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Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems. (English) Zbl 1022.35022

An initial-boundary value problem for a one-dimensional system of conservation laws with a small parameter \( \varepsilon > 0,\) is considered \[ u^{\varepsilon}_t + F(u^{\varepsilon})_x = \varepsilon (B(u^{\varepsilon})u^{\varepsilon}_x)_x , \quad x,t > 0, \] where \(u^{\varepsilon} \in \mathbb{R}^n\), \(F: U\rightarrow \mathbb{R}^n\), \(B: U\rightarrow \mathbb{R}^{n\times n}\); \(F\), \(B\) are smooth. An initial condition and some dissipative boundary conditions are added to this system. The eigenvalues of \(B \) have nonnegative real parts and the rank of \(B\) does not depend on \(U.\) The author proves that if a boundary layer is weak, then \(u^{\varepsilon}\) converges to an inviscid solution as \( \varepsilon \) tends to zero. An application of this result to the isentropic gas dynamics is given.

MSC:

35K65 Degenerate parabolic equations
35L65 Hyperbolic conservation laws
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