锥形坐标系中3维定常位势流方程的椭圆性原理

The ellipticity principle for the 3-D steady potential flow in conical coordinates

本文研究状态方程满足p′(ρ)=ρ^(γ-1)的3维定常可压缩位势流方程,其中p和ρ分别表示压强和密度,γ≥-1是一个已知常数.在锥形坐标系中,位势流方程是一个典型的混合型方程,其类型可由一个拟Mach数L决定:方程是椭圆型的当且仅当L<1,是双曲型的当且仅当L>1.本文首先证明一个椭圆性原理:当γ≥-1时,此方程在抛物-椭圆区域的内部是椭圆型的.特别地,当γ>-1时,可找到一个与区域无关的函数δ0>0使得L≤1-δ0.然后利用该椭圆性原理,可以更进一步地说明,当γ≥-1时,此方程在严格远离退化边界的任一子区域上是一致椭圆型的.

In this paper,we consider the 3-D steady potential flow for a compressible gas with pressure satisfying p′(ρ)=ρ^(γ-1),whereρis the density andγ≥-1 is a constant.The potential equation is of mixed type in conical coordinates,and the type is completely determined by a pseudo-Mach number L with L<1(resp.L>1)corresponding to elliptic(resp.hyperbolic)regions.We first establish an ellipticity principle:forγ≥-1,the equation is elliptic in the interior of a parabolic-elliptic region;in particular,whenγ>-1,there exists a domaindependent functionδ0>0 such that L≤1-δ0.Then applying this ellipticity principle,we further prove that forγ≥-1,the equation is uniformly elliptic in any subregion that is strictly away from the degenerate boundary.