The Analytical Solutions for the N‐Dimensional Damped Compressible Euler Equations

Abstract

Employing matrix formulation and decomposition technique, we theoretically provide essential necessary and sufficient conditions for the existence of general analytical solutions for N -dimensional damped compressible Euler equations arising in fluid mechanics. We also investigate the effect of damping on the solutions, in terms of density and pressure. There are two merits of this approach: First, this kind of solutions can be expressed by an explicit formula u = b(t) + A(t)x and no additional constraint on the dimension of the damped compressible Euler equations is needed. Second, we transform analytically the process of solving the Euler equations into algebraic construction of an appropriate matrix A(t). Once the required matrix A(t) is chosen, the solution u is obtained directly. Here, we overcome the difficulty of solving matrix differential equations by utilizing decomposition and reduction techniques. In particular, we find two important solvable relations between the dimension of the Euler equations and the pressure parameter: γ = 1 − 2/N in the damped case and γ = 1 + 2/N for no damping. These two cases constitute a full range of solvable parameter 0 ≤ γ < +∞. Special cases of our results also include several interesting conclusions: (1) If the velocity field u is a linear transformation on the Euclidean spatial vector x ∈ R N , then the pressure p is a quadratic form of x.(2) The damped compressible Euler equations admit the Cartesian solutions if A(t) is an antisymmetric matrix. (3) The pressure p possesses radially symmetric forms if A(t) is an antisymmetrical orthogonal matrix.

Quantum walks and non-Abelian discrete gauge theory

“…From this Boltzmann equation, one can derive 16 [171, 172, 173] mesoscopic-scale equations (such as the standard Navier-Stokes equation), by taking the so-called hydrodynamical limit 17 [174], which essentially consists in a "patching together of equilibria which are varying slowly in space and time" [56]. A recent paper (2016) by Chow et al [175] gives an analytical solution for the N -dimensional compressible Euler equations with damping, for a barotropic (pressure only function of the density) fluid, with the pressure as function of the density given by a power law.…”

The Cartesian analytical solutions for the N-dimensional compressible Navier–Stokes equations with density-dependent viscosity

“…Based on the new matrix theory and decomposition technique, An, Fan and Yuen proved the existence of the Cartesian solutions for the compressible Euler equations (1.6) [22]. Then Chow, Fan and Yuen further generalized to the damped Euler equations [33].…”

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

“…for the multi-component CH system. This kind solution is part of a long history of finding exact solutions for fluid flows, especially for the Euler and Navier-Stokes (NS) equations [44][45][46][47][48][49][50][51]. A principle result in this direction is the work of Craik and Criminale [49] which gave a comprehensive analysis of solutions to the incompressible NS equations.…”