数理解析研究所講究録
第 1892 巻 2014 年 13-20

13

On a semilinear evolution equation
Kei Matsuura
Research Institute for Science and Engineering,
Waseda University
This talk is based on a joint work with Professor Mitsuharu Otani fromWaseda University.

1

Problem

We shall consider a semilinear parbolic evolution equation arising from fluid
mechanics. More specificaJly, we deal with the following system of equations of
the motion of incompressible micropolar fluids.
$\frac{\partial u}{\partial t}-\mu\Delta u+(u\cdot\nabla)u+\nabla p=^{-}f+2\chi\nabla\cross(\omega-\frac{1}{2}\nabla\cross u)$,$\frac{\partial\omega}{\partial t}-\alpha\triangle\omega-\beta\nabla(\nabla\cdot\omega)+(u\cdot\nabla)\omega=g-4\chi(\omega-\frac{1}{2}\nabla\cross u)$,$\nabla\cdot u=0,$

where the unknown functions are the velocity vector field , the microrotation
vector and the pressure , and and are given vector fields,
areconstants satisfying $\alpha>0,$ $\alpha+\beta>0,$ $\mu>0$ and $\chi>0.$
Micropolar fluid model is one of generalizations of the classical Navier-Stokes
model. We shall discuss the solvability of the initial-boundary value problem ofthis system of equations in a bounded domain with smooth boundaryin$u$$\omega$

$p$

$f$

$g$

$\alpha,$$\beta,$$\Omega$$\mu,$$\chi$$\partial\Omega$$\mathbb{R}^{3}.$As for the boundary conditions, we a,ssume that$u|_{\partial\Omega}=0,$$\omega|_{\partial\Omega}=\frac{\theta}{2}\nabla\cross u|_{\partial\Omega},$

where is a constant belonging to the interval $[0,1]$ (see [1], [3]).
For the case where $\theta=0$ , i.e., both and yield the homogeneous Dirichletboundary conditions, there have been many results concerning this system of
equations [4]. On the other hand, the case whereis not fully pursued yet.
Furthermore, this boundary condition is not based on any physical principles,
the well-posedness of the initial-boundary value problem is not clear. That is
why we consider this problem.
$\theta$$u$$\omega$$\theta\neq 0$

willbeLetusintro.

$duceanewvariab1ev:=\omega-\frac{\theta}{2,f}\nabla\cross uandchangethevariables$$(u, \omega)to(u,v)Thentheorigina1systemoequationsforuand\omega$

14
rewritten in the following form.
$\frac{\partial u}{\partial t}-(\mu+\chi)\Delta u+(u\cdot\nabla)u+\nabla p=f+2\chi\nabla\cross\tau, -\theta\chi\Delta u,$$\frac{\partial\iota}{\partial t}-\alpha\Delta v-\beta\nabla(\nabla\cdot v)+(u\cdot\nabla)v+4\chi v+\theta\chi\nabla\cross(\nabla\cross v)$$=g- \frac{\theta}{2}\nabla\cross f-\frac{\theta}{2}\{(\mu+(1-\theta)\chi-\alpha)\nabla\cross\Delta u$$+((\nabla\cross u)\cdot\nabla)u\}+2(1-\theta)\chi\nabla\cross u,$$\nabla\cdot u=0,$$v|_{\partial t\}}=0,$$u|_{\partial\ddagger l}=0,$$u(\cdot, 0)=u_{0}(\cdot)$,$v(\cdot, 0)=v_{0}(\cdot)$.

Notice that the boundary condition for the new unknown function is reduced
to the homogeneous boundary condition.
Our aim is to show the existence of a solution $(u, v)$ local in time to thissystem of equations in the -framework.
$v$$L^{2}$

2

Function spaces and operators

We refer the details of mathematical facts to be mentioned below to the books[2], [4] and [5].
withLetbe the set of all smooth solenoidal vector functions inincompact support, andthe closure ofand inandthe norm of
the inner productrespectively. We denote by
the inner product of
theof
the norm of,
productofnorm ofthe inner
the norm of. Due to the Poincar\’e inequality, we equip
the inner product ofwith the norm
and$\Omega$$\mathbb{C}_{\sigma}^{\infty}(\Omega)$$\mathbb{H}_{\sigma}^{i}(\Omega)$$\mathbb{L}_{\sigma}^{2}(\Omega)$$\mathbb{H}^{1}(\Omega)$$\mathbb{L}^{2}(\Omega);|\cdot|_{\sigma}$$\mathbb{H}_{0}^{1}(\Omega),$$((\cdot, \cdot))$$((\cdot, \cdot))_{\sigma}$$(\cdot, \cdot)$$\mathbb{L}_{\sigma}^{2}(\Omega);\Vert\cdot\Vert$$(\cdot, \cdot)_{\sigma}$$\mathbb{L}_{\sigma}^{2}(\Omega),$$\mathbb{L}^{2}(\Omega)$$\mathbb{C}_{\sigma}^{\infty}(\Omega)$$\mathbb{L}^{2}(\Omega),$$|\cdot|$$\mathbb{H}_{0}^{1}(\Omega);\Vert\cdot\Vert_{\sigma}$$\mathbb{H}_{\sigma}^{1}(\Omega)$$\mathbb{H}_{\sigma}^{1}(\Omega)$$\mathbb{H}_{\sigma}^{1}(\Omega)$$\mathbb{H}_{0}^{1}(\Omega)$$\Vert u\Vert_{\sigma}:=(\int_{tl}|\nabla u|^{2}dx)^{1/2}$$\Vert\tau,\Vert:=(\int_{\Omega}|\nabla v|^{2}dx)^{1/2}$, respectively.
andtois denoted by $P$ . The Stokes
The orthogonal projection from
$A:=-P\Delta$.
operator is defined bywith. The differLetanddenote the duality pairing between, is defined bytoential operator , which mapsfor$v\in \mathbb{H}_{0}^{1}(\Omega)$$u\in \mathbb{H}_{\sigma}^{1}(\Omega)$$\mathbb{L}^{2}(\Omega)$$A$$\langle\cdot,$$D(A)=\mathbb{H}^{2}(\Omega)\cap \mathbb{H}_{\sigma}^{1}(\Omega)$$\mathbb{H}^{-1}(\Omega)$$\cdot\rangle$$L$$\langle Lv,$

for all$w\rangle$$\mathbb{L}_{\sigma}^{2}(\Omega)$$\mathbb{H}_{0}^{1}(\Omega)$$\mathbb{H}_{0}^{1}(\Omega)$$\mathbb{H}^{-1}(\Omega)$$:= \alpha\int_{\Omega}\nabla v\cdot\nabla wdx+\beta\int_{\Omega}(\nabla\cdot v)(\nabla\cdot w)dx+4\chi\int_{\Omega}$$v,$ $w\in \mathbb{H}_{0}^{1}(\Omega)$

vwdx

(1)

. By integration by parts, we obtain$\int_{\Omega}|\nabla w|^{2}dx=\int_{\Omega}|\nabla\cdot w|^{2}dx+\int_{\Omega}|\nabla\cross w|^{2}dx$

(2)

15
for

$w\in \mathbb{H}_{0}^{1}(\Omega)$. Hence there are positive constants$\alpha_{\star}$and$\alpha^{\star}$such that(3)$\alpha_{\star}|\nabla w|^{2}+4\chi|w|^{2}\leq\langle Lw, w\rangle\leq\alpha^{\star}|\nabla w|^{2}+4\chi|w|^{2}$for$w\in \mathbb{H}_{0}^{1}(\Omega)$

For.

$v\in \mathbb{L}^{2}(\Omega)$, we also define$\nabla\cross?,$$\in \mathbb{H}^{-1}(\Omega)$

by

(4)

$\langle\nabla\cross\uparrow,, w\rangle:=\int_{\Omega}v\nabla\cross wdx$

with.
We identify the spacewith its dual. Then
, wherestands for the norm of the space$w\in \mathbb{H}_{0}^{1}(\Omega)$$\mathbb{L}^{2}(\Omega)$$v\in \mathbb{L}^{2}(\Omega)$

3$\Vert\nabla\cross v\Vert_{\star}\leq|v|$$\Vert\cdot\Vert_{\star}$$\mathbb{H}^{-1}(\Omega)$.

holds for

Main result

In the above settings, our system can be regarded as a system of abstract evolution equations of the following form.
$\frac{du}{dt}+(\mu+\chi)Au=Pf+b_{1}(u, v)$

in$\frac{dv}{dt}+Lv=g-\frac{\theta}{2}\nabla\cross f+b_{2}(u, v)$

in$\mathbb{L}_{\sigma}^{2}(\Omega)$,$\mathbb{H}^{-1}(\Omega)$,$u(0)=u_{0}, v(0)=e,0,$

where $b_{1}(u, v)$ and $b_{2}(u, v)$ are defined by$b_{1}(u, v)$ $:=2\chi\nabla\cross v+\theta\chi Au-P(u\cdot\nabla)u,$$b_{2}(u, v):=- \frac{\theta}{2}\{(\mu+(1-\theta)\chi-\alpha)\nabla\cross\Delta u+((\nabla\cross u)\cdot\nabla)u\}$$-(u\cdot\nabla)\tau, +2(1-\theta)\chi\nabla\cross u.$

Notice thatand if$\nabla\cross w\in \mathbb{L}_{\sigma}^{2}(\Omega)$$w_{0}\in \mathbb{C}_{c}^{\infty}(\Omega)$andwheneversince
is dense in, the integration by parts gives$w\in \mathbb{H}_{0}^{1}(\Omega)$$\mathbb{C}_{c}^{\infty}(\Omega)$$\mathbb{H}_{0}^{1}(\Omega)$$q\in \mathbb{H}^{1}(\Omega)$$(\nabla\cross w_{0}, \nabla q)=(-\nabla\cdot(\nabla\cross w_{0}), q)=0.$

Now our main result reads as follows.

Theorem There exists a constantsatisfying the following property:
Given$T>0,$
and, there exist a $T_{\star}\in(0, T] and a$ unique solution $(u, v)$ to
our system on the time interval
with$\theta_{0}\in(0,1]$$\theta\in[0, \theta_{0}],$$(u_{0}, v_{0})\in \mathbb{H}_{\sigma}^{1}(\Omega)\cross \mathbb{L}^{2}(\Omega),$$g\in L^{2}(0, T;\mathbb{H}^{-1}(\Omega))$$(0, T_{\star})$$u\in C([0, T_{\star}];\mathbb{H}_{\sigma}^{1}(\Omega))\cap W^{1,2}(0, T_{\star};\mathbb{L}_{\sigma}^{2}(\Omega))$,$v\in C([0, T_{\star}];\mathbb{L}^{2}(\Omega))\cap W^{1,2}(0, T_{\star};\mathbb{H}^{-1}(\Omega))$.

$f\in L^{2}(0, T;\mathbb{L}^{2}(\Omega))$16

Sketch of a proof of Theorem

4

Our idea for proof is the following: First, solve a linear problem
in$\frac{du}{dt}+(\mu+\chi)Au=Pf+h$$\frac{dv}{dt}+Lc,$ $=g- \frac{\theta}{2}\nabla\cross f+k$$u(O)=u_{0},$$L_{\sigma}^{2}(\Omega)$,$in\mathbb{H}^{-1}(\Omega)$,$v(O)=v_{0},$andfor given
there is a unique solution $(u, v)$ such that$k\in L^{2}(0, T;\mathbb{H}^{-1}(\Omega))$$h\in L^{2}(0, T;\mathbb{L}_{\sigma}^{2}(\Omega))$$u\in C([O, T);\mathbb{H}_{\sigma}^{1}(\Omega))\cap W^{1,2}(0, T;\mathbb{L}_{\sigma}^{2}(\Omega))$. It is well-known that,$v\in C([O, T);\mathbb{L}^{2}(\Omega))\cap W^{1,2}(0, T;\mathbb{H}^{-1}(\Omega))$.

into
Then one can define a mapping from
itself as $S(h, k)$ $:=(b_{1}(u, v), b_{2}(u, v))$ . If has a fixed point, then $(u, v)$ is aandare sums of terms which are linear
solution of our problem. Sinceor quadratic in and , it is natural to expect that would be a contraction
mapping. This conjecture turns out to be true with the following trick. Let$\eta\in(0,1]$ be a constant to be fixed later and$:=\eta u$ .Then our system becomes
as follows.
$S$$L^{2}(0, T;\mathbb{L}_{\sigma}^{2}(\Omega))\cross L^{2}(0, T;\mathbb{H}^{-1}(\Omega))$$S$$b_{2}$$b_{1}$

$u$

$S$

$v$

$U$

in

$\frac{dU}{dt}+(\mu+\chi)AU=\eta Pf+B_{1}(U_{t},)$$\frac{d\tau}{dt}+L\tau,$$=g- \frac{\theta}{2}\nabla\cross f+B_{2}(U, \tau,)$$U(O)=\eta u_{0},$in$\mathbb{L}_{\sigma}^{2}(\Omega)$,$\mathbb{H}^{-1}(\Omega)$,$v(O)=v_{0},$where $B_{1}(U, v)$ and $B_{2}(U, v)$ are defined by$B_{1}(U, c,)$$:=2\eta\chi\nabla\cross\tau,$$+ \theta\chi AU-\frac{1}{\eta}P(U\cdot\nabla)U,$$B_{2}(U, v):=- \frac{\theta}{2\eta}(\mu+(1-\theta)\chi-\alpha)\nabla\cross\Delta U-\frac{\theta}{2\eta^{2}}((\nabla\cross U)\cdot\nabla)U$$- \frac{1}{\eta}(U\cdot\nabla)\uparrow, +\frac{2(1-\theta)\chi}{\eta}\nabla\cross U.$

We are going to show this modified system of equations has a solution $(U, v)$ .
and be given and take a positive number
Let data $T>0,$satisfying$u_{0},$$v_{0},$

$f$

$g$

$R$

$R \geq\max\{\Vert u0\Vert_{\sigma}, |v0|, \Vert f\Vert_{L^{2}(0,T;L^{2}(\Omega))}, \Vert g\Vert_{L^{2}(0,T;\mathbb{H}^{-1}(\Omega))}\}.$

Let be a positive number in $(0, T]$ and is also to be fixed later. Denote byandthe set of functions $(h, k)$ such that
andwith
It is well-known that there is a unique solution $(U, v)$ to the problem$\tau$$h\in L^{2}(0, \tau;\mathbb{L}_{\sigma}^{2}(\Omega))$$\Vert h\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(t1))}\leq R$$\frac{d\tau}{dt}+Lv=g-\frac{\theta}{2}\nabla\cross f+k$$v(O)=v_{0}$$k\in L^{2}(0, \tau;\mathbb{H}^{-1}(\Omega))$$\Vert k\Vert_{L^{2}(0_{;^{\tau;\mathbb{H}^{-1}(\Omega))}}}\leq R.$$\frac{dU}{dt}+(\mu+\chi)AU=\eta Pf+h$$U(O)=\eta u_{0},$$\mathcal{B}_{R}$,$\mathbb{L}_{\sigma}^{2}(\Omega)$,(5)$\mathbb{H}^{-1}(\Omega)$

,

(6)

in

in

(7)

17
which satisfies,$U\in C([0, \tau);\mathbb{H}_{\sigma}^{1}(\Omega))\cap L^{2}(0, \tau;\mathbb{H}^{2}(\Omega)\cap \mathbb{H}_{\sigma}^{1}(\Omega))\cap W^{1,2}(0, \tau;\mathbb{L}_{\sigma}^{2}(\Omega))$$v\in C([0, \tau);\mathbb{L}^{2}(\Omega))\cap L^{2}(0, \tau;\mathbb{H}_{0}^{1}(\Omega))\cap W^{1,2}(0, \tau;\mathbb{H}^{-1}(\Omega))$

Multiplying (5) by.

, we get$U$$\frac{1}{2}\frac{d}{dt}|U|_{\sigma}^{2}+(\mu+\chi)\Vert U\Vert_{\sigma}^{2}\leq C(|f|+|h|_{\sigma})\Vert U\Vert_{\sigma}.$

Here and henceforth
may depend only on
we have$C$$\mu,$

or$\chi,$$\alpha,$( is a positive number) denotes a constant which
and may take different values line by line. Then$i$

$C_{i}$$\beta,$$\Omega$$\Vert U\Vert_{L^{\infty}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))}\leq C_{1}R,$$\Vert U\Vert_{L^{2}(0,\tau;\mathbb{H}_{\sigma}^{1}(\Omega))}\leq C_{2}R.$

Multiplying (5) by $AU$ , we have$\frac{1}{2}\frac{d}{dt}\Vert U\Vert_{\sigma}^{2}+(\mu+\chi)|AU|_{\sigma}^{2}\leq C(|f|+|h|_{\sigma})|AU|_{\sigma},$whence follows$\Vert U\Vert_{L\infty(0,\tau;\mathbb{H}_{\sigma}^{1}(\zeta)))}\leq C_{3}R,$$\Vert U\Vert_{L^{2}(0,\tau;\mathbb{H}_{\sigma}^{1}(f1)\cap \mathbb{H}^{2}(tl))}\leq C_{4}R.$

We here use the estimate from the elliptic regularity theory:$\Vert w\Vert_{\mathbb{H}^{2}(\Omega)}\leq C_{0}\Vert Aw\Vert_{\sigma},$which holds for $w\in D(A)$ .
Taking the duality pairing between (6) and , we obtain$v$$\frac{1}{2}\frac{d}{dt}|v|^{2}+\alpha_{\star}\Vert v\Vert^{2}+4\chi|v|^{2}\leqC(\Vert g\Vert_{\star}+|f|+|k|)\Vert v\Vert.$

From this it follows that$\Vert v\Vert_{L^{\infty}(0,\tau;\mathbb{L}^{2}(\Omega))}\leq C_{5}R,$$\Vert v\Vert_{L^{2}(0,\tau;\mathbb{H}_{0}^{1}(\Omega))}\leq C_{6}R.$Now we shall show
and can be chosen so that $(B_{1}(U, v), B_{2}(U, v))$ also
belongs to the set
Let and be scalar functions.
denotes any one of the partial derivativeof . We need the following well-known inequalities in order to estimate thenonlinear termes.
If, we have$\theta$$\eta,$$\tau$$\mathcal{B}_{R}.$$\phi$$\psi$$D\phi$$\phi$$\phi\in H_{0}^{1}(\Omega)$$\Vert\phi\Vert_{L^{3}(\Omega)}\leq C\Vert\phi\Vert_{L^{2}(\Omega)}^{1/2}\Vert\phi\Vert_{H^{1}(\Omega)}^{1/2},$$\Vert\phi\Vert_{L^{6}(\Omega)}\leq C\Vert\phi\Vert_{H^{1}(\Omega)}.$18
If we assume further that$\Vert\phi$I$\phi\in H^{2}(\Omega)$, then$\phi\in L^{\infty}(\Omega)$and$L^{\infty}(\Omega)\leq C\Vert\phi\Vert_{H^{1}(\Omega)}^{1/2}\Vert\phi\Vert_{H^{2}(\Omega)}^{1/2}.$

Ifbelongs to$\phi\in H_{0}^{1}(\Omega)$and$L^{2}(\Omega)$$\psi\in H^{2}(\Omega)$or$\phi\cdot\in H^{2}(\Omega)$and$\psi\in H_{0}^{1}(\Omega)$, the product$\phi D\psi$and$\Vert\phi D\psi\Vert_{L^{2}}\leq\{\begin{array}{l}C\Vert\phi\Vert_{H^{1}}^{1/2}\Vert\phi\Vert_{H^{2}}^{1/2}\Vert\psi\Vert_{H^{1}} for \phi\in H^{1}(\Omega), \psi\in H^{2}(\Omega) ,C\Vert\phi\Vert_{H^{1}}\Vert\psi\Vert_{H^{2}}^{1/2}\Vert\psi\Vert_{H^{1}}^{1/2} for \phi\in H^{2}(\Omega), \psi\in H^{1}(\Omega) .\end{array}$(8)

From these estimate, we obtain$\int_{0}^{\tau}|P(U\cdot\nabla)U(s)|_{\sigma}^{2}ds\leq\int_{0}^{\tau}|(U\cdot\nabla)U(s)|^{2}ds$$\leq C_{7}\int_{0}^{\tau}\Vert U(s)\Vert^{3}\Vert U(s)\Vert_{E^{2}}ds$$\leq C_{3}^{3}C_{7}R^{3}\int_{0}^{\tau}\Vert U(s)\Vert_{\mathbb{H}^{2}}ds$$\leq C_{3}^{3}C_{7}R^{3}\tau^{1/2}\Vert U\Vert_{L^{2}(0,\tau;\mathbb{H}^{2}(\Omega)\cap \mathbb{H}_{\sigma}^{1}(\Omega))}$$\leq C_{3}^{3}C_{4}C_{7}R^{4}\tau^{1/2}.$

Therefore$\Vert B_{1}(U, v)\Vert_{L^{2}(0,\tau;L_{\sigma}^{2}(t1))}$$\leq 2\eta\chi\Vert\nabla\cross v\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))}+\theta\chi\Vert AU\Vert_{L^{2}(0,\tau;L_{\sigma}^{2}(\Omega))}+\frac{1}{\eta}\Vert(U\cdot\nabla)U\Vert_{L^{2}(0,\tau;L^{2}(\Omega))}$$\leq(2C_{6}\eta\chi+C_{4}\theta\chi+\frac{C_{3}^{3/2}C_{4}^{1/2}C_{7}^{1/2}R\tau^{1/4}}{\eta})R.$

Suppose thatvanishes on the boundary$w_{1},$ $w_{2},$$w_{3}\in \mathbb{H}^{1}(\Omega),$$\partial\Omega$$\nabla\cdot w_{1}=0$. Thenand at least one of these functions
is well-defined and it holds$((w_{1}\cdot\nabla)w_{2}, w_{3})$that $((w_{1}\cdot\nabla)w_{2}, w_{3})=-((w_{1}\cdot\nabla)w_{3}, w_{2})$ .
we haveFor$w\in \mathbb{H}_{0}^{1}(\Omega)$$|\langle\nabla\cross\Delta U,w\rangle|=|(\Delta U,\nabla\cross w)|$$\leq C_{8}\Vert U\Vert_{\mathbb{H}^{2}(t2)}||w\Vert,$$|\langle((\nabla\cross U)\cdot\nabla)U,w\rangle|=|(((\nabla\cross U)\cdot\nabla)U,w)|$$=|-(((\nabla\cross U)\cdot\nabla)w,U)|$$\leq C\Vert\nabla U\Vert_{L^{3}(\Omega)}\Vert\nabla w\Vert_{L^{2}(\Omega)}\VertU\Vert_{\mathbb{L}^{6}(\Omega)}$$\leq C_{9}\Vert U\Vert_{\sigma}^{3/2}\Vert U\Vert_{\mathbb{H}^{2}}^{1/2}\Vert w\Vert,$$|\langle(U\cdot\nabla)\tau,,w\rangle|=|((U\cdot\nabla)v,w)|$$=|-((U\cdot\nabla)w,v)|$$\leq C\Vert U\Vert_{\mathbb{L}^{6}(\Omega)}\Vert\nabla w\Vert_{\mathbb{L}^{2}(\Omega)}\Vert\tau,\Vert_{L^{3}(\Omega)}$$\leq C_{10}\Vert U\Vert_{\sigma}|v|^{1/2}\Vert v\Vert^{1/2}\Vert w\Vert$$|\langle\nabla\cross U,w\rangle|\leq|U|_{\sigma}\Vert w\Vert,$

19
and further$\Vert B_{2}(U, v)\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(\Omega))}$$\leq[\frac{\theta}{2\eta}C_{4}C_{8}(\mu+\chi+\alpha)+\frac{\theta}{2\eta^{2}}C_{3}^{3/2}C_{7}^{1/2}C_{9}R\tau^{1/4}$$+ \frac{C_{3}C_{5}^{1/2}C_{6}^{1/2}C_{10}R\tau^{1/4}}{\eta}+\frac{2\chi C_{1}\tau^{1/2}}{\eta}]R.$

Next, let $(h_{i}, k_{i})(i=1,2)$ be taken fromsolution of$\mathcal{B}_{R}$and $(U_{i}, v_{i})(i=1,2)$ be the$\frac{dU_{i}}{dt}+(\mu+\chi)AU_{i}=\eta Pf+h_{i},$$\frac{dv_{i}}{dt}+Lv_{i}=g-\frac{\theta}{2}\nabla\cross f+k_{i},$$U_{i}(0)=\eta u_{0}, \tau_{i}(0)=c_{0}.$Then it is easy to see that the differences
estimated as$\tilde{U}$$:=U_{1}-U_{2}$and$\tilde{v}:=v_{1}-v_{2}$can be$\Vert\tilde{U}\Vert_{L^{\infty}(0,\tau;\mathbb{L}_{\sigma}^{2}(Il))}\leq C_{11}\Vert\tilde{h}\Vert_{L^{2}(0,\tau;L_{\sigma}^{2}(1l))},$$\Vert\tilde{U}\Vert_{L^{2}(0,\tau;\mathbb{H}_{\sigma}^{1}(\Omega))}\leq C_{12}\Vert\tilde{h}\Vert_{L^{2}(0,\tau;L_{\sigma}^{2}(\Omega))},$$\Vert\tilde{U}\Vert_{L\infty(0,\tau;\mathbb{H}_{\sigma}^{1}(\Omega))}\leq C_{13}\Vert\tilde{h}\Vert_{L(0,\tau;Il_{\sigma}(\Omega))}22\lrcorner,$$\Vert\tilde{U}\Vert_{L^{2}(0,\tau;\mathbb{H}_{\sigma}^{1}(\Omega)\cap \mathbb{H}^{2}(\Omega))}\leq C_{14}\Vert\tilde{h}\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))},$$\Vert\tilde{v}\Vert_{L\infty(0,\tau;\mathbb{L}^{2}(\Omega))}\leq C_{15}\Vert\tilde{k}\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(\Omega))},$$\Vert\tilde{v}\Vert_{L^{2}(0,\tau;\mathbb{H}_{0}^{1}(\zeta\}))}\leq C_{16}\Vert\tilde{k}\Vert_{L^{2}(0_{\mathcal{T}};\mathbb{H}^{-1}(\Omega))}.$

Since$B_{1}(U_{1}, v_{1})-B_{1}(U_{2}, v_{2})=2\eta\chi\nabla\cross\tilde{t,}+\theta\chi A\tilde{U}-\frac{1}{\eta}[P(U_{1}\cdot\nabla)\tilde{U}+P(\tilde{U}\cdot\nabla)U_{2}],$then$\Vert B_{1}(U_{1}, v_{1})-B_{1}(U_{2}, v_{2})\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))}$$\leq 2\eta\chi C_{16}\Vert\tilde{k}\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(t1))}+[\theta\chi C_{14}+\frac{2C_{3}C_{13}^{1/2}C_{14}^{1/2}R\tau^{1/4}}{\eta}]\Vert\tilde{h}\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))}.$

Similarly we obtain$\Vert B_{2}(U_{1}, v_{1})-B_{2}(U_{2}, v_{2})\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(t\}))}$$\leq[\frac{\theta}{2\eta}C_{14}(\mu+\chi+\alpha)+\frac{C_{2}^{1/2}C_{6}^{1/2}C_{13}R\tau^{1/4}}{\eta}+\frac{\theta C_{3}^{1/2}C_{4}^{1/2}C_{13}R\tau^{1/4}}{\eta^{2}}$$+ \frac{2\chi C_{11}\tau^{1/2}}{\eta}]\Vert\tilde{h}\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))}+\frac{C_{3}C_{15}^{1/2}C_{16}^{1/2}R\tau^{1/4}}{\eta}\Vert\tilde{k}\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(\ddagger l))}.$

Now, set the number $\eta\in(0,1]$ so that the following inequalities hold:$2C_{6} \chi\eta\leq\frac{1}{2},$$2C_{16} \chi\eta\leq\frac{1}{4}.$20After that, chose$\theta\in(0,1]$

and $\tau\in(0, T]$ so small that$\Vert B_{1}(U_{8},)\Vert_{L^{2}(0,\tau;L_{\sigma}^{2}(t)))}\leq R,$$\Vert B_{2}(U, \tau)\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(t1))}\leq R,$$||B_{1}(U, v)-B_{1}(U, v) \Vert_{L^{2}(0,\tau;L_{\sigma}^{2}(\Omega))}\leq\frac{1}{2}\Vert\tilde{h}\Vert_{L^{2}(0,\tau;\mathbb{L}_{\sigma}^{2}(\Omega))},$$\Vert B_{2}(U, \not\in,)-B_{2}(U, \tau,).\Vert_{L^{2}(0_{\mathcal{T}};\mathbb{H}^{-1}(t1))}\leq\frac{1}{2}\Vert\tilde{k}\Vert_{L^{2}(0,\tau;\mathbb{H}^{-1}(t1))}.$

Thus the mapping $(h, k)\mapsto(B_{1}(U, v), B_{2}(U, v))$ turns out to be a contraction,and the existence of a solution to our problem follows. The uniqueness of solution $(U, v)$ logically follows from the above argument.

References[1] E. L. Aero, A. N. Bulygin and E. V. Kuvshinskii, Asymmetric hydromechanics, Prikl. Mat. Mec., Vol.29, no.2 (1965), 297-308.
[2] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
[3] D. W. Condiff and J. S. Dahler, Fluid mechanical aspects of antisymmetricstress, Phys. Fluids 7 (1964), 842-854;[4] G. Lukaszewicz, Micropolar fluids, Theory and applications. Modelingand Simulation in Science, Engineering and Technology, Birkh\"auser, Inc.,
Boston, MA, 1999.
[5] R. Temam, Navier-Stokes equations. Theory and numerical analysis. Reprintof the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001.