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 ${\Vert \pi \Vert }_{L^{\alpha ,\mathrm{\infty }}(0,T;L^{\beta ,\mathrm{\infty }}({\mathbb{R}}^3))}$ with $\frac{2}{\alpha }+\frac{3}{\beta }=2$ and $\frac{3}{2}<\beta <\mathrm{\infty }$ or ${\Vert \mathrm{\nabla }\pi \Vert }_{L^{\alpha ,\mathrm{\infty }}(0,T;L^{\beta ,\mathrm{\infty }}({\mathbb{R}}^3))}$ with $\frac{2}{\alpha }+\frac{3}{\beta }=3$ and $1<\beta <\mathrm{\infty }$ is sufficiently small.

In the existing article, the existence of solutions to nonlinear fractional differential inclusions in the sense of the Atangana-Baleanu-Caputo ( $\mathcal{A}\mathcal{B}\mathcal{C}$) fractional derivatives in Banach space is studied. The investigation of the main results relies on the set-valued issue of Mönch fixed point theorem incorporated with the Kuratowski measure of non-compactness. A simulated example is proposed to explain the obtained results.

Rubio de Francia's extrapolation in generalized grand Morrey spaces is derived. This result is applied to the investigation of the regularity of solutions for the second order partial differential equations with discontinuous coefficients in the framework of generalized grand Morrey spaces under the Muckenhoupt condition on weights. Density properties for these spaces are also investigated.