# Time-Varying Vector Fields and Their Flows (SpringerBriefs in Mathematics)

## Saber Jafarpour

Language: English

Pages: 119

ISBN: 3319101382

Format: PDF / Kindle (mobi) / ePub

This short book provides a comprehensive and unified treatment of time-varying vector fields under a variety of regularity hypotheses, namely finitely differentiable, Lipschitz, smooth, holomorphic, and real analytic. The presentation of this material in the real analytic setting is new, as is the manner in which the various hypotheses are unified using functional analysis. Indeed, a major contribution of the book is the coherent development of locally convex topologies for the space of real analytic sections of a vector bundle, and the development of this in a manner that relates easily to classically known topologies in, for example, the finitely differentiable and smooth cases. The tools used in this development will be of use to researchers in the area of geometric functional analysis.

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(V0x E). E Now recall that E ζ ∗ VE, where ζ : M → E is the zero section [7, p. 55]. Let us abbreviate ˆιE = T ιE |ζ ∗ VE. We then have the following diagram (2.7) 26 2 Fibre Metrics for Jet Bundles describing a monomorphism of real analytic vector bundles over the proper embedding ιM , with the image of ˆιE being Eˆ . Among the many ways to prescribe a linear connection on the vector bundle E, we will take the prescription whereby one defines a mapping K : TE → E such that the two diagrams

there exist a real analytic aﬃne connection ∇ on M and a real analytic vector bundle connection ∇0 on E. 3. The estimates of Lemma 2.5 hold if the Riemannian metric G and the vector bundle metric G0 are only smooth. This is true because, in the proof, G and G0 are not diﬀerentiated; one only requires their values. Therefore, the seminorms defined in Sect. 3.1 can be made sense of for holomorphic sections. Proposition 4.2 (Cauchy Estimates for Vector Bundles). Let π : E → M be an holomorphic

compact. We shall also use this local coordinate notation for seminorms of local representatives of vector fields. Let K ⊆ M be compact and let a ∈ c0 (Z≥0 ; R>0 ). Let x ∈ K and let (U x , φ x ) be a chart for M about x with the property that the coordinate functions x j , j ∈ {1, . . . , n}, are restrictions to U x of globally defined real analytic functions f xj , j ∈ {1, . . . , n}, on M. This is possible by the lemma above. Let X : φ x (U x ) → Rn be the local representative of X ∈ Γ ω (M).

induction on m. In doing this, we will need to understand how diﬀerential equations depending diﬀerentiably on state also have solutions depending diﬀerentiably on initial condition. Such a result is not readily found in the textbook literature, as this latter is typically concerned with continuous dependence on initial conditions for cases with measurable time dependence, and on diﬀerentiable dependence when the dependence on time is also diﬀerentiable. However, the general case (much more

(1989) 36. Schaefer, H.H., Wolﬀ, M.P.: Topological Vector Spaces, 2 edn. No. 3 in Graduate Texts in Mathematics. Springer-Verlag, New York/Heidelberg/Berlin (1999) 37. Schuricht, F., von der Mosel, H.: Ordinary diﬀerential equations with measurable right-hand side and parameter dependence. Tech. Rep. Preprint 676, Universit¨at Bonn, SFB 256 (2000) 38. Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2 edn. No. 6 in Texts in Applied Mathematics. Springer-Verlag,