Blow-up criteria for Boussinesq system and MHD system and Landau-Lifshitz equations in a bounded domain

In this paper, we prove some blow-up criteria for the 3D Boussinesq system with zero heat conductivity and MHD system and Landau-Lifshitz equations in a bounded domain.

Next, we consider the following 3D density-dependent MHD equations: (1.17) For this problem, in [5], Wu proved that if the initial data ρ 0 , u 0 , and b 0 satisfy for some (π 0 , g) ∈ H 1 × L 2 , then there exists a positive time T * and a unique strong solution (ρ, u, b) to the problem (1.12)-(1.17) such that And when b = 0, Kim [2] proved the following regularity criterion: Here L s w denotes the weak-L s space and L ∞ w = L ∞ . The aim of this paper is to refine (1.20), we will prove Theorem 1.3. Let ρ 0 , u 0 , and b 0 satisfy (1.18). Let (ρ, u, b) be a strong solution of the problem (1.12)-(1.17) in the class (1.19). If u satisfies one of the following two conditions: with 0 < T < ∞, then the solution (ρ, u, b) can be extended beyond T > 0.
Finally, we consider the 3D Landau-Lifshitz system: (1.25) Carbou and Fabrie [6] showed the existence and uniqueness of local smooth solutions. When Ω := R n (n = 2, 3, 4), Fan and Ozawa [7] proved some regularity criteria. The aim of this paper is to prove a logarithmic blow-up criterion for the problem (1.23)-(1.25) when Ω is a bounded domain. We will prove

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and 0 < T < ∞, then the solution can be extended beyond T > 0.
In the following section 2, we give some preliminary Lemmas which will be used in the following sections. The proof of Theorem 1.1 of problem (1.1) -(1.5) will be given in section 3. The new regularly criterion of Theorem 1.2 for the 3D MHD problem (1.7) -(1.11) will be proved in section 4. In section 5 is the proof of the Theorem 1.3, and in the next section 6 we give the main proof of final Theorem 1.4.

Preliminary Lemmas
In the following proofs, we will use the following logarithmic Sobolev inequality [8]: (2.1) and the following three lemmas.

Proof.
When Ω := R 3 , (2.4) has been proved in Ogawa [12]. When Ω is a bounded domain in R 3 . We can definef Then we have [10, Page 71] and it is obvious that

Thus (2.4) is proved.
Finally, when b satisfies b · ν = 0 on ∂Ω, we will also use the identity for any sufficiently smooth vector field b.

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. Since it is easy to prove that the problem (1.1) -(1.5) has a unique local-in-time strong solution, we omit the details here. We only need to establish a priori estimates. First, thanks to the maximum principle, it follows from (1.1) and (1.3) that Testing (1.2) by u and using (1.1) and (3.1), we see that Applying curl to (1.2) and setting ω := curl u, we find that Testing (3.3) by ω and using (1.1) and (3.1), we infer that and y(t) := sup u H 3 for any 0 < t 0 ≤ t ≤ T and C 0 is an absolute constant.

Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2, we only need to prove a priori estimates. First, testing (1.8) by u and using (1.7), we see that Testing (1.9) by b and using (1.7), we find that Summing up (4.1) and (4.2), we get the well-known energy inequality with the same y and ǫ as that in (3.5).

Proof of Theorem 1.4
This section is devoted to the proof of Theorem 1.4, we only need to establish a priori estimates.
First, using the formula a × (b × c) = (a · c)b − (a · b)c, and the fact that |d| = 1 implies d∆d = −|∇d| 2 , we have the following equivalent equation Testing (6.1) by d t and using (a × b) · b = 0 and d · d t = 0, we get Testing (1.23) by −∆d t and using |d| = 1, we find that for any 0 < δ < 1. Here we have used the Gagliardo-Nirenberg inequalities: Applying ∂ i to (1.23), we get Testing the above equation by ∆∂ i d, summing over i, and using (6.4) and (6.5) and |d| = 1, we obtain (6.6) Plugging (6.6) into (6.3) and taking δ small enough, we have L q 1 + log(e + ∇d L q ) ∆d 2 L 2 log(e + ∇d L q ) ≤ C + C ∇d d H 3 for any 0 < t 0 ≤ t ≤ T and C 0 is an absolute constant.