On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach.
The goal of the paper is to show using relatively elementary ODE methods that for a class of PDEs for which there exist the compact trapping region and fast decay of Fourier coefficients, the partial derivatives with respect to initial conditions for Galerkin projections converge to the partial derivatives with respect to initial conditions of the semiflow induced by PDE. This also proves smooth dependence on initial conditions for full PDE on a trapping region. An abstract part of present paper depends only on the content of Sections 4 and 5 in [\textit{P. Zgliczynski}, Trapping regions and an ODE-type proof of an existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane, Zesz. Nauk. Uniw. Jagiell. 1269, Univ. Iagell. Acta Math. 41, 89--113 (2003; Zbl 1067.35068)]. An application of the main result -- Theorem 2 -- to the planar Navier-Stokes equations in Section 3 depends on the trapping regions, which are given in the paper cited above. It should be pointed out that in the literature there exist much more general results about regularity of solutions of dissipative PDEs. For example, it is shown that solutions of Navier-Stokes equations in dimension two depend analytically on initial conditions. But still the author believes that the simplicity of the tools used in the paper under review will make this paper interesting for researches with a dynamical system theory and ODE background, especially in the context of a computer-assisted program of rigorous investigation of dynamics of dissipative PDE. It is rather obvious that the presented method should be easily applicable to any PDE for which there exists a compact attractor and Gevrey class regularity results have been established. Here is a list of a few examples for which Theorem 2 holds: -- Navier-Stokes equations with periodic boundary conditions in three dimensions with small initial data and small force; -- Kuramoto-Sivashinsky equations in 1D with periodic boundary conditions; -- Navier-Stokes equation in dimension three with the Laplacian term replaced by \({\left| {\nabla} \right| }^{\alpha} \) for \(\alpha \) big enough \(\left({\alpha > 2.5} \right)\). The reader should notice that the proof of \(C^1\)-convergence of Galerkin projections presented in this paper is not effective from the point of view of rigorous numerics, no error bounds are given, so it cannot be used for rigorous computer assisted study of dynamics of PDEs.
scientific article; zbMATH DE number 2032990