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A regularity criterion for the Cahn-Hilliard-Boussinesq system with zero viscosity
Journal of Inequalities and Applications volume 2014, Article number: 287 (2014)
This paper studies a coupled Cahn-Hilliard-Boussinesq system with zero viscosity. We prove a regularity criterion in terms of vorticity in the homogeneous Besov space .
MSC:35Q30, 76D03, 76D05, 76D07.
In this paper, we study the following Cahn-Hilliard-Boussinesq system with zero viscosity :
with u the fluid velocity field, θ the temperature, ϕ the order parameter, and π the pressure, are the unknowns. . μ is the chemical potential. is the double well potential.
When , (1.1) and (1.2) are the well-known Euler system; Kozono, Ogawa and Taniuchi  proved the following regularity criterion:
Here denotes the homogeneous Besov space.
When , (1.1), (1.2), and (1.3) are the well-known Boussinesq system with zero viscosity; Fan and Zhou  also showed the regularity criterion (1.7).
When , (1.4) and (1.5) are the well-known Cahn-Hilliard system.
It is easy to show that the problem (1.1)-(1.6) has a unique local smooth solution, thus we omit the details here. However, the global regularity is still open. The aim of this paper is to study the blow up criterion. We will prove the following.
Theorem 1.1 Let and , , with in . Suppose that is a local smooth solution to the problem (1.1)-(1.6). Then is smooth up to time T provided that (1.7) is satisfied.
We will use the following logarithmic Sobolev inequality :
and the bilinear product and commutator estimates due to Kato and Ponce :
with , and .
2 Proof of Theorem 1.1
First, it follows from (1.2), (1.3), and the maximum principle that
Testing (1.3) by θ, using (1.2), we see that
Testing (1.4) by ϕ, using (1.2) and (1.5), we find that
Testing (1.1) and (1.4) by u and μ, respectively, using (1.2), (1.5), and (2.2), summing up the result, we deduce that
In the following calculations, we will use the following Gagliardo-Nirenberg inequality:
It follows from (2.6), (1.5), (2.3), (2.4), and (2.7) that
Applying to (1.1), testing by , using (1.2), (1.9), (1.10), and noting that
for any .
Taking Δ to (1.3), testing by Δθ, using (1.2), (1.10), and (2.1), we obtain
Here we have used the Gagliardo-Nirenberg inequalities:
Taking ∇Δ to (1.4), testing by , using (1.5), (1.2), and (1.10), we have
Using (2.4), we get
Now we use the following Gagliardo-Nirenberg inequalities:
Combining (2.10), (2.11), (2.12), and (2.14), taking ϵ small enough, using (1.8), (2.5), Gronwall’s inequality and
we conclude that
This completes the proof.
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Fan J, Zhou Y: A note on regularity criterion for the 3D Boussinesq system with partial viscosity. Appl. Math. Lett. 2009,22(5):802–805. 10.1016/j.aml.2008.06.041
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This project is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant No. 27-130-35-HiCi. The authors, therefore, acknowledge technical and financial support of KAU.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Ma, C., Alzahrani, F.S., Hayat, T. et al. A regularity criterion for the Cahn-Hilliard-Boussinesq system with zero viscosity. J Inequal Appl 2014, 287 (2014). https://doi.org/10.1186/1029-242X-2014-287
- regularity criterion
- zero viscosity