Abstract and Applied AnalysisVolume 2012 (2012), Article ID 802876, 13 pagesdoi:10.1155/2012/802876

Research Article

Blow-Up Criteria of Smooth Solutions for the Cahn-Hilliard-Boussinesq System with Zero Viscosity in a Bounded Domain

Yong Zhou¹ and Jishan Fan²

¹Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China²Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

Received 30 November 2011; Accepted 2 February 2012

Academic Editor: D. Anderson

Copyright © 2012 Yong Zhou and Jishan Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove that a smooth solution of the 3D Cahn-Hilliard-Boussinesq system with zero viscosity in a bounded domain breaks down if a certain norm of vorticity blows up at the same time. Here, this norm is weaker than bmo-norm.

1. Introduction

Let be a bounded, simply connected domain with smooth boundary , and is the unit outward normal vector to . We consider the following Cahn-Hilliard-Boussinesq system with zero viscosity in [1]: where , the fluid velocity field, , the temperature, , the order parameter, , and the pressure are the unknowns. . is a chemical potential. is the double well potential.

When , (1.1) and (1.2) are Euler equations. Ogawa-Taniuchi [3] proved that a smooth solution breaks down if a certain norm of vorticity blows up at the same time. Here this norm is weaker than bmo-norm.

Before presenting our results, we introduce some function spaces and some notations.

Let be the Littlewood-Paley dyadic decomposition of unity that satisfies for any , where denotes the ball centered at of radius .

We first recall the space of Besov type introduced by Vishik [2].

Definition 1.1 (see [2]). Let (≥1) be a nondecreasing function on . is introduced by the norm where and denote the Fourier and inverse Fourier transforms.

We note that

Now let us introduce the space of bmo type in [3].

Definition 1.2. Let be a positive function on , and is a domain with .(1) is the space defined as a set for an function such that where (2)On we define as restrictions of the above space : where is the restriction of on . The norm of this space is defined by

In particular if , we write and . Obviously, if .

Definition 1.3. Let (≥1) be a nondecreasing function on where where

We note that these spaces have the following relations:

From now on we impose the following assumptions.

Assumption 1.4. Let .(H1) is a positive and nondecreasing function on satisfying(H2) For all there exists such that(H3) is a nonincreasing function on .Then Ogawa-Taniuchi [3] proved the following blowup criterion: where and for all and or for all and all with in and in . is a small positive constant depending only on .Since , we see By this inequality and (1.17), (1.20) implies The aim of this paper is to prove a similar result for the problem (1.1)–(1.7). It is easy to show that the problem (1.1)–(1.7) has a unique local smooth solution. Thus, we omit the details here. However, the global regularity is still open, which this paper aims to study. We will prove that.

Theorem 1.5. Let in , on . Suppose that is a local smooth solution to the problem (1.1)–(1.7) on . If is maximal, then (1.20) and (1.21) hold true.

In Section 2, we will give some preliminaries. Section 3 is devoted to the proof of Theorem 1.5.

2. Preliminaries

Lemma 2.1 (see [4]). For any with in and on , there holds for any and .

Lemma 2.2 (see [5]). Let . (1) If , then (2)If and , then for ,

Lemma 2.3 (see [3]). For all , there holds for all with in and on .

Lemma 2.4 (see [3]). There exists a constant depending only on such that the following holds. For all , and for all with in and in , there exists constant depending only on and such that for all with in and on .

Lemma 2.5 (see [6]). Let be nonnegative function on with , let be a positive and nondecreasing for and Assume that and Then, .

3. Proof of Theorem 1.5

In this section, all the integrations with respect to spacial variable are on the domain (we omit it for simplicity).

Since the proof of (1.21) is similar to that of (1.20), we only need to prove (1.20). By the standard argument of continuation of local solutions, it suffices to prove that if then

First, by the maximum principle, it follows from (1.2) and (1.3) that

Testing (1.3) by , using (1.2), we see that whence

Testing (1.4) by , using (1.2) and (1.6), we find that which gives

Testing (1.1) and (1.4) by and , respectively, using (1.2), (1.5), (1.6), and (3.3), we infer that which yields

In the following calculations, we will use the following Gagliardo-Nirenberg inequality:

It follows from (1.5), (3.11), (3.7), (3.9) and (3.12) that which implies

Testing (1.4) by , using (3.9), (3.10), and (3.14), we deduce that which leads to Testing (1.3) by , using (1.2) and (1.6), we infer that Equations (1.3) and (1.6) can be rewritten as

By the classical regularity theory of elliptic equation, using (3.10), we get

Now using the following Gagliardo-Nirenberg inequalities: we obtain

Taking to (1.1), using (1.2) and (1.5), we have

Taking to (3.22), testing by , using (1.2) and (1.6), we derive

Using (1.2), (1.6), Lemma 2.2, and can be bounded as follows:

Using Lemma 2.2, can be bonded as follows

Inserting the above estimates into (3.23), we obtain for any .

Testing (1.1) by , using (1.2), (1.6), (3.3), (3.10), and (3.16), and noting that we reach Here, we have used the Gagliardo-Nirenberg inequality:

Taking to (1.3), we see that

Testing the above equation by , using (1.2), (1.6) and (3.3), we have whence

By the classical regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6), (3.16), and (3.10) that which implies Here, we have used the Gagliardo-Nirenberg inequality:

Taking to (1.4) and (1.5), we have

Testing (3.36) by , using (3.37), (1.4), (1.6), (3.16), and (3.28), we arrive at whence Here, we have used

Combining (3.17), (3.21), (3.26), (3.32), (3.34), and (3.39), using Lemmas 2.1, 2.3, and 2.5, we conclude that It follows from (1.3), (3.41), and (3.42) that It follows from (1.4), (1.5), (3.41), and (3.43) that

This completes the proof.

Acknowledgment

This paper is partially supported by the Zhejiang Innovation Project (Grant no. T200905), the ZJNSF (Grant no. R6090109), and the NSFC (Grant no. 11171154).

References

F. Boyer, “Mathematical study of multi-phase flow under shear through order parameter formulation,” Asymptotic Analysis, vol. 20, no. 2, pp. 175–212, 1999.

M. Vishik, “Incompressible flows of an ideal fluid with unbounded vorticity,” Communications in Mathematical Physics, vol. 213, no. 3, pp. 697–731, 2000.

T. Ogawa and Y. Taniuchi, “On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain,” Journal of Differential Equations, vol. 190, no. 1, pp. 39–63, 2003.

J. P. Bourguignon and H. Brezis, “Remarks on the Euler equation,” Journal of Functional Analysis, vol. 15, pp. 341–363, 1974.

A. B. Ferrari, “On the blow-up of solutions of the 3-D Euler equations in a bounded domain,” Communications in Mathematical Physics, vol. 155, no. 2, pp. 277–294, 1993.

P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass, USA, 2nd edition, 1982.