Search

In this paper, we establish a new regularity criterion in terms of the magnetic field for the local-in-time smooth solution of the two-dimensional non-resistive MHD equations. In particular, our result settles an open problem remarked by Lei et al. (2010), [11]. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 5 3 < p < 11 5 provided that the velocity field belongs to the critical space L 2 p − 1 (0 , T ; B M O (R 3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO -norm. In particular, when 2 < p < 11 5 , we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖ u ‖ B M O and ‖ | D u | p − 2 2 ∇ 2 u ‖ 2. [ABSTRACT FROM AUTHOR]

We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q -tenser. Our results yield that for any positive constant γ > 0 , if either the negative part of the associated pressure Π satisfies Π − [ ln ⁡ (1 + Π −) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , or the quantity 2 Π + | u | 2 + | ∇ Q | 2 satisfies (2 Π + + | u | 2 + | ∇ Q | 2) [ ln ⁡ (1 + 2 Π + + | u | 2 + | ∇ Q | 2) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , then the weak solution (u , Q) , to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript "−" and "+" denote the negative and the nonnegative part, respectively. L 3 2 , ∞ (R 3) denotes the standard weak Lebesgue space. If Q ≡ 0 , then our results extend some previous known results from the theory of the 3d Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to the Beale–Kato–Majda (BKM) regularity criterion of the 2D micropolar equations in the critical fractional diffusion situation α + β = 1 for α > 0 and β > 0. With the use of the Littlewood–Paley theory, the global existence of the unique solution (u , w) to the equations is found to be completely determined by the velocity field u. The dynamical behaviour of the scalar angular velocity w is controlled by the velocity u. More precisely, it is shown that the strong solution (u , w) exists globally if ∫ 0 T ‖ ∇ × u (τ) ‖ L ∞ 2 d τ < ∞ for any T < ∞. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the two-dimensional incompressible generalized magnetohydrodynamics equations. We obtain some regularity criteria to the corresponding equations in terms of the current density j or the vorticity ω. Our approach is based on a key observation on the nice structure of the incompressible magnetohydrodynamics equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the axisymmetric Navier–Stokes equations with non-zero swirl component. It is proved that if r u r ≥ M for some M > − 2 , then the solution actually is smooth. This extends the result in Pan (2017) [14] . [ABSTRACT FROM AUTHOR]

The paper is dedicated to study of the regularity criterion for weak solutions to the 3D incompressible MHD equations. Employing the Littlewood–Paley decomposition, we show that if ∇ ˜ u ˜ = ( ∂ 1 u ˜ , ∂ 2 u ˜ ) ∈ L s 1 ( [ 0 , T ) ; B ˙ r 1 , 2 r 1 3 0 ( R 3 ) ) , 2 s 1 + 3 r 1 = 2 , 3 2 < r 1 ≤ ∞ and ∇ ˜ b ˜ = ( ∂ 1 b ˜ , ∂ 2 b ˜ ) ∈ L s 2 ( [ 0 , T ) ; B ˙ r 2 , 2 r 2 3 0 ( R 3 ) ) , 2 s 2 + 3 r 2 = 2 , 3 2 < r 2 ≤ ∞ , then the solutions to the MHD actually is smooth on ( 0 , T ) . [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized Hall-MHD system in dimension three. A new regularity criterion is established for β ∈ ( 1 2 , 1 ] . [ABSTRACT FROM AUTHOR]

In this note, we prove a logarithmic regularity criterion for the 2D MHD equations without magnetic diffusion in terms of the magnetic field in homogeneous Besov space B . ∞ , ∞ 0 . [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]

In this paper, we consider the classical three-dimensional (3D) incompressible Navier–Stokes equations. By making use of a new anisotropic Sobolev inequality, we improve a regularity criterion for the weak solution to the 3D Navier–Stokes equations involving the gradient of one velocity component. This result improves the previous work of Zhou and Pokorný (2010). [ABSTRACT FROM AUTHOR]

Smooth solutions to the axially symmetric Navier–Stokes equations obey the following maximum principle: ‖ r u θ ( r , z , t ) ‖ L ∞ ≤ ‖ r u θ ( r , z , 0 ) ‖ L ∞ . We first prove the global regularity of solutions if ‖ r u θ ( r , z , 0 ) ‖ L ∞ or ‖ r u θ ( r , z , t ) ‖ L ∞ ( r ≤ r 0 ) is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang [10] . As a corollary, we also prove the global regularity under the assumption that | r u θ ( r , z , t ) | ≤ | ln ⁡ r | − 3 / 2 , ∀ 0 < r ≤ δ 0 ∈ ( 0 , 1 / 2 ) . [ABSTRACT FROM AUTHOR]

In this article, we study the global regularity of the 3D Navier–Stokes equations in terms of one velocity component. In particular, we establish a new version of regularity criterion in the framework of the anisotropic Lebesgue space. It is a generalization of the notable work of C. Cao and E.S. Titi (2011) [3] . [ABSTRACT FROM AUTHOR]

In this paper, we establish the global weak solutions for two-dimensional Boussinesq equations with mixed partial dissipation and thermal diffusivity. Moreover, we give global regularity criterion for weak solutions to the two-dimensional Boussinesq equations with mixed partial dissipation and thermal diffusivity. [ABSTRACT FROM AUTHOR]

In this paper, we establish some regularity criteria for both the simplified and the general Ericksen–Leslie system in R 3 . [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularity criterion of solutions to the incompressible Navier-Stokes equations in three-dimensional infinite channel in terms of the derivatives of the vertical component of the velocity field. More precisely, we show that the strong solution exists on the time interval [0,T] provided that one of the following conditions holds: ∂u3∂xi∈Lq([0,T];Lp(Ω)), 1≤i≤3 where p>3, 1≤q<∞, 3p+2q≤12+32p for i=1,2, and p>2, 1≤q<∞, 3p+2q≤34+32p for i=3. [ABSTRACT FROM AUTHOR]

In this paper we prove the global-in-time existence of smooth solutions of the 2D generalized magnetohydrodynamic equations with zero viscosity, when the power of the Laplacian operator -Δ in the dissipation term of the induction equation is between 1 and 2. We also show a regularity criterion on the direction of the magnetic field b0:=b|b| when β=1. [ABSTRACT FROM AUTHOR]

In this paper, two regularity criteria to the two-dimensional incompressible generalized magnetohydrodynamics equations with zero magnetic diffusivity are given. It is proved that the solution remains smooth on [0,T] provided that the current j×b satisfies 0Tj(.,t) L 2λpλp-1 4αλp(2α-1) p + 2 dt<∞ with 1<α<2, 12≤λ≤1, 2≤p≤2λ, or the vorticity w×u satisfies0w(t,.)BMOdt<∞ with 1<α<2. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the three-dimensional density-dependent incompressible magnetohydrodynamic equations. First, a regularity criterion is proved which allows the initial density to contain vacuum. Then we establish another blow-up criterion in the Besov space when the positive initial density is bounded away from zero. Third, we prove a global nonexistence result for initial density with high decreasing at infinity. Fourth, we obtain a regularity criterion to the density-dependent incompressible magnetohydrodynamic equations in a bounded domain. Finally, we also give some remarks on the regularity criteria for the three-dimensional full compressible magnetohydrodynamic equations in a bounded domain and for the incompressible homogeneous magnetohydrodynamic equation in the whole space . [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions ( ) and in two dimensions ( ). We prove that the solution is smooth up to time provided that there exists a positive constant such that (i) for , and (ii) for , [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Cauchy problem for the 3D viscous MHD equations, and provide some regularity criteria involving only one directional derivative of the pressure, say . In particular, if then the solution remains smooth on . [Copyright &y& Elsevier]

Abstract: In this paper we study the global regularity issue for the 3D incompressible MHD equations. It is proved that if the pressure satisfies , then the corresponding weak solution is regular on . [Copyright &y& Elsevier]

Abstract: This paper studies the Cauchy problem of the Euler–Landau–Lifshitz system arising in the nematic liquid crystal flows. We prove two logarithmically improved regularity criteria for local smooth solutions in the homogeneous Besov space. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a hyperbolic–parabolic system arising from chemotaxis. A blow up criterion of local strong solutions to the model in is established based on the scaling invariance of the system. As a by-product, we prove a global-in-time well-posedness result when for large initial data. [Copyright &y& Elsevier]

Abstract: In the study of regularity criteria for the weak solutions of the 3D Navier–Stokes equations, an improved regularity criterion is obtained. More precisely, it is proved that if the pressure satisfies the critical growth condition then the weak solution is regular on . The finding is mainly based on the innovative function decomposition methods together with Besov space techniques. [Copyright &y& Elsevier]

Abstract: In this article we consider the following generalized quasi-geostrophic equation where , , and . We first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case and we obtain the global well-posedness of the system with smooth initial data; and for the case and we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data. [Copyright &y& Elsevier]

Abstract: This note proves a regularity criterion for the 2D MHD system with zero magnetic diffusivity. [Copyright &y& Elsevier]

Abstract: In this paper, we consider regularity criterion for the three-dimensional incompressible magnetohydrodynamic equations. We present some sufficient integrability conditions on some components of the velocity and magnetic fields for the regularity of the weak solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we study regularity criteria for the Navier–Stokes–Landau–Lifshitz system. Using delicate estimates, the regularity criteria for smooth solution of Navier–Stokes–Landau–Lifshitz system in Besov spaces and the multiplier spaces are obtained. The Navier–Stokes–Landau–Lifshitz system is coupled system of the Navier–Stokes equation and Landau–Lifshitz system, our results generalize the related results for Navier–Stokes equation and Landau–Lifshitz system to our system. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to establish some logarithmically improved regularity criteria in term of the multiplier spaces to the Navier–Stokes equations. [Copyright &y& Elsevier]

Abstract: In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier–Stokes equation in . More precisely, let be a weak solution in . Then u becomes a classical solution if any two functions of , and belong to provided with , . [Copyright &y& Elsevier]

Abstract: In this paper we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions. By means of a commutator estimate based on frequency localization and Bony''s paraproduct decomposition, we obtain a regularity criterion which is an obvious extension of the result in [B. Dong, Z. Chen, A remark on regularity criterion for the dissipative quasi-geostrophic equations, J. Math. Anal. Appl. 329 (2007) 1212–1217] where their restrictive condition on p is . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the regularity criterion of Leray–Hopf weak solutions to the 3D Navier–Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution is regular on if there exist two solution components, for example, and , satisfying the condition [Copyright &y& Elsevier]

Abstract: We consider the regularity of axisymmetric weak solutions to the Navier–Stokes equations in . Let u be an axisymmetric weak solution in , , and be the azimuthal component of w in the cylindrical coordinates. Chae–Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier–Stokes equations, Math. Z. 239 (2002) 645–671] proved the regularity of weak solutions under the condition , with , . We deal with the marginal case which they excluded. It is proved that u becomes a regular solution if . [Copyright &y& Elsevier]

Abstract: This paper concerns with a regularity criterion of solutions to the 2D dissipative quasi-geostrophic equations. Based on a logarithmic Sobolev inequality in Besov spaces, the absence of singularities of θ in is derived for θ a solution on the interval satisfying the condition This is an extension of earlier regularity results in the Serrin''s type space . [Copyright &y& Elsevier]

In this paper, we prove a regularity criterion ψ , A ∈ W : = { (ψ , A) | ∇ ψ ∈ L 2 (0 , T ; L n) , A ∈ L ∞ (0 , T ; L n) ∩ L 2 p p − n (0 , T ; L p) } with n < p ≤ ∞ , for the nD (n ≥ 3) time-dependent Ginzburg-Landau model in superconductivity. Finally, we show that this criterion holds true when n = 3 or 4. [ABSTRACT FROM AUTHOR]

We consider the regularity criterion for the incompressible MHD equations. We show that the weak solution is regular, provided that (0.1) (∂ 3 w + , ∂ 3 w −) ∈ L 5 r 4 r − 6 (0 , T ; L r (R 3)) with 4 ≤ r ≤ ∞ , or (0.2) (∂ 3 w + , ∂ 3 w −) ∈ L 11 r 7 r − 12 (0 , T ; L r (R 3)) with 5 2 ≤ r ≤ 4. [ABSTRACT FROM AUTHOR]