Research Article | Open Access
Jishan Fan, Tohru Ozawa, "Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System", International Journal of Analysis, vol. 2013, Article ID 718173, 5 pages, 2013. https://doi.org/10.1155/2013/718173
Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System
We study a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors). We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.
We consider the following coupled Navier-Stokes and -tensor system [1–4]: Here the unknowns , and denote the velocity field of the fluid, the pressure, and the order parameter, respectively. A -tensor is a symmetric and traceless -matrix, and are physical constants, is the space dimension, , and thus .
When , (2) and (3) are the well-known Navier-Stokes system, for which Kozono et al.  and Kozono and Shimada  proved the well-known regularity criteria where denotes the homogeneous Besov spaces .
Very recently, Paicu and Zarnescu  proved the existence of global-in-time weak solutions in 3-dimensional space and of smooth solutions in 2-dimensional space. The aim of this paper is to study the regularity criteria.
Theorem 1. Let in with . Let be a unique strong solution in with .(i)If and satisfies one of the conditions (5), (6), or (7) and satisfies for some finite , then the solution can be extended beyond .(ii)If and satisfies for some finite , then the solution can be extended beyond .
Remark 2. By the well-known inequality , the condition (9) can be replaced by
It has been proved in  that the system (1)–(4) has a Lyapunov functional: which satisfies from which we easily obtain  When , (13) give  thus (5) and (8) hold true; this proves the existence of global-in-time strong solutions when . In , this result was proven by complicated Littlewood-Paley theory, Bony’s paraproduct decomposition, and the logarithmic Sobolev inequality. The purpose of this paper is to make the argument in  much simpler. However, in , they obtained in addition the rate of increase of high norms.
Our proof uses an energy method and relies on a simple estimate of and the following cancellation property:
Lemma 3 (see ). Let be symmetric matrix-valued functions and let be smooth and decaying and sufficiently fast at infinity (so that one can integrate by parts without boundary terms). Then
2. Proof of Theorem 1
Let us observe that for , a traceless, symmetric, matrix, we have
(i) Let (8) hold true
Using the integration by parts, can be bounded as Here and satisfy the relation (8), and we have used the Gagliardo-Nirenberg inequality
Similarly, we get
By using (16), is simply bounded as Here we treat the term by the Gagliardo-Nirenberg inequality
Inserting the above estimates into (23), we derive
Now we estimate as follows.
(1) Let (5) hold true
We will use the following inequality : and the Gagliardo-Nirenberg inequality
This completes the proof.
(2) Let (6) hold true
This completes the proof.
(3) Let (7) hold true
Let be the Littlewood-Paley dyadic decomposition of unity that satisfies , and for any , where is the Fourier transform and is the ball with radius centered at the origin.
We decompose as follows: where is a positive integer to be chosen later. Plugging this decomposition into , we derive
Recalling Bernstein’s inequality, with being a positive constant independent of and , we apply Hölder’s inequality to deduce that
This completes the proof of part (i).
(ii) Let (9) hold true
We still have (23).
is simply bounded as Here we have used the Gagliardo-Nirenberg inequality
is simply bounded as Here we have used (43) and the Gagliardo-Nirenberg inequality Similarly, , and can be bounded as follows:
is bounded as above.
This completes the proof.
This paper is supported by NSFC (no. 11171154). The authors are indebted to the referee for nice suggestions which improved the paper.
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