The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric fluids

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.

Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations

This paper is concerned with a class of nonlocal dispersive models -- the $theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {it Acta Math. Appl. Sin.} Engl. Ser. {bf 24}(3)(2008)423--440]: $$ (1-partial_x^2)u_t+(1-thetapartial_x^2)(frac{u^2}{2})_x =(1-4theta)(frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $theta=1/3$, and the Degasperis-Procesi equation, $theta=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $theta in mathbb{R}$. It is shown that as $theta$ increases regularity of solutions improves: (i) $0 <theta < 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 leq theta < 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} leq theta leq 1$ and $theta=frac{2n}{2n-1}, nin mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $thetain mathbb{R}$ global smoothness of the corresponding solution is proved. Proofs are either based on the use of some global invariants or based on exploration of favorable sign conditions of quantities involving solution derivatives. Existence and uniqueness results of global weak solutions for any $theta in mathbb{R}$ are also presented. For some restricted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {it J. reine angew. Math.}, {bf 624} (2008)51--80.]

Global Well-Posedness for the Microscopic FENE Model with a Sharp Boundary Condition

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 intensively, 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.