论文部分内容阅读
The equations governing incompressible and compressible inviscid flows and written in the physical frame ( t,x,y,z ) are known to be linearly well posed and exhibit elliptic or hyperbolic nature. The linear well posedness is considered here for these equations under a space time transformation ( t,x,y,z)→(τ,ξ,η,ζ ), where the pseudo time τ and the new space coordinate ( ξ,η,ζ ) all depend on ( t,x,y,z ). Such a transformation could be useful for uniformly treating problems in which the flow is fast unsteady somewhere and slow unsteady or steady elsewhere. It is found that the transformation may alter the ellipticity, the hyperbolicty, and even the well posedness of the original equations. In one dimension, the transformed incompressible flow equations become weakly hyperbolic and the compressible ones could degenerate to elliptical equations. In high dimensions there are conditions such that the transformed equations become ill posed.
The equations governing incompressible and compressible inviscid flows and written in the physical frame (t, x, y, z) are known to be linearly well posed and exhibit elliptic or hyperbolic nature. The linear well posedness is considered here for these equations under a space where pseudo time τ and the new space coordinate (ξ, η, ζ) all depend on (t, x, y, z) → (τ, ξ, η, ζ) Such a transformation could be useful for uniformly treating problems in which the flow is fast unsteady somewhere and slow unsteady or steady elsewhere. It is found that the transformation may alter the ellipticity, the hyperbolicty, and even the well posedness of the original equation In one dimension, the transformed incompressible flow equations become weakly hyperbolic and the compressible ones could could be generative to elliptical equations. In high dimensions there are conditions such that the the transformed equations became ill posed.