Blow-up criterion for 3D compressible viscous magneto-micropolar fluids with initial vacuum
Boundary Value Problemsvolume 2013, Article number: 160 (2013)
In this paper, the author establishes a blow-up criterion of strong solutions to 3D compressible viscous magneto-micropolar fluids. It is shown that if the density and the velocity satisfy , where and , then the strong solutions to the Cauchy problem can exist globally over . The initial density may vanish on open sets, that is, the initial vacuum is allowed.
MSC:76N10, 35B44, 35B45.
In this paper, we consider the following 3D compressible viscous magneto-micropolar fluids:
where is the spacial coordinate and is the time. The unknown functions , , , and (, ) are the fluid density, velocity, micro-rotational velocity, magnetic field and pressure, respectively. The constants μ, λ, ξ, , and σ are the viscosity coefficients of the fluid satisfying
System (1.1)-(1.2) describing the motion of aggregates of small solid ferromagnetic particles relative to viscous magnetic fluids, such as water, hydrocarbon, ester, fluorocarbon, etc., in which they are immersed, covers a wide range of heat and mass transfer phenomena, under the action of magnetic fields, and is of great importance in practical and mathematics applications (see ). Indeed, (1.1) is composed of the balance laws of mass, momentum, moment of momentum and magnetohydrodynamic, respectively. Due to its importance in mathematics and physics, there is a lot of literature devoted to the mathematical theory of the compressible viscous magneto-micropolar system (see [2–4]).
For the incompressible magneto-micropolar fluid models where , Rojas-Medar  established local existence and uniqueness of strong solutions by the Galerkin method. Ortega-Torres and Rojas-Medar  proved global existence of strong solutions for small initial data. A BKM type blow-up criterion for smooth solution that relies on the vorticity of velocity only was obtained by Yuan . For regularity results, refer to Yuan  and Gala .
In particular, if the effect of angular velocity field of the particle’s rotation is omitted, i.e., , then (1.1) reduces to compressible magnetohydrodynamic equations (MHD). There are numerous important progress on compressible MHD (see [10–12] and the references therein). The local strong solutions to the compressible MHD with large initial data were respectively obtained by Vol’pert-Khudiaev  and Fan-Yu  in cases that the initial density is strictly positive and the initial density may vanish. Xu-Zhang  proved a blow-up criterion that if is the maximal time of existence of a strong solution, then
where is the weak space and r, s satisfy
If , (1.1) reduces to compressible micropolar fluid equations. Mujakovic [13, 14] considered the one-dimensional motion of compressible viscous micropolar fluids and studied the local/global existence. The global existence of strong solutions to the 1D model with initial vacuum was also obtained in . For multi-dimensional compressible magneto-micropolar equations, Amirat and Hamdache  proved the global existence of weak solutions with finite energy and the adiabatic constant for , which generalized Lions’ pioneering work  and the work by Feireisl et al. . Chen  established the local existence and uniqueness of strong solutions under the assumption that the initial density may vanish, and in  Chen et al. proved a blow-up criterion that
where r, s satisfy (1.3).
If and , (1.1) reduces to isentropic compressible Navier-Stokes equations. In , the authors established a Serrin-type blow-up criterion that
where r, s satisfy (1.3).
In this paper, our main purpose is to establish a blow-up criterion of strong solutions for system (1.1) with the following conditions:
To proceed, we introduce the following notations. For , we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:
To present the main result, we first give the following local existence and uniqueness of strong solutions to the Cauchy problem (1.1), (1.2) and (1.4) with initial vacuum (without proof), which can be obtained by the same method developed by Choe-Kim in  (see also Fan-Yu  and Chen  for MHD and compressible micropolar fluids, respectively).
Theorem 1.1 Assume that for some , the initial data satisfy
and the compatibility conditions
with some . Then there exists a positive time such that the problem (1.1), (1.2) and (1.4) has a unique strong solution in satisfying, for some ,
Motivated by [20, 21] and , we have the main purpose in this paper to prove a blow-up criterion for the problem (1.1), (1.2) and (1.4). More precisely, the main result in this paper reads as follows.
Theorem 1.2 Assume that the initial data satisfies (1.5)-(1.7). Let be a strong solution of the Cauchy problem (1.1), (1.2) and (1.4) with the regularities (1.8). If is the maximal time of existence, then
for any r and s satisfying (1.3).
Remark 1.3 Theorem 1.1 proves that the strong solutions of (1.1), (1.2) and (1.4) can exist only in a small time , which means that if is the maximal time of existence, then there must be some component of the fluid mechanics blow-ups. Theorem 1.2 points out one kind of blow-up mechanics.
Remark 1.4 There is no any additional growth condition on the micro-rotational velocity w and magnetic field H. This reveals that the density and the linear velocity play a more important role compared to the angular velocity of rotation of particles and the magnetic field in the regularity theory of solutions to 3D compressible magneto-micropolar fluid flows.
The rest of the paper is devoted to completing the proof of Theorem 1.2.
2 Proof of Theorem 1.2
First, we give the following well-known Gagliardo-Nirenberg inequality that will be used frequently.
Lemma 2.1 For , and , there exists some generic constant , which may depend on p, q and r, such that for any and , we have
The following BKM’s type inequality which will be used to estimate and with can be found in .
Lemma 2.2 For , there is a constant , depending only on q, such that the following estimate holds for all :
The proof of Theorem 1.2 is based on the contradiction arguments. Let be a strong solution of the problem (1.1), (1.2) and (1.4) as described in Theorem 1.1. Suppose that (1.9) is false, that is,
where r, s satisfy (1.3) and is a constant.
One can easily deduce from the following energy estimate (1.1), (1.2) and (1.4).
Lemma 2.3 It holds that
Here and hereafter, C denotes a generic positive constant which may depend on μ, , λ, , ξ, σ, A, γ, , , , , , , T and .
We denote the material derivative of f by and set
Since due to (1.1)5, we have from (1.1)2 and (1.1)3 that
Thus, from the standard -estimate of an elliptic system, we have the following lemma.
Lemma 2.4 Under the condition (2.4), it holds that
Proof In view of standard -estimates of elliptic system (2.7), one immediately obtains (2.8). By (2.1) and (2.4), we get that
which, combined with (2.8), yields (2.9) immediately. □
The next lemma is concerned with the higher integrability of H under the assumption (2.4).
Lemma 2.5 Under the condition (2.4), it holds for any that
where is a positive constant depending on q.
The proof is similar to Lemma 3.3 in  and is omitted here.
With the help of (2.4) and Lemmas 2.3-2.5, we can prove the following key lemma.
Lemma 2.6 Under the condition (2.4), it holds that for any ,
Proof Multiplying (1.1)2, (1.1)3 and (1.1)4 by , and , respectively, and integrating the resulting equations by parts, one obtains after summing up that
To estimate the first term on the right-hand side of (2.12), we observe that P satisfies
Hence, using (2.4), (2.5) and (2.10) yields that
where we have used Young’s inequality and (2.1).
For the second term, we have, after integration by parts, that
and by the Cauchy-Schwarz inequality, we have
Similarly, integrating by parts and using the fact , one has
For the last three terms on the right-hand side of (2.12), one has from (2.4) that
Thus, putting (2.13)-(2.17) into (2.12) and choosing suitably small, we infer from (2.8) that
For any r, s satisfying (1.3), we have by the Hölder and Sobolev inequalities that
for some .
Taking , H, , ∇w, ∇H into (2.19) and using (2.10), we obtain
By the standard -estimate, one can deduce from (2.1), (2.4), (2.8) and (2.10) that
Furthermore, it follows from (1.1)4 and Sobolev’s embedding inequality that
putting (2.21) and (2.22) into (2.20), such that
which, together (2.21) and (2.18), choosing suitably small, gives
It is easily seen that
Taking this into account, we conclude from (2.4), (2.24) and Gronwall’s inequality that part of (2.11) holds for any . Note that the estimate of is a consequence of (2.4), (2.22) and (2.23). The proof of this lemma is completed. □
Next we prove the boundedness of , , and by the compatibility conditions (1.6) and (1.7).
Lemma 2.7 Under the condition (2.4), it holds that for any ,
Proof Applying the operator and to both sides of (1.1)2 and (1.1)3, respectively, and using (1.1)1, one can obtain, after a straightforward calculation, that
We get after integration by parts that
Similarly, we also have
After integration by parts, using (1.1)1 and (2.11), we obtain
Using the definition of the material derivation and integrating by parts, we deduce from (2.1), (2.5) and (2.11) that
The ninth term on the right-hand side of (2.26) can be estimated as follows, integrating by parts, using (2.1), (2.5), (2.10), (2.11) and Hölder’s inequality:
In a similar manner, one also has
Putting (2.27)-(2.34) into (2.26), using the Cauchy-Schwarz inequality and choosing suitably small, we get
To estimate , one can differentiate (1.1)4 with respect to t, multiply the resulting equations by in , and integrate by parts over to get
Integrating by parts and using (1.1)5, (2.1), (2.10) and (2.11), then we deduce
for some positive constants . For the second term on the right-hand side of (2.36), integrating by parts and using (2.1) give
Putting the estimates of , into (2.36) and choosing small enough, one has
Then, combining (2.35) and (2.37), using Young’s inequality, and choosing suitably small yield that
Firstly, we use (2.4)-(2.6), (2.9), (2.10), (2.11), (2.1) and (2.2) to infer from the standard -estimate that
Moreover, by the standard -estimate of an elliptic system, we infer from (1.1)4, (2.1), (2.2) and (2.11) that
Combining (2.39)-(2.41), we obtain
Now, putting (2.41) and (2.42) into (2.38), one has
from which and (2.11), we immediately obtain (2.25) by Gronwall’s inequality, (1.6) and (1.7). As a result of (2.41), we can also deduce the boundedness of . □
The next lemma is used to bound the density gradient and .
Lemma 2.8 Under the condition (2.4), it holds that for any ,
for any .
Proof Differentiating (1.1)1 with respect to and multiplying it by () in , we obtain, after integrating by parts and summing up, that
It follows from (2.1), (2.4), (2.6)-(2.9), (2.25) and the interpolation inequality that for any ,
where (2.2) and (2.25) were also used to get that . So, putting this into (2.44) yields
We now estimate . To do this, we first observe that
Hence, using the standard -estimate of an elliptic system leads to
From (1.1)3 and the standard -estimate of the elliptic system, we have that
This, together with Lemmas 2.2 and 2.6, gives
Now, if we set and let
then it is seen from (2.45) and (2.48) that
due to . Thus,
On the other hand, since
we thus deduce from (2.11), (2.25), (2.4), (2.8), (2.9) and (2.2) that
As a result, it follows from (2.49) and Gronwall’s inequality that
From this and (2.25), (2.48), (2.50), one obtains
Taking in (2.45), we get, by using (2.52) and (2.25) and Gronwall’s inequality, that
Moreover, the standard -estimate of an elliptic system and (1.1)2, together with (2.4), (2.11) and (2.25), implies
Similar to the proof of (2.47), there are
where we have used (2.1), (2.11), (2.46), (2.51) and (2.54). From this, together with (2.25), (2.46) and (2.51)-(2.54), we can deduce (2.43). □
As a consequence of Lemmas 2.6-2.8, we have the following lemma.
Lemma 2.9 Under the condition (2.4), it holds that for any ,
The proof is the same as that of Lemma 3.6 in  and is omitted here.
With the help of Lemmas 2.3, 2.6-2.9 and the local existence theorem, we can complete the proof of Theorem 1.2 by the contradiction arguments. In fact, in view of Lemmas 2.3, 2.6-2.9, it is easy to see that the functions have the same regularities imposed on the initial data (1.5) at the time . This implies that the compatibility conditions (1.6) and (1.7) are satisfied at the time . Thus, we can take as the initial data and apply the local existence theorem to extend the local strong solutions beyond . This contradicts the assumption that is the maximal time of existence.
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This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 11QZR16), the National Natural Science Foundation of China (Grant No. 11001090).
The author declares that they have no competing interests.