Viscosity approximation of the solution to Burgers' equations with shock layers

ABSTRACT

Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, $u^{ϵ}$, as a viscosity parameter ε tends to zero. The inviscid model, i.e. when $ϵ=0$, possesses the structure of scalar hyperbolic conservation laws, hence our studies deliver an important idea that arises in the field of shock discontinuities of nonlinear hyperbolic waves. The heart of the paper is to establish asymptotic expansions and utilize inner solutions of sharp transition, which are called a corrector function. With aid of corrector functions and energy estimates, we improve the convergence rate of $u^{ϵ}$ to $u^0$ as $\mathcal{O}\left({ϵ}^{1/2}\right)$ in $L^2\left(\mathbb{R}\right)$ ($\mathcal{O}\left(ϵ\right)$ in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1\left(\mathbb{R}\right)$) in the regions including shocks under an entropy condition.

KEYWORDS:

• Interior layers
• singular perturbations
• Burgers' equation
• viscosity limit
• shocks

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B25
• 35C20
• 76D10
• 35L67
• 34K28

Acknowledgments

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(2018R1D1A1B07048325), the Research Fund (1.190136.01) of UNIST, and the Research Foundation of San Diego State University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Research Foundation of Korea (NRF) [2018R1D1A1B07048325] and San Diego State University.