Stability of boundary layers for the inflow compressible Navier-Stokes equations
Jing Wang Lining Tong
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2595-2613 doi: 10.3934/dcdsb.2012.17.2595
In this paper, we consider the boundary layer stability of the one-dimensional isentropic compressible Navier-Stokes equations with an inflow boundary condition. We assume only one of the two characteristics to the corresponding Euler equations is negative up to some small time. We prove the existence of the boundary layers, then instead of using the skew symmetric matrix, we give a higher convergence rate of the approximate solution than the previous results by a standard energy method as long as the strength of the boundary layers is suitably small.
keywords: inflow boundary condition asymptotic analysis energy estimate. Compressible Navier-Stokes equations degenerate viscosity matrix noncharacteristic boundary layers
Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers
Jing Wang Lining Tong
Communications on Pure & Applied Analysis 2019, 18(2): 887-910 doi: 10.3934/cpaa.2019043

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.

keywords: Two-point boundary value problem multiple boundary layers weak boundary layer strong boundary layer matched asymptotic expansions stability analysis

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