Inertial manifolds and finite-dimensional reduction for dissipative PDEs.

*(English)*Zbl 1343.35039Quoting the first paragraph of the introduction: “This paper is an extended version of the lecture notes taught by the author as part of a crash course in the Analysis of Nonlinear Partial Differential Equations (PDEs) at the Centre for Analysis and Nonlinear PDEs, Edinburgh, on 8–9 November 2012. It is devoted to one of the central problems of the modern theory of dissipative systems generated by PDEs, namely, whether or not the underlying dynamics is effectively finite dimensional and can be described by a system of ordinary differential equations (ODEs).”

The subject is restricted to dissipative systems governed by the following abstract parabolic equation in a Hilbert space H: \[ \frac{\partial u}{\partial t} + Au=F(u),\quad u|_{t=0}=u_0,\eqno{(1)} \] where \(A\) is a linear self-adjoint operator and \(F:H \rightarrow H\) is the nonlinearity, assumed to be globally bounded and globally Lipschitz continuous in \(H\) with the Lipschitz constant \(L\).

The classical theory of inertial manifolds is the subject of section 2. This theory relies on the so-called spectral gap assumption: for some natural number \(N\), \[ \lambda_{N+1} - \lambda_N>2L, \] where \(\{\lambda_j: j \in \mathbb N\}\) are the eigenvalues of the operator \(A\).

The content of Section 3 is an alternative approach to the finite-dimensional reduction based on the Mañé Projection Theorem. Counter-examples of [N. Fenichel, Indiana Univ. Math. J. 21, 193–226 (1971; Zbl 0246.58015)] are discussed in section 4. In Subsection 4.1, it is shown that there exists a globally continuous and smooth nonlinearity \(F\) such that (1) does not possess a finite-dimensional inertial manifold if the spectral gap condition is violated. In the Subsection 4.2, the counter-example of the previous section is refined to show that if the spectral gap condition is not satisfied, then even Lipschitz invariant manifolds may not exist. In Subsection 4.3, the authors outline the proof of a Theorem (of [loc. cit.]) on the construction of the smooth nonlinearity \(F(u)\) such that the corresponding attractor is not embedded in any finite-dimensional log-Lipschitz manifold. The final Section 5 has concluding remarks and the description of open problems.

The subject is restricted to dissipative systems governed by the following abstract parabolic equation in a Hilbert space H: \[ \frac{\partial u}{\partial t} + Au=F(u),\quad u|_{t=0}=u_0,\eqno{(1)} \] where \(A\) is a linear self-adjoint operator and \(F:H \rightarrow H\) is the nonlinearity, assumed to be globally bounded and globally Lipschitz continuous in \(H\) with the Lipschitz constant \(L\).

The classical theory of inertial manifolds is the subject of section 2. This theory relies on the so-called spectral gap assumption: for some natural number \(N\), \[ \lambda_{N+1} - \lambda_N>2L, \] where \(\{\lambda_j: j \in \mathbb N\}\) are the eigenvalues of the operator \(A\).

The content of Section 3 is an alternative approach to the finite-dimensional reduction based on the Mañé Projection Theorem. Counter-examples of [N. Fenichel, Indiana Univ. Math. J. 21, 193–226 (1971; Zbl 0246.58015)] are discussed in section 4. In Subsection 4.1, it is shown that there exists a globally continuous and smooth nonlinearity \(F\) such that (1) does not possess a finite-dimensional inertial manifold if the spectral gap condition is violated. In the Subsection 4.2, the counter-example of the previous section is refined to show that if the spectral gap condition is not satisfied, then even Lipschitz invariant manifolds may not exist. In Subsection 4.3, the authors outline the proof of a Theorem (of [loc. cit.]) on the construction of the smooth nonlinearity \(F(u)\) such that the corresponding attractor is not embedded in any finite-dimensional log-Lipschitz manifold. The final Section 5 has concluding remarks and the description of open problems.

Reviewer: Jauber C. Oliveira (Florianopolis)