Optimal transport :
Theory and Applications
to cosmological Reconstruction and Image processing
Mathematics of particles and flows
Wolfgang Pauli Institute
Vienna, Austria, 28 May–2 June 2012
This web pages contains abstracts of talks and links to
relevant material whenever available.
Selected photos of the meeting (taken by Anna Pomyalov) are now posted on Picasa.
Many problems concerning convergence of solutions of the Navier–Stokes and Boltzmann equations to the Euler equation are wide open… In this presentation I want to emphazise some similarity between the two problems with several examples: Eternal solutions of the Boltzmann equation, Boundary effect both for Boltzmann and Euler etc… This will be a “pot pourri,” but as such it is based on several papers with several authors: Irene Gamba, Francois Golse, Dave Levermore, Lionel Paillard and Edriss Titi.
Joint work with Rehab Bitane and Giorgio Krstulovic.
A mass ejection model in a time-dependent random environment with both temporal and spatial correlations is introduced. The collective dynamics of diffusing particles reaches a statistically stationary state, which is characterized in terms of a fluctuating mass density field. The probability distribution of density is studied for both smooth and non-smooth scale-invariant random environments. A competition between trapping in the regions where the ejection rate of the environment vanishes and mixing due to its temporal dependence leads to large fluctuations of mass. These mechanisms are found to result in the presence of intermediate power-law tails in the probability distribution of the mass density. For spatially differentiable environments, the exponent of the right tail is shown to be universal and equal to −3/2.
Slides of the talk: download (7 MB)
Reference: R. Bitane, J. Bec, and H. Homann, Timescales of turbulent relative dispersion (preprint; download 177 KB).
The purpose of the talk is two-fold:
Reference: Matania Ben-Artzi, Jean-Pierre Croisille, and Dalia Fishelov, Navier–Stokes equations in planar domains, Kluwer, 2012.
Joint work with R. Scatamacchia, M. Cencini, A. S. Lanotte and F. Toschi.
We present a high-statistics numerical study of particle dispersion from point-sources in Homogeneous and Isotropic turbulence (HIT) at Reynolds number Re ∼ 300. Particles are emitted in bunches from very localized sources (smaller than the Kolmogorov scale) in different flows locations. We present a quantitative and systematic analysis of the deviations from Richardson's picture of relative dispersion; these deviations correspond to extreme events either of particle pairs separating faster than usual (worst-case) or of particle pairs separating slower than usual (best-case, i.e. particles which remain close for long time). A comparison with statistics collected in surrogate delta-correlated velocity field at the same Reynolds numbers allow us to assess the importance of temporal correlations along particles trajectories.
Reference: presentation (PDF)
We present some results of computer simulations for the complex-valued solutions of the 2-d Burgers equations on the plane in absence of external forces. The existence of singularities at a finite time for some class of initial data, with divergence of the total energy, was proved by Li and Sinai. The simulations show that the blowup takes place in a very short time, and near the blowup time the support of the solution in Fourier space moves out to infinity along a straight line. In x-space the solution concentrates in a finite region, with large space derivatives, as one would expect for physical phenomena such as tornadoes. The blowup time turns out to be remarkably stable with respect to the computation methods.
Joint work with Miguel D. Bustamante.
Numerical simulations of the incompressible Euler equations are performed using the Taylor–Green vortex initial conditions and resolutions up to 40963. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of the time-decay of the width of the analyticity strip is observed at the highest-resolution for 3.7 < t < 3.85. The BKM criterium on the growth of supremum of the vorticity, applied on the same time-interval, does not rule out the occurrence of a singularity around time 4. These new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if the analyticity strip width vanishes sufficiently fast at the singularity time. If a power-law is assumed then its exponent must be greater than some critical value, thus providing a new test that is applied to our 40963 Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time-interval on which a well-resolved power-law behavior of the analyticity strip width takes place, and check whether the new regime is genuine and not simply a crossover to a faster exponential decay.
Background material: U. Frisch, T. Matsumoto, J. Bec, Singularities of Euler Flow? Not Out of the Blue!, J. Stat. Phys. 113:5 (2003) 761–781, doi10.1023/A:1027308602344.
Slides of the talk: download (3.8 MB)
Surprisingly enough, there are several connections between “adhesion dynamics” and the motion of inviscid incompressible fluids. It has been already established by A. Shnirelman that one can construct a weak (and not smooth at all) solution to the Euler equations of incompressible fluids, based on adhesion dynamics. In this talk, we establish another connection through the concept of approximate geodesics along the group of volume-preserving diffeomorphisms. In particular, we recover some dissipative solutions of the Zeldovich gravitational model.
Reference: preprint hal-00488716
The usual notion of “magnetic flux-freezing” breaks down in the Kazantsev dynamo model with only Hoelder-in-space velocities. Due to “spontaneous stochasticity”, infinitely many field lines are advected to the same point, even in the limit of vanishing resistivity. Their contribution to magnetic energy growth can be obtained in the Kazantsev model both numerically and by WKBJ asymptotics. Numerical results for kinematic dynamo in real hydrodynamic turbulence show remarkable similarity to the solution of the Kazantsev model.
In particular, we present numerical evidence for “spontaneous stochasticity” in hydrodynamic turbulence. Finally, the Kazantsev model in the “failed-dynamo” regime (Hoelder exponent h < 1/2) is a toy problem with many analogies to incompressible Euler equations, e.g. an analogue of the Kelvin circulation theorem. We conjecture that weak solutions of relevance to turbulence are characterized by a backward-martingale property of the circulations. We outline a Hilbert-space approach to this problem in the Kazantsev model based on ideas of Brenier.
I will describe two new analytic derivations, one probably right, another probably wrong, done for the Navier–Stokes equation in Lagrangan coordinates. Mathematical insight into these derivations is called for.
Joint work with E. Aurell, C. Mejia-Monasterio, R. Mohayaee and P. Muratore-Ginanneschi.
Stochastic modelization of mesoscopic systems in interaction with thermal environment permits to revist links between statistical and thermodynamical concepts in simple out of equilibrium situations. I shall discuss in such a setup a finite-time refinement of the 2nd Law of Thermodynamics. The refinement is related to the Monge–Kantorovich optimal mass transport and the underlying inviscid Burgers equation.
Slides of the talk: download (423 KB)
Reference: E. Aurell, K. Gawedzki, C. Mejia-Monasterio, R. Mohayaee and P. Muratore-Ginanneschi, Refined Second Law of Thermodynamics for fast random processes, J. Stat. Phys. 147 (2012) 487-505 (arXiv:1201.3207).
Joint work with Yuri Bakhtin and Eric Cator.
We construct stationary solutions for Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. In particular, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average velocity. We also discuss connections with the theory of directed polymers in dimension 1 + 1 and the KPZ scalings.
I will discuss some recent joint work with Ya. G. Sinai on the construction of bifurcation of solutions to several models in fluid dynamics such as the 2D Navier–Stokes system and 2D quasi-geostrophic equations.
A multi-precision software library enables the spectral method for a three-dimensional Euler flow with, in principle, arbitrary high precision rather than the standard double precision. Such a numerical attempt is reported with emphasis on the short-time behavior of the flow, starting from analytic initial data.
Reference : presentation (PDF)
I will describe some analytical and numerical studies of turbulence in this model. The focus will be on so-called LH-transition feedback loop, in which turbulence forced at small scales generates a zonal flow via an anisotropic inverse cascade which then suppresses the small-scale turbulence, thereby eliminating anomalous transport.
Joint work with U. Frisch, S. S. Ray, G. Sahoo, and D. Banerjee.
The bottleneck effect—an abnormally high level of excitation of the energy spectrum, for three-dimensional, fully developed Navier–Stokes turbulence, that is localized between the inertial and dissipation ranges—is shown to be present for a simple, non-turbulent, one-dimensional model, namely, the Burgers equation with hyperviscous dissipation. This bottleneck is shown to be the Fourier-space signature of oscillations in the real-space velocity. These oscillations are amenable to quantitative, analytical understanding, as we demonstrate by using boundary-layer-expansion techniques. Pseudospectral simulations are then used to show that such oscillatory features are also present in velocity correlation functions in one- and three-dimensional hyperviscous hydrodynamical models that display genuine turbulence.
Slides of the talk: download (10.2 MB).
Joint work with S. Chakraborty, U. Frisch, S. S. Ray.
It has been shown by Nelkin that studying moments of velocity gradients as a function of the Reynolds number represents an alternative approach to obtaining information about properties of turbulent flows in the inertial range. We have used the one-dimensional Burgers equation to verify the utility of this approach in a case which can be treated in detail numerically as well as theoretically. As we have shown, scaling exponents can be reliably identified already at Reynolds numbers of the order of 100 (or even lower when combined with a suitable extended self-similarity technique). It turns out that at moderate Reynolds numbers, for the accurate determination of scaling exponents, it is crucial to use higher than double precision. In particular, from the computational point of view increasing the precision is definitely more efficient than increasing the resolution. We conjecture that similar issues also arise for three-dimensional NavierStokes simulations.
Slides of the talk: download (2.7 MB)
In theoretical physics, a number of results have been obtained by extending the dimension d of space from directly relevant values such as 1, 2, 3 to noninteger values. The main difficulty in carrying out such an extention for hydrodynamics in d < 2 is to ensure the conservation of energy and enstrophy. We discovered a new way of fractal decimation in Fourier space, appropriate for hydrodynamics. Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to kD for large k. At the critical dimension Dc = 4/3 there is an equilibrium Gibbs state with a k−5/3 spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89 (2002) 064501. Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D = 2 with a rapidly rising Kolmogorov constant, likely to diverge as (D − Dc)−2/3.
I will review recent advances in understanding the nature of the plastic instabilities in amorphous solids, identifying them with eigenvalues of the Hessian matrix hitting zero via a saddle node bifurcation. This simple singularity determines exactly interesting exponents of system size dependence of average stress and energy drops in elasto-plastic flows. Finally I will tie these insights to the existence of a static length scale that increases rapidly with the approach to the glass transition.
Reference: Presentation (PPT).
Joint work with U. Frisch, S. Nazarenko, and T. Matsumoto.
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wavenumbers in excess of a threshold KG exhibit unexpected features. At large KG, for smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation, is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features. These oscillations are weak and strongly localized at first—in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to KG−2/3 and KG−1/3 respectively—but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T. D. Lee in 1952.
Reference: S. S. Ray, U. Frisch, S. Nazarenko, and T. Matsumoto, Physical Review E 84 (2011) 016301 and arXiv:1011.1826.
Slides of the talk: download (6.7 MB).
We fill in all details in the proof of Lanford's theorem. This provides a rigorous derivation of the Boltzmann equation as the thermodynamic limit of a d-dimensional Hamiltonian system of particles interacting via a short-range potential, obtained as the number of particles N goes to infinity and the characteristic size of the particles e simultaneously goes to 0, in the Boltzmann-Grad scaling Ned − 1 ≡ 1. The time of validity of the convergence is a fraction of the mean free time between two collisions, due to a limitation of the time on which one can prove the existence of the BBGKY and Boltzmann hierarchies. The propagation of chaos is obtained by a precise analysis of pathological trajectories involving recollisions. We show in particular that the microscopic interaction potential occurs only via the scattering
Slides of the talk: download (14.4 MB).
We show that a necessary condition for T to be a potential blow up time is that the spatial L3 norm of the velocity goes to infinity as the time t approaches T from below.
Reference: G. Seregin, A Certain Necessary Condition of Potential Blow up for Navier-Stokes Equations, CMP 312:3 (2012) 833–845, doi 10.1007/s00220-011-1391-x.
Joint work with J. Delon, S. Nechaev, J. Salomon, and O. Valba.
We review two discrete random growth models whose formal continuous limits are related to PDEs.
Joint work with Camillo De Lellis.
We construct Hölder-continuous weak solutions of the 3D incompressible Euler equations, which dissipate the total kinetic energy. The construction is based on the scheme introduced by J. Nash for producing C1 isometric embeddings, which was later developed by M. Gromov into what became known as convex integration. Weak versions of convex integration (e.g. based on the Baire category theorem) have been used previously to construct bounded (but highly discontinuous) weak solutions. The current construction is the first instance of Nash's scheme being applied to a PDE which one might classify as “hard” as opposed to “soft”. The solution obtained by our scheme can be seen as a superposition of infinitely many perturbed and weakly interacting Beltrami flows. The existence of Hölder-continuous solutions dissipating energy was conjectured by L. Onsager in 1949.
References: arXiv:1202.1751 and 1205.3626.
Joint work with Claude Bardos.
A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limits of solutions of Euler equations may, in some cases, fail to be a solution of the Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to C1, α. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin–Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.
Joint work with Gregory Falkovich and Konstantin S. Turitsyn.
A passive scalar field was studied under the action of pumping, diffusion and advection by a 2D smooth flow with Lagrangian chaos. We present theoretical arguments showing that the scalar statistics are not conformally invariant and formulate a new effective semi-analytic algorithm to model scalar turbulence. We then carry out massive numerics of scalar turbulence, focusing on nodal lines. The distribution of contours over sizes and perimeters is shown to depend neither on the flow realization nor on the resolution (diffusion) scale, for scales exceeding this scale. The scalar isolines are found to be fractal/smooth at scales larger/smaller than the pumping scale. We characterize the statistics of isoline bending by the driving function of the Loewner map. That function is found to behave like diffusion with diffusivity independent of the resolution yet, most surprisingly, dependent on the velocity realization and time (beyond the time on which the statistics of the scalar is stabilized).
Reference: J. Stat. Phys. 147 (2012) 424–435, doi:10.1007/s10955-012-0474-1.
Slides of the talk: download (6.3 MB).
Joint work with Carlos Colosqui, Hudong Chen, Kamil Ekinci, Devrez Karabacak.
Oscillating nano-particles (resonators) can serve as sensors detecting impurities, viruses etc in various simple fluids like air and, in principle, water. In the low frequency limit their dynamics are a viscous process obeying the Navier–Stokes (diffusion) equations. It will be shown that when the Weissenberg number Wi >> 1 this process becomes visco-elastic described by the generic telegrapher equation. The relation representing the particle dynamics in the entire range of the Weissenberg number and/or pressure variation is universal, independent on the particle shape and size. A detailed experimental, theoretical and numerical studies supporting this conclusion will be presented.
Joint work with U. Frisch, O. Podvigina, and B. Villone.
One of the simplest models used in studying the dynamics of large-scale structure in cosmology, known as the Zeldovich approximation, is equivalent to the three-dimensional inviscid Burgers equation for potential flow. For smooth initial data and sufficiently short times it has the property that the mapping of the positions of fluid particles at any time t1 to their positions at any time t2 > t1 is the gradient of a convex potential, a property we call omni-potentiality. We show that, in both two and three dimensions, there exist flows with this property, that are not straightforward generalizations of Zeldovich flows. How general are such flows? In two dimensions, for sufficiently short times, there are omni-potential flows with arbitrary smooth initial velocity. Mappings with a convex potential are known to be associated with the quadratic-cost optimal transport problem. Implications for the problem of reconstructing the dynamical history of the Universe from the knowledge of the present mass distribution are discussed.
Reference: J. Math. Phys., 53 (2012) 033703 and arXiv:1111.2516.
Background material: Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee, A. Sobolevski, Reconstruction of the early Universe as a convex optimization problem, Mon. Not. R. Astron. Soc. 346:2 (2003) 501–524; download (3.6 MB).
Slides of the talk: download (0.9 MB).