American Institute of Mathematical Sciences

March  2019, 18(2): 887-910. doi: 10.3934/cpaa.2019043

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

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

* Corresponding author

Received  April 2018 Revised  June 2018 Published  October 2018

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

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.

Citation: Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043
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