Volume 48, Issue 6, June 2007
 SPECIAL ISSUE: MATHEMATICAL FLUID DYNAMICS


 Regularity Results and Analytical Estimates for NavierStokes and Related Systems

Universal bounds on the attractor of the NavierStokes equation in the energy, enstrophy plane
View Description Hide DescriptionWe continue our examination of the projection of the global attractor of the twodimensional NavierStokes equations in a normalized, dimensionless energy, enstrophy plane. In this plane the attractor must lie above the line of slope 1 through the origin (an immediate consequence of the Poincaré inequality). We showed in Dascaliuc et al. [J. Dyn. Differ. Equ.17, 629–649 (2005)] that the attractor must also lie below a parabola (the square root curve). Here we optimize this parabola to obtain a sharper bound. These bounds are universal; they are independent of viscosity, domain size, and strength of the force. The effectiveness of the bounds is demonstrated on Galerkin approximations of modest size (up to 1372 modes).

Estimates for the twodimensional Navier–Stokes equations in terms of the Reynolds number
View Description Hide DescriptionThe tradition in Navier–Stokes analysis of finding estimates in terms of the Grashof number , whose character depends on the ratio of the forcing to the viscosity, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias [C. R. Doering and C. Foias, J. Fluid Mech.467, 289 (2002)] to the twodimensional Navier–Stokes equations on a periodic domain by estimating quantities of physical relevance, particularly longtime averages , in terms of the Reynolds number, where and is the forcing scale. In particular, the Constantin–Foias–Temam upper bound [P. Constantin, C. Foias, and R. Temam, Physica D30, 284 (1988)] on the attractor dimension converts to , while the estimate for the inverse Kraichnan length is , where is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency: it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time averages, which are themselves interspersed by longer, more quiescent, intervals of time.

NavierStokes equations with regularity in one direction
View Description Hide DescriptionWe consider sufficient conditions for the regularity of LerayHopf solutions of the NavierStokes equations. We prove that if the third derivative of the velocity belongs to the space , where and , then the solution is regular. This extends a result of Beirão da Veiga [Chin. Ann. Math., Ser. B16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math.321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative can be substituted with any directional derivative of .

A posteriori regularity of the threedimensional Navier–Stokes equations from numerical computations
View Description Hide DescriptionIn this paper we consider the role that numerical computations—in particular Galerkin approximations—can play in problems modeled by the threedimensional (3D) Navier–Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem. We then consider Galerkin approximations, and show that if a strong solution exists the Galerkin approximations will converge to it; thus if one is prepared to assume that the Navier–Stokes equations are regular one can justify this particular numerical method rigorously. Combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time. We thus introduce the possibility of rigorous computations of the solutions of the 3D Navier–Stokes equations (despite the lack of rigorous existence and uniqueness results), and demonstrate that numerical investigation can be used to rule out the occurrence of possible singularities in particular examples.

A blowup problem of a class of axisymmetric NavierStokes equations with infinite energy
View Description Hide DescriptionWe consider a class of axisymmetric NavierStokes equations that has geophysical applications. After reviewing stationary solutions studied by DonaldsonSullivan (1960), we study nonstationary solutions. We compare behavior of this class of solution with that of the Fujita equation, a typical example of reactiondiffusion systems. We prove blowup for our system.

Analytical behavior of twodimensional incompressible flow in porous media
View Description Hide DescriptionIn this paper we study the analytic structure of a twodimensional mass balance equation of an incompressible fluid in a porous medium given by Darcy’s law. We obtain local existence and uniqueness by the particletrajectory method and we present different global existence criterions. These analytical results with numerical simulations are used to indicate nonformation of singularities. Moreover, blowup results are shown in a class of solutions with infinite energy.

Further properties of steadystate solutions to the NavierStokes problem past a threedimensional obstacle
View Description Hide DescriptionWe show that the weak formulation of the steadystate NavierStokes problem in the exterior of a threedimensional compact set (closure of a bounded domain), corresponding to a nonzero velocity at infinity and subjected to a given body force, is equivalent to a nonlinear equation in appropriate Banach spaces. We thus show that the relevant nonlinear operator enjoys a number of fundamental properties that allow us to derive many significant results for the original problem. In particular, we prove that the manifold constituted by the pairs , with the nondimensional speed at infinity (Reynolds number) and weak solution corresponding to and to a given body force , is, for “generic” , a onedimensional manifold, and that, for almost any , the number of solutions is finite. We also show that, for any given in the appropriate function space and any given , the corresponding solutions can be “controlled” by their specification only in a suitable neighborhood, , of the boundary. The “size” of depends only on and . Furthermore, we analyze the steady bifurcation properties of branches of these solutions and prove that, in some important cases, the sufficient conditions for bifurcation formally coincide with the analogous ones for flow in a bounded domain. Finally, the stability of these solutions is analyzed. The paper ends with a section on relevant open questions.

Energy dissipation in fractalforced flow
View Description Hide DescriptionThe rate of energy dissipation in solutions of the bodyforced threedimensional incompressible NavierStokes equations is rigorously estimated with a focus on its dependence on the nature of the driving force. For square integrable body forces, the high Reynolds number (low viscosity) upper bound on the dissipation is independent of the viscosity, consistent with the existence of a conventional turbulent energy cascade. On the other hand, when the body force is not square integrable, i.e., when the Fourier spectrum of the force decays sufficiently slowly at high wave numbers, there is significant direct driving at a broad range of spatial scales. Then the upper limit for the dissipation rate may diverge at high Reynolds numbers, consistent with recent experimental and computational studies of “fractalforced” turbulence.
 Vanishing Viscosity Limits and the Relation Between NavierStokes and Euler Systems

Vanishing viscosity limit for an incompressible fluid with concentrated vorticity
View Description Hide DescriptionWe study an incompressible fluid with a sharply concentrated vorticity moving in the whole space. First, we review some known results for the inviscid case, which prove, in particular situations, the local induction approximation. Then, we discuss the vanishing viscosity limit and we state a new result valid for a smoke ring with a very large radius.

Discrete Katotype theorem on inviscid limit of NavierStokes flows
View Description Hide DescriptionThe inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics. Here we present a somewhat surprising result related to numerical approximation of the problem. More precisely, we show that numerical solutions of the incompressible NavierStokes equations converge to the exact solution of the Euler equations at vanishing viscosity and vanishing mesh size provided that small scales of the order of in the directions tangential to the boundary are not resolved in the scheme. Here is the kinematicviscosity of the fluid and is the typical velocity taken to be the maximum of the shear velocity at the boundary for the inviscid flow. Such a result is somewhat counterintuitive since the convergence is ensured even in the case that small scales predicted by the conventional theory of turbulence and boundary layer are not resolved since underresolution (which is allowed in our theorem) in advection dominated problem usually leads to oscillation which inhibits convergence in general. The result also indicates possible difficulty in terms of numerical investigation of the vanishing viscosity problem if rigorous fidelity of the numerics is desired since we have to resolve at least small scales of the order of which is much smaller than any small scales predicted by the conventional theory of turbulence. On the other hand, such a result can be viewed as a discrete version of our result [X. Wang, Indiana Univ. Math. J.50, 223 (2001)] which generalized earlier the result of Kato [in Seminar on PDE, edited by S. S. Chern (Springer, NY, 1984)] where the relevance of a scale proportional to the kinematicviscosity to the problem of vanishing viscosity is first discovered.
 Mathematical Issues Related to Discrete Vortex and Dynamical Systems Representations of Fluid Flows

Point vortex dynamics: A classical mathematics playground
View Description Hide DescriptionThe idealization of a twodimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the threevortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstraß function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community.

Studies of perturbed three vortex dynamics
View Description Hide DescriptionIt is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated with completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half plane, three coaxial slender vortex rings in three space, and “restricted” four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using KolmogorovArnoldMoser and PoincaréBirkhofftype arguments and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics.

Dynamic interaction of point vortices and a twodimensional cylinder
View Description Hide DescriptionIn this paper we consider the system of an arbitrary twodimensional cylinder interacting with point vortices in a perfect fluid. We present the equations of motion and discuss their integrability. Simulations show that the system of an elliptic cylinder (with nonzero eccentricity) and a single point vortex already exhibits chaotic features and the equations of motion are nonintegrable. We suggest a Hamiltonian form of the equations. The problem we study here, namely, the equations of motion, the Hamiltonian structure for the interacting system of a cylinder of arbitrary crosssection shape, with zero circulation around it, and vortices, has been addressed by Shashikanth [Regular Chaotic Dyn.10, 1 (2005)]. We slightly generalize the work by Shashikanth by allowing for nonzero circulation around the cylinder and offer a different approach than that by Shashikanth by using classical complex variable theory.

Lagrangian coherent structures in dimensional systems
View Description Hide DescriptionNumerical simulations and experimental observations reveal that unsteady fluid systems can be divided into regions of qualitatively different dynamics. The key to understanding transport and stirring is to identify the dynamic boundaries between these almostinvariant regions. Recently, ridges in finitetime Lyapunov exponent fields have been used to define such hyperbolic, almost material, Lagrangian coherent structures in twodimensional systems. The objective of this paper is to develop and apply a similar theory in higher dimensional spaces. While the separatrix nature of these structures is their most important property, a necessary condition is their almost material nature. This property is addressed in this paper. These results are applied to a model of RayleighBénard convection based on a threedimensional extension of the model of Solomon and Gollub.

Reduced models for fluid flows with strong constraints
View Description Hide DescriptionThe presence of a dominant balance in the equations for fluid flow can be exploited to derive an asymptotically exact but simpler set of governing equations. These permit semianalytical and/or numerical explorations of parameter regimes that would otherwise be inaccessible to direct numerical simulation. The derivation of the resulting reduced models is illustrated here for (i) rapidly rotating convection in a plane layer, (ii) convection in a strong magnetic field, and (iii) the magnetorotational instability in accretion disks and the results used to extend our understanding of these systems in the strongly nonlinear regime.
 Statistical Fluid Dynamics and Turbulence Models

Note on the emergence of large scale coherent structure under small scale random bombardments: The discrete case
View Description Hide DescriptionWe continue our study on mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments. We treat the case of small scale random bombardments at discrete times which is different from our earlier work [Commun. Pure Appl. Math.59, 467 (2006)], where we approximated the small scale random kicks by a continuous in time random process. In the absence of geophysical effects, the large scale structure emerging out of the small scale random forcing is the same as the case of continuous in time forcing that we studied before.

Transparent boundary conditions as dissipative subgrid closures for the spectral representation of scalar advection by shear flows
View Description Hide DescriptionWe consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space. When the velocity field has small support, the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves. We close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves. This closure is studied numerically in terms of energy dissipation rate and energy spectrum, both for a timeindependent velocity field and for a timedependent synthetic velocity field whose Fourier coefficients follow independent OrnsteinUhlenbeck stochastic processes.

Inviscid dyadic model of turbulence: The fixed point and Onsager’s conjecture
View Description Hide DescriptionProperties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the threedimensional scaling of the quadratic nonlinearity. It is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in norm. Onsager’s [Nuovo Cimento6, 279–287 (1949)] conjecture is confirmed for the model system.

Analytical study of certain magnetohydrodynamic models
View Description Hide DescriptionIn this paper we present an analytical study of a subgrid scale turbulencemodel of the threedimensional magnetohydrodynamic(MHD)equations, inspired by the NavierStokes (also known as the viscous CamassaHolm equations or the Lagrangianaveraged NavierStokesmodel). Specifically, we show the global wellposedness and regularity of solutions of a certain MHDmodel (which is a particular case of the Lagrangian averaged magnetohydrodynamicmodel without enhancing the viscosity for the magnetic field). We also introduce other subgrid scale turbulencemodels, inspired by the Leray and the modified Leraymodels of turbulence. Finally, we discuss the relation of the MHDmodel to the MHDequations by proving a convergence theorem, that is, as the length scale tends to zero, a subsequence of solutions of the MHDequations converges to a certain solution (a LerayHopf solution) of the threedimensional MHDequations.
