We consider a blow-up of smooth, local-in-time solutions of the 3-D incompressible Euler equations. We give a localized analogy of the Beale-Kato-Majda-type criterion that if the solution blows up in an isolated set in our sense, the blow-up is carried with the blow-up of vorticity in that set. Besides, we show that in general the blow-up process is controlled by a suitable norm of any two components of vorticity in Cartesian coordinates.

We use dynamical-systems techniques to study the positive solutions of the scalar curvature equation in $R^n$, $n \geq 3$. In certain cases we prove the existence of a Cantor-like set of singular ground states with slow decay.

We will show that if $u$ is the solution of $u_t=\Delta$ log $u$, $u>0$, in $\Omega\times (0,\infty)$, $u=c_1$ on $\partial\Omega\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ on $\Omega\subset R^n$ where $\Omega$ is a smooth convex bounded domain, then for $c_1=\infty$ the rescaled function $w=$log $(u/t)$ will converge uniformly on every compact subset of $\Omega$ to the unique solution $\psi$ of the equation $\Delta\psi-e^{\psi}=0$, $\psi >0$, in $\Omega$ with $\psi=\infty$ on $\partial\Omega$ as $t\to\infty$. When $0 <c_1 <\infty$, $n=1,2$, or $3$, and $u_0\ge c_1$ on $\Omega$, then the function $w=e^{(\lambda_1/c_1)t}$log $(u/c_1)$ will converge uniformly on $\overline{\Omega}$ to $A\phi_1$ as $t\to\infty$ where $\lambda_1>0$ and $\phi_1$ are the first positive eigenvalue and positive eigenfunction of the Laplace operator $-\Delta$ on $\Omega$ with $\|\phi_1\|_{L^2(\Omega)}=1$ respectively and $A=\lim_{t\to\infty}\|w(\cdot ,t)\|_{L^2(\Omega )}$.

Using variational techniques, we study the properties of the director field of a nematic liquid crystal in steady cylindrical flow, assuming that such a steady flow exists. In particular, we construct an energy functional of which a steady director field would be a stationary point, and we show that, when strong anchoring is imposed at the boundary, this functional has a lower bound. A minimizing sequence in an appropriate Hilbert space is not necessarily bounded in that space, and we show that, as a consequence, a finite nontrivial line energy density may be found along the axis; it can take only discrete values. In contrast, when weak anchoring is imposed at the boundary, there is a critical shear rate beyond which the associated energy has no lower bound. This may result in a tumbling regime for the liquid crystal.

In this paper we prove local existence and uniqueness of regular solutions for a quasistatic model arising in micromagnetism theory. Moreover we show global existence of regular solutions for small data in the 2D case for the Landau-Lifschitz equation. These results extend those already obtained by the authors in the whole space.

In this paper we study positive solutions for the system \begin{align*} -(r^{N-1}\phi(u'))' & \, =\, \lambda r^{N-1}f(v);\ a\, <\,r\, <\,b; \\ -(r^{N-1}\phi(v'))' & \, =\, \lambda r^{N-1}g(u);\ a\, <\,r\, <\,b; \\ u(a)\, =\, 0 & \,=\,u(b)\,;\, v(a)\,=\,0\,=\,v(b), \end{align*} where $ \lambda > 0 $ is a parameter and $ \phi $ is an odd, increasing homeomorphism on $ \Bbb R $. Here $ f,\, g \in C[0,\infty) $ belong to a class of superlinear functions at $\infty$. In particular we allow $ f(0) $ or $ g(0) $ or both to be negative (semipositone system). We discuss the existence of a positive solution for $ \lambda $ small. Our proof is based on degree theory.

We study the existence and uniqueness of classical solutions to the following nonautonomous higher-order Cauchy problem, \begin{equation*} \begin{cases} u^{(n+1)}(t)= A(t)u^{(n)}(t)+B_1(t)u^{(n-1)}(t) \\ \hspace{45pt} + \cdots +B_n(t)u(t)+f(t), \ \ \ \ 0 \le s \le t \le T, \\ u^{(i)}(0)=x_{i} \in E , \hspace{4pt} i=0,1,\dots, n, \end{cases} \end{equation*} by using operator matrices. The results cover some of the known results about the existence and uniqueness of the higher-order Cauchy problem. An example and applications are also given.