Boundary layers for self-similar viscous approximations of nonlinear hyperbolic systems (Q2850025)

The article adresses the self-similar viscous approximation (or Dafermos regularization) of a system of conservation laws NEWLINE\[NEWLINE U^\varepsilon_t + F(U^\varepsilon)_x = \varepsilon t U^\varepsilon_{xx} NEWLINE\]NEWLINE for \(t, x \geq 0\) with initial condition \(U^\varepsilon(0, x) = U_0\) and boundary condition \(U^\varepsilon(t, 0) = U_b\) where \(|U_0 - U_b|\) is small. Moreover, the noncharacteristic case is considered where all eigenvalues of \(DF(U)\) remain bounded away from zero. Looking for self-similar solutions of the form \(U^\varepsilon(t, x) = Q^\varepsilon(x/t) = Q^\varepsilon(\xi)\) it turns out that these solutions converge for \(\varepsilon\searrow 0\) to some function \(Q(x/t)\) for which \(\lim_{\xi\to 0} Q(\xi) = \bar{U}\neq U_0\). The connection between this residual boundary condition and the actual boundary condition is given via a boundary layer. This boundary layer can be found as a solution of the ordinary differential equation \(V'' = DF(V)V'\) with \(V(0) = U_b\) and \(\lim_{y\to\infty} V (y) = \bar{U}\).NEWLINENEWLINEThe main result of the article states that for \(|U_0 - U_b|\) sufficiently small the residual boundary condition is determined by the fact that \(U_b\) lies in the center-stable manifold of \(\bar{U}\) and that the boundary layers generated by the self-similar viscous approximation are the same as those for the viscous approximation. Since the method of proof can also be applied in the non-conservative case, all arguments are presented for equations of the form \(U_t + A(U)U_x = 0\).