In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of finite-time b
low-up solutions in terms of the initial data, and describe the asymptotic behavior of the solutions near blow up times. We also identify a rich set of the initial data which yields global bounded solutions.
The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original
equation using the solutions of the subproblems. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.
The FENE dumbbell model consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation for the polymer distribution. In such a model, the polymer elongation cannot exceed a limit $sqrt{b}$, yielding all interesting features near
the boundary. In this paper we establish the local well-posedness for the FENE dumbbell model under a class of Dirichlet-type boundary conditions dictated by the parameter $b$. As a result, for each $b>0$ we identify a sharp boundary requirement for the underlying density distribution, while the sharpness follows from the existence result for each specification of the boundary behavior. It is shown that the probability density governed by the Fokker-Planck equation approaches zero near boundary, necessarily faster than the distance function $d$ for $b>2$, faster than $d|ln d|$ for $b=2$, and as fast as $d^{b/2}$ for $0<b<2$. Moreover, the sharp boundary requirement for $bgeq 2$ is also sufficient for the distribution to remain a probability density.
We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement. The well-posedness of the FENE model that consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensiv
ely, mostly with the zero flux boundary condition. Recently it was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl. Math., 68(5):1304--1315] that any preassigned boundary value of a weighted distribution will become redundant once the non-dimensional parameter $b>2$. In this article, we show that for the well-posedness of the microscopic FENE model ($b>2$) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function. Under this condition, it is shown that there exists a unique weak solution in a weighted Sobolev space. Moreover, such a condition still ensures that the distribution is a probability density. The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as the distance function.
