Abstract
Full Text
UDC 517.944+531/534
MATHEMATICS
Yu. A. DUBINSKII
ON AN OPERATOR SCHEME AND THE SOLVABILITY OF A NUMBER OF QUASILINEAR EQUATIONS OF MECHANICS
(Presented by Academician A. Yu. Ishlinskii, 2 XII 1966)
The paper proves theorems on the solvability of a certain class of nonlinear equations in Banach spaces, making it possible to consider from a general point of view questions of the existence of solutions of quasilinear equations and systems occurring in mechanics.
§ 1. A general scheme for stationary (elliptic) equations. Let \(V, X, Y\) be a triple of Banach spaces; \(A, B\) nonlinear operators. The interrelation of these spaces and operators is expressed by the following diagram
\[ \begin{array}{ccccc} & \xrightarrow{\ i\ } & X & \xrightarrow{\ A\ } & X^* \\ & & & & \\ V & & & & \xrightarrow{\ i\ } V^*,\\ & & & & \\ & \xrightarrow{\ i\ } & X & \xrightarrow{\ B\ } & Y \end{array} \]
where \(i\) is the embedding operator; \(V^*, X^*\) are the conjugate spaces.
Assume that \(X\) is separable and that the embedding \(V \subset X\) is dense everywhere. Consider the equation
\[ A(u)+B(u)=h, \tag{1} \]
where \(u \in X\) is the unknown solution, and \(h \in X^*\) is given.
Definition. A solution of equation (1) is an element \(u \in X\) such that for every \(v \in V\) the identity
\[ \langle A(u),v\rangle+\langle B(u),v\rangle=\langle h,v\rangle \tag{2} \]
holds.
(Here \(\langle w,v\rangle\) is the value of \(w \in V^*\) on \(v \in V\).)
Assumptions. I. Ellipticity of \(A(u)\). For every \(u \in V\)
\[ \lim_{\|u\|_X\to\infty}\frac{\langle A(u),u\rangle}{\|u\|_X}=+\infty . \]
II. Orthogonality of \(B(u)\). For every \(u \in V\), \(\langle B(u),u\rangle=0\).
III. The operators \(A\) and \(B\) are weakly continuous as operators from \(X\) into \(V^*\), i.e., if \(u_n \to u\) weakly in \(X\), then \(\langle A(u_n),v\rangle \to \langle A(u),v\rangle\) and \(\langle B(u_n),v\rangle \to \langle B(u),v\rangle\) for every \(v \in V\).
Theorem 1. If conditions I–III are satisfied, then for every \(h \in X^*\) equation (1) has at least one solution in the sense of (2).
The proof of the theorem is carried out by B. G. Galerkin’s method; here the solvability of the moment equations and the a priori estimate follow from conditions I, II, while the possibility of passage to the limit follows from condition III.
§ 2. Applications to stationary problems.
- The Navier–Stokes system of equations. Let \(G \subset \mathbb{R}^n\), \(u=(u_1,\ldots,u_n)\),
\[ -\Delta u+\sum_{k=1}^{n}u_k\frac{\partial u}{\partial x_k}+\operatorname{grad} p=h(x),\qquad \operatorname{div}u=0,\qquad u\big|_{\partial G}=0 . \tag{3} \]
We apply Theorem 1 to prove solvability of system (3); accordingly, set
\[ X=\{u(x)\mid u(x)\in \dot W_2^{(1)},\ \operatorname{div}u=0\}, \]
\[ V=\{u(x)\mid u(x)\in \dot W_p^{(1)},\ p>n,\ \operatorname{div}u=0\},\quad Y=L_1,\quad A(u)=-\Delta u, \]
\[ B(u)=\sum_{k=1}^n u_k\frac{\partial u}{\partial x_k}. \]
Theorem 2. If \(h(x)\in X^*\), then system (3) is solvable.
This result is known; see, for example, \((^1)\).
Remark. It is not difficult to show that \(X^*=W_2^{(-1)}/\Pi\), where \(\Pi\) is the subspace of potentials of the form \(u(x)=\operatorname{grad}p(x)\), \(p(x)\in L_2\) (cf. \((^1)\)).
2. The system of large deflection of an A. Föppl plate \((^2)\)
\[ \Delta^2\xi+b_1(\chi,\xi) = \Delta^2\xi+ \frac{\partial^2\chi}{\partial y^2}\frac{\partial^2\xi}{\partial x^2} - 2\frac{\partial^2\chi}{\partial x\partial y}\frac{\partial^2\xi}{\partial x\partial y} + \frac{\partial^2\chi}{\partial x^2}\frac{\partial^2\xi}{\partial y^2} = P(x,y), \]
\[ 2\Delta^2\chi+b_2(\xi) = 2\Delta^2\chi - 2\frac{\partial^2\xi}{\partial x^2}\frac{\partial^2\xi}{\partial y^2} + 2\left(\frac{\partial^2\xi}{\partial x\partial y}\right)^2 = 0. \]
In the domain \(G\subset \mathbf R^2\) it is required to find a solution of the first boundary-value problem. Set
\[ X=\{u=(\xi,\chi)\mid u\in \dot W_2^{(2)}\},\quad Y=L_1,\quad V=\dot W_2^{(2)},\quad A(u)=(\Delta^2\xi,2\Delta^2\chi),\quad B(u)=(b_1(\xi,\chi),b_2(\xi)). \]
Theorem 3 (see also \((^3,^4)\)). If \(P(x,y)\in W_2^{(-2)}\), then the system of A. Föppl equations is solvable in the space \(\dot W_2^{(2)}\).
Remark. From the smoothness theorem for the biharmonic equation it follows that if \(P(x,y)\) is smooth, then the solution is also smooth.
§ 3. General scheme for nonstationary equations
Notation:
\[ L_p(X)=\left\{u(t):[0,T]\to X\mid \|u\|^p=\int_0^T\|u\|_X^p\,dt<\infty\right\},\quad 1\leq p<\infty. \]
Consider the Cauchy problem for the equation
\[ u'+A(u)+B(u)=h(t),\quad u(0)=0, \tag{4} \]
where the operators \(A\) and \(B\) act in the following scheme:
\[ \begin{array}{ccccc} & \xrightarrow{\ i\ } & L_p(X) & \xrightarrow{\ A\ } & L_{p'}(X^*) \\ L_\infty(V) & & & & \downarrow i \\ & & & & L_1(V^*),\quad p'=\dfrac{p}{p-1}.\\ & \xrightarrow{\ i\ } & L_p(X) & \xrightarrow{\ B\ } & Y \\ & & & & \uparrow i \end{array} \]
Definitions:
\[ F_1=\{u(t)\mid u\in L_\infty(V),\ u'\in L_1(V^*),\ u(0)=0\}, \]
\[ F_2=\{u(t)\mid u\in L_p(X),\ u'\in L_1(V^*),\ u(0)=0\}; \quad [w,v]=\int_0^T \langle w,v\rangle\,dt, \]
\[ w\in L_1(V^*),\quad v\in L_\infty(V). \]
Assumptions. I. Parabolicity. \(u'+A(u)\). For any function \(u(t)\in F_1\),
\[ \lim_{\|u\|\to\infty}\frac{[u'+A(u),u]}{\|u\|}=+\infty. \]
II. Orthogonality of \(B(u)\). If \(u(t)\in F_1\), then
\[ [B(u),u]=0. \]
III. The operators \(A\) and \(B\) are weakly continuous from \(F_2\) into \(L_1(V^*)\).
Theorem 4. If conditions I–III are fulfilled, then for every function \(h(t)\in L_{p'}(X^*)\) there exists at least one solution \(u(t)\in F_2\) of equation (4), in the sense of the identity
\[ [u'+A(u)+B(u),v]=[h,v],\quad \forall v(t)\in L_\infty(V). \]
The proof is carried out by the Galerkin method with the introduction of a vanishing “viscosity” (see, for example, \((^5)\) or \((^6)\)).
§ 4. Applications to Nonstationary Problems
- The Euler equations of motion of a rigid body in the principal axes of inertia have the form
\[ I_m \frac{d\Omega_m}{dt}+(I_{m+2}-I_{m+1})\Omega_{m+1}\Omega_{m+2}=K_m(t),\qquad \Omega_m(0)=0, \tag{5} \]
where \(m\) runs through the cyclic permutation of the indices \(1,2,3\).
Put \(G=\cdot;\ X=V=\mathbf{R}^3;\ A\equiv 0;\ B(\vec{\Omega})=\{(I_{m+2}-I_{m+1})\Omega_{m+1}\Omega_{m+2}\},\ m=1,2,3;\ p=2\). Then \(L_2(X)=L_2(0,T)\), \(L_1(V^*)=L_1(0,T)\); further, \(F_1=F_2=\{\vec{\Omega}(t)\mid \vec{\Omega}'(t)\in L_1(0,T),\ \vec{\Omega}(0)=0\}\).
Theorem 5. If \(K_m(t)\) are summable functions, then system (5) has at least one solution \(\vec{\Omega}(t)\in F_1\).
Remark. Since \(F_1\subset C(0,T)\), for continuous \(K_m(t)\) it follows from equations (5) that \(\vec{\Omega}'(t)\) is also continuous.
- The nonstationary Navier—Stokes system. Let \(Q=G\times[0,T]\), \(G\subset \mathbf{R}^n\), \(S=\partial G\times[0,T]\);
\[ u'-\Delta u+\sum_{k=1}^{n}u_k\frac{\partial u}{\partial x_k}+\operatorname{grad} p=h(x,t),\qquad \operatorname{div} u=0, \tag{6} \]
\[ u(x,0)=0,\qquad u|_S=0. \tag{7} \]
Put \(X=\{u(x)\mid u\in \mathring{W}_2^{(2)}(G),\ \operatorname{div}u=0\}\), \(V=\{u(x)\mid u\in \mathring{W}_q^{(1)}, q>n,\ \operatorname{div}u=0\}\); \(p=2;\ Y=L_1(Q)\), \(A(u)=-\Delta u\), \(B(u)=\sum_{k=1}^{n}u_k\frac{\partial u}{\partial x_k}\). Then
\[ F_2=\{u(x,t)\mid u\in L_2(\mathring{W}_2^{(1)}),\ u'\in L_1(W_q^{(-1)}),\ u(x,0)=0\}. \]
Theorem 6. If \(h(x,t)\in L_2(X^*)\), then system (6), (7) has at least one solution in \(F_2\).
This theorem refines the theorem of E. Hopf (9) (see also (8)); in particular, the finiteness of \(u'\) in \(L_1(W_q^{(-1)})\) ensures the continuity of \(u(x,t)\) in \(t\) in the sense of the space \(F_2\).
- Analogy with example 2 of § 2. Consider in the cylinder \(Q=G\times[0,T]\), \(G\in \mathbf{R}^{2n}\), the first boundary-value problem for the system
\[ \xi' + \Delta^2\xi-\sum_{i=1}^{n}b_1^{i}(\xi,\chi) =P_1(x_1,y_1,\ldots,x_n,y_n,t), \]
\[ \chi' + \Delta^2\chi-\sum_{i=1}^{n}b_2^{i}(\xi) =P_2(x_1,y_1,\ldots,x_n,y_n,t), \tag{8} \]
where \(b_1^{i}(\xi,\chi)\), \(b_2^{i}(\xi)\) are the same as in § 2, with \((x,y)\) replaced by \((x_i,y_i)\).
Putting \(X=\mathring{W}_2^{(2)}(G)\), \(V=\mathring{W}_p^{(2)}\), \(p>n/2\), \(Y=L_1(Q)\), we obtain:
Theorem 7. If \(P_i(\ldots)\in L_2(W_2^{(-2)})\), \(i=1,2\), then system (8) is solvable in the space \(F_2=\{u=(\xi,\chi)\mid u\in L_2(\mathring{W}_2^{(2)}),\ u'\in L_1(W_p^{(-2)}),\ u(0)=0\}\).
Moscow Power Engineering Institute
Received
25 XI 1966
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