J},{\mathcal{L}}_{ij})(x,t)=0forx\notin B_{2R}andt\in [0,T]$ (37)

By (36) and (37),

${\Vert (a_{sm}(T){\mathcal{L}}_{is}(t){\partial }_t{v}_i)\Vert }_{L_q \left(\Omega \right)}C{[v]}_T+<t{>}^{-b+\frac{1}{p^{\prime }}}{\Vert {\partial }_t{v}_i(t)\Vert }_{L_{q_2} \left(\Omega \right)}$

for any $q\in \left[1,q_2\right]$ . Since $\frac{1}{p^{\prime }}<b-\frac{N}{2q_2}$ as follows from (11), we have

${\Vert <t{>}^b{a}_{sm}(T){\mathcal{L}}_{is}(t){\partial }_t{v}_i)\Vert }_{L_p((0,T),L_q \left(\Omega \right))} \le C(\mathcal{I}+{[v]}_T^2)$

for any $q\in \left[1,q_2\right]$

Next, by Holder’s inequality,

$<t{>}^b{\Vert v·\nabla v(·,t)\Vert }_{L_{q_1}/2}(\Omega )\le <t{>}^{\frac{N}{2q_1}}{\Vert \nabla v(·,t)\Vert }_{L_{q_1}(\Omega )}$

so that by (36), we have

${\Vert <t{>}^b{a}_{sm}(T){\mathcal{l}}_{is}{a}_{kj}{v}_j\frac{\partial v_i}{\partial {\xi }_k}\Vert }_{L_p((0,T),L_{q_1}/2(\Omega ))} \le C{[v]}_T^2.$

Since

$<t{>}^b{\Vert v·\nabla v(·,t)\Vert }_{L_{q_2}(\Omega )}\le <t{>}^{\frac{N}{2q_2}}{\Vert \nabla v(·,t)\Vert }_{{}_{L_{\infty }(\Omega )}}<t{>}^{b- \frac{N}{2q_2}}{\Vert \nabla v(·,t)\Vert }_{L_{q_2}(\Omega )},$

by (36)

${\Vert <t{>}^b{a}_{sm}(T){\mathcal{l}}_{is}{a}_{kj}{v}_j\frac{\partial v_i}{\partial {\xi }_k}\Vert }_{L_p((0,T),L_{q_2} \left(\Omega \right))} \le C(\mathcal{I}+{[v]}_T)[v]T\le C(\mathcal{I}+{[v]}_T^2)$

Since

$\frac{\partial }{\partial {\xi }_k}(D_{ij}(t)\nabla v)={\displaystyle \sum _{m=1}^N({\mathcal{A}}_{mj}(t)\frac{{\partial }^2{v}_i}{\partial {\xi }_k\partial {\xi }_m}+{\mathcal{A}}_{mi}(t)\frac{{\partial }^2{v}_j}{\partial {\xi }_k\partial {\xi }_m})}+{\displaystyle \sum _{m=1}^N\text{((}\frac{\partial }{\partial {\xi }_m}{\mathcal{A}}_{mj}(t)\frac{\partial v_i}{\partial {\xi }_m}+\frac{\partial }{\partial {\xi }_k}{\mathcal{A}}_{mi}(t))\frac{\partial v_j}{\partial {\xi }_m})}$

by (36)

$<t{>}^b{\Vert \frac{\partial }{\partial {\xi }_k}(D_{ij}(.)\nabla v)\Vert }_{L_q(\Omega )}\le C{[v]}_T\{<t{>}^{b- \frac{N}{2q_2}}{\Vert {\nabla }^{2}v(.t)\Vert }_{L_q(\Omega )}+<t{>}^b{\Vert \nabla v(·,t)\Vert }_{{}_{L_{\infty }(\Omega )}}\}$ (38)

for any $q\in \left[1,q_2\right]$ and therefore

${\Vert <t{>}^b{a}_{sm}(T){\mathcal{l}}_{is}(T)a_{kj}(T)\frac{\partial }{\partial {\xi }_k}(D_{ij}(.)\nabla v)\Vert }_{L_p((0,T),L_q \left(\Omega \right))}\le C{[v]}_T^2$

for any $q\in \left[1,q_2\right]$ . Since

$\frac{\partial }{\partial {\xi }_k}(D_{ij,T}(v))={\displaystyle \sum _{m=1}^N(a_{mj}(T)\frac{{\partial }^2{v}_i}{\partial {\xi }_k\partial {\xi }_m}+a_{mi}(T)\frac{{\partial }^2{v}_j}{\partial {\xi }_k\partial {\xi }_m})}+{\displaystyle \sum _{m=1}^N\text{((}\frac{\partial }{\partial {\xi }_m}{a}_{mj}(T)\frac{\partial v_i}{\partial {\xi }_m}+\frac{\partial }{\partial {\xi }_k}{a}_{mi}(T))\frac{\partial v_j}{\partial {\xi }_m})},$

by (36)

$<t{>}^b{\Vert a_{sm}(T){\mathcal{l}}_{is}(T){\mathcal{A}}_{kj}(t)\frac{\partial }{\partial {\xi }_k}(D_{ij,T}(v)\Vert }_{L_q \left(\Omega \right)}\le C{[v]}_T\{<t{>}^{b- \frac{N}{2q_2}}{\Vert {\nabla }^{2}v(.t)\Vert }_{L_q(\Omega )}+<t{>}^b{\Vert \nabla v(·,t)\Vert }_{{}_{L_{\infty }(\Omega )}}\}$ ,

so that

$<t{>}^b{\Vert a_{sm}(T){\mathcal{l}}_{is}(T){\mathcal{A}}_{kj}(T)\frac{\partial }{\partial {\xi }_k}(D_{ij,T}(v)\Vert }_{L_p((0,T),L_q \left(\Omega \right))}\le C{[v]}_T^2$

for any $q\in \left[1,q_2\right]$ . Analogously, we have

${\Vert <t{>}^b{a}_{sm}(T){\mathcal{L}}_{is}{a}_{kj}\frac{\partial }{\partial {\xi }_k}(D_{ij,T}(v)\Vert }_{L_p((0,T),L_q \left(\Omega \right))}\le C{[v]}_T^2$

for any $q\in \left[1,q_2\right]$ . 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 $f\in H_{\infty }^1(ℝ,L_{\infty }(\Omega ))$ and $g\in H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))$ . Assume that $f\left(x,t\right)=0for\left(x,t\right)\notin B_R\times ℝ$ .

Then,

${\Vert fg\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}\le C{\Vert f\Vert }_{H_{\infty }^1(ℝ,L_{\infty }(\Omega ))}{\Vert g\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_{q_2}(\Omega ))}$ (39)

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

$H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))={(L_p(ℝ,L_q(\Omega )),H_p^1(ℝ,L_q(\Omega )))}_{\left[\frac{1}{2}\right]},$ (40)

where ${(·,·)}_{[\sigma ]}$ denotes a complex interpolation functor. Let $q\in \left[1,q_2\right]$ . Noting that $f\left(x,t\right)=0$ for $\left(x,t\right)\notin B_R\times ℝ$ , we have

${\Vert {\partial }_t(fg)\Vert }_{L_q(\Omega )}\le {\Vert {\partial }_{t}f\Vert }_{L_{\infty }(\Omega )}{\Vert g\Vert }_{L_{q_2}(\Omega )}+{\Vert f\Vert }_{L_{\infty }(\Omega )}+{\Vert {\partial }_{t}g\Vert }_{L_{q_2}(\Omega )},$

and therefore

${\Vert {\partial }_t(fg)\Vert }_{L_p(ℝ,L_q(\Omega ))}\le C{\Vert f\Vert }_{H_{\infty }^1(ℝ,L_{\infty }(\Omega ))}{\Vert g\Vert }_{H_p^1(ℝ,L_{q_2}(\Omega ))}$ .

for any $q\in \left[1,q_2\right]$ . Moreover, we easily see that

${\Vert (fg)\Vert }_{L_p(ℝ,L_q(\Omega ))}\le C{\Vert f\Vert }_{L_{\infty }(ℝ,L_{\infty }(\Omega ))}{\Vert g\Vert }_{L_p(ℝ,L_{q_2}(\Omega ))}.$

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

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

Lemma 6

Let $1 < p < \infty , 1 \le q \le q_2$ and $0 \le a \le b$ . Let $f\in H_{\infty }^1(0,T,L_{\infty }(\Omega ))$ and $g\in H_p^1((0,T),L_{q_2}(\Omega ))\cap L_p((0,T)H_q^2(\Omega ))$ . Assume $f{|}_{t=T} =0$ and $f=0for\left(x,t\right)\notin B_R\times ℝ$ . Let $<t>={(1+t^2)}^{1/2}$ . Then we have

$\begin{array}{l}{\Vert {\tilde{e}}_T (<t{>}^{a}f\nabla g)\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}\le C<t{>}^{\frac{N}{2q_2}}f\Vert {}_{H_{\infty }^1((0,\infty ),L_{\infty }(\Omega ))}\\ \times \left(\Vert <{g\Vert }_{L_p((0,T)H_q^2(\Omega ))}+\Vert <t{>}^{b- \frac{N}{2q_2}}{{\partial }_{t}g\Vert }_{L_p((0,T),L_{q_{{}_2}}\left(\Omega \right))}+{\Vert {g|}_{t=0}\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}\right)\end{array}$ (42)

Proof: Let $f_0(t)=<t{>}^{a-b+\frac{N}{2q_2}}f(t)$ and $g_0 \left(t\right) =< t {>}^{b+\frac{N}{2q_2}}g \left(t\right)$ , and then $< t {>}^a f\nabla g =f_0\nabla g_0$ . Let $\mathcal{T}\left(t\right)$ 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 $h={g|}_{t=0}in\Omega$ and $h={g|}_{t=0}{\Vert h\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}\le {\Vert {g|}_{t=0}\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}$ . Recall the operator eT defined in (32) and note that ${g_0 |}_{t=0}={g |}_{t=0}=\mathcal{T}(t)h|{}_{t=0in\Omega }$ . Let $\tilde{g}(t)=e_T [g_0 -\mathcal{T}(·)h](t)+\mathcal{T}(t)h$

for t > 0 and let

$[\iota g](t)=\{\begin{array}{l}\tilde{g}(t)(t>0),\\ \tilde{g}(-t)(t<0),\end{array}[\iota f](t)=\{\begin{array}{l}0(t>T),\\ f_0(t)(0<t<T),\\ f_0(-t)(-T<t<0),\\ 0(t<-T).\end{array}$

Since $\tilde{g}(t)=g_0 \left(t\right)for0<t<T$ , we have

$\begin{array}{l}{\tilde{e}}_T [<t{>}^a f\nabla g](t)=\{\begin{array}{l}0(t>T),\\ f_0(t)\nabla g_0(t)(0<t<T),\\ f_0(-t)\nabla g_0(-t)(-T<t<0),\\ 0(t<-T).\end{array}=\{\begin{array}{l}0(t>T),\\ f_0(t)\nabla \tilde{g}(t)(0<t<T),\\ f_0(-t)\nabla \tilde{g}(-t)(-T<t<0),\\ 0(t<-T).\end{array}\\ =[\iota f](t)\nabla [\iota g](t).\end{array}$

By Lemma 5,

${\Vert {\tilde{e}}_T [<t{>}^a f\nabla g]\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}={\Vert [\iota f]\nabla [\iota g]\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}\le C{\Vert [\iota f]\Vert }_{H_{\infty }^1(ℝ,L_q(\Omega ))}{\Vert \nabla (\iota g)\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}$

Since, $f_0{|}_{t=T} =0$ we have

${\Vert [\iota f]\Vert }_{H_{\infty }^1(ℝ,L_q(\Omega ))}=2{\Vert [f_0]\Vert }_{H_{\infty }^1((0,T),L_{\infty }(\Omega ))}\le \Vert <t{>}^{\frac{N}{2q_2}}{f\Vert }_{{}_{H_{\infty }^1((0,T),L_{\infty }(\Omega ))}},$

Because $a-b\le 0$ .

To estimate ${\Vert \nabla (\iota g)\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}$ , we use the fact that $H_p^1(ℝ,L_{q_2}(\Omega ))\cap L_p(ℝ,H_{q_2}^2(\Omega ))$ is continuously imbedded into $H_p^{\frac{1}{2}}(ℝ,H_{q_{{}_2}}^1(\Omega ))$ 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

${\Vert \nabla (\iota g)\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_{q_2}(\Omega ))}\le C{\Vert (\iota g)\Vert }_{H_p^1(ℝ,L_{q_2}(\Omega ))}+{\Vert (\iota g)\Vert }_{L_p(ℝ,H_{q_2}^2(\Omega ))}$

$\le C\left({\Vert \tilde{g}\Vert }_{H_p^1((0,\infty ),L_{q_2}(\Omega ))}={\Vert \tilde{g}\Vert }_{L_p((0,\infty ),H_{q_2}^2(\Omega ))}\right)$

$\begin{array}{l}\le C({\Vert g_0\text{-}\mathcal{T}(·)h\Vert }_{H_p^1((0,T),L_{q_2}(\Omega ))}+{\Vert g_0\text{-}\mathcal{T}(·)h\Vert }_{L_p((0,T),H_{q_2}^2(\Omega ))}\\ +{\Vert \mathcal{T}(·)h\Vert }_{H_p^1((0,T),L_{q_2}(\Omega ))}+{\Vert \mathcal{T}(·)h\Vert }_{L_p((0,\infty ),H_{q_2}^2(\Omega ))})\end{array}$

$\le C(\Vert <t{>}^{b- \frac{N}{2q_2}}{{\partial }_{t}g\Vert }_{L_p((0,T),L_{q_{{}_2}}\left(\Omega \right))})+\Vert <t{>}^{b- \frac{N}{2q_2}}{{\partial }_{t}g\Vert }_{L_p((0,T)H_q^2(\Omega ))}+{\Vert {g|}_{t=0}\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}$

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)

${\Vert {\tilde{e}}_T (<t{>}^{a}g\text{(v)}\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}$

$\begin{array}{l}\le {\displaystyle \sum _{j,k=1}^N\Vert }<t{>}^{\frac{N}{2q_2}}(J(T)A_{kj} (·)+T(·)a_{kj} (·))\parallel H_{\infty }^1 ((0,T ),L_{\infty }(\Omega ))\\ \times (\Vert <t{>}^{b- \frac{N}{2q_2}}{\text{v}\Vert }_{L_p((0,T)H_{q_2}^2(\Omega ))}+\Vert <t{>}^{b- \frac{N}{2q_2}}{{\partial }_t\text{v}\Vert }_{L_p((0,T),L_{q_{{}_2}}\left(\Omega \right))}+{\Vert {\text{u}}_0\Vert }_{B_{q_2,p}^{2(1-1/p)}(\Omega )}\\ \le C(\mathcal{I}\text{+}{[}_v^{]})\end{array}$ (43)

for any $a\in \left[0,b\right]$ and $q\in \left[1,q_2\right]$ . Analogously, we have

${\Vert {\tilde{e}}_T (<t{>}^{a}h\text{(v)}\Vert }_{H_p^{\frac{1}{2}}(ℝ,L_q(\Omega ))}\le C(\mathcal{I}\text{+}{[v]}_T^2)$ (44)

for any $a\in \left[0,b\right]$ and $q\in \left[1,q_2\right]$ . Analogously, we have

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

$\begin{array}{l}{\Vert {\tilde{e}}_T [<t{>}^{a}g\text{(v)]}\Vert }_{L_p(ℝ,H_q^1(\Omega ))}\\ \le {\displaystyle \sum _{j,k=1}^N\Vert }<t{>}^{\frac{N}{2q_2}}(J(T)A_{kj}(·)+\mathcal{J}(·)a_{kj} (·)){\Vert }_{L_{\infty }((0,T),L_{\infty }\left(\Omega \right))}\Vert <t{>}^{b- \frac{N}{2q_2}}{\text{v}\Vert }_{L_p((0,T)H_{q_2}^2(\Omega ))}\\ \le C{[}_v^{]}\end{array}$ (45)

for any $a\in \left[0,b\right]$ and $q\in \left[1,q_2\right]$ . Analogously, we have

${\Vert {\tilde{e}}_T (<t{>}^b\text{h(v)}\Vert }_{L_p((0,T)H_q^1(\Omega ))}\le C{[v]}_T^2$ (46)

for any $a\in \left[0,b\right]$ and $q\in \left[1,q_2\right]$ . Since

${\partial }_t{g}_k(\text{v})={\displaystyle \sum _{j=1}^N\Vert }(J(T){\partial }_t{A}_{kj}(t)+({\partial }_t\mathcal{J}(t)a_{kj} (t)+\mathcal{J}(t){\partial }_t{a}_{kj} (t))v_j+{\displaystyle \sum _{j=1}^N\Vert }(J(T)A_{kj}(t)+\mathcal{J}(t)a_{kj} (t))v_j$

and since ${\Vert J(T)a_{kj}(t)\mathcal{J}(t)\Vert }_{L_{\infty }\left(\Omega \right)}\le C$ as follows from (36), by (37) we have

$\begin{array}{l}{\Vert {\tilde{e}}_T [<t{>}^a{\partial }_t{g}_k\text{(v)]}\Vert }_{L_p((0,T),L_q(\Omega ))}\\ \le {\displaystyle \sum _{j=1}^N}(<t{>}^{\frac{N}{2q_2}}{\partial }_t(A_{kj},\mathcal{J},a_{kj}){\Vert }_{L_{\infty }((0,T),L_{\infty }\left(\Omega \right))}\Vert <t{>}^{b- \frac{N}{2q_2}}{\text{v}\Vert }_{L_p((0,T)L_{q_2}\left(\Omega \right)}\\ +(<t{>}^{\frac{N}{2q_2}}(A_{kj},\mathcal{J}){\Vert }_{L_{\infty }((0,T),L_{\infty }\left(\Omega \right))}\Vert <t{>}^{b- \frac{N}{2q_2}}{{\partial }_t\text{v}\Vert }_{L_p((0,T)L_{q_2}\left(\Omega \right)},\end{array}$

which, combined with (36), furnishes that

${\Vert {\tilde{e}}_T [<t{>}^{a}g\text{(v)]}\Vert }_{L_p(ℝ,L_q(\Omega ))}\le C(\mathcal{I}\text{+}{[v]}_T^2)$ (47)

for any $a\in \left[0,b\right]$ and $q\in \left[1,q_2\right]$ .