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

Burago, Dmitri; Ivanov, S. V.; Novikov, Alexei

doi:10.1002/cpa.21878

Cited by 10 publications

(22 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…) is globally controllable (this result has been further extended in [3] to nonautonomous ODEs). Roughly speaking, the assumption of vanishing mean drift means that the average value of the flow velocity over big boxes vanishes with the corresponding limit uniform with respect to the selection of those big boxes.…”

Section: Introductionmentioning

confidence: 80%

Constructive controllability for incompressible vector fields

Kryzhevich¹,

Stepanov²

2022

Preprint

View full text Add to dashboard Cite

We give a constructive proof of a global controllability result for an autonomous system of ODEs guided by bounded locally Lipschitz and divergence free (i.e. incompressible) vector field, when the phase space is the whole Euclidean space and the vector field satisfies so-called vanishing mean drift condition. For the case when the ODE is defined over some smooth compact connected Riemannian manifold, we significantly strengthen the assertion of the known controllability theorem in absence of nonholonomic constraints by proving that one can find a control steering the state vector from one given point to another by using the observations of only the state vector, i.e., in other words, by changing slightly the vector field, and such a change can be made small not only in uniform, but also in Lipschitz (i.e. C 1 ) topology.

show abstract

Section: Introductionmentioning

confidence: 80%

Constructive controllability for incompressible vector fields

Kryzhevich¹,

Stepanov²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…V (x + y) dy = 0, then the above controllability problem in R d is solvable for every couple of points x 0 and x 1 (this result has been further extended in [5] to nonautonomous equations). The respective proof is however by contradiction and hence strongly nonconstructive.…”

Section: Introductionmentioning

confidence: 82%

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

Kryzhevich

Stepanov

2019

Journal of Differential Equations

View full text Add to dashboard Cite

In the paper by D. Burago, S. Ivanov and A. Novikov, "A survival guide for feeble fish", it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative (in the sense of not having wandering sets of positive measure) dynamics. In particular, this leads to the extension of C. Pugh's closing lemma to incompressible vector fields over unbounded domains. The results are based on an extension of the Poincaré recurrence theorem to some σ-finite measures and on specially constructed Newtonian potentials.Date: December 25, 2017.

show abstract

“…The waiting time estimate (3) has been proved previously in space-time periodic [5,13], stationary ergodic (time independent) [6,11]. Recently, Burago, Ivanov and Novikov [3] proved (3) in space-time uniformly ergodic environments, a class which at least includes periodic, almost periodic, and some finite range dependence random velocity fields with special structure. We give the first proof of (3) in the most general setting for homogenization theory, space-time stationary ergodic random environments, building on the new ideas of [3].…”

Section: Introductionmentioning

confidence: 88%

“…In the time independent case V (t, x) = V (x), by some simple manipulations of the control formula (2) using (3), it follows that solutions of the G-equation are Lipschitz continuous at length/time scales larger than the waiting time…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Burago, Ivanov and Novikov [3] proved (3) in space-time uniformly ergodic environments, a class which at least includes periodic, almost periodic, and some finite range dependence random velocity fields with special structure. We give the first proof of (3) in the most general setting for homogenization theory, space-time stationary ergodic random environments, building on the new ideas of [3]. This also gives a new proof of finite waiting time in time independent media, which was proved some time ago by Cardaliaguet and Souganidis [6].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Recovering coercivity for the G-equation in general random media

Feldman

2019

Preprint

View full text Add to dashboard Cite

The G-equation is a popular model for premixed turbulent combustion. Mathematically it has attracted a lot of interest in part because it is a simple example of a Hamilton-Jacobi equation which is only coercive 'on average'. This paper shows that, after an almost surely finite waiting time, coercivity is recovered for the G-equation in a small mean, incompressible, spacetime stationary ergodic velocity field. The argument follows ideas from recent work of Burago, Ivanov and Novikov [3,4], while significantly weakening the assumption on the velocity field. The waiting time is explicitly characterized in terms of the space-time means of the velocity field and so mixing estimates on the waiting time can easily be derived. Examples are provided.

show abstract

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

Cited by 10 publications

References 13 publications

Constructive controllability for incompressible vector fields

Constructive controllability for incompressible vector fields

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

Recovering coercivity for the G-equation in general random media

Contact Info

Product

Resources

About