MATHEMATICS

P. E. SOBOLEVSKII

INVESTIGATION OF THE NAVIER—STOKES EQUATIONS BY METHODS OF THE THEORY OF PARABOLIC EQUATIONS IN BANACH SPACES

(Presented by Academician S. L. Sobolev, 22 I 1964)

We consider the first boundary-value problem for the nonstationary nonlinear system

\[ \frac{\partial \mathbf v}{\partial t} -\nu \Delta \mathbf v + v_k \frac{\partial \mathbf v}{\partial x_k} = \operatorname{grad} p+\mathbf f, \qquad \operatorname{div}\mathbf v=0 \quad (0<t\le T;\ x\in\Omega); \tag{1} \]

\[ \mathbf v=0 \quad (0<t\le T;\ x\in S); \qquad \mathbf v(0,x)=\mathbf v_0(x) \quad (x\in\overline{\Omega}). \]

Here \(\Omega\) is an open bounded domain of \(m\)-dimensional space with boundary \(S\), and \(\overline{\Omega}=\Omega+S\); \(\mathbf v(t,x)=(v_1(t,x),\ldots,v_m(t,x))\) and \(p(t,x)\) are the unknown velocity and pressure, defined on \([0,T]\times \overline{\Omega}\) and satisfying (1); \(\mathbf f(t,x)\) and \(\mathbf v_0(x)\) are the prescribed force and initial velocity. If in equations (1) we discard the nonlinear terms \(v_k\partial \mathbf v/\partial x_k\), then we obtain a linearized problem, which we shall call problem \((*)\).

Denote by \(H_q\) the closure in the metric \(L_q(\Omega)\) \((1<q<\infty)\) of the set of all smooth solenoidal vector-functions defined on \(\overline{\Omega}\) and vanishing near \(S\). In \((^4)\) an operator \(P\) of orthogonal projection in \(L_2\) onto \(H_2\) was constructed. It can be shown that it is a bounded projection operator also in \(L_q\) onto \(H_q\). Projecting problems (1) and \((*)\) onto \(H_q\), we arrive at the nonlinear

\[ \frac{d\mathbf v}{dt} +\nu A\mathbf v +P\left(v_k\frac{\partial \mathbf v}{\partial x_k}\right) =P\mathbf f, \qquad \mathbf v(0)=\mathbf v_0 \quad (A=-P\Delta) \tag{1'} \]

and the corresponding linear \((*)\) Cauchy problems for ordinary differential equations of first order in the Banach space \(H_q\). A solution of problem \((1')\) is declared to be a solution of problem (1).

In works \((^{1-4})\) problem \((1')\) was studied in the Hilbert space \(H_2\) in the case when \(m=2,3\). Passing to a Banach space has made it possible to study problem \((1')\) for arbitrary \(m\) and to prove the existence of its unique solution for arbitrary \(\mathbf v_0\in H_m\). If the force \(\mathbf f\) and the boundary \(S\) are sufficiently smooth, then by means of the methods developed in \((^3)\) it can be shown that this solution of problem \((1')\) will be a classical solution of problem (1).

1. In \((^5)\) estimates were established for solutions of problem \((*)\) or \((*')\). If in these inequalities one sets

\[ \mathbf u(t,x)=e^{\lambda t}\mathbf u_0(x), \qquad p(t,x)=e^{\lambda t}p_0(x), \qquad \mathbf f(t,x)=e^{\lambda t}\mathbf f_0(x), \]

where

\[ \lambda \mathbf u_0-\Delta \mathbf u_0=\operatorname{grad}p_0+\mathbf f_0, \qquad \operatorname{div}\mathbf u_0=0 \ (x\in\Omega); \qquad \mathbf u_0(x)=0 \ (x\in S), \]

then we obtain that for any \(q\in(1,\infty)\), sufficiently large \(\sigma_0=\sigma_0(q)>0\), \(\operatorname{Re}\lambda\ge \sigma_0\), and integer \(l\ge0\), the inequality

\[ |\lambda|\,\|\mathbf u_0\|_{W_q^{2l}} +\|\mathbf u_0\|_{W_q^{2l+2}} +\|p_0\|_{W_q^{2l+1}} \le c(q,l)\,\|P\mathbf f_0\|_{W_q^{2l}} \qquad (W_q^0=L_q). \]

Hence it follows

Theorem 1. For \(\operatorname{Re}\lambda \geqslant \sigma_0\) the operator \(A+\lambda I\) has an inverse in \(H_q^{2l}\) and

\[ \left\|(A+\lambda I)^{-1}\right\|_{H_q^{2l}\to H_q^{2l}} \leqslant c(q,l)(|\lambda|+1)^{-1},\qquad \left\|(A+\lambda I)^{-1}\right\|_{H_q^{2l}\to H_q^{2l+2}} \leqslant c(q,l), \tag{2} \]

where \(H_q^{2l}=H_q\cap W_q^{2l}\).*

The first of inequalities (2) means that the operator \(A\) generates in the spaces \(H_q^{2l}\) the analytic semigroup \(e^{-tA}\) \((^{8,9})\), and therefore the theory of equations of parabolic type in Banach spaces \((^{10})\) is applicable to the investigation of problem \((1')\).

1. From the results \((^{2,11})\) it follows (cf. \((^{4})\))

Lemma 1. For arbitrary \(q\in(1,\infty)\), \((\alpha,\beta)\in G(m)\), \(\mathbf v\in D[(-\Delta)^\alpha]\), \(\mathbf w\in D[(-\Delta)^\beta]\) (and \(\operatorname{div}\mathbf v=0\), if \(\alpha+\beta>m/2q\)) the inequality holds

\[ \left\|(-\Delta)^{\alpha+\beta-(m+q)/2q} \frac{\partial}{\partial x_k}(v_k w)\right\|_{L_q} \leqslant c(q,\alpha,\beta) \left\|(-\Delta)^\alpha \mathbf v\right\|_{L_q} \left\|(-\Delta)^\beta \mathbf w\right\|_{L_q}. \]

Here \(G(m)\) is the set of points of the plane \((\alpha,\beta)\) lying on the triangle with vertices \((0,0)\), \(((m+q)/4q,(m+q)/4q)\), and \((0,(m+q)/2q)\), except for the points \((0,0)\), \((0,(m+q)/2q)\), and \((\alpha,\beta)\) with \(\beta\leqslant 1/2\) and \(\alpha>m/2q-1/2\); \(-\Delta\) is the operator in \(L_q\) defined by the differential expression \(-\partial^2/\partial x_i^2\) on \(\overset{\circ}{W}{}_{q}^{2}\); \(D[(-\Delta)^\gamma]\) is the domain of definition of the operator \((-\Delta)^\gamma\).

Obviously, without loss of generality one may assume that the first of inequalities (2) for \(l=0\) is satisfied for \(\operatorname{Re}\lambda\geqslant 0\). Then all inequalities (2) will also be satisfied for \(\operatorname{Re}\lambda\geqslant 0\). Therefore fractional powers \(A^\gamma\) are defined (see \((^{9})\)), and the inequality

\[ \left\|A^\gamma e^{-tA}\right\|_{H_q^{2l}\to H_q^{2l}} \leqslant c(\gamma,q,l)t^{-\gamma}\qquad (t>0). \tag{3} \]

holds.

An operator \(B\), acting in a Banach space \(E\) and having an everywhere dense domain of definition \(D(B)\), is called weakly positive if for every \(\lambda\geqslant 0\) the operator \(A+\lambda I\) has a bounded inverse whose norm satisfies the inequality

\[ \left\|(A+\lambda I)^{-1}\right\|_{E\to E} \leqslant c(E)(\lambda+1)^{-1}. \]

The operator \(-\Delta\) is weakly positive in \(L_q\) (see, for example, \((^{10})\)). As noted above, the operator \(A\) is weakly positive in \(H_q\).

In \((^{12})\) fractional powers of a weakly positive operator were defined. By the methods developed in \((^{13})\), the following is proved.

Lemma 2. Let \(B\) be weakly positive in \(E\), \(F\in(E\to E_1)\) and admit a closure, \(D(F)\supset D(B)\), and for every \(z\in D(B)\) and some \(\alpha\in(0,1)\)

\[ \|Fz\|_{E_1}\leqslant c(E,E_1)\|Bz\|_E^\alpha\|z\|_E^{1-\alpha}. \]

Then for arbitrary \(0<\varepsilon\leqslant 1-\alpha\), \(0<\eta\leqslant \alpha\) and \(z\in D(B^{\alpha+\varepsilon})\)

\[ \|Fz\|_{E_1} \leqslant c(E,E_1,\varepsilon,\eta) \left\|B^{\alpha+\varepsilon}z\right\|_E^{\eta/(\varepsilon+\eta)} \left\|B^{\alpha-\eta}z\right\|_E^{\varepsilon/(\varepsilon+\eta)}. \]

For weakly positive operators the moment inequality \((^{12})\) is valid.

* In particular, we obtain the result of \((^{6})\). We note that the estimates of problem \((*)\), obtained in \((^{7})\), make it possible analogously to obtain estimates of the resolvent of \(A\) in Hölder norms.

Therefore

\[ \|(-\Delta)^\alpha \mathbf v\|_{L_q} \leq c(\alpha,q)\|(-\Delta)\mathbf v\|_{L_q}^{\alpha}\|\mathbf v\|_{L_q}^{1-\alpha} \quad (\mathbf v\in D[(-\Delta)]). \]

By virtue of (6)

\[ \|(-\Delta)\mathbf v\|_{L_q}\leq c\|A\mathbf v\|_{H_q}. \]

It then follows from Lemma 2 that

\[ \|(-\Delta)^\alpha \mathbf v\|_{L_q} \leq c(\alpha,\varepsilon,\eta,q) \|A^{\alpha+\varepsilon}\mathbf v\|_{H_q}^{\eta/(\varepsilon+\eta)} \|A^{\alpha-\eta}\mathbf v\|_{H_q}^{\varepsilon/(\varepsilon+\eta)} \]

\[ (0<\alpha<1;\ 0<\varepsilon\leq 1-\alpha;\ 0<\eta\leq \alpha;\ \mathbf v\in D[A^{\alpha+\varepsilon}]). \]

Hence, and from (3), in turn it follows that for any \(t>0\), \(-1\leq \alpha\leq 1\), \(-1\leq \beta\leq 1\), \(\alpha+\beta>0\), the operators \((-\Delta)^\alpha e^{-tA}P(-\Delta)^\beta\) and \((-\Delta)^\alpha e^{-tA}A^\beta\) admit closures and

\[ \|(-\Delta)^\alpha e^{-tA}P(-\Delta)^\beta\|_{L_q\to L_q},\quad \|(-\Delta)^\alpha e^{-tA}A^\beta\|_{H_q\to L_q} \leq c(\alpha,\beta,q)t^{-(\alpha+\beta)}. \]

1. The results of the preceding item make it possible to study problem \((1')\) in any \(H_q\), as was done in (4) in \(H_2\). Separating in \((1')\) the linear part and setting \(\mathbf w(t,\mu,\gamma)=t^{\mu-\gamma}(-\Delta)^\mu \mathbf v(t)\) \((\mu=\alpha,\beta;\ (\alpha,\beta)\in G(m);\ 1<q\leq m;\ (m-q)/2q\leq \gamma<\alpha)\), we obtain the system

\[ \mathbf w(t,\mu,\gamma)= t^{\mu-\gamma}(-\Delta)^\mu e^{-tvA}\mathbf v_0 +t^{\mu-\gamma}(-\Delta)^\mu \int_0^t e^{-(t-s)vA}P\mathbf f\,ds \]
\[ - t^{\mu-\gamma}\int_0^t \{(-\Delta)^\mu e^{-(t-s)vA}P(-\Delta)^{[(m+q)/2q]-(\alpha+\beta)}\} \times \]
\[ \times (-\Delta)^{\alpha+\beta-(m+q)/2q} \frac{\partial}{\partial x_k} \left[(-\Delta)^{-\alpha}w_k(s,\alpha,\gamma)(-\Delta)^{-\beta}\mathbf w(s,\beta,\gamma)\right] s^{2\gamma-(\alpha+\beta)}\,ds. \tag{4} \]

Denote by \(\vec{\varphi}_i\) the \(i\)-th term on the right-hand side of (4). If \(\mathbf v_0\in D(A^\gamma)\), then from the estimates of item 2 it follows that \(\vec{\varphi}_1\) is continuous on \([0,T]\) and \(\vec{\varphi}_1\to 0\) as \(t\to 0\). If \(\mathbf w(t,\mu,\gamma)\) is continuous on \([0,T]\), then \(\vec{\varphi}_3\) is continuous on \([0,T]\) and \(\vec{\varphi}_3\to 0\) as \(t\to 0\) (Lemma 1 and the corollary to Lemma 2).

Let \(\mathbf f\) be such that \(\vec{\varphi}_2\) is continuous on \([0,T]\) and \(\vec{\varphi}_2\to 0\) as \(t\to 0\). Put

\[ M=M(T,\alpha,\beta,\gamma)=\max_{t,\mu}\|\vec{\varphi}_1+\vec{\varphi}_2\|_{L_q}, \]

\[ N=N(T,\alpha,\beta,\gamma)= \]

\[ =\max_{t,\mu} t^{\mu-\gamma}\int_0^t \|(-\Delta)^\mu e^{-(t-s)vA}P(-\Delta)^{(m+q)/2q-(\alpha+\beta)}\|_{L_q\to L_q} s^{2\gamma-(\alpha+\beta)}\,ds\ c(\alpha,\beta,q)^* . \]

The following holds (cf. (4)).

Theorem 2. The system (4) has a unique solution in \(C[0,T]\), if

\[ MN<\frac14, \tag{5} \]

and this solution can be found by the method of successive approximations.

Here \(C[0,T]\) denotes the Banach space of continuous vector-functions \([\mathbf w_1(t),\mathbf w_2(t)]\) with values in \(L_q\times L_q\). Condition (5) is always satisfied on a small interval \([0,T]\), i.e., a local existence theorem holds. Note that it has been proved for any \(\mathbf v_0\in D(A^{(m-q)/2q})\) \((1<q\leq m)\), i.e., for any \(\mathbf v_0\in H_m\).

\[ {}^*\ c\text{ is the same constant as in Lemma 1.} \]

If (5) is satisfied for large \(T>0\), or for \(T=\infty\), then a nonlocal existence theorem holds.

Theorem 3. Under the conditions of Theorem 2,

\[ \|w(t,\mu,\gamma)\|_{L_q}\leq M. \]

This theorem makes it possible to study the stability of the trivial solution with respect to perturbations of the initial velocities \(v_0\) and forces \(f\).

1. If one uses the known a priori estimate in \(L_2\) for solutions of the Navier—Stokes equations, then one can obtain an existence theorem and give an estimate for solutions under weaker restrictions on \(v_0\) and \(f\) (cf. \(\left({}^{4}\right)\)).

Received
15 I 1964

REFERENCES

\({}^{1}\) S. G. Krein, UMN, 12, 1 (1957).
\({}^{2}\) P. E. Sobolevskii, DAN, 128, No. 1 (1959).
\({}^{3}\) P. E. Sobolevskii, DAN, 131, No. 4 (1960).
\({}^{4}\) P. E. Sobolevskii, DAN, 155, No. 1 (1964).
\({}^{5}\) V. I. Yudovich, DAN, 130, 1214 (1960).
\({}^{6}\) V. A. Solonnikov, DAN, 130, 988 (1960).
\({}^{7}\) V. A. Solonnikov, Materials for the Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, August, 1963.
\({}^{8}\) K. Yosida, Proc. Japan Acad., 34, No. 6 (1958).
\({}^{9}\) M. Z. Solomyak, DAN, 122, No. 5 (1958).
\({}^{10}\) P. E. Sobolevskii, Tr. Mosk. Mat. Obshch., 10, 297 (1961).
\({}^{11}\) P. E. Sobolevskii, DAN, 142, No. 4 (1962).
\({}^{12}\) M. A. Krasnosel’skii, P. E. Sobolevskii, DAN, 129, No. 3 (1959).
\({}^{13}\) P. E. Sobolevskii, Theory of Fractional Powers of Operators in a Banach Space and Its Application to the Study of Equations of Parabolic Type, Doctoral dissertation, Moscow State University, 1962.