A regularity criterion for the Cahn-Hilliard-Boussinesq system with zero viscosity
© Ma et al.; licensee Springer. 2014
Received: 20 May 2014
Accepted: 15 July 2014
Published: 18 August 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.
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.
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.
with , and .
2 Proof of Theorem 1.1
for any .
This completes the proof.
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.
- Boyer F: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 1999, 20: 175–212.MathSciNetMATHGoogle Scholar
- 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/s002090100332MathSciNetView ArticleMATHGoogle Scholar
- 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.2008.06.041MathSciNetView ArticleMATHGoogle Scholar
- Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1988, 41: 891–907. 10.1002/cpa.3160410704MathSciNetView ArticleMATHGoogle Scholar
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