Inviscid boundary conditions and stability of viscous boundary layers (Q2730782)

The author studies the set of residual boundary conditions for a hyperbolic system of conservation laws which remain after passing to the limit as the diffusion coefficient \(\varepsilon\) in a parabolic regularization NEWLINE\[NEWLINE u_t+f(u)_x=\varepsilon (B(u)u_x)_x, \quad x>0, \;t>0 NEWLINE\]NEWLINE vanishes. This set is shown to be a submanifold in a vicinity of a point where the Evans function \(D(\lambda)\) of the associated boundary layer profile is such that \({D(0)\neq 0}\). Then the author considers the associated linearized problems and shows that the Evans functions for the viscous problem reduces in the long-wave limit to the Lopatinsky determinant. This implies that inviscid well-posedness is necessary for stability of the boundary layer.