Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term

We prove a Harnack inequality for the positive solutions of ultraparabolic equations of the type L u + V u= 0, where L is a linear second order hypoelliptic operator and V belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem

A new regularity criterion for the Navier-Stokes equations in terms of the two components of the velocity

This paper establishes a new regularity criterion for the Navier-Stokes equation in terms of two velocity components. We show that if the two velocity components $\widetilde{u}=\left( u_{1},u_{2},0\right)$ satisfy \begin{equation*} \int_{0}^{T}\Vert \tilde{u}(s)\Vert _{\dot{B}_{\infty ,\infty }^{0}}^{2}ds<\infty , \end{equation*} then the solution can be smoothly extended after $t=T$. This gives an aswer to an open problem in [B. Q. Dong, Z. Zhang, Nonlinear Anal. Real World Appl. 11(2010), 2415-2421]

Interpolation inequalities for weak solutions of nonlinear parabolic systems

Abstract The authors investigate differentiability of the solutions of nonlinear parabolic systems of order 2 m in divergence form of the following type ∑ | α | ≤ m ( - 1 ) | α | D α a α X , D u + ∂ u ∂ t = 0 . The achieved results are inspired by the paper of Marino and Maugeri 2008, and the methods there applied. This note can be viewed as a continuation of the study of regularity properties for solutions of systems started in Ragusa 2002, continued in Ragusa 2003 and Floridia and Ragusa 2012 and also as a generalization of the paper by Capanato and Cannarsa 1981, where regularity properties of the solutions of nonlinear elliptic systems with quadratic growth are reached. Mathematics Subject Classification (2000) Primary 35K41, 35K55. Secondary 35B65, 35B45, 35D1

PAC Fields over Finitely Generated Fields

We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$.Comment: 7 pages, Math.

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in {\mathbb{R}^{3}}

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015)

ESTIMATES OF THE DERIVATIVES OF MINIMIZERS OF A SPECIAL CLASS OF VARIATIONAL INTEGRALS

The note concerns on some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals of the form $$\int_\Omega F (x,u,Du) dx$$ where the integrand has the following special form $$F(x,u,Du)\, =\, A(x,u, g^{\alpha\beta}(x) h_{ij}(u) \frac{\partial u^i}{\partial x^\alpha} \frac{\partial u^i }{\partial x^\beta}),$$ where $(g^{\alpha\beta})$ and $(h_{ij})$ symmetric positive definite matrices. We are not assuming the continuity of $A$ and $g$ with respect to $x$. We suppose that $A(\cdot, u,t)/(1+t)$ and $g(\cdot)$ are in the class $L^\infty\cap VMO$