Smooth invariant foliations and Koopman eigenfunctions about stable equilibria of semiflows
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| Format: | Preprint |
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2024
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| _version_ | 1866910890481156096 |
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| author | Buza, Gergely |
| author_facet | Buza, Gergely |
| contents | We consider a $C^r$ semiflow $\{ φ_t \}_{t \geq 0}$ on a Banach space $X$ admitting a stable fixed point $x$. We show, along the lines of the parameterization method (Cabré et al., 2003), the existence of a $C^r$ invariant foliation tangent to $X_1$ at $x$, for an arbitrary $D φ_t(x)$-invariant subspace $X_1 \subset X$ satisfying some additional spectral conditions. Uniqueness ensues in a subclass of sufficiently smooth invariant foliations tangent to $X_1$ at $x$. We then draw relations to Koopman theory, and thereby establish the existence and uniqueness, in some appropriate sense, of $C^r$ Koopman eigenfunctions. We demonstrate that these results apply to the case of the Navier-Stokes system, the archetypal example considered by the modern upheaval of applied 'Koopmanism'. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smooth invariant foliations and Koopman eigenfunctions about stable equilibria of semiflows Buza, Gergely Dynamical Systems Analysis of PDEs Fluid Dynamics We consider a $C^r$ semiflow $\{ φ_t \}_{t \geq 0}$ on a Banach space $X$ admitting a stable fixed point $x$. We show, along the lines of the parameterization method (Cabré et al., 2003), the existence of a $C^r$ invariant foliation tangent to $X_1$ at $x$, for an arbitrary $D φ_t(x)$-invariant subspace $X_1 \subset X$ satisfying some additional spectral conditions. Uniqueness ensues in a subclass of sufficiently smooth invariant foliations tangent to $X_1$ at $x$. We then draw relations to Koopman theory, and thereby establish the existence and uniqueness, in some appropriate sense, of $C^r$ Koopman eigenfunctions. We demonstrate that these results apply to the case of the Navier-Stokes system, the archetypal example considered by the modern upheaval of applied 'Koopmanism'. |
| title | Smooth invariant foliations and Koopman eigenfunctions about stable equilibria of semiflows |
| topic | Dynamical Systems Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2401.01475 |