Diffusive Stability of Convective Turing Patterns

Following the approach of Eckhaus, Mielke, and Schneider for reaction diffusion systems, we justify rigorously the Eckhaus stability criterion for stability of convective Turing patterns, as derived formally by complex Ginzburg-Landau approximation. Notably, our analysis includes higher-order, nonlocal, and even certain semilinear hyperbolic systems.