Abstract
Full Text
MATHEMATICS
M. G. DZHAVADOV
A MIXED PROBLEM FOR A HYPERBOLIC EQUATION WITH A SMALL PARAMETER AT THE HIGHEST DERIVATIVES
(Presented by Academician I. G. Petrovskii, 23 IV 1963)
The aim of our note is to construct an asymptotic expansion, with respect to the small parameter, of the solution of the following mixed problem for a second-order hyperbolic equation in the cylinder \(Q=\Omega \times [0 \leq x_0 \leq T]\):
\[ \mathscr{L}_{\varepsilon}u \equiv \varepsilon \mathscr{L}_{1}u+\mathscr{L}_{2}u=f; \tag{1} \]
\[ u(x_0,\mathbf{x})\big|_{x_0=0}=0,\qquad \frac{\partial u(x_0,\mathbf{x})}{\partial x_0}\bigg|_{x_0=0}=0; \tag{2} \]
\[ u\big|_{F}=0, \tag{3} \]
where \(\varepsilon>0\) is a small parameter; \(\Omega\) is an \(n\)-dimensional domain with boundary \(S\); \(F=S\times [0\leq x_0\leq T]\) is the lateral surface of the cylinder \(Q\), \(\mathbf{x}=(x_1,\ldots,x_n)\);
\[ \mathscr{L}_{1}u \equiv \sum_{i,j=0}^{n} a_{ij}(x_0,\mathbf{x})\, \frac{\partial^2 u}{\partial x_i\partial x_j}; \qquad \mathscr{L}_{2}u \equiv \sum_{i=0}^{n} a_i(x_0,\mathbf{x})\, \frac{\partial u}{\partial x_i}+b(x_0,\mathbf{x})u, \]
\(a_{00}=1;\ a_{ij}=a_{ji}\), and the hyperbolicity condition is satisfied.
For \(\varepsilon=0\), equation (1) degenerates into the equation \(\mathscr{L}_{2}w=f\). Having specified the positive direction of the characteristics \(l(x_0,\mathbf{x})\) of the equation \(\mathscr{L}_{2}w=f\), we split \(F\) into two parts \(F_1\) and \(F_2\). Let \(F_1\) be the part into which the characteristics enter, and \(F_2\) the part from which they leave.
We shall seek solutions of (1), (2), and (3) in the form
\[ u=\sum_{i=0}^{m}\varepsilon^i w_i+ \sum_{i=0}^{m}\varepsilon^{i+1}(v_i^1+v_i^2)+ \varepsilon^{m+1}Z_m, \tag{4} \]
where the functions \(w_i\) are determined by the first iterative process, and the functions \(v_i^1\) and \(v_i^2\) by the second iterative processes; \(Z_m\) is the remainder term.
Before defining the functions occurring in (4), let us give the second splitting of the operator \(\mathscr{L}_{\varepsilon}\) near the boundary \(x_0=0\) and \(F_2\).
In the first case we make the change of variables \(x_0=\varepsilon\tau\) and expand the coefficients of \(\mathscr{L}_{\varepsilon}\) in powers of \(\varepsilon\tau\). Substituting this expansion into the expression \(\mathscr{L}_{\varepsilon}\) and grouping terms with equal powers of \(\varepsilon\), we obtain
\[ \mathscr{L}_{\varepsilon}\equiv \varepsilon^{-1}\left(M_0^1+\sum_{s=1}^{m+1}\varepsilon^s M_s^1\right), \qquad \text{where}\quad M_0^1\equiv \frac{\partial^2}{\partial \tau^2} +a_0^0(\mathbf{x})\frac{\partial}{\partial \tau} \]
(\(a_0^0(\mathbf{x})\) is the first term in the expansion of the coefficient \(a_0(x_0,\mathbf{x})\) in powers of \(\varepsilon\tau\)).
In order to write the analogous splitting of the operator \(\mathscr{L}_{\varepsilon}\) near the boundary \(F_2\), in a \(\rho_0\)-neighborhood of \(F_2\), we introduce local coordinates \((x_0,\mathbf{y},\rho)\) as follows. Let \(M_1\) be an arbitrary point of the surface \(F_2\), and \(M_2\) an interior point of \(Q\). For \(\rho\) we take the length of such a vector \(\overrightarrow{M_1M_2}\) that
\[ \left|\cos(\vec n,\overrightarrow{M_1M_2})\right|\geq \delta>0, \]
where \(\delta\) is a fixed number for all points \(M_1\); \(\vec n\) is the inward normal. By \(\mathbf{y}=(y_1,\ldots,y_{n-1})\) we denote coordinates on the surface \(S_2\), which is the projection of \(F_2\) onto \(x_0=0\).
In the new coordinates the operator \(\mathscr{L}_{\varepsilon}\) has the form
\[ \mathscr{L}_{\varepsilon}\equiv \varepsilon A(x_0,\mathbf{y},\rho)\frac{\partial^2}{\partial \rho^2} +B(x_0,\mathbf{y},\rho)\frac{\partial}{\partial \rho} +h\left(\varepsilon,x_0,\mathbf{y},\rho,\frac{\partial}{\partial \rho},\ldots\right), \]
where \(h(\varepsilon,x_0,\mathbf{y},\rho,\partial/\partial\rho,\ldots)\) is a known function.
In the last expression \(\mathscr L_\varepsilon\) we make the change of variables \(\rho=\varepsilon\eta\) and expand the coefficients in powers of \(\varepsilon\eta\). Analogously to the preceding, we obtain:
\[ \mathscr L_\varepsilon \equiv \varepsilon^{-1}\left(M_0^2+\sum_{i=1}^{m+1}\varepsilon^i M_i^2\right), \qquad \text{where } M_0^2 \equiv A^0\frac{\partial^2}{\partial \eta^2}+B^0\frac{\partial}{\partial \eta} \]
(\(A^0\) and \(B^0\) are the first terms of the expansions of \(A(x_0,y,\rho)\) and \(B(x_0,y,\rho)\) in powers of \(\varepsilon,\eta\), respectively, and \(A^0B^0>0\)).
In the first iterative process the approximate solution of equation (1) is sought in the form
\[ u=w_0+\sum_{i=1}^{m}\varepsilon^i w_i . \]
Substituting this expression for \(u(x_0,\mathbf x)\) into equation (1) and comparing terms with equal powers of \(\varepsilon\), we obtain the following system of equations:
\[ \mathscr L_2 w_0=f; \tag{5} \]
\[ \mathscr L_2 w_i=-\mathscr L_1 w_{i-1}, \qquad i=1,2,\ldots,m. \tag{6} \]
In the second iterative process, near the boundary \(x_0=0\), the approximate solution of the homogeneous equation \(\mathscr L_\varepsilon v=0\) is sought in the form
\[ v=\varepsilon v_0' + \sum_{i=1}^{m}\varepsilon^{i+1}v_i^1 . \]
Substituting the expression for \(v\) into the equation and comparing terms with equal powers of \(\varepsilon\), we obtain
\[ M_0^1v_0^1=0; \tag{7} \]
\[ M_0^1v_i^1=-\sum_{j=1}^{i}M_j^1v_{i-j}^1,\qquad i=1,\ldots,m. \tag{8} \]
In an analogous manner, near the boundary \(F_2\) we obtain the system
\[ M_0^2v_0^2=0; \tag{9} \]
\[ M_0^2v_i^2=-\sum_{j=1}^{i}M_j^2v_{i-j}^2,\qquad i=1,\ldots,m. \tag{10} \]
Obviously, the results of the iterative processes at each stage are connected with one another by initial and boundary conditions. To reveal this connection, we substitute the expression \(u(x_0,\mathbf x)\) from (4) into (2) and (3) and compare terms with equal powers of \(\varepsilon\). As a result we obtain
\[ w_0\big|_{x_0=0}=0,\qquad w_i\big|_{x_0=0}=-v_{i-1}^1\big|_{\tau=0}-v_{i-1}^2\big|_{x_0=0}, \qquad i=1,2,\ldots,m; \tag{11} \]
\[ Z_m\big|_{x_0=0}=\varphi_1, \tag{12} \]
where \(\varphi_1\) is a known function,
\[ \frac{\partial w_0}{\partial x_0}\bigg|_{x_0=0} = -\frac{\partial v_0^1}{\partial \tau}\bigg|_{\tau=0}; \tag{13} \]
\[ \frac{\partial v_i^1}{\partial \tau}\bigg|_{\tau=0} = -\left(\frac{\partial w_i}{\partial x_0}+ \frac{\partial v_{i-1}^2}{\partial x_0}\right)\bigg|_{x_0=0}, \qquad i=1,\ldots,m; \tag{14} \]
\[ \frac{\partial Z_m}{\partial x_0}\bigg|_{x_0=0} =\varphi_2, \tag{15} \]
where \(\varphi_2\) is a known function, and, finally,
\[ w_i=-(v_{i-1}^1+v_{i-1}^2)\big|_{F_1}, \qquad i=0,1,\ldots,m; \tag{16} \]
\[ v_i^2\big|_{\eta=0}=-(w_i+v_{i-1}^1)\big|_{F_2}, \qquad i=0,1,\ldots,m; \tag{17} \]
\[ Z_m\big|_{F}=\varphi_3, \tag{18} \]
where \(\varphi_3\) is a known function. Here it is assumed that if \(r>0\), then \(v_{-r}^i=0\) \((i=1,2)\).
Now define the functions \(w_i, v_i^1\), and \(v_i^2\) \((i=0,1,\ldots,m)\). The function \(w_0\) is the solution of the problem
\[ \mathcal L_2 w_0=f;\qquad w_0\big|_{x_0=0}=0,\qquad w_0\big|_{F_1}=0. \]
Let \(f\), together with its derivatives up to the order needed for us, vanish along the characteristic \(l((0,0))\). Then the problem for \(w_0\) has a unique smooth solution.
As soon as \(w_0\) becomes known, \(v_0^1\) is determined from the following problem:
\[ M_0^1 v_0^1 \equiv \frac{\partial^2 v_0^1}{\partial \tau^2} +a_0^0(x)\frac{\partial v_0^1}{\partial \tau}=0,\qquad \frac{\partial v_0^1}{\partial \tau}\bigg|_{\tau=0} =-\,\frac{\partial w_0}{\partial x_0}\bigg|_{x_0=0}. \]
Obviously,
\[ v_0^1=c_0^1(x)e^{-a_0^0(x)\tau} =c_0^1(x)e^{-a_0^0(x)\frac{x_0}{\varepsilon}}, \]
where
\[ c_0^1(x)=\frac{1}{a_0^0(x)} \frac{\partial w_0}{\partial x_0}\bigg|_{x_0=0}. \]
Let \(a_0^0(x)>0\). Under this condition \(v_0^1\) is a boundary-layer type function near the boundary \(x_0=0\).
Now define the function \(v_0^2\). It is the solution of the following problem:
\[ M_0^2 v_0^2 \equiv A^0\frac{\partial^2 v_0^2}{\partial \eta^2} +B^0\frac{\partial v_0^2}{\partial \eta}=0,\qquad v_0^2\big|_{F_2}=-w_0\big|_{F_2}. \]
Hence
\[ v_0^2=c_0^2(x_0,y)e^{-\frac{B^0}{A^0}\eta} =c_0^2(x_0,y)e^{-\frac{B^0}{A^0}\frac{\rho}{\varepsilon}}, \]
where \(c_0^2(x_0,y)=-w_0\big|_{F_2}\).
After we have found \(w_0, v_0^1\), and \(v_0^2\), the following functions \(w_i, v_i^1\), and \(v_i^2\) are defined by induction. Suppose that for some \(i\) \((i\le m)\) all functions \(w_s, v_s^1, v_s^2\) \((s<i)\) have already been defined. Then, in the \(i\)-th equation (6) and in the conditions (11) and (16), the right-hand sides are already known. Under the assumptions made concerning the coefficients and the right-hand side of equation (1), the problems (6), (11), and (16) have a unique smooth solution.
Now define \(v_i^1\). In equations (8) for \(v_i^1\) and in the initial conditions (14), the right-hand sides are known functions. Obviously, the solutions of these problems, i.e. \(v_i^1\), are functions of boundary-layer type near \(x_0=0\).
Similarly, in equations (10) for \(v_i^2\) and the boundary conditions (17), the right-hand sides are known functions. Then, solving these problems successively, we determine \(v_i^2\) as a boundary-layer type function near the boundary \(F_2\).
Multiply the functions \(v_i^1\) and \(v_i^2\) by the smoothing functions \(\psi_1(x_0/\sigma)\) and \(\psi_2(\rho/\sigma)\), respectively, and denote the new functions again by \(v_i^1\) and \(v_i^2\). Thus we have determined all the functions \(w_i, v_i^1\), and \(v_i^2\) entering the expansion (4).
It remains to estimate the remainder term \(Z_m\). To this end, apply the operator \(\mathcal L_\varepsilon\) to both parts of (4). We obtain
\[ f=\mathcal L_\varepsilon u \equiv (\varepsilon \mathcal L_1+\mathcal L_2) \left(w_0+\sum_{i=1}^{m}\varepsilon^i w_i\right) +\left(M_0^1+\sum_{i=1}^{m+1}\varepsilon^i M_i^1\right) \left(\sum_{i=0}^{m}\varepsilon^i v_i^1\right)+ \]
\[ +\left(M_0^2+\sum_{i=1}^{m+1}\varepsilon^i M_i^2\right) \left(\sum_{i=0}^{m}\varepsilon^i v_i^2\right) +\varepsilon^{m+1}\mathcal L_\varepsilon Z_m. \tag{19} \]
Taking into account the equations obtained by the iterative processes, from (19) we obtain
\[ \mathscr L_\varepsilon Z_m=g(x_0,\mathbf x). \tag{20} \]
Thus, for the remainder term we obtain the following problem:
\[ \mathscr L_\varepsilon Z_m=g(x_0,\mathbf x),\qquad Z_m\big|_{x_0=0}=\varphi_1,\qquad \frac{\partial Z_m}{\partial x_0}\bigg|_{x_0=0}=\varphi_2,\qquad Z_m\big|_F=\varphi_3. \]
Let us note that the functions \(\varphi_1,\varphi_2\), and \(\varphi_3\) satisfy the compatibility condition, i.e.
\[ \varphi_1\big|_S=\varphi_3\big|_{x_0=0},\qquad \varphi_2\big|_S=\varphi_3\big|_{x_0=0}. \]
We seek the function \(Z_m\) in the form
\[ Z_m=Z_m^{(1)}+Z_m^{(2)}, \]
where \(Z_m^{(2)}=\varphi_1+x_0\varphi_2+\varphi_3-(\varphi_1+x_0\varphi_2)\big|_F\) (here it is assumed that the functions \(\varphi_i\) have already been multiplied by smoothing functions).
Obviously, \(Z_m^{(1)}\) is the solution of the following problem:
\[ \mathscr L_\varepsilon Z_m^{(1)}=g_1, \tag{21} \]
\[ Z_m^{(1)}\big|_{x_0=0}=0,\qquad \frac{\partial Z_m^{(1)}}{\partial x_0}\bigg|_{x_0=0}, \tag{22} \]
\[ Z_m^{(1)}\big|_F=0. \tag{23} \]
Lemma. If the coefficients and the right-hand side of equation (21) are smooth functions and the operator \(\mathscr L_2\) separates \(\mathscr L_1\), then the estimate
\[ \|Z_m^{(1)}\|\leq M\|g_1\|, \]
holds, where \(M\) is a constant independent of \(\varepsilon\), and the norm is understood in the sense of the metric of the space \(\mathscr L_2(Q)\).
Let us note that a more general fact holds. If one considers the Cauchy problem for the hyperbolic equation \(\mathscr L_\varepsilon u\equiv \varepsilon\mathscr L_1u+\mathscr L_2u=f\), where \(\mathscr L_1\) is a hyperbolic operator of order \(m+1\), \(\mathscr L_2\) is a hyperbolic operator of order \(m\), and the operator \(\mathscr L_2\) separates \(\mathscr L_1\), then the estimate
\[ |D^{m-1}u,v|\leq M|f,v|, \]
holds, where \(v\) is the variable strip \(0\leq x_0\leq t\);
\[ |D^{m-1}u,v|^2=\int_v\sum |D^\alpha u|^2\,dv,\qquad \alpha\leq m+1. \]
Taking into account the lemma and the estimate
\[ \|Z_m\|\leq \|Z_m^{(1)}\|+\|Z_m^{(2)}\|, \]
we assert that \(Z_m\) is bounded in the metric of \(\mathscr L_2(Q)\).
Thus, the following has been proved.
Theorem. If the coefficients and the right-hand side of equation (1) are smooth functions, \(f(x_0,\mathbf x)\) vanishes together with its derivatives up to the order needed by us along the characteristic \(l(0,0)\), \(a_0^0(\mathbf x)>0\), and the operator \(\mathscr L_2\) separates \(\mathscr L_1\), then the solution of problem (1), (2), (3) is representable in the form (4), where the remainder term \(\varepsilon^{m+1}Z_m\) tends to zero as \(\varepsilon\to0\) like \(\varepsilon^{m+1}\) in the sense of the metric of \(\mathscr L_2(Q)\).
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
17 IV 1963
CITED LITERATURE
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- V. A. Trenogin, Proceedings of the Moscow Physico-Technical Institute, issue 9 (1962).