L2-flow of elastic curves with clamped boundary conditions

In this paper we investigate the $L^2$-flow of elastic curves with clamped boundary conditions in n-dimensional Euclidean spaces. The $L^2$-flow corresponds to a fourth-order parabolic equation. In the case of closed curves, the long-time existence of solutions of this evolution equation has been derived in the literature. We extend this result to the case of open (i.e., non-closed) curves with clamped boundary conditions.     

Lower bounds for blow-up in a model of chemotaxis

Many special cases of the classical Keller–Segel system for modeling chemotaxis have been investigated in the literature, and typically the solution of the governing equations will blow up at some finite time. However, the question of establishing lower bounds for this blow-up time has been largely ignored. This paper derives such a lower bound in a parabolic–parabolic model in both ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

On the nonexistence of parabolic boundary points of certain domains in C2

In this paper, the nonexistence of parabolic boundary points of infinite type of certain domains in ${\mathbb{C}}^2$ is showed.