Stochastic least-action principle for the incompressible Navier-Stokes equation

We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton’s principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in the framework of the stochastic variational principle. In particular, we discuss the connection of the stochastic Kelvin Theorem with our previous “martingale hypothesis” for fluid circulations in turbulent solutions of the incompressible Euler equations.     

Hydrodynamics in two dimensions and vortex theory

We consider the Navier-Stokes equation for a viscous and incompressible fluid inR ². We show that such an equation may be interpreted as a mean field equation (Vlasov-like limit) for a system of particles, called vortices, interacting via a logarithmic potential, on which, in addition, a stochastic perturbation is acting. More precisely we prove that the solutions of the Navier-Stokes equation may be approximated, in a suitable way, by finite dimensional diffusion processes with the diffusion constant related to the viscosity. As a particular case, when the diffusion constant is zero, the finite dimensional theory reduces to the usual deterministic vortex theory, and the limiting equation reduces to the Euler equation.Partially supported by Italian CNR     

Note on loss of regularity for solutions of the 3—D incompressible euler and related equations

One of the central problems in the mathematical theory of turbulence is that of breakdown of smooth (indefinitely differentiable) solutions to the equations of motion. In 1934 J. Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3—D incompressible Navier-Stokes equations. The problem is still open. We show in this report that breakdown of smooth solutions to the 3—D incompressible slightly viscous (i.e. corresponding to high Reynolds numbers, or highly turbulent) Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. We prove then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041, and partially supported by the National Science Foundation under Grant No. MCS-82-01599     

Applications of Nambu Mechanics to Systems of Hydrodynamical Type II

Abstract

In this paper we further investigate some applications of Nambu mechanics in hydrodynamical systems. Using the Euler equations for a rotating rigid body Névir and Blender [J. Phys. A 26 (1993), L1189–L1193] had demonstrated the connection between Nambu mechanics and noncanonical Hamiltonian mechanics. Nambu mechanics is extended to incompressible ideal hydrodynamical fields using energy and helicity in three dimensional (enstrophy in two dimensional). In this paper we discuss the Lax representation of systems of Névir-Blender type. We also formulate the three dimensional Euler equations of incompressible fluid in terms of Nambu-Poisson geometry. We discuss their Lax representation. We also briefly discuss the Lax representation of ideal incompressible magnetohydrodynamics equations.     

Concentration-cancellation for the velocity fields in two dimensional incompressible fluid flows

We show that the weak-L ² limit of a sequence of solutions of the two dimensional incompressible Euler equation is still a solution, provided that a (strong) concentration set for the reduced defect measure has locally finite one dimensional Hausdorff measure in space and time.     

Energetic variational approaches in modeling vesicle and fluid interactions

In this paper, we establish a hydrodynamic system to study vesicle deformations under external flow fields. The system is in the Eulerian formulation, involving the coupling of the incompressible flow system and a phase field equation. The phase field mixing energy can be viewed as a physical approximation/regularization of the Helfrich energy for an elastic membrane. We derive a self-consistent system of equations describing the dynamic evolution of vesicles immersed in an incompressible, Newtonian fluid, using an energetic variational approach. Numerical simulations of the membrane deformations in flow fields can be conducted based on the developed model.     

Three-dimensional multi-relaxation-time lattice Boltzmann front-tracking method for two-phase flow

We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop,and the drop deformation in a shear flow, were simulated to validate the present model.     

Singularities of Euler Flow? Not Out of the Blue!

Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.     

Euler's fluid equations: Optimal control vs optimization

An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, an optimal control problem and an optimization problem for incompressible ideal fluid flow both yield the same Euler fluid equations, although their Lagrangian parcel dynamics are different. This is a result of the gauge freedom in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabeling of their Lagrangian coordinates. Similar ideas are also illustrated for SO(N) rigid body motion.     

Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

We review and connect different variational principles that have been proposed to settle the dynamical and thermodynamical stability of two-dimensional incompressible and inviscid flows governed by the 2D Euler equation. These variational principles involve functionals of a very wide class that go beyond the usual Boltzmann functional. We provide relaxation equations that can be used as numerical algorithms to solve these optimization problems. These relaxation equations have the form of nonlinear mean field Fokker-Planck equations associated with generalized “entropic” functionals [P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)].     

Exact Local Solutions for the Formation of Singularities on the Free Surface of an Ideal Fluid

A classical problem of the dynamics of the free surface of an ideal incompressible fluid with infinite depth has been considered. It has been found that the regime of motion of the fluid where the pressure is a quadratic function of the velocity components is possible in the absence of external forces and capillarity. It has been shown that equations of plane potential flow for this situation are linearized in conformal variables and are then easily solved analytically. The found solution includes an arbitrary function specifying the initial shape of the surface of the fluid. The developed approach makes it possible for the first time to locally describe the formation of various singularities on the surface of the fluid—bubbles, drops, and cusps.     

Weak-Strong Uniqueness for Measure-Valued Solutions

We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by DiPerna and Majda in their landmark paper (Commun Math Phys 108(4):667–689, 1987), where in particular global existence to any L ² initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open.     

Statistical equilibrium solutions of the shallow water equations

A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2D fluid flow with a free surface, is described. The model contains a competing acoustic turbulent direct energy cascade, and a 2D turbulent inverse energy cascade. It is shown, nonetheless that, just as in the corresponding theory of the inviscid Euler equation, the infinite number of conserved quantities constrains the flow sufficiently to produce nontrivial large-scale vortex structures which are solutions to a set of explicitly derived coupled nonlinear partial differential equations.     

Essential Variational Poisson Cohomology

In our recent paper “The variational Poisson cohomology” (2011) we computed the dimension of the variational Poisson cohomology for any quasiconstant coefficient ℓ × ℓ matrix differential operator K of order N with invertible leading coefficient, provided that is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the -graded Lie superalgebra is isomorphic to the finite dimensional Lie superalgebra . We also prove that the subalgebra of “essential” variational Poisson cohomology, consisting of classes vanishing on the Casimirs of K, is zero. This vanishing result has applications to the theory of bi-Hamiltonian structures and their deformations. At the end of the paper we consider also the translation invariant case.     

A Lagrangian and canonical formalism in the hydrodynamics of an incompressible fluid

The geometrical formulation of the variational principle in the hydrodynamics of an incompressible fluid is given. The Lagrange-Euler equations in the language of differential forms are formulated. The canonical formalism in the hydrodynamics in the general framework of multi-phase spaces ([4]) is considered and the construction of the phase space for hydrodynamics is given.     

Application of the method of a self-consistent field to the calculation of the local density and surface tension of small drops in a simple fluid

An approach based on the perturbation theory combined with the method of a self-consistent field was used to investigate the dimensional dependence of the specific free surface energy (surface tension) of small drops in a simple fluid. The surface tension decreases with decreasing drop size. The approach proposed also allows one to find the radial density distribution in a small object. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–61, August, 2007.     

Glauber Dynamics for the Quantum Ising Model in a Transverse Field on a Regular Tree

Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space.     

Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painlevé I transcendent

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers Lee et al. (2009, 2008) [8] and [9] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painlevé I equation. We argue that the Painlevé I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painlevélinear problem.     

How do singularities move in potential flow?

The equations of motion of point vortices embedded in incompressible flow go back to Kirchhoff. They are a paradigm of reduction of an infinite-dimensional dynamical system, namely the incompressible Euler equation, to a finite-dimensional system, and have been called a “classical applied mathematical playground”. The equation of motion for a point vortex can be viewed as the statement that the translational velocity of the point vortex is obtained by removing the leading-order singularity due to the point vortex when computing its velocity. The approaches used to obtain this result are reviewed, along with their history and limitations. A formulation that can be extended to study the motion of higher singularities (e.g. dipoles) is then presented. Extensions to more complex physical situations are also discussed.