Birkhoff billiard is a model problem in Hamiltonian dynamics. Periodic orbits in Hamiltonian systems are second in importance only to the rest points, which are absent in billiards. We will describe recent results on periodic orbits in convex billiards along with some applications. In particular, we will describe the construction of nonintegrable billiards possessing a continuous family of periodic orbits.
The purpose of this work is to locate localized damage in a structure with distributed sensors. Given a configuration of transducers, we assume that a full response matrix for the healthy structure is known. It is used as a basis for comparison with the response matrix that is recorded when there is damage. We have carried out a numerical experiment with the wave equation in two dimensions. The healthy structure is a domain containing many scatterers. We want to image two point-like defects with the help of 12 sensors regularly distributed. Because of the complexity of the environment, the traces have a lot of delay spread and travel time migration does not work well. Instead, the traces are back propagated numerically in the medium, assuming that we have some knowledge of the background. Since the time at which the back propagated field will focus on the targets is unknown, we compute the Shannon entropy of the image and pick the time where it is minimal. The TV norm proves also a good indicator. This works well for distributed sensors networks because the information will dramatically reduce at the time of refocusing. When there are several defects, the Singular Value Decomposition of the response matrix is performed at each frequency to resolve each of them. This optimally compensated time-reversal imaging algorithm gives good and stable results.
I will show how small stochastic perturbations of dynamical systems can lead to not random stable oscillations and equilibriums, which are not available in the system without these perturbations. If the system is generic, for any initial point and a time scale, one can point out a stable attractor (metastable state) such that the perturbed system spends most of the time (in the given time scale) near this attractor. Bifurcations in the metastable states lead to stochastic resonance. This effects are manifestations of laws of large deviations.
The understanding of scale interactions for the incompressible Euler and Navier-Stokes equations has been a major challenge. Here I will present a new multiscale analysis for the 3D incompressible Euler equation with rapidly oscillating initial data. We first present a multiscale analysis based on the Lagrangian formulation. By using a Lagrangian description, we can characterize the nonlinear convection of small scales exactly and turn a convection dominated transport problem into an elliptic problem for the stream function. At the end, we derive a coupled multiscale system for the flow map and the stream function, which is well-posed. Based on our understanding in the Lagrangian formulation, we derive a similar multiscale analysis using the Eulerian formulation, which is more effective for computational purpose. Our multiscale analysis reveals some interesting structures of the Reynolds stress terms and provide a theoretical guidance in developing a systematic multiscale modeling of incompressible flow. Numerical results will be presented to demonstrate the accuracy and the robustness of the multiscale method.
It is well-known that, in the diffusive scaling, the first order expansion of the Boltzmann equation gives rise to the celebrated incompressible Navier-Stokes-Fourier system. We establish the validity of such a diffusive expansion up to any order for all time near a Maxwellian. In particular, our results lead to error estimates for the incompressible Navier-Stokes-Fourier approximation.
When estimating solutions of dissipative partial differential equations in $L^p$-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian $(-\Delta)^\alpha$, the approach of integration by parts no longer applies. In a recent work, we obtain a lower bound for the integral involving $(-\Delta)^\alpha$ by combining a pointwise inequality for $(-\Delta)^\alpha$ with the Bernstein inequality for fractional derivatives. As an application of this lower bound, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations refer to the resulting equations by replacing $-\Delta$ of the Navier-Stokes equations by $(-\Delta)^\alpha$.
Many problems in physics, material sciences, chemistry and biology can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well-known examples include phase transitions of condensed matter, conformational changes of biopolymers, and chemical reactions. The state of these systems is confined for long periods of time in metastable regions in configuration space and only rarely switches from one region to another. The separation of time scale is a consequence of the disparity between the effective thermal energy and typical energy barrier in these systems, and their dynamics effectively reduces to a Markov chain on the metastables regions. The analysis and computation the transition pathways and rates between the metastable states is a major computational challenge, especially when the energy landscape exhibits multiscale features. I will review recent work done by scientists from several disciplines on probing such energy landscapes, introduce concepts such as reaction coordinate and free energy, and show how these concepts can be made mathematically precise. I will then present a new method, the string method, that has proven to be very effective for some complex systems in material science and chemistry.
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