O. A. OLEINIK

ON THE SYSTEM OF PRANDTL EQUATIONS IN THE THEORY OF THE BOUNDARY LAYER

(Presented by Academician L. S. Pontryagin on 2 XI 1962)

In this paper we study the system of boundary-layer equations for an incompressible fluid

\[ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} - \frac{dp}{dx}, \qquad \frac{\partial u}{\partial x} + \frac{\partial v}{\partial x} = 0 \tag{1} \]

in the domain \(D_A\{0 < x < A, 0 < y < \infty\}\), with conditions

\[ u|_{y=0}=0,\qquad v|_{y=0}=v_0(x),\qquad u|_{x=0}=u_0(y),\qquad u(x,y)\to U(x)\quad \text{as } y\to\infty \tag{2} \]

uniformly in \(x\), where \(U^2(x)+2p(x)=C\) (Bernoulli’s law). Under natural assumptions concerning \(v_0(x)\), \(u_0(y)\), \(p(x)\), a solution of the problem (1), (2) in the domain \(D_A\) has been obtained for some \(A>0\), and its uniqueness has been proved. As is known (see \((^1)\), Ch. IV), a solution of the system (1) with conditions (2) may fail to exist in \(D_A\) for large \(A\), as a consequence of separation of the boundary layer.

We shall assume that \(u_0(y)>0\) for \(y>0\); \(u_0(0)=0\); \(u_0'(0)>0\); \(u_0(y)\to U(0)\ne0\) as \(y\to\infty\); \(dp/dx\) and \(v_0(x)\) are continuously differentiable on \([0,A]\); \(u_0\), \(u_0'\), \(u_0''\) are bounded for \(0\le y<\infty\) and satisfy the Hölder condition. We also assume that the compatibility condition at the point \((0,0)\) is fulfilled:

\[ \nu u_0''(y) - p'(0) - v_0(0)u_0'(y) = O(y^2) \tag{3} \]

for small \(y\). Let \(\bar D_A\) be the closure of \(D_A\).

1. Existence theorem. In the domain \(D_A\), for some \(A>0\), there exists a solution \(u,v\) of the system (1) with conditions (2) such that \(u\) is continuous and bounded in \(\bar D_A\), \(u>0\) for \(y>0\), \(k_1y \ge u \ge k_2y\) for \(0<y\le y_0\), where \(y_0\), \(k_1>0\), \(k_2>0\) are certain constants; \(\partial u/\partial y\) and \(\partial^2u/\partial y^2\) are continuous and bounded in \(\bar D_A\); \(\partial u/\partial x\), \(v\), \(\partial v/\partial y\) are continuous and bounded in any finite part of the domain \(\bar D_A\). If \(u_0'(y)\to0\) as \(y\to\infty\) sufficiently rapidly, then \(\partial u/\partial x\), \(\partial v/\partial y\) are bounded in \(D_A\). The value of \(A\) depends on \(u_0\), \(v_0\), \(p\), \(U\). In the case \(dp/dx\le0\), \(v_0(x)\le0\), such a solution of the problem (1), (2) exists in \(D_A\) for any \(A<\infty\).

The proof of this theorem is based on the following proposition.

Lemma. For the existence of a solution \(u,v\) of the problem (1), (2) possessing the properties indicated in the theorem, it is sufficient that there exist, in the domain \(G_A\{0<x<A,\ 0<\psi<\infty\}\), a solution of the equation

\[ \frac{\partial w}{\partial x} + v_0(x)\frac{\partial w}{\partial \psi} = \nu \sqrt{w}\,\frac{\partial^2 w}{\partial \psi^2} - 2\frac{dp(x)}{dx} \tag{4} \]

with conditions:

\[ w|_{\psi=0}=0,\qquad w|_{x=0}=w_0(\psi),\qquad w_0\left(\int_0^y u_0(y)\,dy\right)\equiv u_0^2(y), \tag{5} \]

which has the following properties: \(w(x,\psi)\) is continuous and bounded in \(\bar G_A\); \(w>0\) for \(\psi>0\); \(w(x,\psi)\) has continuous derivatives entering

in equation (4); \(|\partial w/\partial \psi| \leq M\), \(|\sqrt{w}\,\partial^2 w/\partial\psi^2| \leq M\) in \(G_A\); \(|\partial w/\partial x| \leq M\psi^{1-\beta}\), \(\partial w/\partial \psi > m > 0\) for \(0 \leq \psi \leq \psi_0\), \(0<\beta<1/2\). The constants \(M, m, \psi_0\) depend on \(A, v_0, u_0, p\).

The assertion of the lemma follows from the fact that the change of variables
\[ x' = x,\qquad \psi=\psi(x,y),\qquad w=u^2, \tag{6} \]
where \(\partial\psi/\partial y=u\), \(-\partial\psi/\partial x=v-v_0(x)\), \(\psi(x,0)=0\), reduces system (1) with conditions (2) to problem (4), (5) (see \((^2)\)).

The construction of the solution of problem (4), (5) with the required properties is carried out as follows. Consider the domain \(G_A^\varepsilon\{0<x<A,\ 0<\psi<1/\varepsilon\}\). By \(\Gamma_A^\varepsilon\) denote its boundary lying on the straight lines \(x=0\), \(\psi=0\), \(\psi=1/\varepsilon\). In the domain \(G_A^\varepsilon\) consider equation (4) with the conditions on \(\Gamma_A^\varepsilon\):
\[ w|_{\psi=0}=w_0(\varepsilon)e^{V(\varepsilon)x/w_0(\varepsilon)},\qquad w|_{x=0}=w_0(\varepsilon+\psi), \]
\[ w|_{\psi=1/\varepsilon}=w_0(\varepsilon+1/\varepsilon)e^{V(\varepsilon+1/\varepsilon)x/w_0(\varepsilon+1/\varepsilon)}, \tag{7} \]
where \(V(\psi)=v\sqrt{w_0(\psi)}\,w_0''(\psi)-2p'(0)-v_0w_0'(\psi)\). According to condition (3), \(V(\psi)=O(\psi)\) for small \(\psi\), and \(w\geq k\varepsilon\) on \(\Gamma_A^\varepsilon\), where \(k\) does not depend on \(\varepsilon\). If a positive solution \(w_\varepsilon(x,\psi)\) of problem (4), (7) exists in the domain \(G_A^\varepsilon\), then for \(x\leq x_0\) the a priori estimate
\[ w_\varepsilon(x,\psi)\geq w_\varepsilon(x,0)+f(\psi)(1+e^{-\alpha x}) \tag{8} \]
holds, where \(\alpha>0\), \(f(\psi)=A_1\psi^{4/3}+A_2\psi\) for \(\psi\leq 1\), \(A_1>0\), \(A_2>0\); \(A_3>f(\psi)\geq f(1)\), \(|f'(\psi)|\leq A_4\), \(|f''(\psi)|\leq A_5\) for \(\psi>1\). The constants \(A_1,A_2,A_3,1/\alpha\), and \(x_0\) are sufficiently small and do not depend on \(\varepsilon\). If \(dp/dx\leq 0\) and \(v_0(x)\leq 0\), then the estimate
\[ w_\varepsilon(x,\psi)\geq w_\varepsilon(x,0)+f(\psi)e^{-\alpha x} \tag{9} \]
holds throughout the entire domain \(G_A^\varepsilon\) for any \(A<\infty\) and \(\varepsilon>0\).

It is clear that, by virtue of the maximum principle, \(w_\varepsilon\leq A_6\) in \(G_A^\varepsilon\), where \(A_6\) also does not depend on \(\varepsilon\). Using these estimates and theorem 13 of \((^3)\), we obtain the existence of solutions of problem (4), (7) in the domain \(G_A^\varepsilon\) for \(A=x_0\). If \(dp/dx\leq 0\), then \(w_\varepsilon\) exists in any domain \(G_A^\varepsilon\) for \(A<\infty\) and \(\varepsilon>0\).

The solution of problem (4), (5) in \(G_A\), \(A=x_0\), is obtained as the limit as \(\varepsilon\to0\) of the solutions \(w_\varepsilon(x,\psi)\) of problem (4), (7). For this it is sufficient to establish the following estimates for \(w_\varepsilon\) for \(0\leq x\leq A_0\):
\[ 0<M_1\leq \partial w_\varepsilon(x,0)/\partial\psi,\qquad |\partial w_\varepsilon/\partial\psi|\leq M_2,\qquad |\partial w_\varepsilon/\partial x|\leq M_3, \]
\[ |\sqrt{w_\varepsilon}\partial^2 w_\varepsilon/\partial\psi^2|\leq M_4,\qquad |w_\varepsilon^{\beta-1}\partial w_\varepsilon/\partial x|\leq M_5,\qquad 1/2>\beta>0, \]
where the constants \(M_i\) and \(\beta\) do not depend on \(\varepsilon\), and to show that the norm of \(w_\varepsilon\) in \(C^{2+\gamma}\) is bounded by a constant depending only on \(\psi_1\), for \(\psi\geq\psi_1>0\).

The estimates (8) and (9) are established with the aid of the maximum principle for equation (4). It follows from these estimates that \(\partial w_\varepsilon(x,0)/\partial\psi\geq M_1>0\). Applying the maximum principle, we also obtain that
\[ w_\varepsilon(x,\psi)-w_\varepsilon(x,0)\leq (2\psi-\psi^{4/3})e^{\alpha(x+1)} \]
for \(0\leq\psi\leq1\), and \(\partial w_\varepsilon(x,0)/\partial\psi\leq M_2'\), where \(\alpha>0\) does not depend on \(\varepsilon\) and is sufficiently large. Differentiating equation (4) with respect to \(\psi\), we obtain an equation for \(\partial w_\varepsilon/\partial\psi\), for which the maximum principle is valid. Therefore \(|\partial w_\varepsilon/\partial\psi|\leq M_2\) in \(G_A^\varepsilon\). Taking this estimate, estimates (8) and (9) into account, and applying lemma 6 from \((^3)\) to equation (4) for \(\psi\geq\psi_1\), we obtain that \(w_\varepsilon\) satisfies the Hölder condition with constants independent of \(\varepsilon\). Applying further the results of \((^4)\), we obtain that \(w_\varepsilon\in C^{2+\gamma}\) for \(\psi\geq\psi_1\), with norm bounded uniformly in \(\varepsilon\). The estimate \(\partial w_\varepsilon/\partial x\geq -M_3\) is easily obtained by applying the maximum principle to the equation satisfied by \(\partial w_\varepsilon/\partial x\).

It follows from equation (4) that

\[ \sqrt{w_\varepsilon}\,\frac{\partial^2 w_\varepsilon}{\partial \psi^2}>-M_4 . \]

A uniform in \(\varepsilon\) estimate for \(\sqrt{w_\varepsilon}\,\partial^2 w_\varepsilon/\partial\psi^2\) from above holds on the straight lines \(\psi=0,\ x=0,\ \psi=\psi_1\). If \(\psi_1\) is sufficiently small, then

\[ \frac{\partial w_\varepsilon(x,\psi)}{\partial\psi} \geqslant M_1-M_4\int_0^\psi w_\varepsilon^{-1/2}\,d\psi \geqslant M_6>0 \]

for \(0\leqslant \psi\leqslant \psi_1\). We differentiate equation (4) with respect to \(\psi\) and make the substitution \(\partial w_\varepsilon/\partial\psi=\varphi(q_\varepsilon)\). We differentiate the resulting equation for \(q_\varepsilon\) with respect to \(\psi\) and put \(\sqrt{w_\varepsilon}\,\partial q_\varepsilon/\partial\psi=\eta_\varepsilon\). If \(\varphi(q_\varepsilon)=(1+e^{q_\varepsilon})M_6/2\), then for \(0\leqslant\psi\leqslant\psi_1\) we obtain for \(\eta_\varepsilon\) an equation for which the maximum principle is valid. Therefore \(\eta_\varepsilon\leqslant M_7\) and \(\sqrt{w_\varepsilon}\,\partial^2w_\varepsilon/\partial\psi^2\leqslant M_4\). From (4) it follows that \(|\partial w_\varepsilon/\partial x|\leqslant M_3\). To estimate \(w_\varepsilon^{\beta-1}\partial w_\varepsilon/\partial x\), in equation (4) we make the substitution \(w_\varepsilon^\beta=z_\varepsilon\) and differentiate it with respect to \(x\). The equation for \(\partial z_\varepsilon/\partial x\) has the maximum principle for \(0\leqslant\psi\leqslant\psi_2\), where \(\psi_2\) does not depend on \(\varepsilon\) and is sufficiently small. Therefore

\[ |\partial z_\varepsilon/\partial x|\leqslant M_5\beta . \]

It is easy to verify that \(w(x,\psi)=\lim w_{\varepsilon_n}(x,\psi)\) as \(\varepsilon_n\to0\) is the required solution of (4), (5). Let us prove that \(w\to U^2(x)\) as \(\psi\to\infty\). Since \(w_0(\psi)\to U^2(0)\) as \(\psi\to\infty\), then \(s(x,\psi)\equiv w(x,\psi)+2p(x)-C\to0\) for \(x=0\) and \(\psi\to\infty\); \(\nu\sqrt{w}\,\partial^2s/\partial\psi^2=\partial s/\partial x+v_0\partial s/\partial\psi\). On the basis of the maximum principle for the functions \(\pm s+\delta+Me^{-\psi+\alpha x}\geqslant0\) for \(\psi\geqslant1\), where \(\delta>0\) is arbitrary and \(M>0\) and \(\alpha>0\) are sufficiently large. Hence it follows that \(|w(x,\psi)-U^2(x)|<2\delta\) for large \(\psi\).

Thus, the theorem on the existence of a solution of problem (1), (2) is proved.

2. Uniqueness theorem. Let \(u,v\) satisfy system (1) at points of \(D_A\), be continuous in \(\overline{D}_A\), and satisfy conditions (2); \(0<u<k_0\) for \(\psi>0\). Suppose the inequalities

\[ k_1y\geqslant u\geqslant k_2y\quad \text{for }0\leqslant y\leqslant y_0,\quad k_1>0,\quad k_2>0;\quad |\partial^2u/\partial y^2|\leqslant k_3\ \text{in }D_A. \tag{10} \]

are satisfied.

The solution \(u,v\) of problem (1), (2) possessing these properties is unique. In the case \(dp/dx\leqslant0\) and \(v_0(x)\leqslant0\), assumptions (10) may be omitted.

The proof of this theorem, by means of transformation (6), reduces to the proof of uniqueness of a solution of problem (4), (5) such that \(w\) is continuous and bounded in \(\overline{G}_A\); \(k_4\psi\geqslant w\geqslant k_5\psi\) for \(0\leqslant\psi\leqslant1\); \(|\sqrt{w}\,\partial^2w/\partial\psi^2|\leqslant k_6\) in \(G_A\), and \(w>0\) for \(\psi>0\). The difference of two such solutions \(w_1\) and \(w_2\) satisfies the equation

\[ \nu\sqrt{w_1}\,\frac{\partial^2(w_1-w_2)}{\partial\psi^2} -\frac{\partial(w_1-w_2)}{\partial x} -v_0(x)\frac{\partial(w_1-w_2)}{\partial\psi} +c(x,\psi)(w_1-w_2)=0, \]

where

\[ c(x,\psi)=\left(\sqrt{w_1}+\sqrt{w_2}\right)^{-1} \frac{\partial^2w_2}{\partial\psi^2}. \]

By virtue of the assumptions, \(|c(x,\psi)(w_1-w_2)|\leqslant k_7\), \(|c(x,\psi)|\leqslant k_8\psi^{-1}\) for \(0<\psi\leqslant1\), and \(|c(x,\psi)|\leqslant k_9\) for \(\psi\geqslant1\).

Let \(c_\delta\equiv c(x,\psi)\) for \(\psi\geqslant\delta>0\) and \(c_\delta\equiv0\) for \(\psi\leqslant\delta\);

\[ \mathcal L_\delta(F)\equiv \nu\sqrt{w_1}\,\frac{\partial^2F}{\partial\psi^2} -\frac{\partial F}{\partial x} -v_0\frac{\partial F}{\partial\psi} +c_\delta F; \]

\(\Phi=e^{\alpha x}\varphi(\psi)\), where \(\alpha>0\),

\[ \varphi(\psi)=2\psi-\psi^{4/3}\quad \text{for }0\leqslant\psi\leqslant1, \]

\(\varphi(1)\leqslant\varphi(\psi)\leqslant2\) for \(\psi>1\), and \(\varphi'\), \(\varphi''\) are bounded. It is easy to verify that \(\mathcal L_\delta(\Phi)<-k_{10}\psi^{-1/6}\), \(k_{10}>0\), for \(\psi\leqslant\psi_3\), and \(\mathcal L_\delta(\Phi)<0\) for \(\psi\geqslant\psi_3\). Let \(\varepsilon>0\) be arbitrary. For sufficiently small \(\delta\),

\[ \mathcal L_\delta\bigl(\varepsilon\Phi\pm(w_1-w_2)\bigr)<0. \]

Therefore \(|w_1-w_2|\leqslant\varepsilon\Phi\), and consequently \(w_1\equiv w_2\).

In the case \(dp/dx\leqslant0\) and \(v_0(x)\equiv0\), the existence of merely a solution of problem (4), (5) continuous in \(\overline{D}_A\) was obtained in paper (5). Questions of uniqueness of solutions of problems for the boundary-layer equations were studied in the works of Nickel (see (6)).

3. Behavior of the solutions of problem (4), (5) as \(x \to \infty\) in the case \(dp/dx \leqslant 0\). Let \(2P(x)=-2(p(x)-p(0))+K_0\). The following estimate holds:

\[ \int_0^\infty \left|\sqrt{w_1(X,\psi)}-\sqrt{w_2(x,\psi)}\right|\,d\psi \leqslant \frac{\sqrt{P(0)}}{\sqrt{P(X)}} \int_0^\infty \left|\sqrt{w_1(0,\psi)}-\sqrt{w_2(0,\psi)}\right|\,d\psi, \tag{11} \]

where \(K_0=\max_\psi\{w_1(0,\psi),\,w_2(0,\psi)\}\), for any two solutions \(w_1(x,\psi)\) and \(w_2(x,\psi)\) of problems of the form (4), (5). In the case \(v_0(x)\equiv 0\), using (11) and the particular solutions (1), (2) constructed in \(({}^2,{}^7)\), one can show that
\[ \left|\sqrt{w_1(X,\psi)}-\sqrt{w_2(X,\psi)}\right| \leqslant K_1(X+1)^{-1/2}, \]
if \(|dp/dx|\geqslant K_2(1+x)^{-1/3}\) and the integral on the right-hand side of (11) converges, \(K_i=\mathrm{const}\). If
\[ |dp/dx|\leqslant K_3(1+x)^{-(1/3+\varepsilon)},\qquad \varepsilon>0, \]
then \(w(x,\psi)\to 0\) as \(x\to\infty\), uniformly on every finite interval of the \(\psi\)-axis.

Moscow State University
named after M. V. Lomonosov

Received
22 XI 1962

References

\({}^1\) L. D. Landau, E. M. Lifshitz, Fluid Mechanics, Moscow, 1954, Ch. IV.
\({}^2\) G. Schlichting, Boundary Layer Theory, Moscow, 1956.
\({}^3\) O. A. Oleinik, S. N. Kruzhkov, UMN, 16, no. 6, 116 (1961).
\({}^4\) A. Friedman, J. Math. and Mech., 7, no. 5, 771 (1948).
\({}^5\) A. N. Tikhonov, Izv. AN SSSR, ser. matem., 7, 35 (1943).
\({}^6\) K. Nickel, Partial Differential Equations and Continuum Mechanics, Madison, 1961, p. 319—330.
\({}^7\) H. Weyl, Ann. Math., Ser. 2, 43, no. 2, 381 (1942).