UDC 517.946.51

V. P. MIKHAILOV

ON A BOUNDARY-VALUE PROBLEM FOR THE SYSTEM OF NAVIER–STOKES EQUATIONS

(Presented by Academician S. L. Sobolev on 18 III 1968)

Let \(Q\) be a domain in three-dimensional space \(x=(x_1,x_2,x_3)\), bounded above by the plane \(\gamma_0: x_3=0\), and below by the surface \(\gamma_1: x_3=\varphi(x_1,x_2);\ -H\leqslant \varphi(x_1,x_2)\leqslant 0\), where \(H\) is a certain positive constant. The domain \(Q\) may be either bounded in the directions perpendicular to the axis \(Ox_4\), or unbounded. The line of intersection of \(\gamma_0\) and \(\gamma_1\), if it exists, will be denoted by \(S\). Consider in the domain \(Q\) the stationary linearized system of Navier–Stokes equations

\[ \nu \Delta \mathbf v+\mathbf F=\operatorname{grad}p,\qquad \operatorname{div}\mathbf v=0, \tag{1} \]

where the parameter \(\nu>0\), and \(\mathbf F(x)=(F_1(x),F_2(x),F_3(x))\in \mathcal L_2(Q)\) is a given vector. We are interested in the question of unique solvability of the system (1) under the boundary conditions

\[ \mathbf v|_{\gamma_1}=0, \tag{2} \]

\[ v_{1x_3}+v_{3x_1}|_{\gamma_0}=v_{2x_3}+v_{3x_2}|_{\gamma_0}=p-2\nu v_{3x_3}|_{\gamma_0}=0, \tag{3} \]

where \(\mathbf v(x)=(v_1(x),v_2(x),v_3(x))\) is the unknown velocity vector, and \(p(x)\) the unknown pressure.

The corresponding nonstationary problem is posed analogously. In the cylinder \(\Omega_T=Q\times(0,T)\) one is required to find a vector \(\mathbf v(x,t)\) and a function \(p(x,t)\) satisfying in \(\Omega_T\) the nonstationary linearized Navier–Stokes system

\[ \mathbf v_t+\operatorname{grad}p=\nu\Delta \mathbf v+\mathbf F(x,t),\qquad \operatorname{div}\mathbf v=0, \tag{4} \]

satisfying on the boundary the conditions

\[ v|_{\Gamma_1}=0,\qquad v_{1x_3}+v_{3x_1}|_{\Gamma_0}=v_{2x_3}+v_{3x_2}|_{\Gamma_0}=p-2\nu v_{3x_3}|_{\Gamma_0}=0, \tag{5} \]

where \(\Gamma_1=\gamma_1\times(0,T)\), \(\Gamma_0=\gamma_0\times(0,T)\), \(\mathbf F\in\mathcal L_2(\Omega_T)\), and the initial condition at \(t=0\)

\[ \mathbf v|_{t=0}=0. \tag{6} \]

Physically, problem (1)—(3) (as well as the corresponding nonstationary problem (4)—(6)) is the problem of finding the velocity \(\mathbf v\) of the particles of a viscous incompressible fluid and the pressure \(p\), when on the bottom \(\gamma_1\) the no-slip condition is prescribed, while on the free surface \(\gamma_0\) the conditions (3) of vanishing of the three components of the stress tensor are prescribed \((^1)\). Together with problems (1)—(3) and (4)—(6), one may also consider more complicated problems, for example problems with rotation. By this is meant the problem of finding, in the domain \(Q\) (in the stationary case), a solution of the system

\[ \nu\Delta \mathbf v+[\vec\omega\times \mathbf v]+\mathbf F(x)=\operatorname{grad}p,\qquad \operatorname{div}\mathbf v=0, \tag{1′} \]

satisfying conditions (2)—(3), where \(\vec\omega\) is a certain bounded vector measurable in \(Q\). The case when there is no free surface was studied in important papers \((^{2-4,\,7})\)*.

* Note added in proof. Recently the author became acquainted with paper \([8]\), in which related questions are considered.

1. Denote by \(H_1(Q)\) the Hilbert space of vector-functions \(\{\mathbf{u}(x)\}\) obtained by completing, with respect to the norm

\[ \|\mathbf{u}\|_{H_1(Q)}=\left(\iiint_Q \sum_i |\operatorname{grad} u_i|^2\,dx\right)^{1/2} \tag{7} \]

the set of solenoidal vectors \(\mathbf{u}(x)\) that vanish in a boundary strip near the portion of the boundary \(\gamma_1\).

Lemma 1. In the Hilbert space \(H_1(Q)\), instead of the norm defined by (7), one may introduce the equivalent norm

\[ \|\mathbf{u}\|_{H_1(Q)}'=\left(\iiint_Q \sum_{ij}(u_{i x_j}+u_{j x_i})^2\,dx\right)^{1/2}. \]

We define a generalized solution of problem (1)—(2)—(3) as follows. A vector \(\mathbf{v}\) and a function \(p\) are called a generalized solution of problem (1)—(2)—(3) if \(\mathbf{v}\in H_1(Q)\), \(p\in \mathcal{L}_2(Q)\), and for every vector-function \(\mathbf{u}\in H_1(Q)\) the equality

\[ \frac{\nu}{2}\iiint_Q \sum_{ij}(u_{i x_j}+u_{j x_i})(v_{i x_j}+v_{j x_i})\,dx = \iiint_Q \sum_i F_i u_i\,dx \tag{8} \]

holds.

The correctness of this definition follows from Lemma 2.

Lemma 2. If \(\mathbf{v},p\) is an almost everywhere solution of problem (1)—(2)—(3) (and, a fortiori, a classical solution of this problem), then equality (8) holds for every \(\mathbf{u}\in H_1(Q)\). If \(\mathbf{v}\in H_1(Q)\cap \overset{\circ}{W}{}^2_2(Q)\) and (8) holds for every \(\mathbf{u}\in H_1(Q)\), then there exists a function \(p\in W^1_2(Q)\) such that \((\mathbf{v},p)\) is an almost everywhere solution of problem (1)—(2)—(3). Moreover, if \(V_\delta\) is an arbitrary \(\delta\)-neighborhood, \(\delta>0\), of the curve \(S\), and \(Q'\) is any compact part of the closure \(\overline{Q}\) of the domain \(Q\), and if \(\mathbf{v}\in H_1(Q)\cap W^2_2(Q'\setminus V_\delta)\), then there exists a function \(p\in W^1_2(Q'\setminus V_\delta)\) such that in \(Q'\setminus V_\delta\) \((\mathbf{v},p)\) is an almost everywhere solution of problem (1)—(2)—(3).

Using the Riesz representation theorem for a bounded linear functional in a Hilbert space and Lemma 1, we obtain Theorem 1.

Theorem 1. For every vector-function \(\mathbf{F}\in \mathcal{L}_2(Q)\) there exists a unique generalized solution of problem (1)—(2)—(3).

1. The nonstationary case is considered analogously.

Let \(H_2(\Omega_T)\) be the Hilbert space of vector-functions obtained after completing the set of solenoidal vectors \(\{\mathbf{v}(x,t)\}\) satisfying the boundary conditions (5) and the initial conditions (6), with respect to the norm of the space \(W^1_2(\Omega_T)\):

\[ \|\mathbf{v}\|_{H_2(\Omega_T)} = \left( \int_0^T dt\iiint_Q \left( \sum_i |\operatorname{grad} v_i|^2+|v_{it}|^2 \right)\,dx \right)^{1/2}. \]

A vector \(\mathbf{v}(x,t)\in H_2(\Omega_T)\) and a function \(p(x,t)\in \mathcal{L}_2(\Omega_T)\) will be called a generalized solution of problem (4)—(5)—(6) if the equality

\[ \int_0^T dt\iiint_Q (u_t,v_t)e^{-t}\,dx + \frac{\nu}{2}\int_0^T dt\iiint_Q \sum_{ij}(v_{i x_i}+v_{i x_j})(u_{i t x_j}+u_{j t x_i})e^{-t}\,dx = \]

\[ = \int_0^T dt\iiint_Q \sum_i F_i u_{it} e^{-t}\,dx \tag{9} \]

holds for every vector-function \(\mathbf{u}(x,t)\in H_3(\Omega_T)\), where \(H_3(\Omega_T)\) is the Hilbert space of vector-functions belonging to \(H_2(\Omega_T)\) and such that the vector \(\mathbf{u}_t\) has a square-integrable gradient in the domain \(\Omega_T\).

The uniqueness of such a generalized solution follows directly from Lemma 1: for \(\mathbf F \equiv 0\) we take
\[ \mathbf u=\int_0^t e^t \mathbf v(x,t)\,dt \]
and substitute it into (9) as a test function. To prove the existence of a generalized solution, consider the left-hand side of equality (9) as a linear bounded functional on \(\mathbf v\in H_2(\Omega_T)\) for any fixed element \(\mathbf u\in H_3(\Omega_T)\). Then, according to the Riesz theorem, (9) can be rewritten in the form
\[ [\mathbf v,A\mathbf u]=(\mathbf F,\mathbf u), \tag{10} \]
where the parentheses \((\, ,\, )\) denote the scalar product in \(\mathcal L_2(\Omega_T)\) with weight \(e^{-t}\), and \([\, ,\,]\) denotes the scalar product in \(H_2(\Omega_T)\) with the same weight; the operator \(A\) is, generally speaking, an unbounded operator from \(H_3(\Omega_T)\) into \(H_2(\Omega_T)\). However, it can be shown that the operator \(A\) has a bounded inverse; hence equation (10), rewritten in the form \([A^*\mathbf v,\mathbf u]=[\Phi,\mathbf u]\), where \([\Phi,\mathbf u]=(\mathbf F,\mathbf u)\) is the realization of the functional \((\mathbf F,\mathbf u)\) in the metric of \(H_2(\Omega_T)\), has a solution \(\mathbf v=A^{*-1}\Phi\in H_2(\Omega_T)\). Thus, the following holds.

Theorem 2. For any \(\mathbf F\in \mathcal L_2(\Omega_T)\), problem (4)—(5)—(6) has a unique generalized solution from \(H_2(\Omega_T)\).

1. In the case of problem \((1')\)—(2)—(3), the generalized solution is defined as a solution in the space \(H_1(Q)\) of the operator equation \(\mathbf v+A\mathbf v=\Phi\), where the operator \(A\) is defined by the equality
   \[ \{A\mathbf v,\mathbf u\}=\int_Q (\vec\omega\,\mathbf v\mathbf u)\,dx \]
   (where \((\vec\omega\,\mathbf v\mathbf u)\) denotes the mixed product \(([\omega\times \mathbf v]\mathbf u)\)), and
   \[ \{\Phi,\mathbf u\}=\int_Q (\mathbf F,\mathbf u)\,dx. \]
   The operator \(A\) is completely continuous and skew-symmetric (since
   \[ (\vec\omega\,\mathbf v\mathbf u)=-(\vec\omega\,\mathbf u\mathbf v) \]
   ); therefore Theorem 1 remains valid also in this case.

2. The solutions of the stationary and nonstationary problems constructed in the preceding sections in fact possess greater smoothness than that established in Theorems 1 and 2.

Theorem 3. If \(\mathbf F(x)\in W_p^l(Q)\), \(p>1\), and \(l\) is a nonnegative integer, then the generalized solution from \(H_1(Q)\) of problem (1)—(2)—(3) has the following properties: \(\mathbf v\in W_p^{l+2}(Q')\), \(p(x)\in W_p^{(l+1)}(Q')\), where \(Q'\) is any part of \(Q\) such that \(\overline{Q'}\subset Q\). Further, if \(\delta\) is an arbitrary positive number and the function \(\varphi(x_1,x_2)\) occurring in the definition of the surface \(\gamma_1\) is sufficiently smooth, then \(\mathbf v\in W_p^{(l+2)}(Q''\setminus V_\delta)\), \(p(x)\in W_p^{(l+1)}(Q''\setminus V_\delta)\), where \(Q''\) is any compact part of \(\overline Q\), and \(V_\delta\) is the \(\delta\)-neighborhood of the line \(S\) where the boundaries \(\gamma_0\) and \(\gamma_1\) meet.

Theorems 3 and 2 also make it possible to establish the corresponding result for the nonstationary problem.

Theorem 4. If \(\mathbf F(x,t)\in W_{p,x,t}^{2l,l}(\Omega_T)\), \(p>1\), and \(l\) is a nonnegative integer, then the generalized solution from \(H_2(\Omega_T)\) of problem (4)—(5)—(6) has the following properties: \(\mathbf v(x,t)\in W_{p,x,t}^{2l+2,l+1}(\Omega_T')\), \(p(x)\in W_{p,x,t}^{2l+1,l}(\Omega_T')\), where \(\Omega_T'\) is any part of \(\Omega_T\) such that \(\overline{\Omega_T'}\subset \Omega_T\). On the other hand, if \(\delta>0\) is arbitrarily small and the function \(\varphi(x_1,x_2)\) is sufficiently smooth, then \(\mathbf v(x,t)\in W_{p,x,t}^{2l+2,2l+1}(\Omega_T''\setminus (V_\delta\times(0,T)))\), \(p(x,t)\in W_{p,x,t}^{2l+1,l}(\Omega_T''\setminus (V_\delta\times(0,T)))\), where \(\Omega_T''\) is any compact part of \(\overline{\Omega_T}\).

In the proof of Theorem 3 we shall confine ourselves to studying the smoothness of the generalized solution near the portion \(\gamma_0\setminus S\) of the boundary, since the required interior smoothness, as well as the smoothness of the solution near the portion \(\gamma_1\setminus S\) of the boundary, can be established in the same way as in the book (4). We take
\[ \mathbf u(x)=\mathbf u_N^{(1)}(x)-\mathbf u_N^{(2)}(x),\qquad q(x)=q_N^1(x,y)-q_N^2(x,y), \tag{11} \]

where

\[ \mathbf u_{N}^{(1)}=\frac{1}{\nu}\operatorname{rot}\bigl(\zeta_{\delta}(x)(\operatorname{rot}\,\vec{\Phi}_{N,k}(x-y))\bigr); \]

\(y\) is a parametric point; \(\delta\) is a sufficiently small positive number; \(\zeta_{\delta}(x)\) is a smooth function equal to zero in the \(\delta/2\)-neighborhood of \(\gamma_{1}\) and equal to 1 outside the \(\delta\)-neighborhood of \(\gamma_{1}\),

\[ \vec{\Phi}_{N,k}(x)= -\frac{e^{k}}{(2\pi)^{3}} \left\{ \int_{|\alpha'|\geqslant 1}d\alpha'\int_{-\infty}^{+\infty} \frac{\theta_{N}(\alpha)e^{i(\alpha,x)}}{|\alpha|^{4}}\,d\alpha_{3} + \int_{|\alpha'|<1}d\alpha'\int_{l} \frac{\theta_{N}(\alpha)e^{i(\alpha,x)}}{|\alpha|^{4}}\,d\alpha_{3} \right\}, \]

where \(\theta_{N}(\alpha)\) is a certain function analytic in \(\operatorname{Im}\alpha_{3}>0\), sufficiently rapidly decreasing on the real plane, and, as \(N\to\infty\), uniformly on every compact set tending to \(1\) \((^{5})\); \(l\) is a contour in the \(\alpha_{3}\)-plane consisting of pieces of the real axis \((-\infty,-1)\) and \((1,+\infty)\) and of the semicircle \(|\alpha_{3}|=1\), \(\operatorname{Im}\alpha_{3}\geqslant0\); \(\alpha'=(\alpha_{1},\alpha_{2})\), \(\alpha=(\alpha',\alpha_{3})\); \(e^{k}\) is the unit vector directed along the axis \(OX_{k}\). As the function \(q_{N}^{1}(x,y)\) we take the function \(\nu\operatorname{div}(\zeta_{\delta}(x)\Delta\vec{\Phi}_{N,k})\). The vector-function \(\mathbf u_{N}^{2}\) is defined as the vector \(\operatorname{rot}(\zeta_{\delta}(x)\vec{\theta}(x,y))\), where the vector \(\vec{\theta}\) in the half-space \(x_{3}<0\) is specified by its Fourier transform with respect to \(x_{1}\) and \(x_{2}\) as follows:

\[ \widetilde{\theta}_{1}(\alpha_{1},\alpha_{2},x_{3};y) = \left[ \widetilde{\sigma}_{1}\left(\frac{\alpha_{2}^{2}}{|\alpha|^{4}}+\frac{\alpha_{1}^{2}}{2|\alpha|^{3}}x_{3}\right) -\widetilde{\sigma}_{2}\frac{\alpha_{1}\alpha_{2}}{2|\alpha|^{3}}\left(\frac{2}{|\alpha|}-x_{3}\right) + \widetilde{\sigma}_{3}\frac{i\alpha_{1}\alpha_{2}}{2\nu|\alpha|} \left(\frac{x_{3}}{|\alpha|}-\frac{1}{|\alpha|^{2}}\right) \right]e^{|\alpha|x_{3}}, \]

\[ \widetilde{\theta}_{2}(\alpha_{1},\alpha_{2},x_{3};y) = \left[ -\widetilde{\sigma}_{1}\frac{\alpha_{1}\alpha_{2}}{2|\alpha|^{3}}\left(\frac{2}{|\alpha|}-x_{3}\right) + \widetilde{\sigma}_{2}\left(\frac{\alpha_{1}^{2}}{|\alpha|^{4}}+\frac{\alpha_{2}^{2}}{2|\alpha|^{3}}x_{3}\right) + \widetilde{\sigma}_{3}\frac{i\alpha_{2}}{2\nu|\alpha|} \left(\frac{x_{3}}{|\alpha|}-\frac{1}{|\alpha|^{2}}\right) \right]e^{|\alpha|x_{3}}, \qquad \widetilde{\theta}_{3}\equiv0, \]

and \(\widetilde{\sigma}_{1}=\widetilde{\sigma}_{1}(\alpha_{1},\alpha_{2};y)\), \(\widetilde{\sigma}_{2}\), \(\widetilde{\sigma}_{3}\) are the Fourier transforms with respect to \(x_{1},x_{2}\), at \(x_{3}=0\), of the functions respectively

\[ u^{(1)}_{N1x_{3}}+u^{1}_{N3x_{1}},\quad u^{1}_{N2x_{3}}+u^{1}_{N3x_{2}},\quad q_{N}^{(1)}-2\nu u^{1}_{N3x_{3}}. \]

The Fourier transform of the function \(q_{N}^{2}(x,y)\) in the half-space \(x_{3}<0\) is defined by

\[ \widetilde{q}_{N}^{2} = e^{|\alpha|x_{3}} \left(\widetilde{\sigma}_{3} -\nu\left(\frac{\alpha_{1}}{|\alpha|}\widetilde{\sigma}_{1} +\frac{\alpha_{2}}{|\alpha|}\widetilde{\sigma}_{2}\right)\right). \]

The functions (11) may be substituted into (8) as test functions. In doing so, transferring all derivatives to the function \(\mathbf u\) by integration by parts, we obtain

\[ v_{k,N}(y) = \iiint\limits_{Q}\sum_{i}F_{i}(x)\bigl(u^{1}_{iN}-u^{2}_{Ni}\bigr)\,dx + B_{N}(y,\mathbf v), \tag{12} \]

where

\[ \lim_{N\to\infty}\|v_{k,N}(y)-v_{k}(y)\|_{\mathcal L_{p}(Q_{\delta})}=0, \]

and \(B_{N}(y,\mathbf v)\) is a linear functional of \(\mathbf v\in W_{p}^{1}(Q)\), uniformly bounded in \(N\), such that the functional \(D_{y}^{2}B(y,\mathbf v)\) is also uniformly bounded in \(N\) in \(W_{p}^{1}(Q)\), where \(D_{y}^{2}\) is an arbitrary second derivative with respect to \(y\). From equality (12), with the help of the theorems on multipliers \((^{6})\), the required result is obtained.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
5 II 1968

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