Initial-boundary value problems for 2D Navier-Stokes equations posed on rectangles and on a half-strip were considered. The existence and uniqueness of regular global solutions on rectangles and their exponential decay as well as exponential decay of generalized solutions on a half-strip have been established.

Keywords: navier-stokes equations, lipschitz and smooth domains, decay in bounded and unbounded domains

The main goal of this work is establishing of sharp estimates for the exponential decay rates of solutions to initial-boundary value problems for the 2D Navier-Stokes equations:

$u_t+\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo>=</mo>}\nu \Delta u-\nabla p\mathrm{,}in\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}$  (1)

$\nabla u\mathrm{<mo>=</mo>0}in\Omega \mathrm{,}u{\mathrm{<mo>|</mo>}}_{\partial \Omega }\mathrm{<mo>=</mo>0,}$  (2)

$u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,0<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_0\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>,}$  (3)
where $\Omega$ is either a bounded rectangle or a half-strip in ${ℝ}^2$  with the homogeneous Dirichlet condition on the boundary of $\Omega .$

The question of decay of the energy for generalized solutions had been stated by J.Leray¹ and attracts till now attention of many pure and applied mathematicians^2‒11. In all of these papers, the decay rate of $\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}$  was controlled by the first eigenvalue of the operator $A\mathrm{<mo>=</mo>}-P\Delta ,$ where $P$ is the projection operator on solenoidal subspace of $L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$  Associated with stability questions, problems on dimensions of attractors and nonlinear spectral manifolds also have been studied.^2,6,7

It is well-known that solutions of the 2D Navier-Stokes equations posed on smooth bounded domains with the Dirichlet boundary conditions are globally regular.^9,11‒14 On the other hand, the question of regularity is not obvious in the case of bounded Lipschitz domains and unbounded Lipschitz and smooth domains. It has been proved that for Lipschitz domains, bounded and unbounded, there exists a unique global generalized solution.^9,11,14

$u\mathrm{,}{u}_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{L}^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{H}^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}$

but it was not clear whether

$u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{H}^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

at least for bounded Lipschitz domains.

In this work, we have established this fact for rectangles making use of ideas.¹⁵ The following inequality holds for rectangles

$\parallel u{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u_t{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C\parallel u_0{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\exp \mathrm{<mo\; stretchy="false">(</mo>}-\nu \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\pi }^2}{L^2}+\frac{{\pi }^2}{B^2}\mathrm{<mo\; stretchy="false">)</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$

and
$\parallel u{\parallel }_{H^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u_t{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C\parallel u_0{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\exp \mathrm{<mo\; stretchy="false">(</mo>}-\frac{\nu {\pi }^2}{B^2}t\mathrm{<mo\; stretchy="false">)</mo>}$

for a half-strip.

Our paper has the following structure: Chapter I is Introduction. Chapter 2 contains notations and auxiliary facts. In Chapter 3, existence and uniqueness of global generalized solutions on either bounded or unbounded Lipschitz domains have been established. In Chapter 4, regularity and decay of solutions on rectangles and on a half-strip have been studied.

Let $\Omega$ be a domain in ${ℝ}^2\mathrm{.}$  Define as in:¹¹

$D_x\mathrm{<mo>=</mo>}\frac{\partial }{\partial x}\mathrm{,}{D}_y\mathrm{<mo>=</mo>}\frac{\partial }{\partial y}\mathrm{,}{D}^j\mathrm{<mo>=</mo>}{D}_x{}^{j_x}{D}_y{}^{j_y}\mathrm{<mo>=</mo>}\frac{{\partial }^{\mathrm{<mo>|</mo>}{j}_x+j_y\mathrm{<mo>|</mo>}}}{\partial x^{j_x}\partial y^{j_y}}\mathrm{.}$

We denote for scalar functions $f(x,y,t)$  by $L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>,}\mathrm{1<mo><</mo>}p\mathrm{<mo><</mo>}+\infty$  the Banach space with the norm

$\parallel f{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^p\mathrm{<mo>=</mo>}{\displaystyle {\int }_{\Omega }}\mathrm{<mo>|</mo>}f{\mathrm{<mo>|</mo>}}^{p}dxdy\mathrm{,}p\in \mathrm{<mo\; stretchy="false">(</mo>1,}+\infty \mathrm{<mo\; stretchy="false">)</mo>,}\parallel f{\parallel }_{L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>}D\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}esssu{p}_D\mathrm{<mo>|</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>.}$

For $p\mathrm{<mo>=</mo>2,}{L}^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega$  is a Hilbert space with the scalar product

$\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\Omega }}u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>}v\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>}dxdyandthenorm\parallel u{\parallel }^2\mathrm{<mo>=</mo>}{\displaystyle {\int }_{\Omega }}\mathrm{<mo>|</mo>}u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^{2}dxdy\mathrm{.}$

The Sobolev space $W^{m\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$  is a Banach space with the norm
$\parallel u{\parallel }_{W^{k\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{0\le \mathrm{<mo>|</mo>}\alpha \mathrm{<mo>|</mo>}\le k}}\parallel D^{\alpha }u{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$
When $p\mathrm{<mo>=</mo>2,}{W}^{m\mathrm{,2}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{H}^m\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$  is a Hilbert space with the following scalar product and the norm:

${\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}}_{H^m\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{\mathrm{<mo>|</mo>}j\mathrm{<mo>|</mo>}\le m}}\mathrm{<mo\; stretchy="false">(</mo>}{D}^{j}u\mathrm{,}{D}^{j}v\mathrm{<mo\; stretchy="false">)</mo>,}\parallel u{\parallel }_{H^m\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\mathrm{<mo>=</mo>}{\displaystyle \sum _{\mathrm{<mo>|</mo>}j\mathrm{<mo>|</mo>}\le m}}\parallel D^{j}u{\parallel }^2\mathrm{.}$

Let $\mathcal{D}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$ or $\mathcal{D}\mathrm{<mo\; stretchy="false">(</mo>}\overline{\Omega }\mathrm{<mo\; stretchy="false">)</mo>}$ be the space of $C^{\infty }$  functions with compact support in $\Omega$  or $\overline{\Omega }$ . The closure of $C^{\infty }$  functions in $W^{m\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$  is denoted by $W_0^{m\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$  and $\mathrm{<mo\; stretchy="false">(</mo>}{H}_0^m\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}whenp=2).$

Define the auxiliary spaces which are projections for the solenoidal vector functions,

$\mathcal{V}\mathrm{<mo>=</mo><mo>\{</mo>}u\in \mathcal{D}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>,}\nabla u\mathrm{<mo>=</mo>0<mo>\}</mo>,}V\mathrm{<mo>=</mo>}\text{the}\text{closure}\text{of}\mathcal{V}\text{in}{H}_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>,}$

$H\mathrm{<mo>=</mo>}\text{the}\text{closure}\text{of}\mathcal{V}\text{in}{L}^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>,}$

The space $H$ is eqquiped with the natural $L^2$ inner product. The space $V$ will be equipped with the scalar produt
$\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}{D}_{x}u\mathrm{,}{D}_{x}v\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>}{D}_{y}u\mathrm{,}{D}_{y}v\mathrm{<mo\; stretchy="false">)</mo>}$

when $\Omega$ is bounded. If $\Omega$ is unbounded, we define the inner product as the sum of the inner products as following:

$\mathrm{<mo\; stretchy="false">[</mo><mo\; stretchy="false">[</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">]</mo><mo\; stretchy="false">]</mo><mo>=</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>.}$

We use the usual notations of Sobolev spaces $W^{k\mathrm{,}p},$ $L^p$ and $H^k$ for vector functions and the following notations for the norms:
i) For vector functions $u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}{u}_1\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,}t\mathrm{<mo\; stretchy="false">)</mo>,}{u}_2\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,}t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}$

$\parallel u{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^p\mathrm{<mo>=</mo>}{\displaystyle {\int }_{\Omega }}\mathrm{<mo\; stretchy="false">(</mo><mo>|</mo>}{u}_1{\mathrm{<mo>|</mo>}}^p+\mathrm{<mo>|</mo>}{u}_2{\mathrm{<mo>|</mo>}}^p\mathrm{<mo\; stretchy="false">)</mo>}dxdy\mathrm{,}$

$\parallel u{\parallel }_{W^{k\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{0\le \mathrm{<mo>|</mo>}\alpha \mathrm{<mo>|</mo>}\le k}}\parallel D^{\alpha }{u}_1{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}+\parallel D^{\alpha }{u}_2{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}p\in \mathrm{<mo\; stretchy="false">(</mo>1,}+\infty \mathrm{<mo\; stretchy="false">)</mo>.}$

The closures of $\mathcal{V}$ in $L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$ and in $H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$ are the basic spaces in our study. We denote them by $H$ and $V$ respectively.. Obviously $V$ is a subspace of $H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$

Define the operator

$\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>}{u}_1{u}_{1x}+u_1{u}_{2x}+u_2{u}_{1y}+u_2{u}_{2y}\mathrm{<mo\; stretchy="false">)</mo>.}$

Lemma 4.1 (The Steklov Inequality)¹⁶ Let $v\in H_0^1\mathrm{<mo\; stretchy="false">(</mo>0,}L\mathrm{<mo\; stretchy="false">)</mo>.}$  Then

$\frac{{\pi }^2}{L^2}\parallel v{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel v_x{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$  (4)

Proof. Let $v\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\in H_0^1\mathrm{<mo\; stretchy="false">(</mo>0,}\pi \mathrm{<mo\; stretchy="false">)</mo>}$ , then by the Fourier series,

${\displaystyle {\int }_0^{\pi }}{v}_t^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}dt\ge {\displaystyle {\int }_0^{\pi }}{v}^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}dt\mathrm{.}$

Inequality (4) follows by a simple scaling.

Lemma 4.2 (Differential form of the Grownwall Inequality) Let $I\mathrm{<mo>=</mo><mo\; stretchy="false">[</mo>}{t}_0\mathrm{,}{t}_1\mathrm{<mo\; stretchy="false">]</mo>.}$ Suppose that functions $a\mathrm{,}b\mathrm{<mo>:</mo>}I\to ℝ$  are integrable and a function $a\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$ may be of any sign. Let $u\mathrm{<mo>:</mo>}I\to ℝ$  be a differentiable function satisfying

$u^{\prime }\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le a\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+b\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}\text{for}t\in I\text{and}u\mathrm{<mo\; stretchy="false">(</mo>}{t}_0\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_0\mathrm{,}$  (5)

then
$u\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le u_0{e}^{{\displaystyle {\int }_{t_0}^t}a\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}dt}+{\displaystyle {\int }_{t_0}^t}{e}^{{\displaystyle {\int }_{t_0}^s}a\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}dr}b\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}ds\mathrm{.}$  (6)

Proof. Multiply (5) by the integrating factor $e^{{\displaystyle {\int }_{t_0}^s}a\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}dr}$ and integrate from $t_0$ to $t.$

The next Lemmas will be used in estimates:

Lemma 4.3 (See: ^11,14 ) Let $v\in H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$ , then

$\parallel v{\parallel }_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le 2^{\mathrm{1<mo>/</mo>4}}\parallel v{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^{\mathrm{1<mo>/</mo>2}}\parallel \nabla v{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^{\mathrm{1<mo>/</mo>2}}\mathrm{.}$  (7)

Lemma 4.4 (See: ¹⁴ ) Let $b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{,}w\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}v\mathrm{,}w\mathrm{<mo\; stretchy="false">)</mo>,}$  then

$\mathrm{<mo>|</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{,}w\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}\le 2^{\mathrm{1<mo>/</mo>2}}\parallel u{\parallel }^{\mathrm{1<mo>/</mo>2}}\parallel u{\parallel }_V^{\mathrm{1<mo>/</mo>2}}\parallel v{\parallel }_V\parallel w{\parallel }^{\mathrm{1<mo>/</mo>2}}\parallel w{\parallel }_V^{\mathrm{1<mo>/</mo>2}}$

$\forall u\mathrm{,}v\mathrm{,}w\in H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}$ . If $u\in L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}\cap L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}$ , then we can define the operator $Bu$ such that $Bu$ belongs to $L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{V}^{\prime }\mathrm{<mo\; stretchy="false">)</mo>}$  and
$\mathrm{<mo\; stretchy="false">(</mo>}Bu\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo>,}$

$\parallel B_u{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{V}^{\prime }\mathrm{<mo\; stretchy="false">)</mo>}}\le 2^{\mathrm{1<mo>/</mo>2}}\parallel u{\parallel }_{L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}}\parallel u{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$

Let $\Omega$  be a bounded Lipschitz domain. Given $u_0\in H,$ consider the following problem:

$(\begin{array}{c}{u}_t-\nu \Delta u+\nabla p+\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo>=</mo>0}\text{in}\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}\\ \nabla u\mathrm{<mo>=</mo>0}\text{in}\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}\\ u\mathrm{<mo>=</mo>0}\text{on}\partial \Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}t\mathrm{<mo>></mo>0,}\\ u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,0<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_0\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>,}\text{in}\Omega \end{array}$  (8)

equivalent to the variational problem given by,¹¹

$(\begin{array}{c}{u}^{\prime }+Au+Bu\mathrm{<mo>=</mo>0}\text{in}\mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}t\mathrm{<mo>></mo>0}\\ u\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_0\mathrm{,}\end{array}$  (9)

where $Au\in V^{\prime }$ such that $\mathrm{<mo\; stretchy="false">(</mo>}Au\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}-\nu \mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$ for all $v\in V$ and $Bu\in V^{\prime }$ such that

$\mathrm{<mo\; stretchy="false">(</mo>}Bu\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo>.}$  (10)

Theorem 5.1 Given $u_0\in H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}\cap V,$ there exists a unique generalized solution $u$ to (8) such that for all $\Phi \in V\mathrm{,}\Phi {\mathrm{<mo>|</mo>}}_{\partial \Omega }\mathrm{<mo>=</mo>0}$  it satisfies the following integral identity:

${\displaystyle {\int }_{\Omega }}\mathrm{<mo>\{</mo>}{u}_t\Phi +\nu \mathrm{<mo\; stretchy="false">(</mo>}{u}_x{\Phi }_x+u_y{\Phi }_y\mathrm{<mo\; stretchy="false">)</mo>}-u\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}\Phi \mathrm{<mo>\}</mo>}dxdy\mathrm{<mo>=</mo>0,}$  (11)

where

$u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>,}{u}_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0<mo>;</mo>}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>.}$

Proof. The estimates that follow may be established on Gallerkin approximations.14,9 We estimate:

Estimate I - $u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}$ .

Multiply (9) by $u$ to obtain

$\mathrm{<mo\; stretchy="false">(</mo>}{u}_t\mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>}Au\mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0.}$  (12)

It follows from here that

$\frac{d}{dt}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu \parallel u{\parallel }^2{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_V\mathrm{<mo>=</mo>0.}$  (13)

Integrating (13) over $\mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>},$ , we get

$\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu {\displaystyle {\int }_0^t}\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}ds\mathrm{<mo>=</mo>}\parallel u_0{\parallel }^2\mathrm{,}t\mathrm{<mo>></mo>0.}$  (14)
Hence $u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}$ .

Estimate II - $u_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}$ .

Derivating (9) and multiplying by $u_t,$ we get

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu \parallel u_t{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2b\mathrm{<mo\; stretchy="false">(</mo>}{u}_t\mathrm{,}u\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0.}$  (15)

By Lemma 4.4,

$\mathrm{2<mo>|</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}{u}_t\mathrm{,}u\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}\le 2^{\mathrm{3<mo>/</mo>2}}\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }_V\parallel u{\parallel }_V$

$\le \nu \parallel u_t{\parallel }_V^2+\frac{2^3}{\nu }\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$

and (15) becomes

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\nu \parallel u_t{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \varphi \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}$  (16)

where $\varphi \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{2^3}{\nu }\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$ Making use of Lemma 4.2, we obtain

$\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}{e}^{{\displaystyle {\int }_0^t}\varphi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}ds}\mathrm{.}$  (17)

To prove that $\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}$  is in $H,$ multiply equation (9) by $u_t\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$  to get

$\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\nu \mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}u\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0.}$

In particular, for $t\mathrm{<mo>=</mo>0}$  we have

$\parallel u_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}{\parallel }^2\mathrm{<mo>=</mo>}\nu \mathrm{<mo\; stretchy="false">(</mo>}\Delta u_0\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}-b\mathrm{<mo\; stretchy="false">(</mo>}{u}_0\mathrm{,}{u}_0\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}$  (18)

where $u_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>=</mo>}{{\displaystyle \lim }}_{t\to 0}{u}_t\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>},$ [19]. From this

$\parallel u_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}\parallel \le \nu \parallel \Delta u_0\parallel +\parallel B{u}_0\parallel \mathrm{.}$  (19)

By the Hölder inequality,

$\mathrm{<mo>|</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}\le \parallel u{\parallel }_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>2}\parallel \nabla u{\parallel }_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}+\parallel u_{2x}{\parallel }_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}+\parallel u_{1y}{\parallel }_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}\parallel v\parallel$
$\le C\parallel u{\parallel }_V\parallel u{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\parallel v\parallel \mathrm{,}\forall u\in H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>,}\forall v\in L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$  (20)

Hence

$\mathrm{<mo>|</mo>}B{u}_0\mathrm{<mo>|</mo>}\le C\parallel u_0{\parallel }_V\parallel u_0{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le C\parallel u_0{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2$  (21)

and by (18), $u_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}\in H.$  This and (16) imply that

$u_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>.}$

Returning to (12), we calculate

$\nu \parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}u\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}$  (22)

hence $u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>.}$  This and (17) prove validity of (11) and consequently the existence part of Theorem 3.1. Uniqueness of the generalized solution, $u,$ $u_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}$  has been established.^9,14

Remark 5.1 Estimates $u\mathrm{,}{u}_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>}$  were established first for Lipschitz domains9,14 and were valid also for unbounded domains with a natural condition $li{m}_{\mathrm{<mo>|</mo>}x\mathrm{<mo>|</mo>}\to +\infty }u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0.}$  We repeat them because we will need these estimates while establishing decay of solutions in bounded and unbounded Lipschitz domains.

Consider the Poisson problem in a bounded rectangle $\Omega$

$(\begin{array}{c}\Delta u\mathrm{<mo>=</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>,}\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>}\in \Omega \mathrm{,}\\ u{\mathrm{<mo>|</mo>}}_{\partial \Omega }\mathrm{<mo>=</mo>0,}\end{array}$  (23)

Remark 6.1 It has been proved10 that for

${\Omega }_{\pi }\mathrm{<mo>=</mo><mo>\{</mo>}x\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>}{x}_1\mathrm{,...,}{x}_n\mathrm{<mo\; stretchy="false">)</mo>,}\mathrm{0<mo><</mo>}{x}_i\mathrm{<mo><</mo>}\pi \mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>1,...,}n\mathrm{<mo>\}</mo>}$

the following inequality holds

$\parallel u{\parallel }_{W^{\mathrm{2,}p}\mathrm{<mo\; stretchy="false">(</mo>}{\Omega }_{\pi }\mathrm{<mo\; stretchy="false">)</mo>}}\le C\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}\parallel f{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}{\Omega }_{\pi }\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$

It is easy to generalize this result for any rectangle in ${ℝ}^2\mathrm{.}$  

Theorem 6.1 The problem (23) posed in rectangle $\Omega \mathrm{<mo>=</mo><mo>\{</mo><mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>}\in {ℝ}^2\mathrm{,0<mo><</mo>}x\mathrm{<mo><</mo>}L\mathrm{<mo>;</mo>}\mathrm{0<mo><</mo>}y\mathrm{<mo><</mo>}B\mathrm{<mo>\}</mo>},$ where $f\in L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>},$   $\mathrm{1<mo><</mo>}p\le 2,$ has a solution $u\in W^{\mathrm{2,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$  Moreover,

$\parallel u{\parallel }_{W^{\mathrm{2,}p}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le c_{\Omega }\parallel f{\parallel }_{L^p\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$  (24)

Returning to the original problem for the Navier-Stokes equations,

$(\begin{array}{c}{u}_t-\nu \Delta u+\nabla p+\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo>=</mo>0}\text{in}\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>}\\ \nabla u\mathrm{<mo>=</mo>0}\text{in}\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}\\ u\mathrm{<mo>=</mo>0}\text{in}\partial \Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}t\mathrm{<mo>></mo>0,}\\ u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,0<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_0\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>,}\text{in}\Omega \mathrm{,}\end{array}$  (25)

where $u$ is a vector function from ${ℝ}^2$ into ${ℝ}^2$ and $p$ is a real function from ${ℝ}^2$ into $ℝ,$ and making use of Galerkin approximations, we establish the following result.

Theorem 6.2 Given $u_0\in H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}\cap V,$  the problem (25) has a unique solution $\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}p\mathrm{<mo\; stretchy="false">)</mo>}$  such that

$u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\cap H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}{u}_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>}\cap L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}V\mathrm{<mo\; stretchy="false">)</mo>,}$
$\nabla p\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}H\mathrm{<mo\; stretchy="false">)</mo>.}$       (26)

Moreover,

$\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}+\parallel \nabla p\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{1}{2}\chi t}\mathrm{,}$  (27)
where $\chi \mathrm{<mo>=</mo>}\nu \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\pi }^2}{L^2}+\frac{{\pi }^2}{B^2}\mathrm{<mo\; stretchy="false">)</mo>}$  and $C$  depends on $\parallel u_0{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}.$

Proof. Decay of $L^2$ Norm

By definition,

$\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\parallel u_x{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u_y{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$

Since $u{\mathrm{<mo>|</mo>}}_{\partial \Omega }\mathrm{<mo>=</mo>0},$ making use of Lemma 4.1, we get

$\parallel u_x{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\ge \frac{{\pi }^2}{L^2}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}\parallel u_y{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\ge \frac{{\pi }^2}{B^2}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$
This implies $\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\ge \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\pi }^2}{L^2}+\frac{{\pi }^2}{B^2}\mathrm{<mo\; stretchy="false">)</mo>}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$  (28)

Returning to (12), we obtain

$\frac{d}{dt}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\pi }^2}{L^2}+\frac{{\pi }^2}{B^2}\mathrm{<mo\; stretchy="false">)</mo>}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le 0.$  (29)

Define $\chi \mathrm{<mo>=</mo>}\nu \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\pi }^2}{L^2}+\frac{{\pi }^2}{B^2}\mathrm{<mo\; stretchy="false">)</mo>.}$ Then (29) implies
$\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_0{\parallel }^2{e}^{-2\chi t}\mathrm{.}$  (30)

Decay of $H^1$  Norm
Rewrite (15) in the form

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\nu \parallel u_t{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}-\varphi \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{0,}$  (31)

where $\varphi \mathrm{<mo>=</mo>}\frac{2^3}{\nu }\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$ Acting similarly to the proof of (29), we obtain

$\parallel u_t{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\ge \mathrm{<mo\; stretchy="false">(</mo>}\frac{{\pi }^2}{L^2}+\frac{{\pi }^2}{B^2}\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$  (32)

Hence (31) reduces to the form

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>}\chi -\varphi \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le 0.$  (33)

By Lemma 4.2,

$\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}{\parallel }^2{e}^{{\displaystyle {\int }_0^t}\varphi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}ds}{e}^{-\chi t}\mathrm{.}$  (34)

Since $u\in L^2\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{,}V\mathrm{<mo\; stretchy="false">)</mo>},$ then by (14),

${\displaystyle {\int }_0^t}\varphi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}ds\le \frac{2}{\nu }\parallel u_0{\parallel }^2\mathrm{,}t\mathrm{<mo>></mo>0,}$

and it follows from (13) that

$\nu \parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}{u}_t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$

$\le \parallel u_0\parallel \parallel u_{0^{\prime }}\parallel e^{\frac{2}{\nu }\parallel u_0\parallel }{e}^{-\chi t}{e}^{-\frac{1}{2}\chi t}\mathrm{.}$  (35)
Therefore $\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \frac{1}{\nu }\parallel u_0\parallel \parallel u_{0^{\prime }}\parallel e^{\frac{2}{\nu }\parallel u_0\parallel }{e}^{-\frac{3}{2}\chi t}$  (36)
and $\parallel u{\parallel }_{H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{\nu }\parallel u_0\parallel \parallel u_{0^{\prime }}\parallel e^{\frac{2}{\nu }\parallel u_0\parallel }+\parallel u_0{\parallel }^2\mathrm{<mo\; stretchy="false">)</mo>}{e}^{-\frac{3}{2}\chi t}\mathrm{.}$  (37)

Decay of $H^2$ -Norm
In order to estimate $\parallel u{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>},$ we will use Theorem 6.1. First write (8) as

$\Delta u\mathrm{<mo>=</mo>}f\mathrm{<mo>=</mo>}\frac{1}{\nu }\mathrm{<mo\; stretchy="false">(</mo>}{u}_t+\nabla p-\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo\; stretchy="false">)</mo>.}$

We estimate

$\mathrm{<mo>|</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo>|</mo><mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>,}v\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}\le c_2\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\parallel v\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}$
$\le C\parallel u{\parallel }^2{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\parallel v\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}$  (38)

and by (30),

$\parallel \mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le C{e}^{-\frac{3}{2}\chi t}\mathrm{.}$

Returning to (9), we obtain

$\parallel Au{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel Bu{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u_t{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$  (39)

It follows by (38) and (34) that $\parallel Au\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^{\mathrm{4<mo>/</mo>3}}}\le C{e}^{-\frac{1}{2}\chi t}.$  By Theorem of de Rham,¹⁷ one can check that there exists $\nabla p$  such that¹¹

$-\nabla p\mathrm{<mo>=</mo>}{u}_t+Au+Bu$     (40)

and

$\parallel \nabla p{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel Au{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$
$+\parallel Bu{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{1}{2}\chi t}\mathrm{.}$  (41)

Since $f\in L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>},$ due to Theorem 6.1,

$\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{W^{\mathrm{2,}\frac{4}{3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le \parallel u_t{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel \nabla p{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$
$+\parallel \mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$  (42)

and by (42), we get $\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{W^{\mathrm{2,4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le C{e}^{-\frac{1}{2}\chi t}.$  By the Sobolev theorems,

$\parallel u{\parallel }_{L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C\parallel u{\parallel }_{W^{\mathrm{2,4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{1}{2}\chi t}\mathrm{.}$  (43)
This implies

$\parallel Bu\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\in L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$

To prove that the norms $\parallel u_t{\parallel }_{L^2,}$ $\parallel \nabla p{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}$ and $\parallel \mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}$  have exponential decay, we use the equality (10)

$\parallel \mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\parallel Bu\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}$

where $Bu\in L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega {)}^{\prime }$ such that

$\langle Bu\mathrm{,}v\rangle \mathrm{<mo>=</mo>}{\displaystyle {\int }_{\Omega }}\mathrm{<mo\; stretchy="false">(</mo>}{u}_1{u}_{1x}{v}_1+u_1{u}_{1y}{v}_2+u_2{u}_{2x}{v}_1+u_2{u}_{2y}{v}_2\mathrm{<mo\; stretchy="false">)</mo>}d\Omega$

for every $v\in L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$  We calculate

$\mathrm{<mo>|</mo>}b\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}u\mathrm{,}v\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo><mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C\parallel u{\parallel }_{L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u{\parallel }_{H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel v\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$  (44)

Since the right-hand side of (44) has exponential decay for every $v\in L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>},$ it follows

$\parallel \mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{5}{4}\chi t}\mathrm{.}$  (45)

Returning to (9), we obtain the decay rate for the operator $Au$

$\parallel Au\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel Bu\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>.}$

It follows from (34) and (45) that $\parallel Au\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{1}{2}\chi t}.$  By (40),

$\parallel \nabla p{\parallel }_{L^2}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel Au\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel Bu\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{1}{2}\chi t}\mathrm{.}$  (46)

Since now $f\in L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>,}$  substituting (34), (45), (46) into (24) and making use of Theorem 6,1, we prove
$\parallel u{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C{e}^{-\frac{1}{2}\chi t}.$  It means that a unique generalized soliution is regular.

The proof of Theorem 6.2 is complete.

Theorem 7.1 Consider the half-strip $\Omega \mathrm{<mo>=</mo><mo>\{</mo><mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo>}\in {ℝ}^2\mathrm{<mo>;</mo>0<mo><</mo>}x\mathrm{,}\mathrm{0<mo><</mo>}y\mathrm{<mo><</mo>}B\mathrm{<mo>\}</mo>.}$  Given $u_0\in H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}\cap V,$  the following problem:

$(\begin{array}{c}{u}_t-\nu \Delta u+\nabla p+\mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u\mathrm{<mo>=</mo>0}\text{in}\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}\\ \nabla u\mathrm{<mo>=</mo>0}\text{in}\Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}\\ u\mathrm{<mo>=</mo>0}\text{on}\partial \Omega \times \mathrm{<mo\; stretchy="false">(</mo>0,}t\mathrm{<mo\; stretchy="false">)</mo>,}t\mathrm{<mo>></mo>0,}\\ \underset{x\to \infty }{{\displaystyle \lim }}\mathrm{<mo>|</mo>}u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,}t\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo><mo>=</mo>0,}t\mathrm{<mo>></mo>0,}\\ u\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{,0<mo\; stretchy="false">)</mo><mo>=</mo>}{u}_0\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}\text{in}\Omega \end{array}$  (47)

has a unique solution $\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{,}p\mathrm{<mo\; stretchy="false">)</mo>}$ such that

$u\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{H}_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}{u}_t\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{L}^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}$
$\nabla p\in L^{\infty }\mathrm{<mo\; stretchy="false">(</mo>0,}\infty \mathrm{<mo>;</mo>}{L}^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>.}$  (48)

Moreover,

$\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel u{\parallel }_{H_0^1\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel \nabla p{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le C_2{e}^{-\frac{1}{2}\theta t}\mathrm{,}$  (49)

where $\theta \mathrm{<mo>=</mo>}\nu \frac{{\pi }^2}{B^2}$ and $C_2$ depends on $\nu ,$ $\parallel u_0{\parallel }_{H^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}.$

Proof. Obviously, the variational formulation of (47) is also (9). Repeating the proof of Theorem 5.1 (see Remark 3.1), we can proof the existence and uniqueness of the generalized solution18 to problem (47). Note that (14) holds for the problem (47). Using the Steklov inequality with respect to variable $y,$ we obtain

$\parallel u_y{\parallel }^2\ge \frac{{\pi }^2}{B^2}\parallel u{\parallel }^2\mathrm{,}$

hence, similarly to (13),

$\frac{d}{dt}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu \frac{{\pi }^2}{B^2}\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu \parallel u_x{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le 0.$  (50)

By Lemma 5, $\parallel u{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_0{\parallel }^2{e}^{-2\nu \frac{{\pi }^2}{B^2}t}\mathrm{.}$  (51)

Since (31) holds for the problem (47), making use of Lemma 4.4, we estimate

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+2\nu \parallel u_t{\parallel }^2{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_V\le 2\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_V\parallel u_t\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_V$  (52)

which we rewrite as

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\nu \parallel u_t{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}-\frac{2}{\nu }\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le 0.$  (53)

By Lemma 4.1,

$\parallel u_{ty}{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\ge \frac{{\pi }^2}{B^2}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$

and (53) becomes

$\frac{d}{dt}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">[</mo>}\nu \frac{{\pi }^2}{B^2}-\frac{2}{\nu }\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le 0.$  (54)

By Lemma 5, (54) provides

$\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}{e}^{\frac{2^3}{\nu }{\displaystyle {\int }_0^t}\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}ds}{e}^{-\nu \frac{{\pi }^2}{B^2}t}\mathrm{,}$

hence

$\parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t{\parallel }^2\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}{e}^{\frac{2}{\nu }\parallel u_0\parallel }{e}^{-\nu \frac{{\pi }^2}{B^2}t}\mathrm{.}$  (55)

Returning to (35), we estimate

$\parallel u{\parallel }_V^2\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \frac{1}{\nu }\parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\parallel u\parallel \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \frac{1}{\nu }\parallel u_0\parallel \parallel u_t\parallel \mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}{e}^{\frac{2}{\nu }\parallel u_0\parallel }{e}^{-\nu \frac{3{\pi }^2}{2B^2}t}\mathrm{.}$  (56)

Decay for Pressure
In order to obtain decay for $\parallel \nabla p{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>},$  we start with

$\parallel \mathrm{<mo\; stretchy="false">(</mo>}u\cdot \nabla \mathrm{<mo\; stretchy="false">)</mo>}u{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\parallel Bu{\parallel }_{L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega {)}^{\prime }}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}$

where $L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega {)}^{\prime }$ is the dual of the space $L^4\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>.}$  Since

$Au\mathrm{<mo>=</mo>}-u_t-Bu\mathrm{,}$

repeating calculations of (38) and making use of (34), we get

$\parallel Au\parallel {\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\le c_1{e}^{-\frac{1}{2}\theta t}.$  Observing that (40) holds for the problem (47), we obtain

$\parallel \nabla p{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le \parallel u_t{\parallel }_{L^2\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}+\parallel Au{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$
$+\parallel Bu{\parallel }_{L^{\mathrm{4<mo>/</mo>3}}\mathrm{<mo\; stretchy="false">(</mo>}\Omega \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\le c_2{e}^{-\frac{1}{2}\theta t}\mathrm{.}$  (57)

Jointly (55), (56) and (57) prove (48), (49).

In our work, we tried to respond some questions posed by J. Leray,¹ namely, regularity of global solutions of the Navier-Stokes equations and their decay. Therefore, our results can be divided in two parts: the first one concerns decay of global regular solutions of the 2D Navier-Stokes equations posed on rectangles.¹⁹ It is known that there exist global regular solutions for the 2D Navier-Stokes equations posed on smooth bounded domains,^4,10,11,14 but regularity in nonsmooth (Lipschitz) domains, such as rectangles, is not obvious. For bounded rectangles, we have established the existence of an unique global regular solution which decays exponentially as  We demonstrated that the decay rate is different for different norms, see (26), (30), (36), whereis defined by the geometrical characteristics of a domain

The second part of our work concerns decay of solutions for the 2D Navier-Stokes equations posed on a half-strip. In existing publications,^3–11 the decay rate of  is controlled by the first eigenvalue of the operatorwhereis the projection operator on solenoidal subspace of  It is clear that this approach does not work in unbounded domains On the other hand, our approach based on the Steklov inequality with respect toallowed us to estimate the decay rate of a generalized solution for the 2D Navier-Stokes equations posed on a half-strip.

We must emphasize that this estimate is the first in the history which gives an explicit value of the decay rate for unbounded domains. Results established in our work can be used in constructing of numerical schemes for solving initial-boundary value problems for the Navier-Stokes equations appearing in Mechanics of viscous liquid. From the physical point of view, decay estimates show that the decay rate of perturbations of solutions caused by the initial data is bigger for bigger values of viscosityand smaller values of the width and length of the rectangles and the width of a half-strip.

None.

The authors declare that there are no conflict of interest regarding the publication of this paper.

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