We consider the Cauchy problem of 2 dimensional incompressible Euler equations. We study the problem of growth rate of high Sobolev norms of the solutions. A well-known result that is implicit in [J. T. Beals ,T. Kato and A. Majda, Comm. Math. Phys. 94 (1984), 61-66] is the double exponential rate. An open question that has been asked by Tao is wether we can get an exponential rate. In this pap...

One of the simplest and, at the same time, most important invariants of a topological space is the Euler characteristic. A generalization of the notion of the Euler characteristic to the equivariant setting, that is, to spaces with an action of a group (say, finite) is far from unique. An equivariant analogue of the Euler characteristic can be defined as an element of the ring of representation...

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on ...

Recently an implicit method has been presented for solving first order singular initial value problem. The method is extended to solve second or higher order problems having a singular point. The method presents more correct result than those obtained by the implicit Euler and second order implicit Runge-Kutta (RK2) methods. The method is illustrated by suitable examples.

The behavior of difference approximations of hyperbolic partial differential equations as time t -* oo is studied. The rate of convergence to steady state is analyzed theoretically and expe ¡mentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady state calcul...

We consider the long time behavior of heterogeneously interacting diffusive particle systems and their large population limit. The interaction is mean field type with weights characterized by an underlying graphon. limit given a graphon system consisting independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable assumptions, including ce...

The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. This p...