Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions

Abstract

We prove that a general class of nonlinear, non-autonomous ODEs in Fréchet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. In particular, our method empowers us to study approximate centre manifolds, given by solutions of ODEs that are central up to a desired, possibly nonzero precision. The main motivation is the case where the Fréchet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way.

Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems

“…Second, unbounded nonlinearity, such as u • ∇u and as found so often in applications, such as fluid instabilities, appears to be not generally covered by the Lipschitz and/or uniformly bounded requirement of most extant forward theory (Henry 1981, Mielke 1986, Aulbach & Wanner 2000, Chicone & Latushkin 1997, Haragus & Iooss 2011. So, despite many interesting scenarios (such as autonomous Navier-Stokes problems with certain boundary conditions) having rigorous invariant manifolds beautifully established via strongly continuous semigroup operators and by mollifying nonlinearity (Carr 1981, Vanderbauwhede 1989, extant non-autonomous forward theory even for finite-D typically imposes preconditions (e.g., Haragus & Iooss 2011, Hypothesis 3.8(ii)) that we relax here (and relaxed by Hochs & Roberts 2019) via the proposed backward theory: for example, Assumption 3 only requires nonlinearities to be C p for some order p.…”

“…Future research is planned to address stochastic systems and/or partial differential/integral equation systems (a step in this direction is by Hochs & Roberts 2019).…”

“…Further, for the linear operator L in the original system (1.4), the classic definitions of invariant/integral manifolds require e Lt to be analysable in both forward and backward time, that is, the operator L must be bounded. But in extensions of our approach (Hochs & Roberts 2019) to pdes the operator L is typically unbounded. Consequently we need to change some basic definitions to cope.…”

“…Here we establish a first step in complementing extant forward theory with a backward theory applicable to quite general non-autonomous systems and to systems that have center, stable and unstable dynamics-other backward theory has proved very useful in other contexts (Grcar 2011, Ghosh & Alam 2018. A second step in this project is the initial corresponding backward theory for ∞-D systems recently established by Hochs & Roberts (2019), but for the finite-D systems addressed here we present more accessible arguments extended to non-self-adjoint systems (Section 2), and provide a novel lower bound on the finite domain of validity (Section 3).…”

Learning high-order spatial discretisations of PDEs with symmetry-preserving iterative algorithms

“…Firstly, functions f and g on the right-hand side of pde (1) are to be smooth functions of their arguments, and the function g varies relatively slowly in time t. Secondly, in the cases of second-order pdes, the function f, called the flux, is strictly monotonic decreasing in u x : that is, for some positive ν, ∂f/∂u x −ν < 0 with f(x, u, 0) = 0 . Lastly, for strict support by existing theory, we assume f and g are such that the pde (1), with 'edge conditions' (6), satisfies the requirements of the invariant manifold theory by Hochs and Roberts (2019).…”