Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

Tobasco, Ian; Goluskin, David; Doering, Charles R.

doi:10.1016/j.physleta.2017.12.023

Cited by 52 publications

(96 citation statements)

References 26 publications

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“…It relies on computation of an optimal Lyapunov functional which under the sumof-squares approximation reduces to solution of a convex semidefinite optimization problem (Chernyshenko et al, 2014). To date, this approach has been used to obtain new results concerning the average and extreme behavior of some simple models, both in finite and infinite dimension (Tobasco et al, 2018;Goluskin, 2018;Goluskin & Fantuzzi, 2019).…”

Section: D Burgers Instantaneousmentioning

confidence: 99%

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

2020

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This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system -as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy E 0 are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time T . Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of E 0 and T , we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to E 3/2 0 as E 0 becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.

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Section: D Burgers Instantaneousmentioning

confidence: 99%

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

2020

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“…We use the lim sup in this definition because time averages need not converge. As already noted in [24], Φ(a 0 ) could alternatively be defined using the lim inf, with no effect on the results presented in this work. Conditions under which time averages do converge are discussed in [37].…”

Section: Infinite-time Averages and Auxiliary Functionsmentioning

confidence: 92%

“…where a ∈ R n and f : R n → R n is continuously differentiable. Following [24], we assume that there exists a compact set Ω in which all solutions a(t ; a 0 ) of (1) eventually remain, irrespective of the initial conditions a 0 . Such a set may be found in a variety of ways, but one very common approach is to let Ω = {a ∈ R n | W (a) ≤ C}, where W : R n → R is a radially unbounded and continuously differentiable function such that…”

Section: Infinite-time Averages and Auxiliary Functionsmentioning

confidence: 99%

“…The connection between auxiliary functions proving bounds on infinite-time averages and the corresponding extremal trajectories, which are often UPOs, has been recently established by Tobasco et al [24]. Precisely, if an auxiliary function produces nearly sharp bounds, then it can be used to construct a function whose sublevel sets localise extremal and near-extremal trajectories in state space.…”

Section: Introductionmentioning

confidence: 99%

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Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow

Lakshmi

Fantuzzi²,

Fernández-Caballero³

et al. 2020

SIAM J. Appl. Dyn. Syst.

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Tobasco et al. [Physics Letters A, 382:382-386, 2018] recently suggested that trajectories of ODE systems which optimise the infinite-time average of a certain observable can be localised using sublevel sets of a function that arise when bounding such averages using socalled auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimisation is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localise the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations. We then combine these methods with a singleshooting Netwon-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimises the mean energy dissipation rate and maximises the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

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“…To bound the Nusselt number Nu we follow a general approach [24][25][26][27] and look for a so-called auxiliary functional of the dynamical variables, V T , θ , that is differentiable along solutions of (4a)-(4f), uniformly bounded in time, and satisfies…”

Section: Bounding the Nusselt Numbermentioning

confidence: 99%

Rigorous bounds on the heat transport of rotating convection with Ekman pumping

Pachev

Whitehead

Fantuzzi

et al. 2020

Journal of Mathematical Physics

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We establish rigorous upper bounds on the time-averaged heat transport for a model of rotating Rayleigh-Bénard convection between no-slip boundaries at infinite Prandtl number and with Ekman pumping. The analysis is based on the asymptotically reduced equations derived for rotationally constrained dynamics with no-slip boundaries, and hence includes a lower order correction that accounts for the Ekman layer and corresponding Ekman pumping into the bulk. Using the auxiliary functional method we find that, to leading order, the temporally averaged heat transport is bounded above as a function of the Rayleigh and Ekman numbers Ra and Ek according to Nu ≤ 0.3704Ra 2 Ek 2 . Dependent on the relative values of the thermal forcing represented by Ra and the effects of rotation represented by Ek, this bound is both an improvement on earlier rigorous upper bounds, and provides a partial explanation of recent numerical and experimental results that were consistent yet surprising relative to the previously derived upper bound of Nu Ra 3 Ek 4 . * whitehead@mathematics.byu.edu.

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Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

Cited by 52 publications

References 26 publications

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow

Rigorous bounds on the heat transport of rotating convection with Ekman pumping

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