On Regularity Criteria via Pressure for the 3D MHD Equations in a Half Space

In this paper, we give regularity criteria in terms of the magnetic pressure in Lorentz spaces.


Introduction
We study the regularity issues for suitable weak solutions ðu, b, πÞ: Q T ⟶ ℝ 3 + × ℝ 3 + × ℝ of 3D incompressible magnetohydrodynamic (MHD) equations Here, u is the fluid flow, b is the magnetic vector field, and π = p + ðjbj 2 /2Þ is the total scalar pressure. We consider equation (1) with boundary conditions defined as follows: either where n is the outward unit normal vector along boundary ∂ℝ 3 + .
In pioneering works [1,2], it has been shown that global-in time weak solutions to the MHD equations exist in finite energy space and strong solutions can exist locally-in time. In other words, the weak solutions exist globally in time; however, if a weak solution ðu, bÞ are furthermore in L ∞ ð0, T ; H 1 ðΩÞÞ, they become regular. The regular solution means that kuk L ∞ ðQ T Þ + kbk L ∞ ðQ T Þ < ∞. The uniqueness and regularity of weak solutions to (1) have been left the question open. The authors in [3], very recently, the existence of global weak solutions to the 3D MHD equations via new energy control methods are inspired of a recent work [4]. On the other hand, for nonuniqueness, the author in [5] nonunique weak solutions in Leray-Hopf class are constructed for (1) in a whole space based on appreciated convex integration framework developed in a recent work of Buckmaster and Vicol [6]. In the regularity theory of weak solutions to fluid equations, the role of the pressure is very important (see [7,8]); in particular, it is a more important issue for the boundary value problems. In present paper, we obtain the scaling invariant regularity criterion by focusing on the (magnetic) pressure function.
In this respect, the main results in the present paper are stated as follows.
Then, there exists a constant ε > 0 such that uðx, tÞ is a regular solution on ð0, T provided that one of the following two conditions holds: (A) Under the boundary condition (B2), π ∈ L p,∞ ð0, T ; L q,∞ ðℝ 3 + ÞÞ and π k k L p,∞ 0,T;L q,∞ ℝ 3 (B) Under the boundary conditions (B1) or (B2), Remark 2. Theorem 1 is worth to extend the results of Theorem 4.1 in [28] to the Lorentz space in ℝ 3 + . The result of Theorem 1 is naturally expandable for the n-dimensional half space with aid of Sobolev embedding and Calderon-Zygmund inequalities.
Remark 3. Unlike the results in [29], Theorem 1 is valuable as a result of considering boundary conditions. Remark 4. In light of the approach in [30], under the boundary conditions (B2), we can show the regularity condition of weak solutions to (1) with one component of the gradient of pressure, namely, Remark 5. In part (B) of Theorem 1, unfortunately, it does not obtain a similar result as (A) due to the difficulty of controlling the pressure function from the complexity of mixed term for w + and w − (see Remark 11).
For the Navier-Stokes equations with boundary data (B1) or (B2), Theorem 1 immediately implies. Corollary 6. Suppose that ðu, pÞ is a weak solution to the Navier-Stokes equations. Then, there exists a constant ε > 0 such that uðx, tÞ is a regular solution on ð0, T provided that one of the following two conditions holds: The proof of Corollary 6 is same to that in [31] and thus it is omitted.

Notations and Some Auxiliary Lemmas
For p ∈ ½1,∞, the notation L p ð0, T ; XÞ stands for the set of measurable functions f ðx, tÞ on the interval ð0, TÞ with values in X and k f ð⋅ ,tÞk X belonging to L p ð0, TÞ. The space W k,2 ðΩÞ is denoted the standard Sobolev space. For a func- C is a generic constant. We recall first the definition of weak solutions.
Definition 7 (weak solutions). The vector-valued function ðu, bÞ is called a weak solution of (1) on ð0, TÞ × ℝ 3 + if it satisfies the following conditions: Advances in Mathematical Physics Next, we give some basic facts. For p, q ∈ ½1,∞, we define And thus, Followed in [32], the Lorentz space L p,q ðℝ 3 + Þ may be defined by real interpolation methods that is, We list some lemmas for our analysis.

Proof of Theorems: Half Space Case
Proof of Theorem 1. We rewrite equation (1) with w + = u + b and w − = u − b: Part (A): multiplying both side of (19) by w + jw + j 2 , integrating by parts with the divergence-free condition, we conclude that Using the integration by parts and Hölder inequality, we have

Advances in Mathematical Physics
By means of the Hölder, interpolation, and Sobolev embedding inequalities in the Lorentz spaces, On the other hand, for a magnetic pressure, following the approach of Theorem 2.1 in [36], it is easy to check that With the help of the Hölder inequality with estimates (22) and (23), we infer that And thus, estimate (20) becomes Similarly, we have Summing (39) and (40), we obtain d dt Let NðtÞ ≔ kw + k 4 Applying Lemma 10 (with a = b = 2, c 0 = 4), we have where we use the following estimate in [37]: Since the pair ðp κ , q κ Þ also meets 2/p κ + 3/q κ = 2, using estimate (29), (28) becomes And then integrating with respect to time, we get or equivalently, Due to Lemma 9, we complete the Proof of Theorem 1 under the assumption (A) in Theorem 1.
Part (B): for this, we use the argument in [16], which seems like simple method to deal with the pressure term. 4 Advances in Mathematical Physics Multiplying both side of (19) by w + jw + j 3r−4 , we conclude that for r ≥ 1, On the other hand, Note that 0 ≤ I ≤ a and 0 ≤ I ≤ b; then, I ≤ ffiffiffiffiffi ab p . Combining (34) and (35), we get Due to we can know that d dt In a similar fashion, if you do it for equation (20), we have d dt After summing up (38) and (39), using the Sobolev embedding and Young's inequality, we obtain d dt As the previous way, it allows us to finish the Proof of Theorem 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares no conflict of interest.