On the mathematical justification of reduced models for hydrodynamic problems and pattern forming systems

Many mathematical models for hydrodynamic problems or for pattern forming systems are so complicated that a qualitative understanding of the full models being usable for practical applications does not seem within reach for the near future, neither analytically nor numerically.

Therefore, it is reasonable to approximate such models in various parameter regimes by appropriate reduced models whose qualitative properties are more easily accessible. To understand to which extent these reduced models yield correct predictions of the behavior of the original models it is important to justify the validity of these approximations by estimates of the approximation errors in the typical length and time scales.

In this talk, we present generic reduced model equations for slow modulations of typical solutions of hydrodynamic equations and reaction-diffusion systems and discuss mathematically rigorous justifications of these approximations. Special emphasis will be put on the approximations of the water wave equations by the Korteweg-de Vries equation and the Nonlinear Schrödinger equation.