Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].

A New Regularity Criterion for the Three-Dimensional Incompressible Magnetohydrodynamic Equations in the Besov Spaces

Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field π plays an important role in the regularity problem of MHD equations. We derive that the weak solution u , b is regular on 0 , T , which is provided for the scalar pressure field π in the Besov spaces.

A new blow up criterion for the 3D magneto-micropolar fluid flows without magnetic diffusion

This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution $(u,\omega,b)$ ( u , ω , b ) to the magneto-micropolar fluid flows can be extended beyond time $t=T$ t = T , provided if $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ u 3 ∈ L β ( 0 , T ; L α ( R 3 ) ) with $\frac{2}{\beta }+\frac{3}{\alpha }\leq \frac{3}{4}+\frac{1}{2\alpha },\alpha > \frac{10}{3}$ 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 and $\nabla b\in L^{\frac{4p}{3(p-2)}}(0,T;\dot{M}_{p,q}(R^{3}))$ ∇ b ∈ L 4 p 3 ( p − 2 ) ( 0 , T ; M ˙ p , q ( R 3 ) ) with $1< q\leq p<\infty $ 1 < q ≤ p < ∞ and $p\geq 3$ p ≥ 3 .

New Extended Quantitative Local and Global Regularity Index for Single- and Multiframe RC Bridges Based on Modal Vector Correlation

Seismic demand and performance of bridges are highly dependent upon the level of irregularity. Although previous studies have proposed methodologies so as to quantify the irregularity of the bridges in terms of global regularity index, it still remains unclear how to determine the distribution of irregularity along a bridge, as well as to discover the irregularity sources. This research project is intended to develop a quantitative vector regularity criterion for single- and multiframe bridges based on the modified correlation function for spatial locations of scaled mode shapes of deck-alone and whole bridge. The proposed criterion calculates two types of regularity indices, namely, local (LRI) and global regularity indices (GRI). The GRI is a scalar value representing the overall regularity of the entire bridge, whereas the LRI highlights vector irregularity distribution along the bridge. Since the deck discontinuity due to the in-span hinges is one of the leading causes for irregularity, the proposed index has been employed in case of multiframe bridges as well. Furthermore, the current study aims to investigate the correlation between the proposed irregularity indicators and the nonlinear to linear demand ratio. Therefore, the appropriate analysis method can be chosen based on irregularity extent of bridges. Obtained results of the proposed indices reveal that in-span hinge is one of the main parameters affecting the distribution of local irregularity along a bridge. Therefore, multiframe bridges need to be investigated in detail so as to validate the special design requirements recommended by design codes.

Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative

In this paper, we study the regularity of the weak solutions for the incompressible 3D Navier–Stokes equations with the partial derivative of the velocity. By the embedded technology, we prove that the weak solution u is regular on (0, T] if ∂ 3 u ∈ L p 0 , T ; L q R 3 with 2 / p + 3 / q = 70 / 37 + 15 / 37 q , 15 / 4 ≤ q ≤ ∞ , or 2 / p + 3 / q = 34 / 19 + 9 / 19 q , 9 / 4 ≤ q ≤ ∞ .