Wednesdays at 4 PM in Eckhart 206.
The long-time asymptotics are determined for fast nonlinear diffusion, by linearizing Otto's gradient descent model at the Barenblatt profile. The spectrum of the entropy is explicitly determined. The dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied.
I will discuss several $\mathbf{BV}$ and $\mathbf L^1$ stability results for wave patterns arising from the strictly hyperbolic systems of conservation laws. The proposed methods apply to patterns of non-interacting shock and rarefaction waves, generated as solutions to the one-dimensional Riemann and Cauchy problems without any restriction on their amplitude. In particular, I will focus on the analysis of a single strong rarefaction wave.
Episodic memory can be expressed as a memory of causal sequence of individual experience. Neurophysiological studies clarified that the hippocampal formation is a responsible organ for episodic memory. The discovery of chaotic itinerancy in high-dimensional dynamical systems has motivated a new interpretation of this dynamic neural activity, cast in terms of the high-dimensional transitory dynamics among attractor ruins. Based on the intensive studies, a hypothesis regarding the formation of episodic memory is given.
In this talk we consider the conditioning properties of classical perturbative methods for electromagnetic and acoustic scattering, and discuss stabilized alternatives. One of our main observations is that classical shape-perturbation methods rely heavily on significant cancellations for their convergence resulting in ill-conditioned numerics and preventing a straightforward convergence proof. At low- (first- and second-) order we advocate the explicit re-writing of the classical recursions to identify and eliminate the cancellations. At high-order we use the knowledge that a change of independent variables allows a direct demonstration of convergence, and design a stable numerical method based on the resulting equations. We conclude with an algorithm which mimics the stabilizing features of our high-order algorithm while retaining the dimension-reducing properties of classical approaches.
For $2D$ or $3D$ Navier--Stokes equations defined in a bounded domain $\Omega$ we study stabilization of solution near a given steady-state flow $\hat v(x)$ by means of feedback control defined on a part $\Gamma$ of boundary $\partial \Omega$ . New mathematical for\-ma\-li\-za\-tion of feedback notion is proposed. With its help for a prescribed number $\sigma >>0$ and for an initial condition $v_0(x)$ placed in a small neighborhood of $\hat v(x)$ a control $u(t,x'),\; x\in \Gamma$, is constructed such that solution $v(t,x)$ of obtained boundary value problem for 2D Navier--Stokes equations satisfies the inequality: $\|v(t,\cdot)-\hat v\|_{H^1}\le ce^{-\sigma t}\quad \mbox{for}\; t\ge 0.$ To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations. Besides,we introduce a notion of real process that is an abstract analog of approximate solution obtained as a result of numerical calculations. Investigation a structure of real process gives us a possibility to construct feedback stabilization from the boundary $\partial \Omega$ for it and to obtain an estimate for stabilized real process. The result of these investigation is construction of feedback control from the boundary such that it stabilizes solution of 3D Navier-Stokes equations. Moreover, it can react on unpredictable fluctuations of solution when they arise, damping them.
A class of Godunov-type schemes for solving scalar conservation laws will be considered. There are two main steps in such schemes: evolution and projection. In the original Godunov scheme, the projection is onto piecewise constant functions -- the cell averages. In the general Godunov-type method, the projection is onto piecewise polynomials. Many well known methods are nonoscillatory, however, nonoscillation is, in general, not sufficient to prove convergence of such methods to the entropy solution or derive error estimates. For example, MinMod, UNO, ENO, and WENO methods are known to be numerically robust, at least for piecewise smooth initial data, but theoretical results about convergence are still missing. The notion of weakly nonoscillatory schemes (WNO) will be introduced. For example, any Godunov-type scheme with nonoscillatory evolution and projection is WNO. The main result is a convergence theorem and an error estimate for a subclass of WNO schemes which includes simple modifications of MinMod and UNO. In the case of a linear flux, new stability results will be derived. Based on that, it will be shown that the modified MinMod scheme coincides with the original MinMod scheme, and the error estimate can be improved in this case.
The Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation is a classical model used to describe the evolution of a spatially distributed population with local logistic (growth-saturation) dynamics and diffusive spreading. It is a `mean-field' model in the sense that all discreteness and noise effects are neglected. In this talk we describe a rigorous connection between a stochastic FKPP partial differential equation with a particular form of multiplicative noise and a single species birth- coalescence reaction-diffusion particle system. The correspondence is not in terms of a fluctuating hydrodynamic description for the reaction-diffusion model, but rather via the concept of `duality', an idea that has played a major role in the probabilistic analysis of interacting particle systems in recent decades. The idea of duality will be discussed and used to derive an exact formula for the extinction probability of any initial configuration for the stochastic FKPP equation. Duality will also be used to exploit the connection between the diffusion-limited birth-coalescence process and the strong-noise limit of the stochastic FKPP equation to determine the effect of high noise levels on the propagation speed of a wavefront in this stochastic pde. This is joint work with Carl Mueller (University of Rochester) and Peter Smereka (University of Michigan).
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