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UDC 517.945.7
MATHEMATICS
M. S. AGRANOVICH
POSITIVE MIXED PROBLEMS
FOR CERTAIN HYPERBOLIC SYSTEMS
(Presented by Academician I. G. Petrovskii, 28 X 1965)
In this note a mixed problem is considered in a cylindrical domain for a general first-order hyperbolic system, assumed to be symmetric (more precisely, Hermitian) only at the points of the lateral surface of the cylinder. An energy inequality and the existence of a strong solution are established under positivity assumptions on the boundary condition of the same kind as in \({}^{(1-3)}\).
Let \(G\) be a bounded domain in the real \(n\)-dimensional space \(R_x^n\) of points \(x=(x^1,\ldots,x^n)\), having an \((n-1)\)-dimensional boundary \(\Gamma\) of smoothness class \(C^2\). In the space \(R_{t,x}^{n+1}\) of points \((t,x)\) consider the cylindrical domain \(\Omega=I\times G\), where \(I\) is the segment \(\{t: 0\le t\le T\}\). Denote by \(\Omega'\) the lateral part \(I\times\Gamma\) of the boundary of the domain \(\Omega\), by \(G(\tau)\) the intersection of the domain \(\Omega\) with the plane \(t=\tau\), and by \(\Gamma(\tau)\) the boundary of this intersection. By \(S\) denote the unit sphere \(|\xi|=1\) in the space \(R_\xi^n\) of points \(\xi=(\xi_1,\ldots,\xi_n)\).
Suppose that in the closure \(\overline{\Omega}\) of the domain \(\Omega\) there is given a system of first-order partial differential equations
\[ Lu\equiv \frac{\partial u}{\partial t} -\sum_{\nu=1}^{n} A_\nu(t,x)\frac{\partial u}{\partial x^\nu} + C(t,x)u=f. \tag{1} \]
Here \(u=u(t,x)\) and \(f=f(t,x)\) are column vectors of height \(N\) with complex components; \(A_\nu\) and \(C\) are square complex matrices of order \(N\). We shall assume that \(A_\nu\in C^2(\overline{\Omega})\).* It is sufficient to assume that the elements of the matrix \(C\) are bounded measurable functions.
We subject the solution \(u(t,x)\) to the boundary condition
\[ B(t,x)u(t,x)=g(t,x)\quad \text{on } \Omega', \tag{2} \]
where \(B\) is a smooth complex matrix on \(\Omega'\), having \(N\) columns and a fixed number \(q\) of linearly independent rows, and \(g\) is a complex column vector of height \(q\), given on \(\Omega'\), and also to the initial condition
\[ u(0,x)=h(x). \tag{3} \]
We assume system (1) to be hyperbolic in the generalized sense of I. G. Petrovskii \({}^{(5)}\) in \(\Omega\) and Hermitian on \(\Omega'\). More precisely, let
\[ a(t,x,\xi)=\sum_{\nu=1}^{n}\xi_\nu A_\nu(t,x). \tag{4} \]
We shall assume the following to hold.
Condition I. For each point \((t_0,x_0)\in\overline{\Omega}\) there exists a neighborhood \(U_0\) in \(R_{t,x}^{n+1}\), and for each point \(\xi_0\in S\) there exists a neighborhood \(S_0\) on \(S\), such that in
\[ V_0=(U_0\cap\overline{\Omega})\times S_0 \]
there is defined a nonsingular matrix
\[ \text{* Our smoothness assumptions can be weakened.} \]
\(r(t,x,\xi)\) of order \(N\) and smoothness class \(C^{2;\infty}_{t,x;\xi}(V_0)\)*, possessing the following two properties: 1) the matrix \(rar^{-1}\) is diagonal and real in \(V_0\) (it suffices to assume it Hermitian); 2) the matrix \(r\) is unitary for \((t,x)\in\Omega'\) (so that the \(A_\nu\) are Hermitian on \(\Omega'\)).
Let us note two important cases in which Condition I is satisfied.
\(1^\circ\). The matrices \(A_\nu\) are Hermitian everywhere in \(\overline{\Omega}\), so that for \(r\) one may take the identity matrix. In this case the theorems formulated below are well known \((^{1-3,6,7})\).
\(2^\circ\). The system (1) is strictly hyperbolic in the sense of I. G. Petrovskii \((^5)\) (i.e., the eigenvalues of the matrix (4) are real and pairwise distinct for \((t,x)\in\overline{\Omega}\), \(\xi\ne0\)), and, moreover, the matrices \(A_\nu\) are Hermitian on \(\Omega'\). In this case, for \(r^{-1}\) one must take the matrix whose columns are the normalized linearly independent eigenvectors of the matrix \(a\). If, in addition, \(n>2\) and the domain \(G\) is simply connected, then \(r\) is immediately constructed globally \((^5)\).
Put \(a_l(t,x)=a(t,x,l)\), where \(l=(l_1,\ldots,l_n)\) is the unit vector of the inner normal to \(\Gamma(t)\) at the point \((t,x)\in\Omega'\).
Condition II. \(\det a_l(t,x)\ne0\) on \(\Omega'\).
Examples show that there exist strictly hyperbolic systems with infinitely smooth \(A_\nu(t,x)\), satisfying Conditions I and II and such that, for no nonsingular matrix \(T(t,x)\), will the matrices \(TA_\nu T^{-1}\) \((\nu=1,\ldots,n)\) be simultaneously Hermitian in \(\Omega\).
For two vectors \(u=\{u_j\}\), \(v=\{v_j\}\) from \(C^N\) put \(u\cdot v=\sum u_j\overline{v}_j\). Denote by \(\mathfrak B(t,x)\) the kernel of the matrix \(B(t,x)\) in \(C^N\). Following \((^{1-3})\), we formulate two more conditions.
Condition III. For every point \((t,x)\in\Omega'\) the quadratic form \(u\cdot a_l(t,x)u\) is positive on \(\mathfrak B(t,x)\):
\[ u\cdot a_l(t,x)u \ge p_0 u\cdot u \quad \text{for } \quad 0\ne u\in\mathfrak B(t,x), \tag{5} \]
where \(p_0=\mathrm{const}>0\).
Condition IV. Condition III is fulfilled, and, moreover, in any \((N-q+1)\)-dimensional subspace of \(C^N\) containing \(\mathfrak B(t,x)\), there exists a vector \(v\) for which \(v\cdot a_l(t,x)v<0\).
Put
\[ \|u\|^2_{G(t)}=\int_{G(t)} |u(t,x)|^2\,dx, \qquad \|u\|^2_{\Gamma(t)}=\int_{\Gamma(t)} |u(t,x)|^2\,dx', \]
where \(dx'\) is the surface element on \(\Gamma(t)\). We define \(\|g\|_{\Gamma(t)}\) and \(\|h\|_G\) analogously.
Theorem 1. Let Conditions I, II, and III be fulfilled. Then, for solutions \(u(t,x)\in C^1(\overline{\Omega})\) of problem (1)—(3), the energy inequality
\[ \|u\|^2_{G(t)}+\int_0^t \|u\|^2_{\Gamma(\tau)}\,d\tau \le \mathrm{const}\left\{ \int_0^t \left[\|f\|^2_{G(\tau)}+\|g\|^2_{\Gamma(\tau)}\right]\,d\tau +\|h\|^2_G \right\}, \qquad 0\le t\le T, \tag{6} \]
holds, where the constant does not depend on \(t\) or on \(u(t,x)\).
In the proof, approaches proposed for other purposes in \((^{4,8,9})\) are used. After localization one computes the derivative
\[ \frac{d}{dt}\int |R(t)u_\varepsilon(t,x)|^2\,dx, \]
where \(R(t)\) is a singular integral operator in \(R_x^n\), depending on the parameter \(t\), with symbol, roughly speaking, equal to \(r\), and where \(u_\varepsilon(t,x)\) is a smooth
* The derivatives with respect to \(t,x\) of order \(\le 2\) are continuous in \((t,x,\xi)\) together with their derivatives of arbitrary order with respect to \(\xi\).
continuation of the function \(u(t,x)\) to \(R_x^n\), equal to 0 outside the \(\varepsilon\)-neighborhood of the domain \(G(t)\); then one passes to the limit as \(\varepsilon \to 0\).
Denote by \(H(\Omega)\), \(H^{(q)}(\Omega')\), and \(H(G)\) the Hilbert spaces of (vector-) functions in \(\Omega\), \(\Omega'\), and \(G\), respectively, with norms defined by the equalities
\[ \|u\|_{\Omega}^{2}=\int_{0}^{T}\|u\|_{G(t)}^{2}\,dt,\qquad \|g\|_{\Omega'}^{2}=\int_{0}^{T}\|g\|_{\Gamma(t)}^{2}\,dt, \]
and \(\|w\|_{G}\); in the first and third cases the column vectors have height \(N\), and in the second, height \(q\). Let \(f,u\in H(\Omega)\); \(g\in H^{(q)}(\Omega')\); \(v\in H^{(N)}(\Omega')\); \(h,w\in H(G)\). We shall call a triple \((u,v,w)\) a strong solution of problem (1)—(3) if there exists a sequence of functions \(u_\nu(t,x)\in C^1(\bar\Omega)\) such that, as \(\nu\to\infty\),
\[ \|u_\nu-u\|_{\Omega}+\|u_\nu-v\|_{\Omega'}+\|u_\nu(T,x)-w\|_{G}+ \]
\[ +\|Lu_\nu-f\|_{\Omega}+\|Bu_\nu-g\|_{\Omega'}+\|u_\nu(0,x)-h\|_{G}\to0. \tag{7} \]
Thus, for a strong solution, the values on \(\Omega'\), \(G(0)\), and \(G(T)\) are defined.
Theorem 2. If conditions I, II, and IV are satisfied, then for any \(f\in H(\Omega)\), \(g\in H^{(q)}(\Omega')\), \(h\in H(G)\) problem (1)—(3) has a strong solution \((u,v,w)\). For strong solutions
\[ \|u\|_{\Omega}+\|v\|_{\Omega'}+\|w\|_{G}\leq \text{const}\{\|f\|_{\Omega}+\|g\|_{\Omega'}+\|h\|_{G}\}, \tag{8} \]
where the constant does not depend on \(f\), \(g\), or \(h\).
The proof of existence is the same as in \((^{1-3})\): the adjoint problem is used and the theorem on coincidence of strong and weak solutions \((^2)\) is applied.
Inequality (8) shows that the strong solution is unique and depends continuously on the right-hand sides.
Remarks. \(1^\circ\). Theorems 1 and 2 remain valid if, with respect to the boundary condition on \(\Omega'\), one assumes, as in \((^{1-3})\), that it can only be written locally in the form (2). In this case it is convenient to regard \(g(t,x)\) as a section of some vector bundle over \(\Omega'\) with a \(q\)-dimensional fiber. The norm \(\|g\|_{\Omega'}\) is then defined by means of a partition of unity on \(\Omega'\) (see, for example, \((^{10})\), Sec. 6). The corresponding definition of a strong solution is equivalent to that given in (3).
\(2^\circ\). Suppose that the matrices \(A_\nu(t,x)\) are Hermitian not only on \(\Omega'\), but also near \(\Omega'\). Then condition II may be replaced by the condition of constancy of the rank of the matrix \(a_\nu(t,x)\) on \(\Omega'\), and condition III by the condition of nonnegativity of the form \(u\cdot a_\nu u\) on \(\mathfrak{B}(t,x)\). Theorems 1 and 2 remain valid after, for example, the following modifications: \(g=0\); \(\|u\|_{\Gamma(\tau)}^{2}\) on the left in (6) is replaced by \(\int_{\Gamma(\tau)} u\cdot a_\nu u\,dx'\); in the definition of a strong solution \(Bu_\nu=0\). Such theorems are obtained almost directly from the results of \((^{1-3,8,9})\) by localization.
\(3^\circ\). Put, for an integer \(l\geq0\),
\[ |||u|||_{G(t),\,l}^{2} = \sum_{\alpha+|\beta|\leq l}\int_{G(t)} \left| \frac{\partial^{\alpha}}{\partial t^{\alpha}} \frac{\partial^{\beta}u}{\partial x^{\beta}} \right|^{2}dx,\qquad \|h\|_{G,l}^{2} = \sum_{|\beta|\leq l}\int_{G} \left| \frac{\partial^{\beta}u}{\partial x^{\beta}} \right|^{2}dx \]
and define similarly \(|||u|||_{\Gamma(t),l}\) with the aid of a partition of unity and local coordinates on \(\Gamma(t)\). Then, if the smoothness assumptions on the matrices \(A_\nu\), \(B\), and \(C\) are strengthened by order \(l\), then under conditions I—III one establishes...
the following inequality, generalizing (6), is established:
\[ |||u|||_{G(t),\,l}^{2}+\int_{0}^{t}|||u|||_{\Gamma(\tau),\,l}^{2}\,d\tau \leq \]
\[ \leq \operatorname{const}\left\{\int_{0}^{t}\left[|||f|||_{G(\tau),\,l}^{2}+|||g|||_{\Gamma(\tau),\,l}^{2}\right]\,d\tau+\|h\|_{G,\,l}^{2}\right\},\qquad 0\leq t\leq T . \tag{9} \]
Inequality (9) is proved by means of localization, application of Theorem 1 to the system obtained from (1)—(3) by differentiation, and use of the embedding theorems of N. Aronszajn—L. N. Slobodetskii (see, for example, (11)).
Moscow Institute
of Electronic Machine Building
Received
20 X 1965
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