Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers

Jing Wang; Lining Tong

doi:10.3934/cpaa.2019043

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2019, Volume 18, Issue 2: 887-910. Doi: 10.3934/cpaa.2019043

This issue Previous Article A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator Next Article Attractors and their stability on Boussinesq type equations with gentle dissipation

Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers

Jing Wang^1, and
Lining Tong^2, ,

1.
Department of Mathematics, Shanghai Normal University, ShangHai, 200234, China
2.
Department of Mathematics, Shanghai University, ShangHai, 200444, China

* Corresponding author
* Corresponding author

Received: March 31, 2018

Revised: May 31, 2018

Published: October 2018

The first author is supported by NSFC grant 11771297; The second author is supported by NSFC grant 11771274

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Abstract

In this paper, we study the limiting behavior of solutions to a 1D two-point boundary value problem for viscous conservation laws with genuinely-nonlinear fluxes as $\varepsilon$ goes to zero. We here discuss different types of non-characteristic boundary layers occurring on both sides. We first construct formally the three-term approximate solutions by using the method of matched asymptotic expansions. Next, by energy method we prove that the boundary layers are nonlinearly stable and thus it is proved the boundary layer effects are just localized near both boundaries. Consequently, the viscous solutions converge to the smooth inviscid solution uniformly away from the boundaries. The rate of convergence in viscosity is optimal.

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Mathematics Subject Classification: Primary: 35L50, 35K60; Secondary: 58K25.

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