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

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 $$\rho (\partial_t u + u \partial_x u)=\mu \partial_x \left(\left|\partial_x u \right|^{p-2} \partial_x u\right),$$ augmented with the initial condition $u(0, x)=u_0$, $0<x<L$ and the boundary condition $u(t, 0)=u(t, L)=0$, where $\rho$ is the density, $\mu$ the viscosity, $u$ the velocity of the fluid and $p$, $1<p<2$, $L, T>0$. Moreover, numerical solutions to the problem are constructed by applying the high-level modeling and simulation package COMSOL Multiphysics at small and large Reynold's numbers.