Search

In this paper, we prove the local well-posedness and regularity criterion of strong solutions to the 3D density-dependent incompressible Maxwell–Navier–Stokes system in a bounded and simply connected domain with vacuum. [ABSTRACT FROM AUTHOR]

In this paper, we show that a weak solution to 3D shear-thinning flows becomes strong provided that the gradient of one component of velocity belongs to a certain Lebesgue space class. [ABSTRACT FROM AUTHOR]

In this article, we study the regularity criteria of the weak solutions to the Boussinesq equations involving the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. This result reveals that the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

The question of whether the two-dimensional inviscid Boussinesq equations can develop a finite-time singularity from general initial data is a challenging open problem. In this paper, we obtain two new regularity criteria for the local-in-time smooth solution to the two-dimensional inviscid Boussinesq equations. Similar result is also valid for the nonlocal perturbation of the two-dimensional incompressible Euler equations. [ABSTRACT FROM AUTHOR]

In this work, we investigate the regularity of a weak solution to a model for unsteady motion of an incompressible non-Newtonian fluid. Our main result is a regularity criterion for 3D flows of shear-thinning fluids via one component of velocity. [ABSTRACT FROM AUTHOR]

We consider the three-dimensional inviscid Boussinesq–Voigt system which is a regularization model for the inviscid Boussinesq equations. We prove a regularity criterion for the weak solutions (in particular, for the time-derivative of the velocity), in L p -spaces, involving first derivatives of the pressure. [ABSTRACT FROM AUTHOR]

A one-component regularity criterion for the non-resistive axially symmetric Hall-MHD system is given in this paper. More precisely, we show that strong solutions to a family of axially symmetric initial data, whose velocity and current density have trivial swirl components, could be smoothly extended beyond a possible blow-up time T ∗ if and only if the swirl component of the magnetic field h θ satisfies a Beale-Kato-Majda-type criterion. See (1.5) below. This criterion is not trivial even if the velocity field vanishes since the quantity h θ r satisfies a partial Burgers' equation in this situation. [ABSTRACT FROM AUTHOR]

In this paper, we prove a single-component BKM-type regularity criterion for the inviscid axially symmetric Hall-MHD system. More precisely, we show that if the current density is swirl-free, then the L t 1 L x ∞ boundedness of partial vorticity (corresponding to curl of the swirl part of the velocity) implies regularity of the solution. The novelty of our result is that we only impose a critical regularity criterion on one component of the velocity which is new even if we ignore the magnetic field and consider only the 3D axially symmetric Euler equations. Our results can not be easily extended to the classical L t 1 B M O norm regularity criterion at present, which will be pursued in our further works. [ABSTRACT FROM AUTHOR]

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 , ω , b) to the magneto-micropolar fluid flows can be extended beyond time t = T , provided if u 3 ∈ L β (0 , T ; L α (R 3)) with 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 and ∇ b ∈ L 4 p 3 (p − 2) (0 , T ; M ˙ p , q (R 3)) with 1 < q ≤ p < ∞ and p ≥ 3 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the 3D density-dependent magnetohydrodynamic equations with vacuum in the whole space R 3 , and provide a regularity criterion involving the velocity field in BMO space norm. This work generalizes the regularity criterion of the constant density MHD equations to the density-dependent one. [ABSTRACT FROM AUTHOR]

We prove a regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system with the Coulomb gauge in ℝ3. It is proved that if the velocity field in the Besov space satisfies some integral property, then the solution keeps its smoothness. [ABSTRACT FROM AUTHOR]

Tropical climate model derived by Frierson et al. (Commun Math Sci 2:591–626, 2004) and its modified versions have been investigated in a number of papers [see, e.g., Li and Titi (Discrete Contin Dyn Syst Series A 36(8):4495–4516, 2016), Wan (J Math Phys 57(2):021507, 2016), Ye (J Math Anal Appl 446:307–321, 2017) and more recently Dong et al. (Discrete Contin Dyn Syst Ser B 24(1):211–229, 2019)]. Here, we deal with the 2D tropical climate model with fractional dissipative terms in the equation of the barotropic mode u and in the equation of the first baroclinic mode v of the velocity, but without diffusion in the temperature equation, and we establish a regularity criterion for this system. [ABSTRACT FROM AUTHOR]

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L ∞ (0 , T ; B ˙ ∞ , ∞ − 1) -norm of a suitable low frequency part of (u , ∇ d) is bounded by a scaling invariant constant and the initial data (u 0 , ∇ d 0) . Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020). [ABSTRACT FROM AUTHOR]

The paper deals with the regularity criteria for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution (u , θ) becomes regular provided that (∇ h u ~ , ∇ h θ) ∈ L 1 (0 , T ; B · ∞ , ∞ 0 (R 3)) Our results improve and extend the well-known result of Dong and Zhang (Nonlinear Anal 11:2415–2421, 2010) for the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this work, we prove some regularity criteria for a Ginzburg–Landau–Navier–Stokes system with the Coulomb gauge in a bounded domain Ω ⊂ R 3 . [ABSTRACT FROM AUTHOR]

This work establishes a new logarithmical improved regularity criterion for the 3D Boussinesq equations in terms of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫ 0 T ‖ ∂ 3 P (· , t) ‖ L x 3 γ L x 1 x 2 α β 1 + ln (1 + ‖ θ (· , t) ‖ L 4) d t < ∞ , where 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α < 1 k - 2 , for some T > 0 , then the corresponding solution (u , θ) to the 3D Boussinesq equations is regular on [0, T]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Euler-Boussinesq equations in dimension two. Some new regularity criteria via the temperature are established by making use of the Littlewood-Paley technique and the new logarithmic Gronwall inequality. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of u3 and ω3, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorný (2004). [ABSTRACT FROM AUTHOR]

The paper deals with the 3D incompressible MHD equations and aims at improving a regularity criterion in terms of the horizontal gradient of velocity and magnetic field. It is proved that the weak solution ( u, b) becomes regular provided that The result is an extension of regularity criterion for 3D Navier-Stokes equations in Besov space due to Fang and Qian (Commun Pure Appl Anal 13:585-603, 2014) [see also (Ni et al. in J Math Anal Appl 396:108-118, 2012)]. [ABSTRACT FROM AUTHOR]

In this paper, we prove that suitable weak solution ( u, b) of the 3-D MHD equations can be extended beyond T if u ∈ L (0, T; L (ℝ)) and the horizontal components b of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin. [ABSTRACT FROM AUTHOR]

This paper focuses on the global regularity problem for a family of active scalar equations with fractional dissipation. We obtain an explicit lower bound on the local existence of solutions, the eventual regularity of solutions and the global regularity of solutions for large initial data for the supercritical active scalar equations. In addition, a new regularity criterion is presented for the supercritical case which is used to study the eventual regularity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the two-dimensional (2D) incompressible Bénard equations. On the one hand, we prove global-in-time existence of smooth solutions to the 2D Bénard equations with critical dissipation. On the other hand, we establish several regularity criteria involving temperature for 2D Bénard equations with supercritical dissipation. [ABSTRACT FROM AUTHOR]

We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the axisymmetric Navier–Stokes equations with swirl. By invoking the magic identity of Miao and Zheng, the symmetry properties of Riesz transforms and the Hardy–Sobolev inequality, we establish regularity criterion involving r d ω θ with − 1 ≤ d < 0 . This improves and extends the results of [3,21]. [ABSTRACT FROM AUTHOR]

This paper studies the regularity problem for the 3D incompressible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence [14] predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi–Serrin type criteria) for the 3D Hall-MHD system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a logarithmically improved the regularity criterion in terms of the homogeneous Besov spaceto the Boussinesq equations. We prove the solutionis smooth up to timeprovided that for someand. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following regularity criterion ω : = curl u ∈ L 1 ( 1 , T ; B M O ) for the 3D Boussinesq system with zero heat conductivity in a bounded domain. Here ω is the vorticity of the fluid velocity field u and BMO denotes the function space of bounded mean oscillation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the density-dependent Boussinesq equations, and provide a regularity criterion involving the velocity field in BMO space. [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularity criterion in terms of the temperature for the 2D fractional Boussinesq equations with zero heat conductivity. [ABSTRACT FROM AUTHOR]

In this paper, we provide a regularity criterion for the Navier–Stokes equations involving the diagonal entry of the velocity gradient, which says that if ∂ 3 u 3 ∈ L ∞ ( 0 , T ; L 2 ( R 3 ) ) , then the weak solution is smooth on ( 0 , T ) . This is an important case in the sense that the strong solution is in this class. Moreover, we verify the limiting case of one regularity criterion in Cao and Titi (2011). [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional Boussinesq equations, and obtain a logarithmically improved regularity criterion in terms of pressure. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R, and provide a new regularity criterion involving only two entries of the Jacobian matrix of the velocity field. [ABSTRACT FROM AUTHOR]