Wednesdays at 3:30 PM in Eckhart 202.
Existence theory of global-in-time energy-level weak solutions to a nonlinear fluid-structure interaction model governed by the 3D Navier-Stokes equations coupled with the linear elastic wave equation will be presented. The interaction takes place via an interface--the boundary of the elastic solid immersed in the fluid--and is realized through the continuity of both the velocities and the normal components of the stress tensors across the interface. The essential difficulty is that traces of the elastic component are apriori not well-defined in the setting of the energy-level spaces [the classical trace theory is insufficient], and only after a careful microlocal analysis argument a natural variational formulation is possible. In addition, the aforementioned solutions are shown to be smooth assuming a natural compatibility condition and smooth data.
The Keller-Segel system is a nonlinear advection-diffusion system describing the collective motion of cells under the effect of chemoattraction. Several forms of the system have been proposed depending on the modeling context and are characterized by the blow-up in finite time of the solution to a Dirac mass (aggregation of cells). Several observations, on different scales, have lead to consider other models as kinetic equations and zero diffusion limit with chorum sensing. These models rise new mathematic al questions.
We consider divergence form elliptic operators with bounded ($L^\infty$) coefficients. Although solutions of these operators are only Hölder-continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium is characterized by a continuum of scales. Next we consider the inverse homogenization of these operators which is known to be a non-linear ill posed problem. We show how this problem can be transformed into the search of an optimal solution within a linear space by representing (effective) conductivities as curvatures of (volume averaged) concave functions. (Parts of this talk are joint works with Lei Zhang, Roger Donaldson, Mathieu Desbrun and Yiying Tong)
Consider the planar 3 body problem consisting of large, small, and very small body,e.g. a Sun-Jupiter-Asteroid system. Assume that Asteroid has mass zero. During the talk we discuss analyze variety of unstable motions for this problem. Motions is called oscillatory if as time tends to infinity limsup (resp. liminf) of distance to the origin is infinite (resp. finite). A long standing conjecture by Kolmogorov: oscillatory motions form a set of measure zero. Jointly with A. Gorodetski we show that these motions form a set of the full Hausdorff dimension. Then we discuss phenomenon of Arnold diffusion for this problem, proved jointly with A. Delshams and T. Seara, and existence of nonlocal instabilities proved using Aubry-Mather theory. The latter is related to Herman's oldest open question in dynamical systems.
The flows of 2-dimensional ideal incompressible fluid look paradoxically and counterintuitive. Their peculiarity is loosely described by the term Inverse Cascade. This means that the energy is transferred from small to large scales, eventually accumulating in a large-scale steady flow. Such behaviour has been observed in numerous physical experiments and computer simulations, but is not properly understood. The existence of the inverse cascade is in an apparent contradiction with the time reversibility of the Euler equations, as well as with intuitive considerations of the entropy growth. In fact, closer study shows that there is no contradiction. This problem has many sides, and is connected with interesting questions of analysis, geometry, dynamical systems, nonequilibrium statistical physics and many other. The talk is self-contained and does not require special preliminary knowledge.
The need to take stochastic effects into account for modeling complex systems has now become widely recognized. Stochastic partial differential equations arise naturally as mathematical models for multiscale systems under random influences. We consider macroscopic dynamics of microscopic systems described by stochastic partial differential equations. The microscopic systems are characterized by small scale heterogeneities (spatial domain with small holes or oscillating coefficients), fast scale boundary impact (random dynamic boundary condition), and, random fluctuations. An effective macroscopic model for such a stochastic microscopic system is derived. The homogenized effective model is still a stochastic partial differential equation, but defined on a unified spatial domain and the random impact is represented by an extra term in the effective model. The solutions of the microscopic model is shown to converge to those of the effective macroscopic model in probability distribution, as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of convergence in probability distribution, and in the sense of convergence in energy are also proved.
We have recently found a way to describe large-scale brain activity in terms of non-equilibrium statistical mechanics. This allows us to calculate (perturbatively) the effects of fluctuations and correlations on brain activity. Major results of this formulation include a role for critical branching, and the demonstration that there exist non-equilibrium phase transitions in brain activity which are in the same universality class as directed percolation. This result leads to explanations for the origin of many of the scaling laws found in brain measurements of Local Field Potentials, Electroencephalograms, fMRI BOLD signals, and in Interspike Interval Distributions, and provides a possible explanation for the origin of alpha, beta, gamma, delta and theta brain waves. It also leads to ways of calculating how correlations can affect neocortical activity, and therefore provides a new tool for investigating the connections between brain dynamics, cognition and behavior.
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