Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator

Abstract

In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids 𝜌⁢(∂𝑡𝑢+𝑢∂𝑥𝑢) =𝜇∂𝑥(|∂𝑥𝑢|𝑝−2∂𝑥𝑢), augmented with the initial condition 𝑢⁡(0,𝑥) =𝑢0, 0 <𝑥 <𝐿, and the boundary condition 𝑢⁡(𝑡,0) =𝑢⁡(𝑡,𝐿) =0, where 𝜌 is the density, 𝜇 the viscosity, u the velocity of the fluid, 1 <𝑝 <2, 𝐿 >0, and 𝑇 >0. We show that this initial boundary problem has an unique solution in the Buchner space 𝐿2⁡(0,𝑇;𝑊01,𝑝⁡(0,1)) for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions.