Explicit Solutions to the n-dimensional Semi-stationary Compressible Stokes Problem

Abstract

The n-dimensional semi-stationary compressible Stokes equations, established to model the dynamics of vortices in the Ginzburg-Landau theories in superconductivity, is a simplification of the isentropic compressible Navier-Stokes equations. This model equation can also be derived from the equations for the flows in compressible porous media in petroleum engineering or in compressible tissues. In this work, we investigate the n-dimensional semi-stationary compressible Stokes equations and obtain a family of explicit solutions in which the density does not change with space variables but decays to zero with time increasing exponentially, while the velocity is a quadratic polynomial of the space variables. It is worth pointing out that our results not only include the known results for \(n=2\) and \(n=3\) in Xue and Dong (Int. J. Theor. Phys. 63(6), 151 (2024)), but also present the explicit solutions to the semi-stationary compressible Stokes equations in arbitrary higher dimensional space.