In this paper, based on matrix and curve integration theory, we theoretically show the existence of Cartesian vector solutions u = b(t) + A(t)x for the general N-dimensional compressible Euler equations. Such solutions are global and can be explicitly expressed by an appropriate formulae. One merit of this approach is to transform analytically solving the Euler equations into algebraically constructing an appropriate matrix A(t). Once the required matrix A(t) is chosen, the solution u is directly obtained. Especially, we find an important solvable relation in ƴ = 1 + 2 / N between the dimension of equations and pressure parameter, which avoid additional independent constraints on the dimension N in existing literatures. Special cases of our results also include some interesting conclusions: (1) If the velocity field u is a linear transformation on x ϵ Rᴺ, then the pressure p is a relevant quadratic form. (2) The compressible Euler equations admit the Cartesian solutions if A(t) is an antisymmetric matrix. (3) The pressure p possesses radial symmetric form if A(t) is an antisymmetrically orthogonal matrix. Copyright © 2014 Wiley Periodicals, Inc.
|Journal||Studies in Applied Mathematics|
|Early online date||Sept 2014|
|Publication status||Published - 2015|