Traveling fronts of viscous Burgers’ equations with the nonlinear degenerate viscosity

Abstract

In this paper, we continue the study of viscous Burgers’ equations by Il’in and Oleinik for single model equation with convex nonlinearity [6] by introducing the nonlinear degenerate viscosity. Since the degeneracy of this paper is considered, then there are two conditions for estimate of the traveling fronts U when $m\ge 1$ and $0<m<1$. The following higher-order estimate of traveling fronts U is introduced to overcome the energy estimate

$$ \begin{gathered} U^{{m - 2}} = \left( {\frac{1}{U}} \right)^{{2 - m}} \le Ku_{ + } \le Cu_{ + } ,\;{\text{if}}\;0 < m < 2, \hfill \\ U^{{m - 2}} \le Lu_{ - } \le Cu_{ - } ,\;{\text{if}}\;m \ge 2 \hfill \\ \end{gathered} $$

where $C=max\left\{ K,L \right\} =max\left\{ \frac{b}{m-b}, (m+b)^m \right\}$ for $b>0$ and $m>b$. Moreover, the following Taylor expansion is employed

$$ f(U + \pi _{z} ) - f^{'} (U)\pi _{z} - f(U){\text{ }} = \int_{0}^{1} {\left( {(1 - s)f^{} (U + s\pi _{z} )ds} \right)} \pi _{z}^{2} = \rm{\mathcal{O}}(1)\pi _{z}^{2} $$

to overcome the estimate of term $F_1$, where this term is transformation result of nonlinearity for first order derivative in (1). The stability of traveling fronts U is presented to give the information how close the distance between the solution u of (1) and the traveling fronts U is under the small perturbations. This stability result is based on the energy estimates under the condition $N(t)\le Dm(u_++u_-)$. To validate our works and to illustrate the effect of nonlinear degenerate viscosity, the numerical simulations are provided by using the standard finite difference for discretization steps.