Abstract
We consider sufficient conditions to ensure the smoothness of solutions to 3D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations.
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1 Introduction and the main result
In this paper, we consider the 3D magneto-micropolar fluid equations studied by Galdi and Rionero [5]:
Here u=u(x,t),b=b(x,t),ω=ω(x,t) represent the velocity field, the magnetic field and the micro-rotational velocity respectively; p denotes the hydrodynamic pressure; μ>0 is the kinematic viscosity, χ>0 is the vortex viscosity, κ>0 and γ>0 are the spin viscosities, 1/ν (with ν>0) is the magnetic Reynold; while u 0, b 0, ω 0 are the corresponding initial data with ∇⋅u 0=∇⋅b 0=0.
The global weak solution to system (1.1) is established by Rojas-Medar and Boldrini [10], while the local strong solutions are given by Rojas-Medar [9]. However, whether or not the local strong solutions can exist globally is still an open problem. Thus regularity criteria appears. Notice that:
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(1) Yuan [14] first established the following fundamental regularity criterion in terms of the velocity or its gradient
$$\begin{array}{@{}rcl@{}} {\kern-1.7pc}u\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \ \frac{2}{p}+\frac{3}{q}=1, \ \ \ 3<q\leqslant \infty \end{array} $$(1.2)and
$$\begin{array}{@{}rcl@{}} {\kern-1.7pc}\nabla u\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \ \frac{2}{p}+\frac{3}{q}=2, \ \ \ \frac{3}{2}<q\leqslant \infty. \end{array} $$(1.3)Then Gala [4] extended it to the Morrey–Campanato spaces, Zhang et al [16] improved it to some more general Triebel–Lizorkin spaces.
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(2) When ω=b=0, system (1.1) is just the classical Naiver–Stokes equations. Serrin [11], Prodi [8] and Beirão da Veiga [1] proved regularity if some scaling-invariant norm of u or ∇u is bounded.
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(3) When ω=0, system (1.1) is then the 3D MHD equations, He and Xin [6], and Zhou [19] gave criteria similar to the case for Navier–Stokes equations.
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(4) When b=0, system (1.1) is reduced to the micropolar fluid equations, Yuan [13] gave some criteria in Lorentz spaces.
For later developments, see [2; 3; 15; 18] and references cited therein. Recently, Vasseur [12] proved that if
then the solutions to the Navier–Stokes equations are smooth. Later, Luo [7] extended (1.4) to the MHD equations, but involves the magnetic field also. We now extend the result of Vasseur [12] and Luo [7] to system (1.1). The main result is the following:
Theorem 1.1.
Let \(u_{0},\omega _{0},b_{0}\in H^{1}(\mathbb {R}^{3})\) with ∇⋅u 0 =∇⋅b 0 =0 in the sense of distributions. Suppose that (u,ω,b) is a strong solution to ( 1.1 ) in (0,T) such that
and ∇⋅u=∇⋅b=0. If additionally,
and
then the solution can be extended smoothly beyond t=T.
Remark 1.1.
Theorem 1.1 shows that it is enough to control the rate of change in the direction of the velocity and the norm of b to get full regularity of the solutions. Notice that we add no conditions on the micro-rotational velocity ω.
Remark 1.2.
Our theorem covers the results of Vasseur [12] and Luo [7] for Navier–Stokes and MHD equations, respectively. Observe that the condition (1.6) is a scaling-invariant, but (1.5) is not. Whether or not the 1/2 in (1.5) can increase to 1 is our future work.
Before giving a proof, let us first recall the definition of weak solutions to system (1.1).
DEFINITION 1.1
Let \(u_{0},\omega _{0},b_{0}\in L^{2}(\mathbb {R}^{3})\) with ∇⋅u 0=∇⋅b 0=0. A triple (u,ω,b) of measurable functions on \(\mathbb {R}^{3}\times (0,T)\) is said to be a weak solution of system (1.1) if
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(1) \(u,\omega ,b\in L^{\infty }(0,T;L^{2})\cap L^{2}(0,T;H^{1})\) with ∇⋅u=∇⋅b=0;
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(2) System (1.1) holds in the sense of distributions.
Remark 1.3.
Testing (1.1) 1, (1.1) 2, (1.1) 3 by u,ω,b respectively, after suitable integration by parts, one has the energy inequality:
Throughout the proof in the next section, we shall frequently use the following interpolation inequality (see [17]):
for (p,q) satisfying
In this paper, we shall use standard notations for Lebesgue space \(L^{q}(\mathbb {R}^{3})\) endowed with the norm ∥⋅∥ q , and anisotropic Lebesgue space \(L^{p}(I; L^{q}(\mathbb {R}^{3}))\) endowed with the norm ∥⋅∥ p,q . Here \(I\subset \mathbb {R}^{+}\) is an interval. A constant C (C=C(∗,∗,…) which depends on the parameters) may differ from line to line.
2 Proof of Theorem 1.1
By decreasing p or r if necessary, we may assume that
For an ε>0 to be chosen sufficiently small (see (2.12)), choose t 1∈(0,T) such that
and
Hereafter, the integrals are over \(\mathbb {R}^{3}\times (t_{1}, T )\).
Utilizing the regularity criteria (1.2), we complete the proof of Theorem 1.1 provided
To this end, denote by
Multiplying (1.1) 1,(1.1) 3 by \(\left |u\right |^{2}u, \left |b\right |^{2}b\) respectively and integrating over \(\mathbb {R}^{3}\times (t_{1}, T )\), we find that
as well as
Using (2.5) and (2.6), notice that (see [12])
and (see [13])
Thus we have
Here C is a constant depending only on μ,χ,ν.
Using Cauchy–Schwartz inequality, I 1 can be bounded as
By generalized Hölder inequality and (1.8), it follows that
where
and we have used (1.8). In fact, we can choose
where r,s are as in (1.6). Thus
For I 2, let us first take divergence of (1.1) 1 to see
thus classical Calder\(\acute {\mathrm {o}}\)n–Zygmund estimates imply
invoking again the generalized Hölder inequality and (1.8),
where
In fact, we can choose
where p,q are as in (1.5).
Finally, using (1.8), I 3 is treated as
Combining the estimates for I 1,I 2,I 3, i.e. (2.9), (2.10), (2.11), and substituting into (2.7), we find
where C is the generic constant appearing in (2.9), (2.10) and (2.11). Thus, we see that
provided
Consequently, by (2.4), we have
The proof is completed.
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ZHANG, Z. Regularity criteria for the 3D magneto-micropolar fluid equations via the direction of the velocity. Proc Math Sci 125, 37–43 (2015). https://doi.org/10.1007/s12044-015-0213-z
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DOI: https://doi.org/10.1007/s12044-015-0213-z