MATHEMATICS

P. E. SOBOLEVSKII

ON THE APPLICATION OF THE METHOD OF FRACTIONAL POWERS OF OPERATORS TO THE INVESTIGATION OF THE NAVIER—STOKES EQUATIONS

(Presented by Academician M. A. Lavrent’ev, 2 IX 1963)

1. The present paper* continues the investigation begun in \((^{2,3})\). The system of Navier—Stokes equations is reduced to the integral equation

\[ \mathbf v(t)=e^{-t\nu A}\mathbf v(0)+\int_0^t e^{-(t-s)\nu A}P\mathbf f\,ds -\int_0^t e^{-(t-s)\nu A}P\frac{\partial}{\partial x_k}(v_k\cdot \mathbf v)\,ds, \tag{1} \]

in the Hilbert space \(H\)—the closure in the metric \(L_2(\Omega)\) of the set of smooth solenoidal \(m\)-dimensional vector functions in a bounded \(m\)-dimensional domain \(\Omega\), vanishing near the boundary \(S\) of the domain \(\Omega\). Here \(P\) is the operator of orthogonal projection in \(L_2\) onto \(H\); \(A\) is the Friedrichs extension of the operator \(-P\Delta\), initially defined on \(W_2^2\cap H\) \((^{1-3})\).

Equation (1) has unbounded nonlinearities. Having made, in §§ 3, 4, and 5, a substitution, we shall pass to an integral equation with continuous nonlinearities and establish existence theorems and estimates for solutions of these equations. The estimates of \(A\) and of the nonlinearities needed for this are given in § 2 (cf. \((^{2,3})\)).

2. Lemma 1. For every \(\alpha\in[0,1]\) the inequalities

\[ \|(-\Delta)^\alpha \mathbf v\|\le C(\alpha)\|A^\alpha \mathbf v\| \quad (\mathbf v\in D[A^\alpha]); \tag{2} \]

\[ \|A^\alpha P\mathbf v\|\le C(\alpha)\|(-\Delta)^\alpha \mathbf v\| \quad (\mathbf v\in D[(-\Delta)^\alpha]). \tag{3} \]

hold.

Proof. In \((^2)\), (2) was established for \(\alpha=2^{-n}\), \(n=1,2,\ldots\), with \(C(\alpha)=1\). It is not hard to see that
\(P\mathbf u=\mathbf u-\operatorname{grad}\Delta^{-1}\operatorname{div}\mathbf u-\operatorname{grad}q\), where \(\Delta q=0\) and

\[ \left.\frac{\partial q}{\partial n}\right|_S = \left.(\mathbf u-\frac{\partial}{\partial n}\Delta^{-1}\operatorname{div}\mathbf u)\right|_S . \]

By virtue of the known properties of solutions of the Neumann problem (see \((^4)\)),

\[ \|q\|_{W_2^i}\le C_i\|\mathbf u-\operatorname{grad}\Delta^{-1}\operatorname{div}\mathbf u\|_{W_2^{i-1}}, \quad i=1,2,\ldots . \]

Therefore \(P\mathbf u\in W_2^i\cap H\), if \(\mathbf u\in W_2^i\), and

\[ \|P\mathbf u\|_{W_2^i}\le C_i\|\mathbf u\|_{W_2^i}. \]

By virtue of \((^5)\),

\[ \|A\mathbf u\|\le C\|\mathbf u\|_{W_2^2} \quad (\mathbf u\in D[A]). \]

By virtue of \((^6)\),

\[ \|\mathbf u\|_{W_2^2}\le C\|(-\Delta)\mathbf u\| \quad (\mathbf u\in D[(-\Delta)]). \]

Hence (3) holds for \(\alpha=1\). This means that

\[ \|(-\Delta)^{-1}\mathbf u\|\le C(1)\|A^{-1}\mathbf u\|\quad(\mathbf u\in H). \]

Proceeding as in the proof of Lemma 2 of \((^2)\), we obtain

\[ \|(-\Delta)^{-\alpha}\mathbf u\|\le [C(1)]^\alpha\|A^{-\alpha}\mathbf u\| \quad(\mathbf u\in H) \]

for \(\alpha=2^{-n}\), \(n=1,2,\ldots\). Thus, we have (3) for \(\alpha=2^{-n}\) with \(C(\alpha)=[C(1)]^\alpha\). With the aid of this one can obtain (2) for all dyadic rational \(\alpha\). Let, for example, \(\alpha=3/4\). Since

\[ I=(A^{-3/4}P[-\Delta]^{3/4}\vec\varphi,\vec\psi) =([- \Delta]^{1/4}\vec\varphi,A^{-1}A^{1/4}\vec\psi), \]

\(\vec\varphi\in D([-\Delta]^{3/4})\), \(\vec\psi\in D(A^{1/4})\), then, taking into account that

\[ \|A^{-1}P\mathbf u\|\le C\|[-\Delta]^{-1}\mathbf u\| \]

and applying Heinz’s inequality \((^7)\) and (3) for \(\alpha=1/4\), we obtain

\[ I\le C\|[-\Delta]^{-3/4}[-\Delta]^{3/4}\vec\varphi\|\cdot \|[-\Delta]^{-1/4}A^{1/4}\vec\varphi\| \le C\|\vec\varphi\|[C(1)]^{1/2}\|\vec\psi\|. \]

\(C(\alpha)\) in (2)

* These results were reported at the First Interzonal Scientific Conference in Voronezh in February 1962.

is thereby uniformly bounded. Passing to the limit we obtain from this (2) for any \(\alpha \in [0,1]\). Similarly, (3) is established.

By \(G(m)\) denote the set of points of the plane \((\alpha,\beta)\) lying in the triangle with vertices \((0,0)\), \(((m+2)/8,(m+2)/8)\), and \((0,(m+2)/4)\), except for the points \((0,0)\), \((0,(m+2)/4)\) and \((\alpha,\beta)\) with \(\beta < 1/2\) and \(\alpha > m/4 - 1/2\).

Lemma 2. For any \((\alpha,\beta)\in G(m)\), \(\mathbf v \in D([1-\Delta]^\alpha)\), \(\mathbf w \in D([-\Delta]^\beta)\), \(\operatorname{div}\mathbf v=0\), if \(\alpha+\beta>m/4\),

\[ \left\|(-\Delta)^{\alpha+\beta-(m+2)/4}\frac{\partial}{\partial x_k}(v_k\mathbf w)\right\| \leq C(\alpha,\beta)\|(-\Delta)^\alpha\mathbf v\|\cdot\|(-\Delta)^\beta\mathbf w\|. \tag{4} \]

Proof. If \(\alpha+\beta \leq m/4\), then for any \(\vec\varphi \in L_2\)

\[ I= \left|\int_\Omega \vec\varphi\,(-\Delta)^{\alpha+\beta-(m+2)/4} \frac{\partial}{\partial x_k}(v_k\mathbf w)\,dx\right| \leq \int_\Omega \left|\frac{\partial}{\partial x_k}(-\Delta)^{\alpha+\beta-(m+2)/4}\vec\varphi\right|\cdot |v_k\mathbf w|\,dx \leq \]

\[ \leq m\left\|\operatorname{grad}(-\Delta)^{\alpha+\beta-(m+2)/4}\vec\varphi\right\|_{L_{\frac{m}{2(\alpha+\beta)}}} \cdot \|\mathbf v\|_{L_{\frac{2m}{m-4\alpha}}} \cdot \|\mathbf w\|_{L_{\frac{2m}{m-4\beta}}}. \]

Since

\[ \|\mathbf v\|_{L_{\frac{2m}{m-4\gamma}}} \leq C_\gamma\|(-\Delta)^\gamma \mathbf v\| \quad(0<\gamma<m/4), \]

\[ \|\operatorname{grad}\mathbf v\|_{L_{\frac{2m}{m+2-4\gamma}}} \leq C_\gamma\|(-\Delta)^\gamma\mathbf v\| \quad(1/2\leq\gamma<(m+2)/4) \quad (\mathbf v\in D[(-\Delta)^\gamma]) \]

(see \((2,8)\)), it follows from this that (4) holds. If \(\alpha+\beta>m/4\), then \(\beta>1/2\), and

\[ I\leq \int_\Omega \left|(-\Delta)^{\alpha+\beta-(m+2)/4}\vec\varphi\right| \left|v_k\frac{\partial \mathbf w}{\partial x_k}\right|\,dx \leq \]

\[ \leq m\left\|(-\Delta)^{\alpha+\beta-(m+2)/4}\vec\varphi\right\|_{L_{\frac{m}{2(\alpha+\beta)-1}}} \cdot \|\mathbf v\|_{L_{\frac{2m}{m-4\alpha}}} \|\operatorname{grad}\mathbf w\|_{L_{\frac{2m}{m+2-4\beta}}}. \]

Hence, and from \((2,8)\), (4) follows.

Remark. From Lemma 1 it follows that in (4) one may replace \((-\Delta)\) by \(A\).

1. Let \(m=2,3\). Making in (1) the substitution
   \(\mathbf w(t,\mu,\gamma)=t^{\mu-\gamma}A^\mu\mathbf v(t)\)
   \((\mu=\alpha,\beta;\;(\alpha,\beta)\in G(m);\;(m-2)/4\leq\gamma\leq\alpha;\;\alpha>(m-2)/4)\), we obtain

\[ \begin{aligned} \mathbf w(t,\mu,\gamma) &=t^{\mu-\gamma}A^\mu e^{-t\nu A}\mathbf v(0) +t^{\mu-\gamma}A^\mu\int_0^t e^{-(t-s)\nu A}P\mathbf f\,ds \\ &\quad -t^{\mu-\gamma}\int_0^t A^{\mu+(m+2)/4-(\alpha+\beta)}e^{-(t-s)\nu A} A^{\alpha+\beta-(m+2)/4}P \frac{\partial}{\partial x_k} \bigl[A^{-\alpha}w_k(s,\alpha,\gamma)\times \\ &\qquad\qquad\qquad\qquad\qquad\qquad \times A^{-\beta}\mathbf w(s,\beta,\gamma)\bigr] s^{2\gamma-(\alpha+\beta)}\,ds. \end{aligned} \tag{5} \]

By \(\vec\varphi_i\) denote the \(i\)-th term of the right-hand side of (5). Let

\[ \mathbf v(0)\in D(A^\gamma). \tag{*} \]

Then \(\vec\varphi_1\) is continuous in \(t\) for \(t\geq 0\). If \(\gamma<\mu\), then \(\vec\varphi_1=0\) at \(t=0\). Let

\[ \vec\varphi_2 \text{ be continuous in }t\in[0,\tau]\text{ for some }\tau>0 \text{ and }\vec\varphi_2=0\text{ at }t=0. \tag{**} \]

If

\[ \mathbf w(t,\mu,\gamma) \text{ are continuous on }[0,\tau]\;(\mu=\alpha,\beta), \tag{***} \]

then \(\vec\varphi_3\) is also continuous on \([0,\tau]\), and \(\vec\varphi_3=0\) at \(t=0\)*.

* To prove the latter for \(\gamma=(m-2)/4\), it suffices to consider only differentiable \(\mathbf w(t,\mu,\gamma)\).

All this makes it possible to consider system (5) as an operator equation in the space \(C[0,\tau]\)—the space of vector functions \([\mathbf w^{1}(t),\mathbf w^{2}(t)]\) continuous on \([0,\tau]\), with values in \(L_{2}\times L_{2}\) and norm
\[ \|[\mathbf w^{1}(t),\mathbf w^{2}(t)]\|=\max_{0,\tau;\,i}\|\mathbf w^{i}(t)\|. \]
Let \(S[R]\) be the ball of the space \(C[0,\tau]\) with center at zero and radius \(R\). The right-hand side of system (5) defines a nonlinear operator \(\mathcal L\), acting and continuous in \(C[0,\tau]\) (see (4)). If \(MN<1/4\), where
\[ M=M(\tau,\alpha,\beta,\gamma)=\max\|\vec\varphi_{1}+\vec\varphi_{2}\|, \]
\[ N=N(\tau,\alpha,\beta,\gamma)=\max t^{\mu-\gamma}\int_{0}^{t}\|A^{\mu+(m+2)/4-(\alpha+\beta)}e^{-(t-s)\nu A}\|s^{2\gamma-(\alpha+\beta)}\,ds\cdot C(\alpha,\beta) \tag{6} \]
(the maximum is taken over \(t\in[0,\tau]\) and \(\mu=\alpha,\beta\)), then in the ball \(S[(1-\sqrt{1-4MN})(2N)^{-1}]\) the operator \(\mathcal L\) is a contraction. Consequently, in this ball there exists a unique solution of system (5), which can be found by the method of successive approximations. The formula \(\mathbf v(t)=t^{\gamma-\mu}A^{-\mu}\mathbf w(t,\mu,\gamma)\) determines a solution of (1) and, consequently, a solution of the system of Navier—Stokes equations (see (\(^{2,3}\))). The solution (5) found is unique in \(C[0,\tau]\). For the proof in the case \(\gamma>(m-2)/4\), see (\(^{2,3}\)). If \(\gamma\in[(m-2)/4,\mu)\), then, by virtue of (), () and (**), \(\mathbf w(0,\mu,\gamma)=0\). Since the nonlinearity in (5) is of second order, uniqueness holds on a small interval. Next one must pass to equation (1) and apply the “gluing lemma” (see, for example, (\(^{9}\))). Thus one proves:

Theorem 1. System (5) has in \(C[0,\tau]\) a unique solution if () , (*) and
\[ MN<1/4. \tag{7} \]

By virtue of (), () and (**) condition (7) is satisfied for small \(\tau>0\)—the local existence theorem. We note that it is established for \(\mathbf v(0)\in D(A^{(m-2)/4})\). If (7) is satisfied for large \(\tau>0\) or for \(\tau=\infty\), then we obtain a nonlocal theorem more general than the existing ones (see (\(^{10,11}\)).

Theorem 2. Under the conditions of Theorem 1
\[ \|\mathbf w(t,\mu,\gamma)\|\le M\qquad (0\le t\le\tau;\ \mu=\alpha,\beta). \tag{8} \]
This makes it possible to study the stability of the trivial solution with respect to perturbations of the initial velocities \(\mathbf v(0)\) and forces \(\mathbf f\). By Lemma 2, this stability can be estimated in the norms of the Aronszajn—Slobodetskii spaces (see (\(^{4}\)).

4. A nonlocal theorem for (5) can be obtained with the aid of the local one if one establishes the existence of such an \(R\) that from the inequality \(\|\mathbf w\|\le R\) on \([0,\tau]\) there follows the inequality \(\|\mathbf w\|\le R_{1}<R\). Let
\[ \vec\varphi_{2}\ \text{for }\mu=1/2\text{ be continuous for }t>0. \tag{****} \]
Then (cf. (\(^{2,3}\))) for solutions of (1) the energy inequality is valid, whence
\[ \|\mathbf v(t)\|\le e^{-\delta t}\|\mathbf v(0)\|+\int_{0}^{t}e^{-\delta(t-s)}\|P\mathbf f\|\,ds\le V=V(\tau) \]
\[ (0\le t\le\tau,\ \delta=\inf\sigma(-\Delta),\ \sigma(-\Delta)\text{ is the spectrum of }(-\Delta)). \tag{9} \]

From (5), the moment inequality (12), and (9) it follows that

\[ W(\tau,\beta,\gamma)\leq M(\tau,\beta,\beta,\gamma) +K(\tau,\alpha,\beta,\gamma)V(\tau)^{1-\alpha/\beta} W(M,\beta,\gamma)^{1+\alpha/\beta}, \tag{10} \]

where

\[ W=W(\tau,\beta,\gamma)=\max_{[0,\tau]}\|w(t,\beta,\gamma)\|, \]

\[ K=K(\tau,\alpha,\beta,\gamma) =\max_{[0,\tau]} t^{\beta-\gamma} \int_0^t \|A^{(t+2)/4-\alpha}e^{-(t-s)\nu A}\| \,s^{(\gamma-\beta)(1+\alpha/\beta)} C(\alpha,\beta)\,ds . \]

From (10) we obtain

Theorem 3. Suppose that \((****)\) is satisfied and that

\[ KV^{1-\alpha/\beta}M^{\alpha/\beta} < \frac{\beta}{\beta+\alpha} \left(\frac{\alpha}{\beta+\alpha}\right)^{\alpha/\beta}. \tag{11} \]

Then (5) has a unique solution in \(C[0,\tau]\), and

\[ \|w(t,\beta,\gamma)\|\leq Mr, \tag{12} \]

where \(r\) is the smallest root of the equation

\[ r=1+KV^{1-\alpha/\beta}M^{\alpha/\beta}r^{1+\alpha/\beta}. \]

1. Theorem 4. Suppose that \((****)\) is satisfied and \(m=2\). Then

\[ \|w(t,\alpha,0)\|\leq Q(\tau,\alpha)\qquad (0<\alpha<1/2), \tag{13} \]

where \(Q(\tau,\alpha)\) depends only on

\[ \|v(0)\|,\qquad \int_0^t \|Pf\|\,ds,\qquad \max_{t\in[0,\tau]} t^\alpha \left\|A^\alpha\int_0^t e^{-(t-s)\nu A}Pf\,ds\right\|. \]

Proof. Using the estimate (9)

\[ \left\|A^{1/2}\int_0^t e^{-(t-s)A}f(s)\,ds\right\| \leq \frac{1}{\sqrt{2}} \left[\int_0^t \|f(s)\|^2\,ds\right]^{1/2}, \]

which is valid for any positive definite self-adjoint operator, the system (5) for \(\beta=1/2\), and estimate (4), we obtain

\[ \begin{aligned} \|w(t,\alpha,0)\| \leq{}& \|t^\alpha A^\alpha e^{-t\nu A}v(0)\| + \left\|t^\alpha A^\alpha\int_0^t e^{-(t-s)\nu A}Pf\,ds\right\| \\ &+ t^\alpha\int_0^{t/2} \|A^{1/2}e^{-(t-s)\nu A}\| C(\alpha,1/2)\|w(s,\alpha,0)\|\, \|A^{1/2}v(s)\|\,s^{-\alpha}\,ds \\ &+ \frac{2^{\alpha-1/2}C(\alpha,1/2)}{\sqrt{\nu}} \left[ \int_{t/2}^t \|w(s,\alpha,0)\|^2 \|A^{1/2}v(s)\|^2 s^{-2\alpha}\,ds \right]^{1/2}, \end{aligned} \]

where \(v(s)\) is the solution of (1). Since, by virtue of the energy inequality,

\[ \int_0^t \|A^{1/2}v\|\,ds \]

is finite, (13) follows from this.

Estimate (13) means that for the system (5) a local existence theorem holds for any \(v(0)\in H\).

Voronezh Agricultural Institute

Received
26 VIII 1963

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