Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems

Rousset, F.

doi:10.1090/S0002-9947-03-03279-3

Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems
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Trans. Amer. Math. Soc. 355 (2003), 2991-3008 Request permission

Abstract:

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity $\varepsilon$, $u^\varepsilon _t+F(u^\varepsilon )_x =\varepsilon (B(u^\varepsilon ) u^\varepsilon _x )_x .$ When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that $u^\varepsilon$ converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for $B$ invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet’s boundary condition.

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