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The Cartesian Vector Solutions for the N ‐Dimensional Compressible Euler Equations
Author(s) -
An Hongli,
Fan Engui,
Yuen Manwai
Publication year - 2015
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12056
Subject(s) - mathematics , cartesian coordinate system , euler equations , matrix (chemical analysis) , antisymmetric relation , mathematical analysis , dimension (graph theory) , orthogonal transformation , orthogonal matrix , pure mathematics , geometry , mathematical physics , physics , orthogonal basis , quantum mechanics , composite material , materials science
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 γ = 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 ∈ R N , 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.