Existence and Uniqueness of the Viscous Burgers’ Equation Based on Ellis Model

In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of Ellis fluids $$\partial_t u + u \partial_x u=\nu\partial_x \left(\phi^{-1}\left (\partial_x u\right)\right)$$ with the initial condition $u(0, x)=u_0$, $0<x<l$, and the boundary condition $u(t, 0)=u(t, l)=0$, $0<t<T$, where $\phi(\tau)=\left[1+c\left|\tau \right|^{\alpha-1}\right] \tau $, $0<c<\infty$, $0<\alpha<\infty$, $\nu$ is the kinematic viscosity, $\tau$ is the shear stress, $u$ is the velocity of the fluid and $l, T>0$. We proved the existence and uniqueness of solution $u \in L^2\left(0, T; {W_0}^{1,p}(0, l)\right)$ for $u_0\in W^{1,p}_0(0,l)$, where $p=1+ \frac{1}{\alpha}$. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL® Multiphysics 6.0 to show the images of the solutions.