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Laure SAINTRAYMOND  WPI, Seminarroom C 714  Mon, 28. May 12, 10:45 
About the BoltzmannGrad limit  
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 ddimensional Hamiltonian system of particles interacting via a shortrange 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 BoltzmannGrad scaling $N e^{d1} equiv 1.$ 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  

Claude BARDOS  WPI, Seminarroom C 714  Mon, 28. May 12, 11:30 
Comparison between the Boltzmann and the Navier Stokes limit for the Euler equation  
Many problems concerning convergence of solutions of the NavierStokes 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...  
Note: [1]Bardos, C. [2]Titi, E. S. Euler equations for an ideal incompressible fluid. (Russian) [3]Uspekhi Mat. Nauk 62 (2007), [4]no. 3(375), 546 Bardos, C. Golse, F. and Paillard, L.: The incompressible Euler limit of the Boltzmann equation with accomodation boundary condition Comm. Math. Sciences, 2012 International Press Vol. 10, No. 1, pp. 159190 KanielShinbrot iteration and global solutions of the Cauchy problem for the Boltzmann equation. Claude Bardos, Irene Gamba and C. David Levermore, Preprint.  

Rahul PANDIT  WPI, Seminarroom C 714  Mon, 28. May 12, 14:15 
Energyspectra Bottlenecks: Insights from Hyperviscous Hydrodynamical Equations  
The bottleneck effect  an abnormally high level of excitation of the energy spectrum, for threedimensional, fully developed NavierStokes turbulence, that is localized between the inertial and dissipation ranges  is shown to be present for a simple, nonturbulent, onedimensional model, namely, the Burgers equation with hyperviscous dissipation. This bottleneck is shown to be the Fourierspace signature of oscillations in the realspace velocity. These oscillations are amenable to quantitative, analytical understanding, as we demonstrate by using boundarylayerexpansion techniques. Pseudospectral simulations are then used to show that such oscillatory features are also present in velocity correlation functions in one and threedimensional hyperviscous hydrodynamical models that display genuine turbulence. The bottleneck effect  an abnormally high level of excitation of the energy spectrum, for threedimensional, fully developed NavierStokes turbulence, that is localized between the inertial and dissipation ranges  is shown to be present for a simple, nonturbulent, onedimensional model, namely, the Burgers equation with hyperviscous dissipation. This bottleneck is shown to be the Fourierspace signature of oscillations in the realspace velocity. These oscillations are amenable to quantitative, analytical understanding, as we demonstrate by using boundarylayerexpansion techniques. Pseudospectral simulations are then used to show that such oscillatory features are also present in velocity correlation functions in one and threedimensional hyperviscous hydrodynamical models that display genuine turbulence.  

Kostya KHANIN  WPI, Seminarroom C 714  Mon, 28. May 12, 15:15 
Spacetime stationary solutions for the random forced Burgers equation  
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 spacetime 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.  

Andrei SOBOLEVSKI  WPI, Seminarroom C 714  Mon, 28. May 12, 16:00 
From particles to Burgers and beyond: some new random growth models  
We review two discrete random growth models whose formal continuous limits are related to PDEs. This talk is based on joint works with Sergei Nechaev and other colleagues [13].  
Note: References [1] K. Khanin, S. Nechaev, G. Oshanin, A. Sobolevski, O. Vasilyev, Phys. Rev. E 82, 061107 (2010) and arXiv:1006.4576 [2] J. Delon, J. Salomon, A. Sobolevski, J. Math. Sciences 181:6 (2012) 782791 and arXiv:1102.1558 [3] S. Nechaev, A. Sobolevski, O. Valba, arXiv:1203.3248  

Edriss TITI  WPI, Seminarroom C 714  Tue, 29. May 12, 9:30 
On the Loss of Regularity for the ThreeDimensional Euler Equations  
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 nongeneric, yet nontrivial, examples concerning the immediate loss of smoothness and illposedness of solutions of the threedimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the threedimensional Euler equations with vorticity that is having a nontrivial density concentrated on nonsmooth surface. This is very different from what has been proven for the twodimensional KelvinHelmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of nonregular solutions of the threedimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture. 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 nongeneric, yet nontrivial, examples concerning the immediate loss of smoothness and illposedness of solutions of the threedimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the threedimensional Euler equations with vorticity that is having a nontrivial density concentrated on nonsmooth surface. This is very different from what has been proven for the twodimensional KelvinHelmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of nonregular solutions of the threedimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.  

Gregory SEREGIN  WPI, Seminarroom C 714  Tue, 29. May 12, 10:45 
On a certain condition of a blow up for the NavierStokes equations  
We show that a necessary condition for $T$ to be a potential blow up time is that the spatial $L_3$ norm of the velocity goes to infinity as the time $t$ approaches $T$ from below.  

Carlo BOLDRIGHINI  WPI, Seminarroom C 714  Tue, 29. May 12, 11:30 
Simulating explosive solutions of hydrodynamic equations  
We present some results of computer simulations for the complexvalued solutions of the 2d 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.  

Marc BRACHET  WPI, Seminarroom C 714  Tue, 29. May 12, 14:15 
Interplay between the BealeKatoMajda theorem and the analyticitystrip method to investigate numerically the incompressible Euler singularity problem  
Numerical simulations of the incompressible Euler equations are performed using the TaylorGreen vortex initial conditions and resolutions up to 4096^3. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A wellresolved acceleration of the timedecay of the width of the analyticity strip is observed at the highestresolution for 3.7  

Walter PAULS  WPI, Seminarroom C 714  Tue, 29. May 12, 15:15 
Nelkin scaling for the Burgers equation and the role of highprecision calculation  
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 onedimensional 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 selfsimilarity 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 threedimensional NavierStokes simulations.  

Takeshi MATSUMOTO  WPI, Seminarroom C 714  Tue, 29. May 12, 16:00 
An attempt at a multiprecision spectral simulation of a threedimensional Euler flow  
A multiprecision software library enables the spectral method for a threedimensional Euler flow with, in principle, arbitrary high precision rather than the standard double precision. Such a numerical attempt is reported with emphasis on the shorttime behavior of the flow, starting from analytic initial data  

Gregory EYINK  WPI, Seminarroom C 714  Wed, 30. May 12, 9:30 
Spontaneous Stochasticity and Turbulent Magnetic Dynamo  
The usual notion of "magnetic fluxfreezing" breaks down in the Kazantsev dynamo model with only Hoelderinspace 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.  
Note: References: G. L. Eyink, Turbulent diffusion of lines and circulations, Phys. Lett. A 368 486‚490 (2007) G. L. Eyink, Stochastic fluxfreezing and magnetic dynamo, Phys. Rev. E. 83 056405 (2011) G. L. Eyink, Turbulent diffusion of lines and circulations, Phys. Lett. A 368 486‚490 (2007) G. L. Eyink, Stochastic fluxfreezing and magnetic dynamo, Phys. Rev. E. 83 056405 (2011)  

Gregory FALKOVICH  WPI, Seminarroom C 714  Wed, 30. May 12, 10:45 
Some new analytic results on Lagrangian statistics  
I will describe two new analytic derivations, one probably right, another probably wrong, done for the NavierStokes equation in Lagrangan coordinates. Mathematical insight into these derivations is called for.  

Marija VUCELJA  WPI, Seminarroom C 714  Wed, 30. May 12, 11:30 
Fractal contours of passive scalar in 2D random smooth flows  
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 semianalytic 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).  
Note: Coauthors: Gregory Falkovich and Konstantin S. Turitsyn  

Krzysztof GAWEDZKI  WPI, Seminarroom C 714  Thu, 31. May 12, 9:30 
2nd Law of Thermodynamics and Optimal Mass transport  
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 finitetime refinement of the 2nd Law of Thermodynamics. The refinement is related to the MongeKantorovich optimal mass transport and the underlying inviscid Burgers equation.  

Yann BRENIER  WPI, Seminarroom C 714  Thu, 31. May 12, 10:45 
Approximate geodesics on groups of volumepreserving diffeomorphisms and adhesion dynamics  
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 volumepreserving diffeomorphisms. In particular, we recover some dissipative solutions of the Zeldovich gravitational model.  

Vladimir ZHELIGOVSKII  WPI, Seminarroom C 714  Thu, 31. May 12, 11:30 
Optimal transport by omnipotential flow and cosmological reconstruction  
One of the simplest models used in studying the dynamics of largescale structure in cosmology, known as the Zeldovich approximation, is equivalent to the threedimensional 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 $t_1$ to their positions at any time $t_2\ge t_1$ is the gradient of a convex potential, a property we call omnipotentiality. 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 omnipotential flows with arbitrary smooth initial velocity. Mappings with a convex potential are known to be associated with the quadraticcost 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.  

Luca BIFERALE  WPI, Seminarroom C 714  Thu, 31. May 12, 14:15 
Turbulent dispersion from pointsources: corrections to the Richardson distribution  
We present a highstatistics numerical study of particle dispersion from pointsources in Homogeneous and Isotropic turbulence (HIT) at Reynolds number $Re \sim 300$. 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 (worstcase) or of particle pairs separating slower than usual (bestcase, i.e. particles which remain close for long time). A comparison with statistics collected in surrogate deltacorrelated velocity field at the same Reynolds numbers allow us to assess the importance of temporal correlations along particles trajectories.  

Jeremie BEC  WPI, Seminarroom C 714  Thu, 31. May 12, 15:15 
Mass fluctuations and diffusion in timedependent random environments  
A mass ejection model in a timedependent 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 nonsmooth scaleinvariant 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 powerlaw 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.  

Sergei NAZARENKO  WPI, Seminarroom C 714  Fri, 1. Jun 12, 9:30 
Turbulence in CharneyHasegawaMima model  
I will describe some analytical and numerical studies of turbulence in this model. The focus will be on socalled LHtransition feedback loop, in which turbulence forced at small scales generates a zonal flow via an anisotropic inverse cascade which then suppresses the smallscale turbulence, thereby eliminating anomalous transport.  

Anna POMYALOV  WPI, Seminarroom C 714  Fri, 1. Jun 12, 10:45 
Turbulence in noninteger dimensions by fractal Fourier decimation  
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 $k^D$ for large $k$. At the critical dimension $D_c=4/3$ there is an equilibrium Gibbs state with a $k^{5/3}$ spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002). Spectral simulations of fractally decimated twodimensional turbulence show that the inverse cascade persists below D = 2 with a rapidly rising Kolmogorov constant, likely to diverge as $(DD_c)^{2/3}$ .  

Samriddhi Sankar RAY  WPI, Seminarroom C 714  Fri, 1. Jun 12, 11:30 
Resonance Phenomena in the Galerkintruncated Burgers and Euler Equations  
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 shortwavelength oscillation, is caused by a resonant interaction between fluid particle motion and truncation waves generated by smallscale 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.  
Note: Coauthors: U. Frisch, S. Nazarenko, and T. Matsumoto  

Victor YAKHOT  WPI, Seminarroom C 714  Fri, 1. Jun 12, 14:15 
Oscillating particles in fluids. Theory and experiment in the entire range of frequency and pressure variation  
Oscillating nanoparticles (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 NavierStokes (diffusion) equations. It will be shown that when the Weissenberg number Wi>>1 this process becomes viscoelastic 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.  

Laszlo SZEKELYHIDI  WPI, Seminarroom C 714  Fri, 1. Jun 12, 15:15 
Dissipative continuous solutions of the Euler equations  
We construct Hoeldercontinuous 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 $C^1$ 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 H\"oldercontinuous solutions dissipating energy was conjectured by L. Onsager in 1949.  
Note: Coauthor: Camillo De Lellis  

Itamar PROCACCIA  WPI, Seminarroom C 714  Sat, 2. Jun 12, 9:30 
Universal Plasticity in Amorphous Solids with Implications to the Glass Transition.  
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 elastoplastic 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.  

Matania BENARTZI  WPI, Seminarroom C 714  Sat, 2. Jun 12, 10:45 
On 2D flows with rough initial data  
The purpose of the talk is twofold:  

Dong LI and Yasha SINAI  WPI, Seminarroom C 714  Sat, 2. Jun 12, 11:30 
Bifurcation of solutions to equations in fluid dynamics  
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 NavierStokes system and 2D quasigeostrophic equations  

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