Symmetry and asymptotic solutions for a Magnetohydrodynamics Darcy-Forchheimer flow with a p-Laplacian operator

Abstract

Fluid flows under a $p$-Laplacian operator formulation have been considered recently in connection with the modelling of non-Newtonian fluid processes. To a certain extent, the main reason behind the interest in $p$-Laplacian operators is the possibility of determining experimental values for the constant $p$ appearing in them. The goal of the present study is to introduce the analysis of solutions of a one-dimensional porous media flow arising in Magnetohydrodynamics (MHD) with generalized initial data under a Lebesgue integrability condition. We present a weak formulation of this problem, and we consider boundedness and uniqueness properties of solutions, and also its asymptotic relation with the standard parabolic $p$-Laplacian equation. Then, we explore solutions arising from classical symmetries (including an explicit kink solution in the $p=3$ case) along with asymptotic stationary and non-stationary solutions. The search of stationary solutions is based on a Hamiltonian approach. Finally, non-stationary solutions are obtained by using an exponential scaling resulting in a Hamilton-Jacobi type of equation.