Linear and weakly nonlinear stability theory of parallel and non-parallel flows

In this work we revisit the mathematical foundation of weakly nonlinear stability theory of infinite-dimensional systems possessing critical phenomena in order to remove ambiguities in the range of applicability of well-known model and amplitude equations.; In the first part, a number of near critical infinite-dimensional normal forms is derived from a general mathematical formulation of an evolution system with the aid of a rigorous and effective approach based upon spectral theory of linear operators and a new attracting manifolds theory for infinite-dimensional integro-differential systems. The spectral formulation allows us to introduce criteria for distinguishing sub- and super-criticality, and attracting manifolds construction naturally gives rise to resonance conditions precluding projections of the space of slave modes onto that of master modes. As an illustration, this approach is implemented in the rigorous derivation of the amplitude equation for the three-dimensional Rayleigh-Bénard convection problem with stress-free horizontal surfaces and to spatially extended shear flow systems where a continuous spectrum is present. This last class of problems reveals the role played by continuous modes in the stability of dynamical systems. As a result, the meanings of low-dimensional modelling and universality of near-critical equations are clarified.; The second part of this work is devoted to revealing the nature of marginal stability phenomena and to exploring the subsequent weakly nonlinear regime in problems with nonparallel effects, usually arising from nontrivial spatially inhomogeneous basic flows. As a generalization of the well-known concept of linear temporal stability in Fourier space for a parallel basic state, we introduce a new concept valid for nonparallel flows as well. The new picture allows one to demonstrate the possible singular limit to the parallel case. Further, on the basis of a generalization of the concept of linear temporal stability in Fourier space for nonparallel flows and with the development of a formal method to arrive at a coarse system, we derive a weakly nonlinear model valid near criticality. The damped Kuramoto-Sivashinsky equation with variable coefficients is used to illustrate the application of the theory.