In this note we study the regularity of weak solutions to a nonlinear steady-state Maxwell's equation in conductive media: $\nabla\times \big[|{\nabla \times }{{\bf H}}|^{p-2} \nabla \times {{\bf H}} \big]={ {\bf F} }(x), p>1, $ where ${{\bf H}}(x)$ represents the magnetic field while ${ {\bf F} }(x)$ is the internal magnetic current. It is shown that the weak solution to the above system is of class $C^{1+\alpha}$, which is optimal. The basic idea is to introduce a suitable potential and then to transform the system into a $p-$Laplacian equation subject to a Neumann type of boundary condition. The desired regularity is established by using the known theory for the scalar p-Laplacian equation.

In this paper we will construct local mild solutions of the Cauchy problem for the incompressible homogeneous Navier-Stokes equations in $d$-dimensional Euclidian space with initial data in uniformly local $ L^{p} $ ($ L^{p}_{uloc}$) spaces where $ p $ is greater than or equal to $d$. For the proof, we shall establish $L^p_{uloc}-L^q_{uloc}$ estimates for some convolution operators. We will also show that the mild solution associated with $ L^{d}_{uloc} $ almost periodic initial data at time zero becomes uniformly local almost periodic ($L^{\infty}$-almost periodic ) in any positive time.

\Abstract

Let $\Omega\subset{{\mathbb{R}}}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The corresponding higher-dimensional result is also given. These results contain both Hardy's and Trudinger's inequalities, and yield a new variational characterization of the maximal solution of the Liouville equation on smooth domains, in terms of a renormalized functional. A global $H^1$ bound on the difference between the maximal solution and the first term of its asymptotic expansion follows.

We study time-dependent flows of incompressible degenerate power-law fluids characterized by the power-law index $p-2$ with $p>2$. In this case, the generalized viscosity vanishes as (the modulus of) the shear rate tends to zero. We prove global-in-time existence of a weak solution if $p>\max\{\frac{3d-4}{d},2\}$. This improves the range $p>\frac{3d+2}{d+2}$ for which the existence result was obtained by O.A.\ Ladyzhenskaya and J.L.\ Lions, via standard monotone operator theory. Since we apply higher differentiability techniques, certain regularity results are also established. The key step of the proof is an estimate of the velocity gradient in a suitable Nikol$'$skĭ space. To make the presentation of the method transparent, we restrict ourselves to the spatially periodic problem. A possible extension of the approach to no-slip boundary conditions is however discussed as well.

The focus of the present study is the standard BBM equation which models unidirectional propagation of small amplitude long waves in dispersive media. The equation is posed on the entire real line and the interest here is the relationship between two different types of solutions. The problem has been studied with initial data in various Sobolev spaces defined on $\mathbb R$ and for periodic initial data, say of period $2l.$ The principal new result is an exact theory of convergence of the periodic solutions to the solutions in Sobolev spaces as $l\to\infty.$