We consider the equation $u_t+(u^nu_{xxx})_x=0$ with $2\le n <3$ and establish an estimate for the finite speed of propagation of the support of compactly supported nonnegative solutions. The estimate depends only on the $L^1$-norm and is valid a posteriorifor strong solutions obtained through a Bernis-Friedman regularization.

This paper studies various qualitative properties, such as exponential stability, spectrum determining growth property, differentiability, of Gevrey class and analyticity, for the semigroup $e^{\mathcal{At}}$ generated by the operator of the form $\mathcal{A}=\begin{pmatrix} -A_0 & {B}\cr {C}& {-A_1} \end{pmatrix}$, where both $-A_0$ and $-A_1$ generate contraction semigroups on some Hilbert spaces, and $B$ and $C$ are certain closed densely defined linear operators.

The author studies the modified Stefan problem in the plane with surface tension and kinetic undercooling when the interfacial curve is a polygon. Existence of local-in-time solutions is shown. Geometric properties of the flow are studied if the Wulff shape is a regular $N$-sided polygon. The author shows that an initial interface being a scaled Wulff shape with sufficiently small perimeter shrinks to a point. Moreover, at each time the interface remains a scaled Wulff shape.

In this paper we derive the quasi-geostrophic potential vorticity equations, starting from primitive type equations, as the Rossby number goes to zero, and prove the convergence as long as the limit system has strong enough solutions. We in particular take account of the boundary layers and investigate the existence of global weak solutions of the limit system with physical boundary conditions.