- Research
- Open access
- Published:
A regularity criterion for the Cahn-Hilliard-Boussinesq system with zero viscosity
Journal of Inequalities and Applications volume 2014, Article number: 287 (2014)
Abstract
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.
1 Introduction
In this paper, we study the following Cahn-Hilliard-Boussinesq system with zero viscosity [1]:
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 [2] 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 [3] 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 [2]:
and the bilinear product and commutator estimates due to Kato and Ponce [4]:
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
whence
Testing (1.4) by Ï•, using (1.2) and (1.5), we find that
which gives
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
which gives
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
which yields
Applying to (1.1), testing by , using (1.2), (1.9), (1.10), and noting that
we derive
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:
We obtain
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.
References
Boyer F: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 1999, 20: 175–212.
Kozono H, Ogawa T, Taniuchi Y: The critical Sobolev inequalities in Besov space and regularity criterion to some semilinear evolution equations. Math. Z. 2002, 242: 251–278. 10.1007/s002090100332
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
Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1988, 41: 891–907. 10.1002/cpa.3160410704
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-287