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

Authors: Jae-Myoung Kim
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

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


---

## Body

## 1. Introduction

We study the regularity issues for suitable weak solutionsu,b,π:QT⟶ℝ+3×ℝ+3×ℝ of 3D incompressible magnetohydrodynamic (MHD) equations
(1)∂u∂t−Δu+u⋅∇u−b⋅∇b=−∇p+b22,∂b∂t−Δb+u⋅∇b−b⋅∇u=0onΩT≔ℝss+3×0,T,divu=divb=0,ux,0=u0xandbx,0=b0x.Here,u is the fluid flow, b is the magnetic vector field, and π=p+b2/2 is the total scalar pressure. We consider equation (1) with boundary conditions defined as follows: either
(2)B1u=0,b⋅n=0,∇×b×n=0,or
(3)B2u⋅n=0,∇×u×n=0,b⋅n=0,∇×b×n=0,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 u,b are furthermore in L∞0,T;H1Ω, they become regular. The regular solution means that uL∞QT+bL∞QT<∞. 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.Note that equation (1) has the following scale:
(4)uλ=λuλx,λ2t,bλ=λbλx,λ2t,πλ=λ2πλx,λ2t,λ>0.For the regularity conditions in Sobolev space, the results in terms of magnetic pressure and the gradient of magnetic pressure for (1) in ℝ3 were obtained by Zhou [9] with some magnetic field condition (see also [10–15]). After that, Duan [16] showed π∈Lp0,T;Lqℝ3 with 2/p+3/q=2,q>3/2 or ∇π∈Lp0,T;Lqℝ3 with 2/p+3/q=3,q>1.On the other hand, for the regularity criteria in Lorentz space, He and Wang [17] proved that a weak solution u,b for 3D MHD equations becomes regular under the scaling invariant conditions, the so-called Serrin’s conditions, u∈Lq,∞0,T;Lp,∞ℝ3 with 3/p+2/q≤1 and p>3 or ∇u∈Lq,∞0,T;Lp,∞ℝ3 with 3/p+2/q≤2 and p>3/2 (compared to [7, 18–26] for Navier-Stokes equations). In particular, for the magnetic pressure, Suzuki [24, 25] proved the regularity criteria to the Navier-Stokes equations in the Lorentz space under the assumption for the pressure via the truncation method introduced by [27]; namely, if π∈Lp,∞0,T;Lq,∞ℝ3 and πLp,∞0,T;Lq,∞ℝ3≤ε with 2/p+3/q=2,5/2<q<∞ or ∇π∈Lp,∞0,T;Lq,∞ℝ3 and ∇πLp,∞0,T;Lq,∞ℝ3≤ε with 2/p+3/q=3,5/3≤q<3, u,b is regular.In this respect, the main results in the present paper are stated as follows.Theorem 1.
Suppose thatu,b,π is a weak solution to (1) with the divergence-free initial data u0,b0∈H2ℝ+3∩W1,qℝ+3, q>3. Then, there exists a constant ε>0 such that ux,t is a regular solution on 0,T provided that one of the following two conditions holds:
(A)
Under the boundary condition (B2),π∈Lp,∞0,T;Lq,∞ℝ+3 and
(5)πLp,∞0,T;Lq,∞ℝ+3≤ε,with2/p+3/q=2,3/2<q<∞(B)
Under the boundary conditions (B1) or (B2),(6)∇π∈Lp0,T;Lq,∞ℝ+3with2/p+3/q=3,1<q<∞Remark 2.
Theorem1 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,
(7)∂x3π∈Lp0,T;Lq,∞ℝ+3with2/p+3/q≤2,1<q<∞.Remark 5.
In part (B) of Theorem1, 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), Theorem1 immediately implies.Corollary 6.
Suppose thatu,p is a weak solution to the Navier-Stokes equations. Then, there exists a constant ε>0 such that ux,t is a regular solution on 0,T provided that one of the following two conditions holds:
(A)
Under the boundary condition (B2),π∈Lp,∞0,T;Lq,∞ℝ+3 and
(8)πLp,∞0,T;Lq,∞ℝ+3≤ε,with2/p+3/q=2,3/2<q<∞(B)
Under the boundary conditions (B1) or (B2),∇π∈Lp,∞0,T;Lq,∞ℝ+3 and
(9)∇πLp,∞0,T;Lq,∞ℝ+3≤ε,with2/p+3/q=3,1<q<∞The proof of Corollary6 is same to that in [31] and thus it is omitted.


## 2. Notations and Some Auxiliary Lemmas

Forp∈1,∞, the notation Lp0,T;X stands for the set of measurable functions fx,t on the interval 0,T with values in X and f⋅,tX belonging to Lp0,T. The space Wk,2Ω is denoted the standard Sobolev space. For a function fx,t, Ω⊂ℝ3, we denote fLx,tp,qΩ×I=fLtqI;LxpΩ=fLxpΩLtqI. C is a generic constant.We recall first the definition of weak solutions.Definition 7 (weak solutions).
The vector-valued functionu,b is called a weak solution of (1) on 0,T×ℝ+3 if it satisfies the following conditions:
(1)
u,b∈L∞0,T;L2ℝ+3∩L20,T;H1ℝ+3(2)
divu=divb=0 in the sense of distribution(3)
For any functionψt,x∈C0∞0,T×ℝ+3 with divψ=0, there hold(10)∫0T∫ℝ+3u⋅ψt−∇u⋅∇ψ+∇ψ:u⊗u−b⊗bdxdt=0,∫0T∫ℝ+3b⋅ψt−∇b⋅∇ψ+∇ψ:u⊗b−b⊗udxdt=0Next, we give some basic facts. Forp,q∈1,∞, we define
(11)fLp,qℝ+3=p∫0∞αqx∈Ω:fx>αq/pdαα1/q,q<∞,supα>0αx∈ℝ+3:fx>α1/p,q=∞.And thus,(12)Lp,qℝ+3=f:fisameasurablefunctiononℝ+3andfLp,qℝ+3<∞.Followed in [32], the Lorentz space Lp,qℝ+3 may be defined by real interpolation methods
(13)Lp,qℝ+3=Lp1ℝ+3,Lp2ℝ+3α,q,with
(14)1p=1−αp1+αp2,1≤p1<p<p2≤∞,that is,
(15)L2p/p−1,2ℝ+3=L2ℝ+3,L6ℝ+33/2p,2.We list some lemmas for our analysis.Lemma 8. ([33]).
Assume1≤p1, p2≤∞, 1≤q1, q2≤∞, u∈Lp1,q1Ω, and v∈Lp2,q2Ω. Then, uv∈Lp3,q3Ω with 1/p3=1/p1+1/p2 and 1/q3≤1/q1+1/q2, and the inequality
(16)uvLp3,q3Ω≤CuLp1,q1ΩvLp2,q2Ωis valid.Lemma 9 ([20, 34, 35]).
LetT>0 and ϕ∈Lloc0,T be nonnegative function. Assume further that
(17)ϕt≤C0+C1∫0tμsϕsds+κ∫0tλs1−εϕs1+Aεds,∀0<ε<ε0,where κ,ε0>0 are constants, μ∈L10,T, and Aε>0 satisfies limε⟶0Aε/ε=c0>0. Then ϕ is bounded on 0,T if λL1,∞0,T<c0−1κ−1.Lemma 10 ([31]).
Assume that the pairp,q satisfies 2/p+3/q=a with a,q≥1 and p>0. Then, for every κ∈0,1 and given b,c0≥1, there exist pκ>0 and minq,b≤qκ≤maxq,b such that
(18)2pκ+3qκ=a,pκqκ=p1−κq+c0κb.


## 3. Proof of Theorems: Half Space Case

Proof of Theorem 1.
We rewrite equation (1) with w+=u+b and w−=u−b:
(19)wt+−Δw++w−⋅∇w+=∇π,divw+=0,wt−−Δw−+w+⋅∇w−=∇π,divw−=0,w+x,0=w0+andw−x,0=w0−.
Part (A): multiplying both side of (19) by w+w+2, integrating by parts with the divergence-free condition, we conclude that
(20)14ddt∫ℝ+3w+4dx+∫ℝ+3∇w+2w+2dx+12∫ℝ+3∇w+22dx=−∫ℝ+3w+⋅∇πw+2dx=I.
Using the integration by parts and Hölder inequality, we have(21)I=∫ℝ+3πw+⋅∇w+2dx≤C∫ℝ+3π2w+2dx+18∫ℝ+3∇w+2w+2dx.
By means of the Hölder, interpolation, and Sobolev embedding inequalities in the Lorentz spaces,(22)w+2L2q/q−1,2ℝ+3≤w+2L2ℝ+31−3/2qw+2L6,2ℝ+33/q≤Cw+2L2ℝ+31−3/2q∇w+2L2ℝ+33/2q.
On the other hand, for a magnetic pressure, following the approach of Theorem 2.1 in [36], it is easy to check that
(23)πLpℝ+3≤Cw+L2pℝ+32+w−L2pℝ+32,1<p<∞.
With the help of the Hölder inequality with estimates (22) and (23), we infer that
(24)∫ℝ+3π2w+2dx≤πLq,∞ℝ+3πL2q/q−1,2ℝ+3w+2L2q/q−1,2ℝ+3≤CπLq,∞ℝ+3w+2L2q/q−1,2ℝ+3+w−2L2q/q−1,2ℝ+3w+2L2q/q−1,2ℝ+3≤CπLq,∞ℝ+32q/2q−3w+2L2ℝ+32+w−2L2ℝ+32+18w+∇w+L2ℝ+32+w−∇w−L2ℝ+32.
And thus, estimate (20) becomes
(25)14ddt∫ℝ+3w+4dx+∫ℝ+3∇w+2w+2dx≤CπLq,∞ℝ+32q/2q−3w+2L2ℝ+32+w−2L2ℝ+32+18w+∇w+L2ℝ+32+w−∇w−L2ℝ+32.
Similarly, we have(26)14ddt∫ℝ+3w−4dx+∫ℝ+3∇w−2w−2dx≤CπLq,∞ℝ+32q/2q−3w+2L2ℝ+32+w−2L2ℝ+32+18w+∇w+L2ℝ+32+w−∇w−L2ℝ+32.
Summing (39) and (40), we obtain
(27)ddt∫ℝ+3w+4+w−4dx+12∫ℝ+3∇w+2w+2+∇w−2w−2dx≤CπLq,∞ℝ+32q/2q−3w+2L2ℝ+32+w−2L2ℝ+32.
LetNt≔w+L4ℝ+34+w−L4ℝ+34, and thus, (27) becomes
(28)ddtNt≤CπLq,∞ℝ+3pNt,p=2q2q−3.
Applying Lemma10 (with a=b=2,c0=4), we have
(29)πLqκ,∞ℝ+3pκ≤πLq,∞ℝ+3p1−κπL2,∞4κ≤CπLq,∞ℝ+3p1−κπL2ℝ+34κ≤CπLq,∞ℝ+3p1−κu2L2ℝ+34κ,where we use the following estimate in [37]:
(30)fLp,q2ℝ+3≤q1p1/q1−1/q2fLp,q1ℝ+3,1≤p≤∞,1≤q1<q2≤∞.
Since the pairpκ,qκ also meets 2/pκ+3/qκ=2, using estimate (29), (28) becomes
(31)ddtNt≤CπLqκ,∞ℝ+3pκu2L2ℝ+32≤CπLq,∞ℝ+3p1−κYt1+2κ.
And then integrating with respect to time, we get(32)Nt≤CN0+C∫0tπLq,∞ℝ+3p1−κNt1+2κds,or equivalently,
(33)w+tL4ℝ+34+w−tL4ℝ+34≤Cw0+L4ℝ+34+w0−L4ℝ+34+C∫0tπLq,∞ℝ+3p1−κw+L4ℝ+34+w−L4ℝ+341+2κds.
Due to Lemma9, 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. Multiplying both side of (19) by w+w+3r−4, we conclude that for r≥1,
(34)13r−2ddt∫ℝ+3w+3r−2dx+43r−43r−22∫ℝ+3∇w+3r−2/22dx+∫ℝ+3∇w+2w+3r−4dx=−∫ℝ+3∇π⋅w+w+3r−4dx≔II.
On the other hand,(35)II≤3r−4∫ℝ+3π∇w+w+3r−4dx≤23r−43r−2∫ℝ+3π2w+3r−4dx1/2∫ℝ+3∇w+3r−2/2dx1/2.
Note that0≤I≤a and 0≤I≤b; then, I≤ab. Combining (34) and (35), we get
(36)II≤C∫ℝ+3∇π⋅w+w+3r−4dx1/2∫ℝ+3p23r−3dx1/4∫ℝ+3∇w+3r−2/2dx1/4≤C∫ℝ+3∇πw+2+w−2dx3r−3/22/3∫ℝ+3πw+2+w−2dx3r−4/21/4+3r−43r−22∫ℝ+3∇w+3r−2/22dx.
Due to(37)∫ℝ+3πw+2+w−3r−4/2dx≤πL3r/22w++w−L3r/23r−4/2,∫ℝ+3∇πw+2+w−3r−3/2dx≤∇πLrw++w−L3r/23r−3/2,we can know that
(38)ddt∫ℝ+3w+3r−2dx+∫ℝ+3∇w+3r−2/22dx+∫ℝ+3∇w+2w+3r−4dx≤∇πLr2/3w++w−L3r/23r−2/2.
In a similar fashion, if you do it for equation (20), we have
(39)ddt∫ℝ+3w−3r−2dx+∫ℝ+3∇w−3r−2/22dx+∫ℝ+3∇w+2w+3r−4dx≤∇πLr2/3w++w−L3r/23r−2/2.
After summing up (38) and (39), using the Sobolev embedding and Young’s inequality, we obtain
(40)ddt∫ℝ+3w+3r−2+w−3r−2dx+∫ℝ+3∇w+3r−2/22+∇w−3r−2/22dx+∫ℝ+3∇w+2w+3r−4dx+∫ℝ+3∇w−2w−3r−4dx≤C∇πLr,∞2/3w+3r−2/2L6r/3r−2,12+w−3r−2/2L6r/3r−2,12≤C∇πLr,∞2/3w+3r−2/2L21−1/r∇w+3r−2/2L22/r+w−3r−2/2L21−1/r∇w−3r−2/2L22/r≤C∇πLr,∞2r/3r−3w+3r−2/2+w−3r−2/2L22+18∫ℝ+3∇w+3r−2/22+∇w−3r−2/22dx≤C∇πLr,∞2r/3r−3w+L3r−2ℝ+33r−2+w−L3r−23r−2+18∫ℝ+3∇w+3r−2/22+∇w−3r−2/22dx.
LetNt≔w+L3r−2ℝ+33r−2+w−L3r−2ℝ+33r−2, and then, (40) becomes
(41)Nt≤C∇πLq,∞ℝ+32q/3q−3Nt.
As the previous way, it allows us to finish the Proof of Theorem1.Remark 11.
In part (B) of Theorem1, adding the following conditions
(42)u∇u,b∇b∈L20,T;L2ℝ+3,we also can obtain ∇π∈Lp,∞0,T;Lq,∞ℝ+3 and
(43)∇πLp,∞0,T;Lq,∞ℝ+3≤ε,with2/p+3/q=3,1<q<∞(see [31] for a detailed proof). Condition (42) is too strong because it is regular condition of weak solutions to (1) (see, e.g., Lemma 7 in [38] or [39]).



---

*Source: 6954802-2022-07-26.xml*