Numerical study on the incompressible Euler equations as a Hamiltonian system: Sectional curvature and Jacobi field

We study some of the key quantities arising in the theory of [Arnold "Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits," Annales de l'institut Fourier 16, 319 (1966)] of the incompressible Euler equations both in two and three dimensions. The sectional curvatures for the Taylor-Green vortex and the ABC flow initial conditions are calculated exactly in three dimensions. We trace the time evolution of the Jacobi fields by direct numerical simulations and, in particular, see how the sectional curvatures get more and more negative in time. The spatial structure of the Jacobi fields is compared to the vorticity fields by visualizations. The Jacobi fields are found to grow exponentially in time for the flows with negative sectional curvatures. In two dimensions, a family of initial data proposed by Arnold (1966) is considered. The sectional curvature is observed to change its sign quickly even if it starts from a positive value. The Jacobi field is shown to be correlated with the passive scalar gradient in spatial structure. On the basis of Rouchon's physical-space based expression for the sectional curvature (1984), the origin of negative curvature is investigated. It is found that a "potential" alpha(xi) appearing in the definition of covariant time derivative plays an important role, in that a rapid growth in its gradient makes a major contribution to the negative curvature. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3407673

Near-invariance under dynamic scaling for the Navier-Stokes equations in critical spaces: a probabilistic approach to regularity problems

We make a detailed comparison between the Navier-Stokes equations and their dynamically-scaled counterpart, the so-called Leray equations. The Navier-Stokes equations are invariant under static scaling transforms, but are not generally invariant under dynamic scaling transforms. We will study how close they can be brought together using the critical dependent variables and discuss the implications on the regularity problems. Assuming that the Navier-Stokes equations written in the vector potential have a solution that blows up at t = 1, we derive the Leray equations by dynamic scaling. We observe: (1) The Leray equations have only one term extra on top of those of the Navier-Stokes equations. (2) We can recast the Navier-Stokes equations as a Wiener path integral and the Leray equations as another Ornstein-Uhlenbeck path integral. By the Maruyama-Girsanov theorem, both equations take the identical form modulo the Maruyama-Girsanov density, which is valid up to t = 2√ 2 by the Novikov condition. (3) The global solution of the Leray equations is given by a finite-dimensional projection R of a functional of an Ornstein-Uhlenbeck process and a probability measure. If R remains smooth beyond t = 1 under an absolute continuous change of the probability measure, we can rule out finite-time blowup by contradiction. There are two cases: (A) R given by a finite number of Wiener integrals, and (B) otherwise. Ruling out blowup in (A) is straightforward. For (B), a condition based on a limit passage in the Picard iterations is identified for such a contradiction to come out. The whole argument equally holds in R d for any d ≥ 2

Study of the Navier–Stokes regularity problem with critical norms

We study the basic problems of regularity of the Navier–Stokes equations. The blowup criteria on the basis of the critical ${H}^{1/2}$-norm, is bounded from above by a logarithmic function, (Robinson et al 2012 J. Math. Phys. 53 115618). Assuming that the Cauchy–Schwarz inequality for the ${H}^{1/2}$-norm is not an overestimate, we replace it by a square-root of a product of the energy and the enstrophy. We carry out a simple asymptotic analysis to determine the time evolution of the energy. This generalises the (already ruled-out) self-similar blowup ansatz. Some numerical results are also presented, which support the above-mentioned replacement. We carry out a similar analysis for the four-dimensional Navier–Stokes equations

Burgers equation with a passive scalar: Dissipation anomaly and Colombeau calculus

A connection between dissipation anomaly in fluid dynamics and Colombeau’s theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the well-known viscosity-independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate ϵθϵθ of the passive scalar on the Prandtl number PrPr is given by a simple analysis: we show, in particular, ϵθ∝1/Pr−−√ϵθ∝1/Pr for large PrPr. The passive scalar profile is shown to have a form of a sum of tanh2n+1 xtanh2n+1 x with suitably scaled xx, thereby implying the necessity to distinguish HH from HnHn when PrPr is large, where HH is the Heaviside function and nn is a positive integer. An incorrect result of ϵθ∝1/Prϵθ∝1/Pr would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole–Hopf-type transform is also given for the case of unit Prandtl number

Numerical study on comparison of Navier-Stokes and Burgers equations

We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematicviscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [A. A. Kiselev and O. A. Ladyzenskaya, “On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid,” Izv. Akad. Nauk SSSR Ser. Mat.21, 655 (1957); A. A. Kiselev and O. A. Ladyzenskaya, Am. Math. Soc. Transl.24, 79 (1957)] unlike the Navier-Stokes equations. It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term −u⋅∇p−u·∇p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e., it can be either positive or negative with fluctuations. Its joint probability density functions with 12|u|212|u|2 and with 12|ω|212|ω|2 are also found to be symmetric, fluctuating at the same times as the probability density function of −u⋅∇p−u·∇p. A power-law relationship is found in the mathematical bound for the enstrophy growth dQdt+2νP∝(Qa,Pb)α,dQdt+2νP∝QaPbα, where Q and P denote the enstrophy and the palinstrophy, respectively, and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis

Asymptotic formulas for the Lyapunov spectrum of fully developed shell model turbulence

The scaling behavior of the Lyapunov spectrum of a chaotic shell model for three-dimensional turbulence is studied in detail. First, we characterize the localization property of the Lyapunov vectors in wave-number space by using numerical results. By combining this localization property with Kolmogorov’s dimensional argument, we deduce explicitly the asymptotic scaling law for the Lyapunov spectrum, which in turn is shown to agree well with the numerical results. This shell model is an example of high-dimensional chaotic systems for which an asymptotic scaling law is obtained for the Lyapunov spectrum. Implications of the present results for the Navier-Stokes turbulence are discussed. In particular, we conjecture that the distribution of Lyapunov exponents is not singular at null exponent

Cole-Hopf--Feynman-Kac formula and quasi-invariance of Navier-Stokes equations

We make a refined comparison between the Navier–Stokes equations and their dynamically-scaled Leray equations solely on the basis of their scaling property. Previously it was observed using the vector potentials that they differ only by one drift term (Ohkitani 2017 J. Phys. A: Math. Theor. 50 045501). The Duhamel principle recasts the equations in path integral forms, which differ by two Maruyama–Girsanov densities. In this brief paper we simplify the concept of quasi-invariance (or, near-invariance) by combining the result with a Cole–Hopf transform and the Feynman–Kac formula. That way, as a multiplicative characterisation we can place those equations just one Maruyama–Girsanov density apart. Furthermore, as an additive characterisation we express the difference in terms of the Malliavin H-derivative