On the Fractional Fractal Analysis of Multivariate Pointwise Lipschitz Oscillating Regularity

Classical Lipschitz regularity does not allow to capture possible different oscillating directional pointwise regularity behaviors in coordinate axes of functions f on Rd, d ≥ 2. To overcome this drawback, we use iterated fractional primitives to introduce a notion of multivariate pointwise Lipschitz oscillating regularity. We show a characterization in hyperbolic wavelet bases. As an application, we obtain the fractal print dimension of a given set of multivariate Lipschitz oscillating regularity, from the knowledge of fractional axes oscillating spaces to which f belongs

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.

Regularity results for solutions of micropolar fluid equations in terms of the pressure

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on $(0, T]$ provided that either the norm $\left\Vert \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R}^{3}))}$ with $\frac{2}{\alpha }+ \frac{3}{\beta } = 2$ and \frac{3}{2} < \beta < \infty or $\left\Vert \nabla \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R} ^{3}))}$ with $\frac{2}{\alpha }+\frac{3}{\beta } = 3$ and 1 < \beta < \infty is sufficiently small

On the blow up criterion for the 3D nematic liquid crystal flows involving the second eigenvalue of the deformation tensor

In this paper, we study the blow up criterion of the smooth solutions to the three-dimensional incompressible nematic liquid crystal flows in terms of $\lambda _{2}^{+}$ in the multiplier space $\dot{X}_{1}$ and $\nabla d$ in $BMO$. It is shown that the solution $(u,d)$ can be extended beyond $t=T$ if T λ + ( · , t ) 2 2 2 X ̇ 1 + ∥ ∇ d ( · , t ) ∥ B M O d t \u3c ∞ . 0 ln(e+∥∇u(·,t)∥X ̇1) ln(e+∥∇d(·,t)∥BMO

Directional Thermodynamic Formalism

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional H&ouml;lder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional H&ouml;lder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets

The Wente problem associated to the modified Helmholtz operator in weighted Sobolev spaces

In this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modified Helmholtz operator∆ + I, where  is a positive constan