Regularity Criteria For Weak Solutions Research Articles

The regularity criteria of weak solutions to 3D axisymmetric incompressible Boussinesq equations

The regularity criteria of weak solutions to 3D axisymmetric incompressible Boussinesq equations

A regularity criterion for the 3D axisymmetric Boussinesq equations with non-zero swirl

In this paper, we are devoted to establishing a new regularity criterion of weak solutions to incompressible axisymmetric Boussinesq equations. More precisely, we prove that for a small ϵ > 0, if the angular component of velocity field uθ(t, r, z) satisfies sup0<t<T‖zuθ‖L∞(Ωδ)≤ϵ, where Ωδ≔x1,x2,z∈R3|x12+x22<δ denotes a thin cylinder with infinite height and radius δ > 0, which is independent of the initial data, then the weak solution (u, ρ) to 3D Boussinesq equations is regular in (0, T].

Regularity criteria for weak solutions to the 3d co-rotational Beris-Edwards system via the pressure

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 satisfiesΠ−[ln⁡(1+Π−)]1+γ∈L∞(R+;L32,∞(R3)), or the quantity 2Π+|u|2+|∇Q|2 satisfies(2Π++|u|2+|∇Q|2)[ln⁡(1+2Π++|u|2+|∇Q|2)]1+γ∈L∞(R+;L32,∞(R3)), 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. L32,∞(R3) 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.

Regularity for 3D inhomogeneous incompressible MHD equations with vacuum

In this paper, we consider the conditional regularity for 3D inhomogeneous incompressible magnetohydrodynamic equations in Vishik spaces and give a regularity criterion of weak solutions in terms of a gradient velocity vector.

Regularity Criteria for Weak Solutions to the Navier–Stokes Equations in Terms of Spectral Projections of Vorticity and Velocity

Regularity Criteria for Weak Solutions to the Navier–Stokes Equations in Terms of Spectral Projections of Vorticity and Velocity

New Regularity Criteria for Weak Solutions to the MHD Equations in Terms of an Associated Pressure

We assume that \(\Omega \) is either a smooth bounded domain in \({{\mathbb {R}}}^3\) or \(\Omega ={{\mathbb {R}}}^3\), and \(\Omega '\) is a sub-domain of \(\Omega \). We prove that if \(0\le T_1<T_2\le T\le \infty \), \(({\mathbf {u}},{\mathbf {b}},p)\) is a suitable weak solution of the initial–boundary value problem for the MHD equations in \(\Omega \times (0,T)\) and either \({\mathscr {F}}_{\gamma }(p_-)\in L^{\infty }(T_1,T_2;\, L^{3/2}(\Omega '))\) or \({\mathscr {F}}_{\gamma } ({\mathcal {B}}_+)\in L^{\infty }(T_1,T_2;\, L^{3/2}(\Omega '))\) for some \(\gamma >0\), where \({\mathscr {F}}_{\gamma }(s)=s\, [\ln {}(1+ s)]^{1+\gamma }\), \({\mathcal {B}}= p+\frac{1}{2}|{\mathbf {u}}|^2+\frac{1}{2}|{\mathbf {b}}|^2\) and the subscripts “−” and “\(+\)” denote the negative and the nonnegative part, respectively, then the solution \(({\mathbf {u}},{\mathbf {b}},p)\) has no singular points in \(\Omega '\times (T_1,T_2)\). If \({\mathbf {b}}\equiv {\mathbf {0}}\) then our result generalizes some previous known results from the theory of the Navier–Stokes equations.

A double-logarithmically improved regularity criterion of weak solutions for the 3D MHD equations

A double-logarithmically improved regularity criterion of weak solutions for the 3D MHD equations

Regularity Criteria for the 3D Magneto-Hydrodynamics Equations in Anisotropic Lorentz Spaces

In this paper, we investigate the regularity of weak solutions to the 3D incompressible MHD equations. We provide a regularity criterion for weak solutions involving any two groups functions (∂1u1,∂1b1), (∂2u2,∂2b2) and (∂3u3,∂3b3) in anisotropic Lorentz space.

An improved regularity criterion for the 3D magneto-micropolar equations in homogeneous Besov space

In this paper we consider the regularity criterion for the weak solutions to the 3D incompressible magneto-micropolar equations. We establish an improved regularity criterion for weak solutions in terms of two pairs of (∂iui,∂ibi)(i=1,2,3). More precisely, we show that for some i,j∈{1,2,3} with i≠j, if(∂iui,∂ibi),(∂juj,∂jbj)∈Lp(0,T;B˙q,∞0(R3)),2p+3q=2,3≤q≤∞, then the solution (u,w,b) to the magneto-micropolar equations is smooth on [0,T].

On the role of pressure in the theory of MHD equations

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin’s integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of π≔p+12|b|2 are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier’s or Navier-type boundary conditions for the velocity u.

Regularity criterion for weak solutions to the 3D Navier–Stokes equations via two vorticity components in [formula omitted

In this paper, we provide a regularity criterion for 3D Navier–Stokes equations in terms of two vorticity components, which extends a recent result established by Guo, Kučera and Skalák (2018). More precisely, we prove that a unique local strong solution u to 3D Navier–Stokes equations does not blow up at time T provided only two components of vorticity belongs to L2(0,T;BMO−1). We also prove that u can be smoothly extended beyond T, if the horizontal gradient of horizontal components of velocity u belong to L2(0,T;BMO−1). The proof relies on the technique of the Bony decomposition and some Fourier multiplier theorems.

An optimal regularity criterion for the 3D MHD equations in homogeneous Besov spaces

In this paper, we establish a new regularity criterion for weak solutions to the 3D incompressible MHD equations in terms of two pairs of (∂iui, ∂ibi) . More precisely, it is proved that the weak solution (u, b) is smooth on (0, T], provided that for some i, j ∈ {1, 2, 3} with i ≠ j, it holds that

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field.

Regularity criteria for weak solutions of the Magneto-micropolar equations

&lt;p style='text-indent:20px;'&gt;In this paper, we show that a weak solution &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ (\mathbf{u},\mathbf{w},\mathbf{b})(\cdot,t) $&lt;/tex-math&gt;&lt;/inline-formula&gt; of the magneto-micropolar equations, defined in &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;$ [0,T) $&lt;/tex-math&gt;&lt;/inline-formula&gt;, which satisfies &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;$ \nabla u_3, \nabla_{h} \mathbf{w}, \nabla_{h} \mathbf{b} $&lt;/tex-math&gt;&lt;/inline-formula&gt; &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;$ \in L^{\frac{32}{7}}(0,T; $&lt;/tex-math&gt;&lt;/inline-formula&gt; &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;$ L^2(\mathbb{R}^3)) $&lt;/tex-math&gt;&lt;/inline-formula&gt; or &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;$ \partial_3 u_3, \partial_3 \mathbf{w}, \partial_3 \mathbf{b} \in L^{\infty}(0,T;L^2(\mathbb{R}^3)) $&lt;/tex-math&gt;&lt;/inline-formula&gt;, is regular in &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;$ \mathbb{R}^3\times(0,T) $&lt;/tex-math&gt;&lt;/inline-formula&gt; and can be extended as a &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;$ C^\infty $&lt;/tex-math&gt;&lt;/inline-formula&gt; solution beyond &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;$ T $&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

A refined regularity criteria of weak solutions to the magneto-micropolar fluid equations

We consider a sufficient condition to ensure the smoothness of solutions to 3D magneto-micropolar fluid equations. It involves partial derivatives of the partial components of the velocity field and magnetic field. We refine and extend the results in Zhang (J Math Anal Appl 346(1):336–339, 2008) for incompressible Navier–Stokes equations, Jia and Zhang et al. (Acta Math Appl Sin Engl Ser 29(4):869–880, 2013) for micropolar fluid equations and Jia (Appl Math Lett 50:1–4, 2015), Xu and Li et al. (Z Angew Math Phys 68(6):125, 2017) for MHD equations simultaneously.

Regularity criteria for the 3D magnetohydrodynamics system involving only two velocity components

This paper is dedicated to the regularity criteria of weak solutions to the three‐dimensional incompressible magnetohydrodynamics system in terms of only two velocity components. It is proved that the weak solution (u,b) is smooth on (0,T], provided that or

A regularity criterion of the 3D MHD equations involving one velocity and one current density component in Lorentz space

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% \begin{equation} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq \alpha \leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2\beta }{2\beta -3}}\left( 0,T;L^{\beta ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq \beta \leq \infty , \label{eq02} \end{equation}% which generalize some known results.

Existence of mild solutions and regularity criteria of weak solutions to the viscoelastic Navier–Stokes equation with damping

Existence of mild solutions and regularity criteria of weak solutions to the viscoelastic Navier–Stokes equation with damping

The anisotropic integrability logarithmic regularity criterion to the 3D micropolar fluid equations

The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by Jia, Zhang and Dong in [ 21 ].

A Regularity Criterion of Weak Solutions to the 3D Boussinesq Equations

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 $$(u,\theta )$$ becomes regular provided that $$\begin{aligned} (\nabla _{h}{\widetilde{u}},\nabla _{h}\theta )\in L^{1}(0,T;\overset{\cdot }{B }_{\infty ,\infty }^{0}({\mathbb {R}}^{3})) \end{aligned}$$Our results improve and extend the well-known result of Dong and Zhang (Nonlinear Anal 11:2415–2421, 2010) for the Navier–Stokes equations.