Wednesdays at 4pm in Eckhart 202.
Recently, Chen-Strain-Tsai-Yau and Koch-Nadirashvili-Seregin-Sverak proved that axially simmetric solutions to the Cauchy problem for the Navier-Stokes equations have no Type I singularity. In this talk, the corresponding local version will be discussed.
We consider the distribution of a long homopolymer in a potential field. The typical shape of the polymer depends on the size of the potential. We shall discuss various phenomena when the temperature parameter is at or near the critical value. Mathematically, the problems concern the study of spectral properties of certain elliptic differential operators.
The talk is concerned with qualitative properties of pulsating travelling fronts in periodic media, for monostable reaction-diffusion equations. First, the uniqueness of pulsating fronts for a given speed is established for Kolmogorov-Petrovsky-Piskunov nonlinearities. These results provide in particular a complete classification of KPP pulsating fronts. To do so, the main tool is to prove the exponential behavior of a front when it approches its unstable limiting state. In the general monostable case, the logarithmic equivalent of the fronts is proved. Further qualitative properties will be discussed: stability of pulsating fronts and a homogenization limit. This talk is based on joint works with L. Roques and M. El Smaily.
We shall discuss statistical properties of global solutions to the random forced Burgers equation. The problem is closely related to analysis of minimizers for random time-dependent Lagrangian systems. We show that for such systems on compact manifolds there exists a unique global minimizer. In the one-dimensional case this global minimizer corresponds is a hyperbolic trajectory of the random Lagrangian flow.
We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the differences between the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.
This is a self-contained introduction to the theory and computation of convex measures of financial risk, which has been a very active area of Financial Mathematics, with a rich history in a short number of years. The axioms specify sensible properties that measures of risk should possess (and which the industry's favourite, value-at-risk, does not). The most common example is related to the expectation of an exponential utility function. A basic application is hedging, that is taking off-setting positions, to optimally reduce the risk measure of a portfolio. In standard continuous-time models with dynamic hedging, this leads to nonlinear PDE problems of HJB type. We discuss so-called static-dynamic hedging of exotic options under convex risk measures, and specifically the existence and uniqueness of an optimal position. We illustrate the computational challenge when we move away from the risk measure associated with exponential utility. Joint work with Aytac Ilhan (Goldman Sachs) and Mattias Jonsson (University of Michigan).
We will analyze the incoherent waves reflected by a random medium in the case in which the medium has three-dimensional rapid random fluctuations and one-dimensional slow variations. We will show that the second-order statistics (cross correlations) of the reflected waves are determined by the slow spatial variations of the background velocity, the scattering coefficient and the absorption coefficient of the medium via a system of transport equations. By observing the reflected waves, it is possible to invert this system and to reconstruct the parameters of the medium. This is a joint work with Knut Solna (University of California at Irvine).
In this talk we first discuss some refinements of the nonexistence results on the asymptotically self-similar blow-ups for the 3D Euler equations. Then, I present new type of inequalities for the solutions derived, using a particular form of functional similarity transform. Finally we discuss observations of complex Riccati structure in the axisymmetric 3D Euler equations.
Due to a supercritical nature of the 3D Navier-Stokes equations, the best known estimates on the enstrophy growth rate do not rule out the existence of finite time singularities. Recently Doering and Lu numerically showed that these estimates are sharp. In this talk I will present some analytical results in this direction. This is a joint work with C. R. Doering and C. Foias.
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