An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

Convergence results for a class of spectrally hyperviscous models of 3-D turbulent flow

We consider the spectrally hyperviscous Navier–Stokes equations (SHNSE) which add hyperviscosity to the NSE but only to the higher frequencies past a cutoff wavenumber _m0$m_0$. In Guermond and Prudhomme (2003) [18], subsequence convergence of SHNSE Galerkin solutions to dissipative solutions of the NSE was achieved in a specific spectral-vanishing-viscosity setting. Our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of Leray solutions. In particular we obtain subsequence convergence of SHNSE strong solutions to Leray solutions of the NSE by fixing the hyperviscosity coefficient μ$\mu$ while the spectral hyperviscosity cutoff _m0$m_0$ goes to infinity. This formulation presents new technical challenges, and we discuss how its motivation can be derived from computational experiments, e.g. those in Borue and Orszag (1996, 1998)  and . We also obtain weak subsequence convergence to Leray weak solutions under the general assumption that the hyperviscous coefficient μ$\mu$ goes to zero with no constraints imposed on the spectral cutoff. In both of our main results the Aubin Compactness Theorem provides the underlying framework for the convergence to Leray solutions.     

Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations

In this paper, we are concerned with the partial regularity for suitable weak solutions of the tri-dimensional magnetohydrodynamic equations. With the help of the De Giorgi iteration method, we obtain the results proved by He and Xin (C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152), namely, the one dimensional parabolic Hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero.     

A note on weakly hyperbolic equations with analytic principal part

In this note we show how to include low order terms in the C^∞$C^{\infty }$ well-posedness results for weakly hyperbolic equations with analytic time-dependent coefficients. This is achieved by doing a different reduction to a system from the previously used one. We find the Levi conditions such that the C^∞$C^{\infty }$ well-posedness continues to hold.     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

An application of weighted Hardy spaces to the Navier–Stokes equations

In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces H^p(w)$H^p(w)$ with w   belonging to Muckenhoupt class _A∞$A_{\infty }$. As a corollary, one obtains decay estimates of heat semigroup on weighted Hardy spaces. After a weighted version of the div–curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier–Stokes equations with the initial data belonging to weighted Hardy spaces.     

Equivalent conditions on periodic feedback stabilization for linear periodic evolution equations

This paper studies the periodic feedback stabilization for a class of linear T  -periodic evolution equations. Several equivalent conditions on the linear periodic feedback stabilization are obtained. These conditions are related to the following subjects: the attainable subspace of the controlled evolution equation under consideration; the unstable subspace (of the evolution equation with the null control) provided by the Kato projection; the Poincaré map associated with the evolution equation with the null control; and two unique continuation properties for the dual equations on different time horizons [0,T]$[0,T]$ and [0,_n0T]$[0,n_{0}T]$ (where _n0$n_0$ is the sum of algebraic multiplicities of distinct unstable eigenvalues of the Poincaré map). It is also proved that a T-periodic controlled evolution equation is linear T-periodic feedback stabilizable if and only if it is linear T-periodic feedback stabilizable with respect to a finite-dimensional subspace. Some applications to heat equations with time-periodic potentials are presented.     

Backward stochastic differential equations associated with the vorticity equations

In this paper we derive a non-linear version of the Feynman–Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2. A particular class of terminal values provide, via these probabilistic methods, solutions for the vorticity equation.