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This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux in such a way that preserves the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes., 34 pages and 1 figure

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $\theta$ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity $u$ is of lower singularity, i.e., $u=-\nabla^{\perp}\Lambda^{\beta-2}p(\Lambda)\theta$, where $p$ is a logarithmic smoothing operator and $\beta \in [0,1]$. We complete this study by considering the more singular regime $\beta\in(1,2)$. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness., Comment: Minor revisions. Accepted to Communications on Pure and Applied Analysis. 18 pages

Given a velocity field $u(x,t)$, we consider the evolution of a passive tracer $\theta$ governed by $\partial_t\theta + u\cdot\nabla\theta = \Delta\theta + g$ with time-independent source $g(x)$. When $\|u\|$ is small, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134) predicted that the tracer spectrum scales as $|\theta_k|^2\propto|k|^{-4}|u_k|^2$. In this paper, we prove that this scaling does indeed hold for large $|k|$, in a probabilistic sense, for random synthetic two-dimensional incompressible velocity fields $u(x,t)$ with given energy spectra. We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of $u$., Comment: 15 pages

The vertically averaged velocity of the 3D Rayleigh–Benard problem is analyzed and numerically simulated. This vertically averaged velocity satisfies a 2D incompressible Navier–Stokes system with a body force involving the 3D velocity. A time average of this force is estimated through time averages of the 3D velocity. Relations similar to those from 2D turbulence are then derived. Direct numerical simulation of the 3D Rayleigh–Be nard is carried out to test how prominent the features of 2D turbulence are for this Navier–Stokes system.

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elementary harmonic analysis. Finally, we provide elementary proofs of the existence of time-periodic solutions, steady state solutions, as well as the existence of finitely many determining parameters for the SQG equation.

We prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic domain with fractional Laplacian in the subcritical case.

We show that the global attractor of a weakly damped and driven Korteweg–de Vries equation (KdV) is embedded in the long-time dynamics of an ordinary differential equation called a determining form. In particular, there is a one-to-one identification of the trajectories in the global attractor of the damped and driven KdV and the steady state solutions of the determining form. Moreover, we analyze a data assimilation algorithm (down-scaling) for the weakly damped and driven KdV. We show that given a certain number of low Fourier modes of a reference solution of the KdV equation, the algorithm recovers the full reference solution at an exponential rate in time.

We develop, analyze, and test an approximate, global data assimilation/synchronization algorithm based on purely local observations for the two-dimensional Navier-Stokes equations on the torus. We prove that, for any error threshold, if the reference flow is analytic with sufficiently large analyticity radius, then it can be recovered within that threshold. Numerical computations are included to demonstrate the effectiveness of this approach, as well as variants with data on moving subdomains. In particular, we demonstrate numerically that machine precision synchronization is achieved for mobile data collected from a small fraction of the domain., Comment: 11 figures, 27 pages

Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is H��lder continuous with exponent $r=1/2$ results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, $r>1/2$ (algebraic in Grashof)., 19 pages, 3 figures

An algorithm for continuous data assimilation for the two-dimensional Benard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier–Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the Benard convection problem associated with the observed (finite dimensional projection of) velocity.

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrodinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on $[0,L]^2$, and a force $g\in\calD(A^{\frac{\alpha-1}{2}})$, we show there is a fixed strip about the real time axis on which a uniform bound $|A^{\alpha}u|< m_\alpha\nu\kappa_0^\alpha$ holds for each $\alpha \in \bN$. Here $\nu$ is viscosity, $\k0=2\pi/L$, and $m_\alpha$ is explicitly given in terms of $g$ and $\alpha$. We show that if any element in $\calA$ is in $\D(A^\alpha)$, then all of $\calA$ is in $\D(A^\alpha)$, and likewise with $\D(A^\alpha)$ replaced by $C^\infty(\Omega)$. We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class., Comment: 40 pages

We construct semi-integral curves which bound the projections of the global attractors of the 3D NS-$\alpha$ and 3D Leray-$\alpha$ sub-grid scale turbulence models in the plane spanned by their energy and enstrophy. We note the dependence of these bounds on the filter width parameter $\alpha$, and determine subregions where each quantity, energy and enstrophy, must decrease, while isolating one which is recurrent.

We study the influence of the energy spectrum on the extent of the cascade range of a passive tracer in turbulent flows. The interesting cases are when there are two different spectra over the potential range of the tracer cascade (in 2D when the tracer forcing is in the inverse energy cascade range, and in 3D when the Schmidt number Sc is large). The extent of the tracer cascade range is then limited by the width of the range for the shallower of the two energy spectra. Nevertheless, we show that in dimension $d=2,3$ the tracer cascade range extends (up to a logarithm) to $\kappa_{d\text{D}}^{p}$, where $\kappa_{d\text{D}}$ is the wavenumber beyond which diffusion should dominate and $p$ is arbitrarily close to 1, provided Sc is larger than a certain power (depending on $p$) of the Grashof number. We also derive estimates which suggest that in 2D, for Sc${}\sim1$ a wide tracer cascade can coexist with a significant inverse energy cascade at Grashof numbers large enough to produce a turbulent flow.

The evolution of a determining form for the 2D Navier–Stokes equations (NSE) which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and a chosen, fixed steady state, with a dynamical convexity parameter $$\theta $$ , which will be called the characteristic determining parameter. That is, we show a separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter $$\theta $$ which is governed by a one-dimensional ODE. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than $${\mathcal {O}}(\tau ^{-1/2})$$ , otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than $${\mathcal {O}}(\tau ^{-1})$$ , as $$\tau \rightarrow \infty $$ , where $$\tau $$ is the evolutionary variable in determining form. The one-dimensional ODE is also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. It is shown that the zeros of the scalar function that governs the dynamics of $$\theta $$ , which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE

Rigorous estimates to support the Batchelor–Kraichnan–Leith theory of 2-D turbulence are made for time-dependent forcing at all length scales. The main estimate, derived under several different assumptions on the smoothness of the force in space and time, bounds the dissipation wavenumber κ η from above and below in terms of a generalized Grashof number. That estimate is shown to be connected to the energy power law, the dissipation law, and the enstrophy cascade. These results impose certain restrictions on the shape of the force, which in several cases is allowed to be discontinuous in time.

We construct semi-integral curves which bound the projection of the global attractor of the 2-D Navier–Stokes equations in the plane spanned by enstrophy and palinstrophy. Of particular interest are certain regions of the plane where palinstrophy dominates enstrophy. Previous work shows that if solutions on the global attractor spend a significant amount of time in such a region, then there is a cascade of enstrophy to smaller length scales, one of the main features of 2-D turbulence theory. The semi-integral curves divide the plane into regions having limited ranges for the direction of the flow. This allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy. We also derive a sharp, universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero.

A rigorous study of universal laws of 2-D turbulence is presented for time independent forcing at all length scales. Conditions for energy and enstrophy cascades are derived, both for a general force, and for one with a large gap in its spectrum. It is shown in the gap case that either a direct cascade of enstrophy or an inverse cascade of energy must hold, provided the gap modes of the velocity has a nonzero ensemble average. Partial rigorous support for 2-D analogs of Kolmogorov's 3-D dissipation law, as well as the power law for the distribution of energy are given.

Rigorous upper and lower bounds are proved for the Taylor and the Kolmogorov wavenumbers for the three-dimensional space periodic Navier–Stokes equations. Under the assumption that Kolmogorov’s two-thirds power law holds, the bounds sharpen to κ T ∼ Gr 1 / 4 and κ ϵ ∼ Gr 3 / 8 respectively, where Gr is the Grashof number. This provides a rigorous proof that the power law implies (1) the energy cascade, (2) Kolmogorov dissipation law, and (3) a connection between κ T and κ ϵ . The portion of phase space where a key a priori estimate on the nonlinear term is sharp is shown to be significant by means of a lower bound on any probability measure associated with an infinite-time average.

We derive upper and lower bounds for ensemble averages of energy, enstrophy, and palinstrophy for the 2D periodic Navier–Stokes equations. This is carried out both in the general case, and in the case where the energy power law for fully developed turbulence holds. In the turbulent case, the bounds are sharp, up to a logarithm, and provide a new lower bound on the Landau–Lifschitz degrees of freedom. We also prove two properties of the inertial term under the turbulence assumption. One is that as the Grashof number is increased, the ensemble average of this term approaches the force. The other is that an estimate of it via the Ladyzhenskaya inequality is sharp on a considerable portion of the global attractor.

We provide an analysis of the error in approximating Lyapunov exponents of dissipative PDEs on inertial manifolds using QR techniques. The reduction in the number of modes needed for an inertial form facilitates the error analysis. Numerical computations on the Kuramoto-Sivashinsky equation illustrate the results.

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators., Comment: 28 pages, referee comments incorporated, references added, abstract and introduction modified, main theorems cover full subcritical range of dissipation, certain boundedness properties of observation operators extended

Several relations from the Kolmogorov theory of fully-developed three-dimensional turbulence are rigorously established for finite-time averages over Leray–Hopf weak solutions of the Navier–Stokes equations. The Navier–Stokes equations are considered with periodic boundary conditions and an external forcing term. The main parameter is the Grashof number associated with the forcing term. The relations rigorously proved in this article include estimates for the energy dissipation rate, the Kolmogorov wavenumber, the Taylor wavenumber, the Reynolds number, and the energy cascade process. For some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone, while for others such as the Kolmogorov energy dissipation law and the energy cascade, the estimates depend also on the Grashof number. As compared with earlier works by some of the authors the more physical concept of finite-time average is replacing the concept of infinite-time average used before.

An iterative Lyapunov-Perron algorithm for the computation of inertial manifolds is adapted for centre manifolds and applied to two test problems. The first application is to compute a known non-smooth manifold (once, but not twice differentiable), where a Taylor expansion is not possible. The second is to a smooth manifold arising in a porous medium problem, where rigorous error estimates are compared to both the correction at each iteration and the addition of each coefficient in a Taylor expansion. While in each case the manifold is 1D, the algorithm is well-suited for higher dimensional manifolds. In fact, the computational complexity of the algorithm is independent of the dimension, as it computes individual points on the manifold independently by discretising the solution through them. Summations in the algorithm are reformulated to be recursive. This acceleration applies to the special case of inertial manifolds as well.

We examine how the global attractor $$\mathcal {A}$$ of the 2-D periodic Navier–Stokes equations projects in the normalized, dimensionless energy–enstrophy plane (e, E). We treat time independent forces, with the view of understanding how the attractor depends on the nature of the force. First we show that for any force, $$\mathcal {A}$$ is bounded by the parabola E = e1/2 and the line E=e. We then show that for $$\mathcal {A}$$ to have points near enough to the parabola, the force must be close to an eigenvector of the Stokes operator A; it can intersect the parabola only when the force is precisely such an eigenvector, and does so at a steady state parallel to this force. We construct a thin region along the parabola, pinched at such steady states, that the attractor can never enter. We show that 0 cannot be on the attractor unless the force is in H m for all m. Different lower bound estimates on the energy and enstrophy on $$\mathcal {A}$$ are derived for both smooth and nonsmooth forces, as are bounds on invariant sets away from 0 and near the line E = e. Motivation for the particular attention to the regions near the parabola and near 0 comes from turbulence theory, as explained in the introduction.

The sets of solutions to the Lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined. The set associated with the middle eigenvalue is shown to project surjectively onto a plane, thereby providing a lower estimate for its dimension. Specific bounds are also found for a cone containing this set.

Part of the Kolmogorov-Kraichnan-Batchelor theory of turbulence concerns the average enstrophy flux across wave numbers. To support that theory, rigorous relations involving both the net and one-way flux are established using ensemble averages in [9]. In this note we show that some of these relations hold recurrently, and provide explicit estimates for the time intervals of recurrence which are independent of the solution.

A Nevanlinna–Pick algorithm is developed and applied to short numerical time series approximating trajectories of ordinary differential equations to determine whether the data are near the global attractor. The algorithm, while ultimately unstable to numerical error, is found to give reliable results for more data points than standard algorithms commonly used today. Connections between the growth of trajectories backward in time and the success in locating the global attractor are explored. The numerical sensitivity to both round-off errors and the integrator used to generate the time series is analysed. Applications are made to the Lorenz system and Kuramoto–Sivashinsky equation.

A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier–Stokes equations. A simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber, up to a logarithmic correction. This is followed by partial rigorous support for Kraichnan's eddy breakup mechanism. A rigorous estimate for the total energy is found to be consistent with Kraichnan's theory. Finally, it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case.

Three approaches for the rigorous study of the 2D Navier-Stokes equations (NSE) are applied to the Lorenz system. Analysis of time averaged solutions leads to a description of invariant probability measures on the Lorenz attractor which is much more complete than what is known for the NSE. As is the case for the NSE, solutions on the Lorenz attractor are analytic in a strip about the real time axis. Rigorous estimates are combined with numerical computation of Taylor coefficients to estimate the width of this strip. Approximate inertial forms originally developed for the NSE are analyzed for the Lorenz system, and the dynamics for the latter are completely described.

The behavior of an ordinary differential equation for the low wave number velocity mode is analyzed. This equation was derived in [5] by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It resembles the NSE in form, except that the kinematic viscosity is replaced by an iterated viscosity which is a partial sum, dependent on the low-mode velocity. The convergence of this sum as the number of iterations is taken to be arbitrarily large is explored. This leads to a limiting dynamical system which displays several unusual mathematical features.

It is shown that the long-time dynamics (the global attractor) of the 2D Navier-Stokes system is embedded in the long-time dynamics of an ordinary differential equation, called a determining form, in a space of trajectories which is isomorphic to C1b (R;RN) for sufficiently large N depending on the physical parameters of the Navier-Stokes equations. A unified approach is presented, based on interpolant operators constructed from various determining parameters for the Navier-Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, and so on. There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, and thus its solutions converge to the set of steady states of the determining form as the time goes to infinity. The second is that these steady states of the determining form can be uniquely identified with the trajectories in the global attractor of the Navier-Stokes system. It should be added that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

The global attractor of a dissipative system of ordinary differential equations can be characterized as the set of solutions which permit an extension to a bounded analytic function on a uniform strip in a complex plane. Using this property, we present two methods for constructing sequences of functions, which may be explicitly computed from the system, and from which one can deduce whether a specific point belongs to the attractor or not. Approximation methods obtained in this way are tested on the Lorenz system and compared with those from Foias and Jolly (1995 Nonlinearity 8 295 - 319).

The set of nonzero external forces for which the zero function is in the global attractor of the 2D Navier-Stokes equations is shown to be meagre in a Fr\'echet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of $C^{\infty}$ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time., Comment: 16 pages

We show that if the Kraichnan theory of fully developed turbulence holds, then the Landau–Lifschitz degrees of freedom is bounded (up to a logarithmic term) by G1/2, where G is the Grashof number.