Wednesdays at 4 PM in Eckhart 202.
We consider a model of non Newtonian fluids: a grade-two fluid, that generalizes the Navier-Stokes equations. According to recent works of Holm, Marsden and Ratiu, it can also be interpreted as a model of turbulence. In two dimensions, it has a global solution without restriction on the size of the data, as in the case of the Navier-Stokes equations. This valuable property is due to the fact that, when the problem is written in a suitable form, solutions can be constructed without requiring a $W^{1,\infty}$ estimate for the velocity. By discretizing the grade-two fluid model with appropriate finite-elements, this property may be preserved. We present here some finite-element schemes for which we prove existence of discrete solutions, their strong convergence, and convergence of a simple algorithm for computing these solutions, without restriction on the size of the data or the shape of the domain. We present a brief comparison with the numerical analysis of general Oldroyd B models, which is much more difficult. The structure of the problem is such that its analysis requires at least a $W^{1,\infty}$ estimate for the velocity. As a consequence, when this problem is discretized, its numerical analysis also requires a uniform $W^{1,\infty}$ a priori estimate for the discrete velocity. This result is difficult and appears to be an open problem.
The singular perturbation of the potential energy $\int(1-u^2)^2$ by $\epsilon^2 |\nabla u|^2$ is a classical model for phase transitions. The extension of this problem from scalar fields $u$ to gradient vector fields has until recently resisted analysis. In this talk we will present some recent developments in the analysis of weak solutions for the variational PDE's associated to this problem. We will also discuss a further extension of this problem to variations over fields which are only locally gradient (director fields) which has relevance to the modelling of pattern formation far from threshold.
We investigate a Boltzmann equation for inelastic scattering in which the relative velocity in the collision frequency is approximated by the thermal speed. We study the homogeneous regime using Fourier analysis methods. We analyze the existence and uniqueness questions, linearized operator around Dirac delta function, self-similar solutions and moment equations. We study the conditions under which self-similar solutions describe the asymptotic behavior of the homogeneous equation. In addition, we obtain formally a hydrodynamic description for near elastic particles under the assumption of constant and variable restitution coefficient. We describe the linear long-wave stability/instability for homogeneous cooling states. Finally, in the case of driven granular media in which particles perform inelastic collisions among themselves while they follow Brownian dynamics in between collisions, we analyze the homogeneous regime showing that existence of homogeneous steady states is plausible. We approximate the steady states by a small energy dissipation expansion showing they are given by a Maxwellian distribution corrected by a Sonine polynomials depending on the order in the expansion parameter.
Although a linear problem, the dispersion of a passive scalar in a steady cellular flow becomes a rather non trivial question in the large Peclet number limit. The final result is an effective diffusion coefficient of order Dm ( molecular diffision) times Pe^(1/2). A rather natural extension of this work is to look at non linear reaction diffusion in the same limit Pe tending to infinity. Borrowing ideas from the passive scalar problem, I'll show that the reaction diffusion zone ( a rather complex geometrical object now) moves on at constant speed of order C Pe^(1/4) in the large Pe limit, C front velocity without fluid motion.
Wavy dynamics of thin films with moderate Reynolds number shows a remarkable feature: there appears to be a natural wavelength between solitary-like pulses far downstream. This wavelength is much larger than the average film thickness, and, for long systems, does not depend on system size. The phenomenon is robust, resembles the inverse energy cascade in turbulence, and probably is the most interesting in the interfacial dynamics of thin films. A new simple approach is proposed for the description of the interfacial dynamics in film flows, which captures such an inverse cascade. The essence of the method is in combination of the pertinent dispersion relation, evaluated exactly by numerics, with the simplest quadratic nonlinearity. It turns out that the exact dispersion relation differs dramatically from the conventional asymptotic dispersion relations (e.g., in KS equation). This is the reason why the conventional evolution equations were not successful in description of the wavy film dynamics for moderate Reynolds numbers.
This talk is a review of the definitive progress that has been made recently on turbulent transport and Burgers turbulence. We will discuss issues on asymptotic behavior of PDFs for velocity and velocity gradients in Burgers turbulence, intermittency in Kraichnan's model of passive scalar transport, Prandtl number effect on the scaling of structure functions and the geometry of the passive scalar field.
Recently methods have been developed with permit one to rigorously justify the use of ``amplitude'' or ``modulation'' equations that arise in a wide variety of physical contexts. In particular, we have recently shown that over the time and length scales commonly used to derive long-wave equations for fluid surfaces, the irrotational motion of an incompressible, inviscid fluid of finite depth can be described by a pair of {\em uncoupled} Korteweg-de Vries equations. In this talk I will review this result, describe the general method of proof, and apply this method in another context To show that in an appropriate scaling, the motion of the Fermi-Pasta-Ulam model of coupled, nonlinear oscillators can also be approximated by a pair of uncoupled KdV equations. This is joint work with Guido Schneider of the University of Bayreuth.
Abstract .
Focusing on the sharpening and dynamics of orientation selectivity, we describe the construction and response properties of a network model of layer 4C in Macaque primary visual cortex (Area V1). The model consists of a large number of integrate-and-fire, conductance based point neurons, both excitatory and inhibitory, which represent dynamics in a small patch of 4C - 1 mm^2$ in lateral area -- which contains four orientation hypercolumns. Convergent feed-forward input sets up a slight orientation preference, and recurrent cortical connections cause the network to sharpen its selectivity. The physiological properties and coupling architectures of the model are constrained by experimental data on layer 4C$\alpha$ of Macaque. In particular, the pattern of local lateral connections is taken as isotropic, with the spatial range of monosynaptic excitation exceeding that of inhibition. The model (i) obtains sharpening, diversity in response, and dynamics of orientation selectivity, each in qualitative agreement with experimental observations of 4C; (ii) predicts more sharpening near orientation pinwheel singularities, than away from them, with inhibitory neurons on average more broadly turned than excitatory; (iii) clarifies mechanisms for the sharpening and dynamics of selectivity; and (iv) suggests new experiments.
In the modelling of electrons as quantum mechanical particles several types of NLS arise. The first class are "weakly nonlinear" models such as the Schroedinger-Poisson equation. Relativistic extensions of this model are given by the Dirac-Maxwell system and selfconsistent Pauli equations. Another nonlinearity arises in the context of the Hartree-Fock equations. It can be approximated by a local function of the density similar to the focusing case of the cubic NLS. We present the models and the open mathematical problems as well as recent progress e.g. on the limit from Schroedinger- Poisson in a crystal to the semiclassical equations of solid state physics using a new variant of Wigner transforms.
The incorporation of dissipative effects is often crucial in obtaining good agreement between experimental observations and the prediction of theoretical models describing the propagation of waves in nonlinear dispersive media. To take account of dissipative mechanisms, a Burgers-type term is often appended in these models. The effects of such a term is partially revealed in the investigation of the zero-dissipation limit for these models. In this talk we report uniform bounds (independent of time and viscosity) and zero-dissipation limit results for the BBM-Burgers, Korteweg-de Vries-Burgers, and a more general class of model equations. This is joint work with Jerry Bona.
Transport equations are used to describe the propagation of high frequency waves in heterogeneous media. The transport model can further be simplified in highly scattering media. There, the transport equation posed in phase space is replaced by its diffusion approximation, posed in the physical domain. In the vicinity of clear regions, such as embedded objects and clear layers, the diffusion regime breaks down. This hampers the use of the diffusion approximation. We propose an alternative diffusion model with suitable non-local boundary conditions at the inclusions boundary that takes into account both the diffusive and non-diffusive regions. Asymptotic convergence of the transport solution to the modified diffusion solution is shown for different types of inclusions, including clear layers. Numerical experiments evidence the accuracy of the novel model in the setting of straight clear layers. Applications include the modeling of clear layers in Near-Infra-Red spectroscopy, which is increasingly used in medical imaging for monitoring certain properties of human tissues. A similar theory may also be used to quantify the distribution of neutrinos in the explosion of type II supernovae. Rayleigh-Taylor instabilities create zones of large and low densities. The latter could be seen as some clear layers where the diffusion does not hold, and could be treated by some modified diffusion equations. Interaction with astrophysicists is clearly needed here.
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