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| On the Initial-Boundary Value Problems for Quasilinear SymmetricHyperbolic System with Characteristic Boundary |
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Citation： |
Chen Shuxing.On the Initial-Boundary Value Problems for Quasilinear SymmetricHyperbolic System with Characteristic Boundary[J].Chinese Annals of Mathematics B,1982,3(2):223~232 |
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Net amount： 789 |
Authors： |
Chen Shuxing; |
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| Abstract： |
In this paper we discuss the initial-boundary value problems for quasilinear symmetric hyperbolic system with characteristic boundary.Suppose \Omega is a bounded domain,its boundary \partial \Omega is sufficiently smooth.We consider the quasilinear symmetric hyperbolic system
$[\sum\limits_{i = 0}^n {{\alpha _i}(x,u)\frac{{\partial u}}{{\partial {x_i}}}} = f(x,u)\]$
in the domain [0, h]*\Omega. The initial-boundary conditions are
$u|_x_0=0$(2)
$Mu|_[0,h]*\partial \Omega=0$(3)
(No loss of generality, the initial condition may be considered as homogeneous one) . We assume the coefficients of (1), (3) are sufficiently smooth, the compatibility condition and the following conditions are satisfied.
1) when t = 0, u=0, the $\alpha_0(x,u)$ is a positive definite matrix.
2) If [\tilde u\] denotes any vector function satisfying the condition (3), the boundary
[0,h]*\partial \Omega is non-characteristic or regular oliarabterigitic for the operator $[\sum\limits_{i = 0}^n {{\alpha _i}(x,\tilde u) \times \frac{\partial }{{\partial {x_i}}}} \]$. and if $v(0,v_1,\cdots,v_n)$ denotes the normal direction to the boundary, the matrix $\beta(x,\tilde u\)=\sum\limits_{i=0}^n v_i \alpha_i(x,\tilde u\)$ is equal to \beta_0, which only depends on x, and Mu=0 is a maximum non-negatiye subspace of quahratio form u\beta_0u.
3) There exists a non-singular matrix Q (x), such that the matrix
$\tilde =beta(x,v)=Q'(x)\beta(x,Q^-1v)Q(x)$
may be reduced to a block diagonal matrix $[\left( {\begin{array}{*{20}{c}}
{{B_1}}&0\0&{{B_2}}
\end{array}} \right)\]$ and the boundary condition may be reduced to $v_1=\cdots=v_L=0$(4)
when (t,x) lies on the boundary [0,h] \times \partial \Omega, and v satisfies (4),the block B_1 will be equal to a non-singular matrix B_10 and B_2 will vanish.
Under these assumptions, we have proved:
Theorem I. There exists a sufficiently small number \delta, such that if h \leq \delta, the local smooth solution of the initial-boundary value problem (1)—(3) uniquely exists.
This theorem has been applied to gas dynamics. For both steady flow and unsteady
flow in three dimentional space we can use Theorem I to obtain the result of the
unique existance of local smooth solution for the correponding system of equations, if
there isn;t any shook wave. |
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