Serrin-type Regularity Criterion Research Articles

Regularity for Ostwald-de Waele type shear thickening fluids

We obtain local in time existence of strong solution for non-Newtonian fluid with shear thickening viscosity. We also obtain the Hausdorff dimension of time singular set, and a Serrin type regularity criterion.

A remark on the regularity criterion of Boussinesq equations with zero heat conductivity

In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by Geng and Fan (2012) [6].

Remarks on regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure

In this article, we establish a Serrin-type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equation in ℝ3. It is proved that if the gradient of pressure belongs to , where is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1, then the weak solution is actually regular. Since this space is wider than , our regularity criterion covers the previous results given by Struwe [M. Struwe, On a Serrin-type regularity criterion for the Navier–Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007), pp. 235–242], Berselli-Galdi [L.C. Berselli and G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations, Proc. Amer. Math. Soc. 130 (2002), pp. 3585–3595] and Zhou [ Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in ℝ3 , Proc. Am. Math. Soc. 134 (2006), pp. 149–156].

A new regularity criterion for the nematic liquid crystal flows

In this article, we establish the Serrin-type regularity criteria for the 3D nematic liquid crystal flows in the terms of the multiplier space . Our criterion may be regarded as an extension of the recent result of Liu–Cui [Regularity of solutions to 3-D nematic liquid crystal flows, Electron. J. Diff. Equ. 2010(173) (2010), pp. 1–5].

A regularity criterion to the biharmonic map heat flow in ℜ4

AbstractWe consider the regularity problem under the critical condition to the biharmonic map heat flow from ℜ4 to a smooth compact Riemannian manifold without boundary. Using Gagliardo‐Nirenberg inequalities and delicate estimates, the Serrin type regularity criterion for the smooth solutions of biharmonic map heat flow is obtained without assuming a smallness condition on the initial energy. Our result improved the results of Lamm in 5 and 6 and generalized the results of Chang, Wang, Yang 1, Strzelecki 11 and Wang 13, 14 to non‐stationary case.

Regularity criterion of the Newton-Boussinesq equations in $R^3$

In this paper, we consider the regularity problem under the criticalcondition to the Newton-Boussinesq equations. The Serrin typeregularity criteria are established in terms of the criticalMorrey-Campanato spaces and Besov spaces.

Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the pressure in the class [formula omitted

The goal of this paper is to establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier–Stokes equations in R 3 . It is proved that if the pressure belongs to L 2 ( 0 , T ; B . ∞ , ∞ − 1 ( R 3 ) ) , then the weak solution is actually regular. This improves a result established in a recent work by the second author Gala (2011) [19].

Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations

We consider the regularity criteria for a weak solution \documentclass[12pt]{minimal}\begin{document}$\bm {u}=(u_1,u_2,u_3)$\end{document}u=(u1,u2,u3) to the Navier-Stokes equations in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3. Denoting by \documentclass[12pt]{minimal}\begin{document}$\bm {\omega }=(\omega _1,\omega _2,\omega _3)=\mbox{curl }\bm {u}$\end{document}ω=(ω1,ω2,ω3)=curlu the vorticity, we then prove that \documentclass[12pt]{minimal}\begin{document}$\bm {u}$\end{document}u is smooth, provided u3, ω3; or ∂3u3, ω3 are in some Serrin-type integrability classes.

Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R d \mathbb {R}^{d} . It is known that if a Leray weak solution u u belongs to L 2 1 − r ( ( 0 , T ) ; L d r ) \ \ for some \ \ 0 ≤ r ≤ 1 , \begin{equation} L^{\frac {2}{1-r}}\left ( \left ( 0,T\right ) ;L^{\frac {d}{r}}\right ) \text { \ \ for some \ \ }0\leq r\leq 1, \end{equation} then u u is regular. It is proved that if the pressure p p associated to a Leray weak solution u u belongs to L 2 2 − r ( ( 0 , T ) ; M . 2 , d r ( R d ) d ) , \begin{equation} L^{\frac {2}{2-r}}\left ( \left ( 0,T\right ) ;\overset {.}{\mathcal {M}}_{2,\frac { d}{r}}\left ( \mathbb {R}^{d}\right ) ^{d}\right ) , \end{equation} where M . 2 , d r ( R d ) \overset {.}{\mathcal {M}}_{2,\frac {d}{r}}\left ( \mathbb {R}^{d}\right ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 > r > 1 0>r>1 , then the weak solution is actually regular. Since this space M . 2 , d r \overset {.}{\mathcal {M}}_{2,\frac {d}{r}} is wider than L d r L^{\frac {d}{r}} and X . r \overset {.}{X}_{r} , the above regularity criterion (0.2) is an improvement of Zhou’s result.

On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey–Campanato space

In this paper, we establish a Serrin-type regularity criterion for the weak solutions to the micropolar fluid equations in R 3 . It is proved that, if the velocity field satisfies one of the following conditions: u ∈ L 2 1 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) or ∇ u ∈ L 2 2 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) , or the associated pressure satisfies one of the following conditions: p ∈ L 2 2 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) or ∇ p ∈ L 2 3 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) , then the solution remains smooth on [ 0 , T ] .

Regularity criteria in weak spaces for $3$-dimensional Navier-Stokes equations in terms of the pressure

In this paper we consider the Cauchy problem for the $3$-dimensional Navier-Stokes equations and we establish some Serrin-type regularity criterion in weak spaces involving the summability of the pressure or the gradient of the pressure.

A note on the regularity criterion in terms of pressure for the Navier–Stokes equations

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier–Stokes equation in R d . Here we call u a Leray weak solution if u is a weak solution of finite energy, i.e. u ∈ L ∞ ( ( 0 , T ) ; L 2 ) ∩ L 2 ( ( 0 , T ) ; H . 1 ) . It is known that if a Leray weak solution u belongs to (0.1) L 2 1 − r ( ( 0 , T ) ; L d r ) for some 0 ≤ r ≤ 1 , then u is regular (see [J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187–195]). We succeed in proving the regularity of the Leray weak solution u in terms of pressure under the condition (0.2) p ∈ L 2 2 − r ( ( 0 , T ) ; X . r ( R d ) d ) , where X . r ( R d ) is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1 . Since this space X . r is wider than L d r , the above regularity criterion (0.2) is an improvement on Zhou’s result [Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in R 3 , Proc. Amer. Math. Soc. 134 (2006) 149–156].

On a Serrin-Type Regularity Criterion for the Navier–Stokes Equations in Terms of the Pressure

We prove a Serrin-type regularity result for Leray–Hopf solutions to the Navier–Stokes equations, extending a recent result of Zhou [28].

On regularity criteria in terms of pressure for the Navier-Stokes equations in ℝ³

In this paper we establish a Serrin-type regularity criterion on the gradient of pressure for the weak solutions to the Navier-Stokes equations in R 3 \mathbb {R}^3 . It is proved that if the gradient of pressure belongs to L α , γ L^{\alpha ,\gamma } with 2 / α + 3 / γ ≤ 3 2/\alpha +3/\gamma \leq 3 , 1 ≤ γ ≤ ∞ 1\leq \gamma \leq \infty , then the weak solution is actually regular. Moreover, we give a much simpler proof of the regularity criterion on the pressure, which was showed recently by Berselli and Galdi (Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585–3595).