Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator - Lyailya Zhapsarbayeva; Dongming Wei; Bagyzhan Bagymkyzy (Mathematics)
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 <semantics>ρ(∂tu+u∂xu)=μ∂x<mfenced separators="" open="(" close=")"><mfenced separators="" open="|" close="|">∂xu</mfenced>p−2∂xu</mfenced>,</semantics> augmented with the initial condition <semantics>u(0,x)=u0</semantics>, <semantics>0<x<L</semantics>, and the boundary condition <semantics>u(t,0)=u(t,L)=0</semantics>, where <semantics>ρ</semantics> is the density, <semantics>μ</semantics> the viscosity, u the velocity of the fluid, <semantics>1<p<2</semantics>, <semantics>L>0</semantics>, and <semantics>T>0</semantics>. We show that this initial boundary problem has an unique solution in the Buchner space <semantics>L2<mfenced separators="" open="(" close=")">0,T;W01,p(0,1)</mfenced></semantics> 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.