Wednesdays at 4pm in Eckhart 202.
The talk will investigate the interactions of circadian rhythms and cell division cycle. This is motivated by chronotherapeutics, which aims at optimizing the therapy delivery hour. We will show how entropy principles allows to prove the relaxation to periodic solution and how growth rates of tumor cells (without circadian control) and healthy cells (subject to circadian control) can be compared. In other words we give a comparison between Floquet and Perron eigenvalues of general positive operators.
We discuss the phenomenon of anomalous transport in quantum mechanical systems displaying intermediate disorder. As a specific example, we consider the Fibonacci Hamiltonian and show how the intermediate disorder manifests itself on many levels, such as critical eigenstates, purely singular continuous spectral measures, as well as fractional spectral dimensions and transport rates.
It is well known from experiments that inviscid fluid motion leads to creation of small scale structures for many smooth initial data. Nevertheless, there are few rigorous quantitative estimates reflecting this tendency of the solutions to classical fluid dynamics equations. We describe an explicit class of initial data for which higher order Sobolev norms of solutions can be shown to exhibit significant growth. The proof is based on the "energy pump" estimate that controls energy transfer to higher modes for this class of solutions.
Geologic processes occur on time scales that introduce non-standard rheologies. In particular, magma migration is modeled as a poro-viscous flow. We will see that such models lead to nonlinear, nonlocal, dispersive wave equations with the potential for degeneracy. This talk will discuss the well-posedness of these equations, the stability of their solitary waves, and associated open problems.
The problem under study is that of a function having zero fractional laplacian on one side of an unknown surface, being zero on the other side, and having a prescribed growth in the vicinity of the surface. This model can be interpreted as the singular limit of a boundary chemical reaction model. Due to the non-locality of the fractional laplacian, rescalings are non trivial operations. We will explain how an extension formula, discovered by Caffarelli and Silvestre, can help us retrieve some local properties of the free boundary: optimal regularity, nondegeneracy, smoothness. Joint work with L. Caffarelli and Y. Sire.
We formulate a criterion for the existence, uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, the weak-* ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting with any initial distribution, is established. The principal assumptions are the lower bound of the ergodic averages of the transition probability function and the e-property of the semigroup. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. We use the weak-* mean ergodicity of the respective invariant probability measure to derive the law of large numbers for the trajectory of the passive tracer. This is a joint work with Sz. Peszat (IM PAN, Cracow) and T. Szarek (Silesian University, Katowice)
We study nonlinear integro-differential equations. Typical examples are the ones that arise from stochastic control problems with discontinuous Levy processes. We can think of these as nonlinear equations of fractional order. Indeed, second order elliptic PDEs are limit cases for integro-differential equations. Our aim is to extend the theory of fully nonlinear elliptic equations to this class of equations. We are able to obtain a result analogous to the Alexandroff estimate, Harnack inequality and C^{1,\alpha} regularity. As the order of the equation approaches two, in the limit our estimates become the usual regularity estimates for second order elliptic pdes. This is a joint work with Luis Caffarelli.
In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics. To better understand these phenomena we study numerically the 2D Oldroyd-B Viscoelastic model at low Reynolds number. A background force is used to create a periodic cell with four-roll mill vertical structure around a hyperbolic fixed point. We consider both steady and time-periodic forcing. For low Weissenberg number (Wi) the elastic stresses are bounded and slave to the forcing, with mixing confined to small sets near the hyperbolic point. At larger Wi an analog to the coil-stretch transition occurs yielding large stresses and stress gradients concentrated on sets of small measure, perhaps indicating the development of singularities. The flow then becomes very sensitive to perturbations in the forcing and there is a transition to global mixing in the fluid.
In their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on a line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of a propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. In this talk I will show that actually in the problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the considered equation is a gradient flow in an exponentially weighted L^2 space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. I will show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a "pulled" front, or a "pushed" front. The choice of the scenario is completely characterized via a minimization problem.
In this talk, we show how the third derivatives of solutions to the 3D Navier-Stokes equations can be bound in weak L^1. The proof uses blow-up techniques and relies on a non-linear scaling of the dissipation of energy. Estimates can be obtained by this means thanks to the Galilean invariance of the transport part of the equation.
I review various aspects of current research, both experimental and theoretical, on stirring and mixing in fluids. Three main threads are followed: 1) How topological features influence mixing effectiveness; 2) How this leads to novel optimization methods; and 3) How one has to be mindful of wall effects, which can dramatically slow down mixing.
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