A survival guide for feeble fish

Abstract

As avid anglers we were always interested in the survival chances of fish in turbulent oceans. This paper addresses this question mathematically. We show that a fish with bounded aquatic locomotion speed can reach any point in the ocean if the fluid velocity is incompressible, bounded, and has small mean drift.2010 Mathematics Subject Classification. 34H05, 49L20.

Feeble Fish in Time‐Dependent Waters and Homogenization of the G‐equation

“…The incompressibility and small mean drift assumptions imply the following lemma, which we borrow from [5]. This is the only place in the proof where the small mean drift assumption is used.…”

“…This lemma is stated in [5] for a time-independent vector field. We apply [5, lemma 3.1] to the vector field V t for every fixed t. The constant L 0 (named A 0 in [5, lemma 3.1]) depends on the vector field, so we need to make sure that it can be chosen independently of t. In the proof in [5] one can see that L 0 depends only on M and on the rate of convergence of the mean drift to 0. Hence the proof works for our Lemma 4.3 as well.…”

The saga of a fish: from a survival guide to closing lemmas

“…The first corollary of the above theorem is the following global point-to-point controllability result, which is a reformulation of the Burago-Ivanov-Novikov controllability theorem 1.1 from [6] (and a partial extension of theorem 4.2.7 in [3] formulated for compact manifolds, although for possibly more general control affine systems), however proven now by a completely different and direct method. Proof.…”

“…Consider the following problem first suggested in [6]: a fish in an unbounded turbulent ocean is able to move with its own velocity u not exceeding in modulus the given value δ > 0. The ocean is assumed to be unbounded and is identified hereinafter with R d , its velocity field V is assumed, as it is customary, to be bounded and incompressible, i.e.…”

“…In case when the phase space of (1.2) instead of R d is just a compact subset of the latter invariant with respect to the flow of (1.1) (in particular, a smooth compact manifold), then the positive answer to the posed question is provided by the global controllability theorem 4.2.7 in [3] (there it is formulated for analytic vector fields on compact Riemannian manifolds), while in the whole R d the incompressibility condition of V is clearly not enough for such a theorem to hold as can be seen just taking V to be constant with sufficiently large modulus. This is in fact the only possible obstacle for controllability in the whole R d : in fact, it has been proven in [6] that if V is incompressible and has vanishing mean drift (called small mean drift in [6]) in the sense…”