MATHEMATICS

Yu. V. EGOROV

ON HYPERBOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician I. G. Petrovskii, May 7, 1960)

In the first part of the note, the solution of a mixed problem is considered for a general linear hyperbolic equation of the second order with discontinuous coefficients (the general nonstationary problem of diffraction). Under particular assumptions, a generalized solution of this problem was constructed in (¹) by the finite-difference method. In the present work a method is used that was proposed by O. A. Oleinik in (², ³) and consists in approximating the coefficients by smooth functions and applying a priori integral estimates and embedding theorems of S. M. Nikol’skii. This method makes it possible not only to solve the problem in the general case, but also to investigate the smoothness of the obtained solution.

In the second part, by means of the same method, a solution of the Cauchy problem is constructed for a linear hyperbolic system of first order with discontinuous coefficients in the plane, and also for certain symmetric systems in \((n+1)\)-dimensional space \((t, x_1, \ldots, x_n)\).

1. The domain \(\Omega\) of the space \(x = (x_1, \ldots, x_n)\) with boundary \(S\) is divided by \((n-1)\)-dimensional surfaces \(\gamma\) into a finite number of domains \(\Omega_1, \ldots, \Omega_m\). In the cylinder \(Q \equiv \Omega \times [0,T]\) of the space \((t,x)\) the equation is given
   \[ \rho(t,x)\frac{\partial^2 u}{\partial t^2} = \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \left( a_{ij}(t,x)\frac{\partial u}{\partial x_j} \right) + \sum_{i=1}^{n} b_i(t,x)\frac{\partial u}{\partial x_i} + c(t,x)u+f(t,x), \tag{1} \]
   where \(\rho(t,x)\ge \rho_0>0,\ a_{ij}(t,x)=a_{ji}(t,x),\ \sum_{i,j=1}^{n} a_{ij}(t,x)\xi_i\xi_j \ge \alpha_0\sum_{i=1}^{n}\xi_i^2\ (\alpha_0>0)\) for \((t,x)\in Q\) and real \(\xi_1,\ldots,\xi_n\); \(\rho(t,x)\) and \(a_{ij}(t,x)\) are continuous in \(Q_r=\Omega_r\times[0,T]\) \((r=1,\ldots,m)\), and on the surfaces \(\Gamma\equiv\gamma\times[0,T]\) may have discontinuities of the first kind.

A classical solution of the mixed problem for equation (1) with the conditions
\[ u(t,x)\big|_{t=0}=\varphi_0(x),\qquad \frac{\partial u(t,x)}{\partial t}\bigg|_{t=0}=\varphi_1(x),\qquad u(t,x)\big|_{\sigma}=0 \tag{2} \]
\((\sigma\equiv S\times[0,T])\) will mean a function \(u(t,x)\) that is twice continuously differentiable in each of the domains \(Q_r\) \((r=1,\ldots,m)\), satisfies equation (1) in \(Q\setminus\Gamma\), the conditions (2), and, for all points \((t,x)\in\Gamma\),
\[ [u(t,x)]_{\Gamma}=0,\qquad \left[k(t,x)\frac{\partial u}{\partial N}\right]_{\Gamma}=0, \tag{3} \]
where the symbol \([v(t,x)]_{\Gamma}\) denotes the “jump” of the function \(v(t,x)\) at the point \((t,x)\in\Gamma\) when passing through the surface \(\Gamma\); \(\frac{\partial u}{\partial N}\equiv \sum_{i,j=1}^{n} a_{ij}(t,x)\cos(\nu,x_i)\frac{\partial u}{\partial x_j}\) (\(\nu\) is the direction of the normal to \(\Gamma\) at the point \((t,x)\)), and \(k(t,x)\) is a piecewise smooth function with discontinuities of the first kind on \(\Gamma\), with \(k(t,x)\ge k_0>0\). General—

a generalized solution of problem (1), (2), (3) we shall call a function \(u(t,x)\in W_2^{(1)}(Q)\) (see (4)) such that \(u|_{t=0}=\varphi_0\), \(u|_\sigma=0\), and

\[ \int_0^T\!\!\int_\Omega \left\{ k\rho\,\frac{\partial u}{\partial t}\frac{\partial F}{\partial t} -\sum_{i,j=1}^n ka_{ij}\frac{\partial u}{\partial x_i}\frac{\partial F}{\partial x_j} +\left[ \sum_{i=1}^n\left(kb_i-\sum_{j=1}^n a_{ij}\frac{\partial k}{\partial x_j}\right)\frac{\partial u}{\partial x_i} +\frac{\partial k\rho}{\partial t}\frac{\partial u}{\partial t} +kcu+kf \right]F \right\}\,dx\,dt +\int_\Omega (k\rho)|_{t=0} F(0,x)\varphi_1(x)\,dx=0 \tag{4} \]

for every function \(F(t,x)\in W_2^{(1)}(Q)\), equal to zero on \(\sigma\), and for \(t\ge t_1\), where \(t_1\in(0,T)\).

Theorem 1. Let \(\rho(t,x)\), \(a_{ij}(t,x)\), \(b_i(t,x)\), \(c(t,x)\), and their generalized derivatives \(\partial\rho/\partial t\), \(\partial^2\rho/\partial t^2\), \(\partial a_{ij}/\partial t\), \(\partial b_i/\partial t\) be measurable bounded functions in \(\overline{Q}_r\); let \(k(t,x)\in C^{(2)}(\overline{Q}_r)\) \((r=1,\ldots,m)\); \(f(t,x)\in L_2(Q)\). Suppose that \(\varphi_0(x)\in W_2^{(1)}(\Omega)\), \(\varphi_0(x)|_S=0\), \(\varphi_1(x)\in L_2(\Omega)\), and that \(S\) and \(\gamma\) are piecewise-smooth surfaces. Then the generalized solution of problem (1), (2), (3) exists and is unique.

For the proof of the existence of a generalized solution, consider the problem:

\[ \rho^h(t,x)\frac{\partial^2 u_h}{\partial t^2} = \sum_{i,j=1}^n \frac{\partial}{\partial x_i}\left(a_{ij}^h(t,x)\frac{\partial u_h}{\partial x_j}\right) + \sum_{i=1}^n b_i^h(t,x)\frac{\partial u_h}{\partial x_i} + c^h(t,x)u_h+f^h(t,x), \tag{\(1^h\)} \]

\[ u_h|_{t=0}=\varphi_0^h(x),\qquad \frac{\partial u_h}{\partial t}\bigg|_{t=0}=\varphi_1^h(x),\qquad u_h|_{\sigma^h}=0, \tag{\(2^h\)} \]

where \(\rho^h\), \(a_{ij}^h\), \(b_i^h\), \(c^h\in C^{(\infty)}(Q)\) and, as \(h\to0\), converge in the mean to the functions

\[ k\rho,\quad ka_{ij},\quad kb_i-\sum_{j=1}^n a_{ij}\frac{\partial k}{\partial x_j},\quad kc, \]

respectively, remaining uniformly bounded with respect to \(h\), together with the derivatives \(\partial\rho^h/\partial t\) and \(\partial a_{ij}^h/\partial t\); \(f^h\in C^{(\infty)}(Q)\), and as \(h\to0\), \(f^h\) converge to \(kf\) in the mean, and \(\partial^\alpha f^h/\partial t^\alpha|_{t=0}=0\) \((\alpha=0,1,\ldots)\); \(\Omega^h\) lie in \(\Omega\) and have an infinitely differentiable boundary, and the measure of \(\Omega\setminus\Omega^h\) tends to zero as \(h\to0\); \(\sigma^h\) is the lateral surface of the cylinder \(Q^h=\Omega^h\times[0,T]\); \(\varphi_0^h(x)\) and \(\varphi_1^h(x)\in C_0^{(\infty)}(\Omega^h)\), and as \(h\to0\)

\[ \|\varphi_0-\varphi_0^h\|_{W_2^{(1)}(\Omega)} + \|\varphi_1-\varphi_1^h\|_{L_2(\Omega)} \to0. \]

Problem \((1^h),(2^h)\) has a solution \(u_h\in C^{(\infty)}(Q^h)\) (see \((5)\)). In \(Q\setminus Q^h\) set \(u_h\equiv0\). For \(u_h(t,x)\) the energy estimate holds:

\[ \|u_h\|_{W_2^{(1)}(Q)}^2 \le C\left( \|\varphi_0^h\|_{W_2^{(1)}(\Omega)}^2 + \|\varphi_1^h\|_{L_2(\Omega)}^2 + \|f^h\|_{L_2(Q)}^2 \right), \tag{5} \]

where \(C\) does not depend on \(h\). From estimate (5) follow the weak compactness of \(\{u_h\}\) in \(W_2^{(1)}(Q)\) and the existence of a generalized solution. Uniqueness is proved as in (5). Passing to the limit in (5) as \(h\to0\), we obtain the proof of the continuous dependence of the generalized solution on the initial conditions and the function \(f(t,x)\).

Theorem 2. Let \(S\) and \(\gamma\) be continuously differentiable up to order \(l+2\) (\(l\ge n+1\) is an integer), and let \(\gamma\) be closed surfaces not intersecting one another or \(S\). Let \(k(t,x)\), \(\rho(t,x)\), \(a_{ij}(t,x)\in C^{(l+1)}(\overline{Q}_r)\); \(b_i(t,x)\), \(c(t,x)\in C^{(l)}(\overline{Q}_r)\) \((r=1,\ldots,m)\); \(f(t,x)\in W_2^{(l)}(Q\setminus\Gamma)\); \(\varphi_0(x)\in W_2^{(l+2)}(\Omega\setminus\gamma)\); \(\varphi_1(x)\in W_2^{(l+1)}(\Omega\setminus\gamma)\), and suppose the compatibility conditions are satisfied:

\[ \varphi_\alpha|_S=0,\qquad [\varphi_\alpha]_\gamma=0,\qquad \left[ \sum_{p=0}^{\alpha} C_p^\alpha \sum_{i,j=1}^n \left(\frac{\partial^p}{\partial t^p}ka_{ij}\right)\bigg|_{t=0} \frac{\partial\varphi_{\alpha-p}}{\partial x_i}\cos(\nu,x_j) \right]_\gamma=0 \tag{6} \]

\[ (\alpha=0,1,\ldots,l), \]

where \(\varphi_\alpha \equiv \partial^\alpha u/\partial t^\alpha\big|_{t=0}\) for \(\alpha \geqslant 2\) is found from \(\varphi_0\) and \(\varphi_1\) with the aid of equation (1). Then the generalized solution of problem (1), (2), (3) will be classical. For \(l \geqslant n+1+k\), \(u(t,x)\in C^{(k)}(\overline Q_r)\) \((r=1,\ldots,m)\).

The general problem (with conditions (2)) can be reduced to the problem with conditions

\[ u\big|_{t=0}=0,\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=0,\qquad u\big|_\sigma=0,\qquad \frac{\partial^\alpha f}{\partial t^\alpha}\bigg|_{t=0}=0 \quad (\alpha=0,1,\ldots,l-2) \tag{7} \]

by replacing \(u(t,x)\) by \(u(t,x)-v(t,x)\), where \(v(t,x)\in W_2^{(l+1)}(Q\setminus \Gamma)\) and is such that \(v|_\sigma=0\), \([v]_\Gamma=0\), \([k\partial v/\partial N]_\Gamma=0\), and \(\partial^\alpha v/\partial t^\alpha\big|_{t=0}=\varphi_\alpha\) \((\alpha=0,1,\ldots,l)\). The construction of such a function is carried out, when (6) is satisfied, with the aid of results of L. N. Slobodetskii \((^6)\).

Let \(u_h(t,x)\) be the solution of equation \((1^h)\) with the conditions

\[ u_h\big|_{t=0}=0,\qquad \frac{\partial u_h}{\partial t}\bigg|_{t=0}=0,\qquad u_h\big|_\sigma=0,\qquad \frac{\partial^\alpha f^h}{\partial t^\alpha}\bigg|_{t=0}=0 \quad (\alpha=0,1,\ldots,l-1). \]

Then \(u_h\in W_2^{(l+1)}(Q)\), and for the derivatives \(\partial^s u_h/\partial t^s\) an estimate of the form (5) is established:

\[ \int_0^T\!\int_\Omega \left\{ \left(\frac{\partial}{\partial t}\frac{\partial^s u_h}{\partial t^s}\right)^2 + \sum_{i=1}^n \left(\frac{\partial}{\partial x_i}\frac{\partial^s u_h}{\partial t^s}\right)^2 \right\}\,dx\,dt \leqslant C_1\|f\|_{W_2^{(s)}(Q\setminus\Gamma)}^2 \quad (s=0,1,\ldots,l-1). \tag{8} \]

For any domain \(\Omega'\subset \Omega\setminus\gamma\) we obtain the estimate

\[ \|u_h\|_{W_2^{(l)}(\Omega'\times[0,T])} \leqslant C_2\|f\|_{W_2^{(l-1)}(Q\setminus\Gamma)}, \tag{9} \]

using (8) and the equations obtained by differentiating \((1^h)\) and multiplying by certain finite functions.

Similarly, in a neighborhood \(\Omega''\) of any point of \(\sigma\) we have the estimate

\[ \|u_h\|_{W_2^{(l)}(\Omega''\times[0,T])} \leqslant C_3\|f\|_{W_2^{(l-1)}(Q\setminus\Gamma)}. \tag{10} \]

Here \(C_1, C_2, C_3\) do not depend on \(h\); therefore, by the embedding theorems of S. L. Sobolev \((^4)\), for \(l \geqslant \left[\frac{n+1}{2}\right]+1+k\), \(u(t,x)\) has continuous derivatives up to order \(k\) outside \(\Gamma\).

Suppose that, in a neighborhood of a point \(A\) on \(\gamma\), local coordinates \(y_1,\ldots,y_n\) with origin at the point \(A\) are introduced, and that in the cube \(\omega(A,\delta)=\{|y_i|\leqslant\delta\ (i=1,\ldots,n)\}\) the surface \(\gamma\) coincides with the plane \(y_n=0\). Taking the coefficients of \((1^h)\) in \(\omega(A,\delta)\times[0,T]\) to be averages of the corresponding limiting functions with an infinitely differentiable averaging kernel in the coordinates \(t,y_1,\ldots,y_n\), we obtain the estimate

\[ \sum_{r=0}^{l} \sum_{\substack{r_0+r_1+\cdots+r_{n-1}=r-1\\ r'_0+r'_1+\cdots+r'_{n-1}=r}} \int_0^T\!\int_{\omega(A,\delta)} \left\{ \left( \frac{\partial^r u_h} {\partial t^{r_0}\,\partial y_1^{r_1}\cdots \partial y_{n-1}^{r_{n-1}}\,\partial y_n} \right)^2 + \left( \frac{\partial^r u_h} {\partial t^{r'_0}\,\partial y_1^{r'_1}\cdots \partial y_{n-1}^{r'_{n-1}}} \right)^2 \right\}\,dy\,dt \leqslant C_4\|f^h\|_{W_2^{(l-1)}(Q\setminus\Gamma)}^2, \tag{11} \]

using the equation obtained by differentiating \((1^h)\) with respect to \(t,y_1,\ldots,y_{n-1}\). Here \(C_4\) does not depend on \(h\). It follows from (11) that the functions \(v_h=u_h\psi\), where

\(\psi(t,y)\)—a function from \(C^{(\infty)}\), equal to zero outside \(\omega(A,\delta)\times[0,T]\), belong to the space \(H^{l;\,l,\ldots,l,1}_{2,t;\,y_1,\ldots,y_{n-1},y_n}\) (see (7)) and have there norms uniformly bounded with respect to \(h\). Hence, by the embedding theorems of S. M. Nikol’skii (7), the continuity of \(u(t,x)\) everywhere in \(\bar Q\) follows if \(l\ge n+1\). Similarly we obtain that, if \(l\ge n+1+k\), then the function \(u(t,x)\in C^{(k)}(\omega(A,\delta_0)\times(0,T))\) \((0<\delta_0<\delta)\) with respect to the variables \(t,y_1,\ldots,y_{n-1}\). Using equation (1), we obtain that \(u(t,x)\in C^{(k)}(\bar Q_r)\) \((r=1,\ldots,m)\).

Remark. In the general case, when the discontinuities \(\gamma\) intersect one another and the boundary \(S\), arguments analogous to those carried out make it possible to obtain the same smoothness of the solution outside the points of intersection.

1. Consider the problem

\[ \frac{\partial u}{\partial t}+\frac{\partial A(t,x)u}{\partial x}+B(t,x)u=f(t,x),\qquad u|_{t=0}=\varphi_0(x)\quad (u=(u_1,\ldots,u_N)), \tag{12} \]

where the elements of the matrix \(A(t,x)=\|a_{ij}(t,u)\|_1^N\) are continuous and bounded with their first derivatives in the strip \(S_T=\{0\le t\le T,\ -\infty<x<\infty\}\) of the plane \((t,x)\), except for a finite number of smooth nonintersecting lines \(\Gamma\); the elements of the matrix \(B(t,x)\) are measurable bounded functions; \(f(t,x)\in L_2(S_T)\), and \(\varphi_0(x)\in L_2(-\infty,+\infty)\). We shall assume that the eigenvalues of the matrix \(A(t,x)\) are real and distinct at every point \((t,x)\in S_T\).

The definition of a generalized solution of problem (12), its determination, and the study of its smoothness are carried out essentially as for a single equation. To obtain a priori estimates, the method of the paper (8) is used.

In a completely analogous way one studies the solution of the problem

\[ \frac{\partial u}{\partial t}+\sum_{i=1}^{n}\frac{\partial A_i(t,x)u}{\partial x_i} +B(t,x)u=f(t,x)\qquad (u=(u_1,\ldots,u_N)), \]

\[ u(t,x)|_{t=0}=\varphi_0(x)\qquad \left(\int(\varphi_0(x),\varphi_0(x))\,dx<\infty\right), \]

where \(A_i(t,x)\) \((i=1,\ldots,n)\) are symmetric matrices of order \(N\), continuously differentiable in the strip
\(S_T=\{0\le t\le T,\ -\infty<x_i<\infty\ (i=1,\ldots,n)\}\), except for a finite number of smooth surfaces \(\Gamma\), and, moreover, on a surface \(\Gamma\) having the form \(\Phi(t,x)=0\), the matrix

\[ \sum_{i=1}^{n}\frac{\partial\Phi}{\partial x_i}(A_i^+-A_i^-) \]

is nonnegative. Here \(A_i^+\) (respectively \(A_i^-\)) are the limiting values of the matrix \(A_i(t,x)\) on the surface \(\Phi(t,x)=0\) from that side toward which the vector
\(\nabla\Phi=(\partial\Phi/\partial x_1,\ldots,\partial\Phi/\partial x_n)\) is directed (respectively, \(-\nabla\Phi\)).

The author expresses deep gratitude to O. A. Oleinik for valuable suggestions.

Moscow State University
named after M. V. Lomonosov

Received
6 V 1960

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