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ON COUPLING OF A FIRST ORDER HYPERBOLIC SYSTEM WITH AN ELLIPTIC EQUATION

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{\it R. Bochorishvili, D. Jaiani}

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{\it I.Vekua Institute of Applied Mathematics}\\
{\it I.Javakhishvili Tbilisi State University}
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\par We investigate boundery conditions for coupling of two simplified
models of gas dynamics: system of acoustics and potential flow equations[1]:
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle\frac{\partial\rho}{\partial t}+div\vec u=0, \\
\\[-0.2cm]
\displaystyle\frac{\partial \vec u}{\partial t}+grad P=0, 
\end{array}\right.
\end{equation}
$$
P=c^2\rho,  c=const,   \vec u=(u_1, u_2, u_3),
$$ 
and
\begin{equation}
\Delta\phi=0.
\end {equation}
The first one is a system of hyperbolic equations and another is a scalar 
elliptic one which are valid in domains $ (0,T)\times\Omega _2 $ and  
$\Omega _1 $  respectively, $ T>0 $,
$\Omega_{12}=\Omega_1\cap \Omega_2$, mes $(\Omega_{12})\ne 0,\;\gamma_{12}
=\partial\Omega_{12}\cap\;\Omega_1,\;\gamma_{21}=\partial\Omega_{12}\cap\Omega_2,\;
\gamma_{12}\cap\gamma_{21}=\O.$ Let us denote 
the remaining part of boundaries of $\Omega_1$ and $\Omega_2$ by
$\gamma_1$ and $\gamma_2$ respectively.
\par Note that the relationship between the velocity vector $\vec u$ and the 
potential function $\Phi$ is given by the formulae
\begin{equation}
\vec u=grad\;\Phi.
\end{equation}
\par In [2] formulae (3) was used as a boundury condition for the overlapping domain
and the existence and uniqueness of solution was proved for coupled second 
order elliptic and hyperbolic problems in one space dimension. The similar
result is valid for (1)-(3) in a general case. Let us specify initial and boundary
conditions that will be used below for equations (1) and (2):
\begin{equation}
\frac{\partial\Phi}{\partial n}|_{\gamma_{21}}=\varphi_{21}|_{\gamma_{21}},
\end{equation}
\begin{equation}
\Phi|_{\gamma_1}=\varphi_1|_{\gamma_1} ,
\end{equation}
\begin{equation}
\rho(0,x)=\rho_0(x),x\in\;\Omega_2,
\end{equation}
\begin{equation}
\vec u(0,x)=\vec u_0(x), x\in\Omega_2,
\end{equation}
\begin{equation}
\rho(t,s)=\rho_2(t,s), s\in\gamma_2 ,
\end{equation}
\begin{equation}
\frac{\partial\rho(t,s)}{\partial n}=\rho_{12}(t,s), s\in\gamma_{12},
\end{equation}
\begin{equation}
u(t,s)=grad\;\phi(t,s), s\in \gamma_{12}\cup \gamma_{21}, t\in(0,T) ,
\end{equation}
where $\varphi_{21},\varphi_1,u_0,\rho_0,\rho_2,\rho_{12}$ are given sufficiently
smooth functions.
\par Note that the system of acoustics can be reduced to the following equation
for the density $ \rho$ :
\begin{equation}
\frac{\partial\rho}{\partial t}=c^2\Delta\rho.
\end{equation}
\par Evidently, after $\rho$ is resolved, one can easily restore the velocity
vector $ \vec u$ from (1). This approach, equation (11) and theory of characteristics
are essentially used in the proof of the following
\par {\bf Proposition.} Let the classical solution of problems (2),(4),(5) and (6), (8), (9),
(11) exist and let it be unique. Then the coupled problem (1),(2),(5)-(8),(10) has
a unique classical solution as well.
\par Finally note that the use of the same technique as for the proof of the
above proposition yields other possibilities for correct coupling of  equations
(1) and (2), e.g. using (3) as an interface boundary condition on $\gamma_{21}$
and introducing a non local boundary condition for the velocity vector ensuring
coincidence of its values on $\gamma_{21}$ and $\gamma_{12}$.

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\footnotesize
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{\bf R e f e r e n c e s}

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\par
[1] Landau L.,  Lipschitz E., Theoretical Physics, VI, Hydromechanics, 
M., Nauka, 1986, 786, (Russian). \par
[2] Bochorishvili R., Jaiani D., On a Coupling of Differential Equations Via
Interface Boundary Conditions, BULLETIN of TICMI, 
2, 1998,  25-27,  (electronic version: 
http://www. viam.hepi.edu.ge/others/TICMI).
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