Serrin-type Regularity Criterion Research Articles

Regularity criteria for the three-dimensional axially symmetric non-resistive incompressible magnetohydrodynamic system

In this paper, we study the three-dimensional axially symmetric non-resistive incompressible magnetohydrodynamic system. By establishing a rough blow-up criteria on ω θ r , we show a Ladyzhenskaya–Prodi–Serrin type regularity criteria with respect to the swirl component u θ of the velocity.

A weak-$$L^{p}$$ Prodi–Serrin type regularity criterion for the micropolar fluid equations in terms of the pressure

A weak-$$L^{p}$$ Prodi–Serrin type regularity criterion for the micropolar fluid equations in terms of the pressure

Serrin–type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

In this paper, we consider the 3D MHD equations in which the viscosity coefficient \(\mu \) and the resistivity coefficient \(\nu \) are equal. We show that the Serrin–type conditions imposed on one component of the velocity \(u_{3}\) and one component of magnetic fields \(b_{3}\) with $$\begin{aligned} u_{3} \in L^{p_{0},1}(-1,0;L^{q_{0}}(B(2))),\ b_{3} \in L^{p_{1},1}(-1,0;L^{q_{1}}(B(2))), \end{aligned}$$\(\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1\) and \(3<q_{0},q_{1}<+\infty \), imply that the suitable weak solution is regular at (0, 0).

Serrin type regularity criterion for the shear thinning fluids via the velocity field

By estimating the nonlinear term in an innovative way, we could obtain the optimal regularity criterion for the shear thinning fluids via the velocity field.

The global regularity for 3D inhomogeneous incompressible fluids with vacuum

In this paper, we investigate the global regularity for 3D inhomogeneous incompressible Navier–Stokes equations with vacuum. More precisely, we establish several Prodi–Serrin type regularity criteria in the Besov space in terms of only velocity or gradient of velocity which allow the initial density to contain vacuum. By using the embedding theory of Besov space, it is shown that our results improve the Prodi–Serrin type regularity condition which is scaling invariant in the time–space Lebesgue space.

On Prodi–Serrin type conditions for the 3D Navier–Stokes equations

In this paper we prove several new Prodi–Serrin type regularity criteria with weak Lebesgue integrability in both space and time for the 3D Navier–Stokes equations in the whole space.

A local Serrin-type regularity criterion for a suitable weak solution to the incompressible liquid crystal system

A local Serrin-type regularity criterion for a suitable weak solution to the incompressible liquid crystal system

On regularity criteria of a weak solution to the three-dimensional magnetohydrodynamics equations in Lorentz space

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

On the 3D Navier–Stokes equations with regularity in pressure

In this paper, we investigate the global regularity of Leray–Hopf weak solutions to the 3D Navier–Stokes equations in terms of pressure. In particular, we establish three Serrin-type regularity criteria in framework of the anisotropic Lebesgue spaces. Our result can be understood as a generalization of the notable works of Y. Zhou (2006) [23], C. Cao and E. Titi (2008) [5], and Y. Zhou and M. Pokorný (2010) [24].

The regularity criterion for the 3D Navier–Stokes equations involving end-point Prodi–Serrin type conditions

We consider the Prodi–Serrin type regularity criterion involving ∂ 3 u h and the third component of velocity (or the gradient of velocity). In particular, if the ∂ 3 u h satisfies the end-point Prodi–Serrin type condition, one can show that Leray’s weak solutions of the three-dimensional Navier–Stokes equations become regular with the help of additional assumption on the third component of velocity (or the gradient of velocity field).

On the local well-posedness and a Prodi–Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion

We prove a Prodi–Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. To the best of our knowledge, this is the first Prodi–Serrin-type criterion for such a 3D hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-Boussinesq equations. We note that, as a special case, these results include the 3D non-diffusive Boussinesq system and the 3D MHD equations. Moreover, they can be extended without difficulty to include the case of a Coriolis rotational term.

On regularity criteria of weak solutions to the 3D viscoelastic Navier–Stokes equations with damping

We give a weak − L p Serrin type regularity criteria of weak solutions to the 3D viscoelastic Navier–Stokes equations with damping, which is described as an incompressible non-Newtonian fluid in the whole space R 3 .

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

A remark on the global regularity for the 3D Navier–Stokes equations

In this paper, we investigate the case of Prodi–Serrin type regularity criterion involving u 3 and ∂ 3 u h . More precisely, it is shown that Leray’s weak solutions of the three-dimensional Navier–Stokes equations become regular if the third component of velocity (or the gradient of the velocity field) satisfies the additional end-point Prodi–Serrin type condition.

Ladyzhenskaya–Prodi–Serrin type regularity criteria for the 3D incompressible MHD equations in terms of 3 × 3 mixture matrices

We prove that a weak solution (u, b) to the MHD equations is smooth on (0, T ] if with , and , where is a mixture matrix (see its definition below). As we will explain later, this kind of regularity criteria is more likely to capture the nature of the coupling effects between the fluid velocity and the magnetic field in the evolution of the MHD flows. Moreover, the condition on is scaling invariant, i.e. it is of Ladyzhenskaya–Prodi–Serrin type.

An almost Serrin-type regularity criterion for the Navier–Stokes equations involving the gradient of one velocity component

This paper concerns about the Cauchy problem for the three-dimensional Navier–Stokes equations and provides a regularity criterion in terms of the gradient of one velocity component. This improves previous results.

Regularity Criteria in Weak &lt;i&gt;L&lt;/i&gt;&lt;sup&gt;3&lt;/sup&gt; for 3D Incompressible Navier-Stokes Equations

We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Navier-Stokes equations. Let Br denote concentric balls in R3 with radius r. We will show that if p ∈ Lm(0,1; L1(B2)), m > 2, and if u is sufficiently small in L∞(0,1; L3,∞(B2)), without any assumption on its gradient, then u is bounded in B1 × (1/10,1). It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.

A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

We give simple proofs that a weak solution u of the Navier–Stokes equations with H 1 initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L s (0, T;L r,∞(Ω)) or if its L s,∞(0, T;L r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.

Regularity criterion for generalized Newtonian fluids in bounded domains

The regularity of a non-Newtonian flow in a domain with a boundary cannot be treated easily because of the nonlinear viscosity term and the global property of the pressure. A distance function was used in a previous study to obtain a strong solution. In the present study, using the distance function idea, we obtain Serrin-type regularity criteria for the vorticity and the velocity with regard to non-Newtonian equations with shear-dependent viscosity in a bounded domain.

A Refinement of the Local Serrin-Type Regularity Criterion for a Suitable Weak Solution to the Navier–Stokes Equations

We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x 0, t 0). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x 0, t 0), intersected with the exterior of a certain space-time paraboloid with vertex at point (x 0, t 0). We make no special assumptions on the solution in the interior of the paraboloid.