# Global Well-Posedness and Analyticity for the Three-Dimensional Incompressible Nematic Liquid Crystal Flows in Scaling Invariant Spaces

Authors: Xuanjiang Liu; Fuyi Xu; Peng Fu
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The Cauchy problem for the three-dimensional incompressible flows of liquid crystals in scaling invariant spaces is considered. In this work, we exhibit three results. First, we prove the global well-posedness of mild solution for the system without the supercritical nonlinearity∇d2d when the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. Our second result is a proof of the global existence of mild solution in the time dependent spaces for the system including the term ∇d2d for small initial data. Lastly, we also get analyticity of the solution.


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## Body

## 1. Introduction and Main Results

Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid and those of a solid crystal. A liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many different types of liquid crystal phases, which can be distinguished based on their different optical properties. One of the most common liquid crystal phases is the nematic, where the molecules have no positional order, but have long-range orientational order. Nematic liquid crystals are aggregates of molecules which possess same orientational order and are made of elongated, rod-like molecules (see [1–3]). The continuum theory of liquid crystals was developed by Ericksen [4] and Leslie [5] during the period of 1958 through 1968, and for more details, see also the book by de Gennes [6]. Since then, there have been remarkable research developments in liquid crystals from both theoretical and applied aspects. When the fluid containing nematic liquid crystal materials is at rest, we have the well-known Ossen-Frank theory (for static nematic liquid crystals, see Hardt-Lin-Kinderlehrer [7]) on the analysis of energy minimal configurations of nematic liquid crystals. In this paper, we mainly study two simplified versions of the hydrodynamics of nematic liquid crystals, but still retain most of the interesting mathematical properties of the original Ericksen-Leslie model (see [4, 5]).In 1989, Lin [8] first proposed the following a simplified three-dimensional Ericksen-Leslie equation modeling incompressible liquid crystal flows
(1)∂tu+u·∇u−μΔu+∇Π=−θdiv∇de∇d,∂td+u·∇d=νΔd+∇d2d,divu=0,u,dt=0=u0,d0,whereu∈ℝ3 is the velocity and d∈S2 (the unit sphere in ℝ3) is the unit-vector field that represents the macroscopic molecular orientations. The scalar function Π∈ℝ is the pressure. The positive constants μ,θ,ν stand for viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time or the Deborah number for the molecular orientation field, respectively. The symbol ⊗ denotes the Kronecker tensor product such that u⊗u=uiuj1≤i,j≤3 and ∇de∇d denotes a matrix whose ij-th entry is ∂xid·∂xjd1≤i,j≤3. Indeed, ∇de∇d=∇dΤ∇d, where ∇dΤ denotes the transpose of the 3×3 matrix ∇d. We set θ=1 since their exact values do not play any role in our analysis.For system (1), the appearance of the nonlinear term ∇d2d with the restriction d=1 causes significant mathematical difficulties. In 1990s, Lin and Liu [9, 10] introduced another simplified three-dimensional Ericksen-Leslie equation modeling incompressible liquid crystal flows:
(2)∂tu+u·∇u−μΔu+∇Π=−div∇de∇d,∂td+u·∇d=νΔd−fd,divu=0,u,dt=0=u0,d0,where fd is a vector valued, smooth, and bounded function defined for all d∈ℝ3.Whend is a constant vector field, systems (1)-(2) reduce to the three-dimensional incompressible Navier-Stokes equations, which is an extremely important system to describe incompressible fluids. It has attracted great interests among many researchers, and there have been many important developments. Leray [11] and Hopf [12] showed the global existence of weak solutions. Fujita and Kato [13] established the local well-posedness for large initial data and the global well-posedness of strong solutions for small initial data in Sobolev space. Similar results have been established in Lnℝn by Kato [14], in critical Besov space Bp,∞−1+n/pℝn1<p<∞ by Cannone [15], and in the larger space BMO−1 by Koch and Tataru [16]. In 2011, Lei and Lin [17] proved global well-posedness results in a new space X−1 (see Definition 4) if u0X−1<μ. Based on it, Benameur [18] proved a large time decay to the Lei-Lin solution, and Bae [19] presented analyticity of the solution, respectively.In the past several decades, there are many fruitful results on the analysis of System (1). In certain Besov spaces, Li-Wang [20] obtained the local strong solution with large initial data and the global strong solution with small data. Hineman and Wang [21] established the global well-posedness of system (1) in dimensions three with small initial data u0,∇d0 in Luloc3, where Luloc3ℝ3 is the space of uniformly locally L3-integrable functions. We would like to mention that Wang [22] has recently obtained the global (or local) well-posedness of system (1.1) for initial data u0,d0 belonging to possibly the largest space BMO−1×BMO with ∇·u0=0, which is a invariant space under parabolic scaling associated with system (1), with small norms. Tan and Yin [23] established local well-posedness with large initial data and the existence of global strong solution to system (1) with small initial-boundary condition. For system (2), there are also lots of important conclusions when the function fd=1/ε21−d2d or fd is identically zero (see [8–10, 24–26]). Especially, when fd=0, although the case is physically irrelevant, one believes it is of interest from the analysis point of view (see [9]). Lin and Liu [9] established the global existence of classical solutions under the additional assumption that the initial data are small in a suitable sense. Hu-Wang [25] obtained the existence and uniqueness of the global strong solution with small initial data. It should be emphasized that the norms in corresponding spaces of the initial data in all these works mentioned above for system (2) are smaller than the viscosity coefficients μ and ν multiplied by a tiny positive constant ε. Then, an interesting question arises, namely, whether it is possible to establish global existence of solutions for the Cauchy problem to the hydrodynamics of nematic liquid crystals in X−1×X0, provided that the norms of the initial data in X−1×X0 are bounded exactly by the minimal value of the viscosity coefficients. The goal of this paper is to give a positive answer to this question for system (1.2) when fd=0. On the other hand, for the more complicated system (1) including the super critical nonlinearity ∇d2d, we also prove the the global existence of mild solution in the time dependent spaces for small initial data in X−1×X0. Furthermore, we also prove analyticity of mild solution to system (1). Compared with the known results for the incompressible Navier-Stokes equations [17–19], the main difficulty of system (1) is much more complicate nonlinear system due to the super critical nonlinearity ∇d2d in the transported heat flow of harmonic map equation and the strong coupling nonlinear term div∇de∇d in the momentum equation. In particular, in order to avoid trouble by directly taking Fourier transformations for these nonlinear terms, we exploit some important nonlinear estimates in some time dependent spaces.Our first main result on system (2) then reads as follows:Theorem 1.
Letfd=0 in system (2). Suppose that u0,d0∈X−1×X0 satisfy
(3)u0X−1+d0X0<minμ,ν.Then, system (2) has a unique global-in-time solution
(4)u,d∈C0,+∞;X−1×X0∩L10,+∞;X1×X2.Moreover,(5)sup0≤t<∞utX−1+dtX0+μ−u0X−1−d0X0∫0∞uτX1dτ+ν−u0X−1−d0X0∫0∞dτX2dτ≤u0X−1+d0X0.Our second main result in the time dependent spaces on system (1) is the following theorem:Theorem 2.
Suppose thatu0,d0∈X−1×X0. There exists a small enough constant η>0 such that if
(6)u0X−1+d0X0≤η.Then, system (1) admits a unique global-in-time solution
(7)u,d∈Edef=L~∞X−1×L~∞X0∩L~1X1×L~1X2.Our third main result on the analyticity of solution for system (1) is the following theorem.Theorem 3.
Under the assumptions in Theorem2, then the solution for system (1) is analytic in the sense that
(8)uL~∞etΛX−1+μuL~1etΛX1+dL~∞etΛX0+νdL~1etΛX2≤Cu0X−1+d0X0.The rest of the paper unfolds as follows. In the next section, we recall some basic notions and useful properties of function spaces. In Section3, we will present the proof of Theorem 1. Section 4 is devoted to the proof of Theorem 2. At last, we show the analyticity of the solution to system (1).


## 2. Preliminaries

In this section, we introduce some common notations and basic theories about function spaces, and present some auxiliary lemmas.Definition 4.
Forα∈ℝ, we define the function spaces Xα to be
(9)Xα=f∈D′ℝ3:fXα<∞with
(10)fXαdef=∫ℝ3ξαf^dξ,where D′ℝ3 represents the space of distributions and f^ represents the Fourier transformation of f.By a straight computation, we have(11)lnlflxXαfxXα=lnllα=α.For alll>0, systems (1)-(2) are invariant under the following transformations:
(12)u~=lul2t,lx,Π~=l2Πl2t,lx,d~=dl2t,lx.We say that a function space is the initial critical space for systems (1)-(2) if the associated norm is invariant under the transformation u0,d0⟶u~0,d~0≔lu0lx,d0lx for all l>0. Obviously, X−1×X0 is the critical space for systems (1.1)-(1.2).In what follows, we present the following time dependent function spaces.Definition 5.
Forα∈ℝ, we define the function spaces L~∞Xα and L~1Xα to be, respectively:
(13)L~∞Xα=f∈D′ℝ+×ℝ3:fL~∞Xα<∞,with
(14)fL~∞Xαdef=∫ℝ3sup0≤t<∞ξαf^t,ξdξ,L~1Xα=f∈D′ℝ+×ℝ3:fL~1Xα<∞,with
(15)fL~1Xαdef=∫ℝ3∫0∞ξαf^t,ξdtdξ.Definition 6.
Forα∈ℝ, we define the function spaces L~∞etΛXα and L~1etΛXα to be, respectively:
(16)L~∞etΛXα=f∈D′ℝ+×ℝ3:etΛfL~∞Xα<∞,with
(17)etΛfL~∞Xαdef=∫ℝ3sup0≤t<∞etξξαf^t,ξdξ,L~1etΛXα=f∈D′ℝ+×ℝ3:etΛfL~1Xα<∞,with
(18)etΛfL~1Xαdef=∫ℝ3∫0∞etξξαf^t,ξdtdξ.Here,etΛ is a Fourier multiplier whose symbol is given by etξ .We next present some important properties of the spaces mentioned the above, which will be frequently used in this paper.Lemma 7 [27].
(i)
Letf,g∈X0, then fg∈X0 and(19)fgX0≤fX0gX0.(ii)
Letf∈Xα∩Xα+2, then f∈Xα+1 and(20)fXα+1≤fXα1/2gXα+21/2.(iii)
Letf∈X−1∩X1,g∈X−1∩X1, then fg∈X0 and(21)fgX0≤12fX−1gX1+fX1gX−1.
In particular, iff=g, we have
(22)f2X0≤fX−1fX1.(iv)
IfDβf∈Xα, then f∈Xα+β and(23)DβfXα=fXα+β.For the time dependent function spaces, we have the following properties.Lemma 8 [27].
(i)
Letf∈L~∞X0,g∈L~1X0, then fg∈L~1X0 and(24)fgL~1X0≤fL~∞X0gL~1X0.(ii)
Letf∈L~∞X−1∩L~1X1,g∈L~∞X−1∩L~1X1, then fg∈L~1X0 and(25)fgL~1X0≤12fL~∞X−1gL~1X1+fL~1X1gL~∞X−1.
In particular, iff=g we have
(26)f2L~1X0≤fL~∞X−1fL~1X1.(iii)
IfDβf∈L~∞Xα, then f∈L~∞Xα+β and(27)DβfL~∞Xα=fL~∞Xα+β.(iv)
IfDβf∈L~1Xα, then f∈L~1Xα+β and(28)DβfL~1Xα=fL~1Xα+β.Lemma 9 [27].
(i)
Letf∈L~∞etΛX0,g∈L~1etΛX0, then fg∈L~1etΛX0, and there exists a positive constant C such that(29)fgL~1etΛX0≤CfL~∞etΛX0gL~1etΛX0.(ii)
Letf∈L~∞etΛX−1∩L~1etΛX1,g∈L~∞etΛX−1∩L~1etΛX1, then fg∈L~1etΛX0, and there exists a positive constant C such that(30)fgL~1etΛX0≤CfL~∞etΛX−1gL~1etΛX1+fL~1etΛX1gL~∞etΛX−1.
In particular, iff=g we have
(31)f2L~1etΛX0≤CfL~∞etΛX−1fL~1etΛX1.(iii)
IfDβf∈L~∞etΛXα, then f∈L~∞etΛXα+β and(32)DβfL~∞etΛXα=fL~∞etΛXα+β.(iv)
IfDβf∈L~1etΛXα, then f∈L~1etΛXα+β and(33)DβfL~1etΛXα=fL~∞etΛXα+β.Lemma 10 [27].
Letu satisfy
(34)∂tu−κΔu=f,ut=0=u0,
withκ>0, f∈L~1Xα and u0∈Xα. Then, for all T>0 and a positive constant C, the following a priori estimate is fulfilled:
(35)uL~∞Xα+κuL~1Xα+2≤Cu0Xα+fL~1Xα.Lemma 11 [27].
Letu be a solution to system (34) with κ>0, f∈L~1etΛXα, and v0∈Xα. Then, for all T>0 and a positive constant C, the following a priori estimate is fulfilled:
(36)uL~∞etΛXα+κuL~1etΛXα+2≤Cu0Xα+fL~1etΛXα.


## 3. The Proof of Theorem 1

In this section, we prove the proof of Theorem 1 and divide it into several steps.Step 1.
The approximate solution sequence. Letρ be the standard mollifier in ℝ3: ρ∈C0∞, 0≤ρ≤1, ∫ρxdx=1. For λ>0, let ρλx=λ−3ρλ−1x and u0λ=ρλ∗u0,d0λ=ρλ∗d0. For u0∈X−1,d0∈X0, since ρ^ξ≤∫ρxdx=1, one has
(37)u0λX−1=∫ξ−1u0^ξρ^λξdξ≤u0X−1,d0λX0=∫d0^ξρ^λξdξ≤d0X0,u0λL∞≲∫ξρ^λξξ−1u0^ξdξ≤Cλu0X−1,d0λL∞≲∫ρ^λξd0^ξdξ≤Cλd0X0.Thus, by slight modifications of the proof of Theorem 3 in [28] or Theorem 2 in [29], we can obtain a unique local existence for smooth solution uλ,dλ on some time internal 0,Tλ for the liquid crystal flow system. Furthermore,
(38)uλX−1=∫ξ≤1ξ−1vλ^dξ+∫ξ>1ξ−2−Δvλ^dξ≤CvλH1,dλX0=∫ξ≤1vλ^dξ+∫ξ>1ξ−2−Δvλ^dξ≤CvλH2∫0TλvλtX1dt=∫0Tλ∫ξ≤1ξvλ^dξdt+∫0Tλ∫ξ>1ξ−2−Δ3vλ^dξdt≤C∫0TλvλH3ds,∫0TλdλtX2dt=∫0Tλ∫ξ≤1ξ2dλ^dξdt+∫0Tλ∫ξ>1ξ−2−Δ4vλ^dξdt≤C∫0TλvλH4ds.Therefore,uλ,dλ∈L∞0,Tλ;X−1∩X0∩L10,Tλ;X1∩X2.Step 2.
Global uniform estimates
Taking the Fourier transformations of system (2), we get
(39)∂tuλ^−iξΠλ^+μξ2vλ^=−iξuλ⊗uλ^−iξ∇dλe∇dλ^,∂tdλ^+νξ2dλ^=−uλ·∇dλ^,ξ·uλ^=0.It follows from (39) and Lemma 7(i) (iii) and (iv) that
(40)ddtuλX−1+dλX0+μuλX1+νdλX2≤uλ⊗uλX0+∇dλe∇dλX0+uλ·∇dλX0≤uλX−1uλX1+∇dλX−1∇dλX1+uλX−1∇dλX1+uλX1∇dλX−1≤uλX−1uλX1+dλX0dλX2+uλX−1dλX2+uλX1dλX0.Note that(41)uλtX−1+dλtX0<minμ,ν,for t∈0,δ with 0<δ<Tλ. Therefore,
(42)ddtuλX−1+dλX0≤0,fort∈0,δ.Thus, we obtain from (37)
(43)uλX−1+dλX0≤u0λX−1+d0λX0≤u0X−1+d0X0<minμ,ν.From a continuity argument in the time variable, we have(44)uλX−1+dλX0≤u0X−1+d0X0<minμ,ν,for allt∈0,Tλ. From (40), we get for ∀t∈0,Tλ(45)uλX−1+dλX0+μ−u0X−1−d0X0uλL10,t;X1+ν−u0X−1−d0X0dλL10,t;X2≤u0X−1+d0X0,which together with (20) and (45) implies that
(46)∫0Tλ∇uλtL∞dt≤∫0TλuλtX1dt≤u0X−1+d0X0μ−u0X−1−d0X0,(47)∫0Tλ∇dλtL∞2dt≤∫0TλdλtX12dt≤∫0TλdλtX0dλtX2dt≤dλL∞0,Tλ;X0∫0TλdλtX2dt≤u0X−1+d0X02ν−u0X−1−d0X0,where we have used ∇gL∞=F−1iξg^L∞≤iξg^L1=gX1 by the definition of the inverse Fourier transformation.On the other hand, the standard energy method in [30] gives that
(48)uλtHs+dλtHs+1≤u0Hs+d0Hs+1expcs∫0Tλ∇uλsL∞+∇dλsL∞2ds.Putting (46) and (47) into (48), we obtain
(49)uλtHs+dλtHs+1≤u0Hs+d0Hs+1expcsminu0X−1+d0X0μ−u0X−1−d0X0,u0X−1+d0X02ν−u0X−1−d0X0,for all 0≤t<Tλ and all s>0. This implies that Tλ=∞. Moreover, we obtain the following global uniform estimates for
(50)sup0≤t<∞uλtX−1+dλtX0+μ−u0X−1−d0X0uλL10,∞;X1+ν−u0X−1−d0X0dλL10,∞;X2≤u0X−1+d0X0.The estimate (50) implies that there exists a subsequence of uλ,dλ (we will still denote it by uλ,dλ) such that as λ⟶0,
(51)uλ⇀uinL1ℝ+;X1,uλ⇀uweakly∗inL∞ℝ+;X−1,dλ⇀dinL1ℝ+;X2,dλ⇀dweakly∗inL∞ℝ+;X0,for some,
(52)u,d∈L∞ℝ+;X−1×X0∩L1ℝ+;X1×X2.Step 3.
Strong converges
By straight computations, we obtain(53)u0λ−u0X−1=∫ξ≤Mξ−1ζ^λξ−1u0^ξdξ+∫ξ>Mξ−1ζ^λξ−1u0^ξdξ≤2supη≤λMζ^η−1∫ξ<Mξ−1u0^ξdξ+∫ξ>Mξ−1ζ^λξ−1u0^ξdξ,d0λ−d0X0=∫ξ≤Mζ^λξ−1d0^ξdξ+∫ξ>Mζ^λξ−1d0^ξdξ≤2supη≤λMζ^η−1∫ξ<Md0^ξdξ+∫ξ>Mζ^λξ−1d0^ξdξ.By takingM=λ−1/2, and using ζ^0=∫ζxdx=1, we conclude that
(54)u0λ−u0X−1⟶0,d0λ−d0X0⟶0asλ⟶0.In what follows, we prove the strong convergences ofuλ and dλ. Similar to (40), we obtain
(55)ddtuλ1−uλ2X−1+dλ1−dλ2X0+μuλ1−uλ2X1+νdλ1−dλ2X2≤uλ1⊗uλ1−uλ2X0+uλ2⊗uλ1−uλ2X0+∇dλ1e∇uλ1−∇uλ2X0+∇dλ2e∇dλ1−∇dλ2X0+uλ1⊗∇dλ1−∇dλ2X0+uλ1−uλ2·∇dλ2X0≤12uλ1X1uλ1−uλ2X−1+12uλ1X−1uλ1−uλ2X1+12uλ2X1uλ1−uλ2X−1+12uλ2X−1uλ1−uλ2X1+12∇dλ1X1∇dλ1−∇dλ2X−1+12∇dλ1X−1∇dλ1−∇dλ2X1+12∇dλ2X1∇dλ1−∇dλ2X−1+12∇dλ2X−1∇dλ1−∇dλ2X1+12uλ1x1∇dλ1−∇dλ2X−1+12uλ1X−1∇dλ1−∇dλ2X1+12∇dλ2X1uλ1−uλ2X−1+12∇dλ2X−1uλ1−uλ2X1≤12uλ1X1uλ1−uλ2X−1+12uλ1X−1uλ1−uλ2X1+12uλ2X1uλ1−uλ2X−1+12uλ2X−1uλ1−uλ2X1+12dλ1X2dλ1−dλ2X0+12dλ1X0dλ1−dλ2X2+12dλ2X2dλ1−dλ2X0+12dλ2X0dλ1−dλ2X2+12uλ1X1dλ1−dλ2X0+12uλ1X−1dλ1−dλ2X2+12dλ2X2uλ1−uλ2X−1+12dλ2X0uλ1−uλ2X1,which together with (50) yieldsd/dtuλ1−uλ2X−1+dλ1−dλ2X0+μ−u0X−1−d0X0uλ1−uλ2X1+ν−u0X−1−d0X0dλ1−dλ2X2≤1/2uλ1X1+uλ2X1+dλ2X2uλ1−uλ2X−1+1/2dλ1X2+dλ2X2+uλ1X1dλ1−dλ2X0.(23)Thus, we obtain(56)uλ1−uλ2X−1+dλ1−dλ2X0≤u0λ1−u0λ1X−1+d0λ1−d0λ1X0exp2u0X−1+d0X0minμ,ν−u0X−1−d0X0,μ−u0X−1−d0X0∫0∞uλ1−uλ2X1dt+ν−u0X−1−d0X0∫0∞dλ1−dλ2X2dt≤u0λ1−u0λ1X−1+d0λ1−d0λ1X0exp2u0X−1+d0X0minμ,ν−u0X−1−d0X0.Combining (54) with (56), we conclude that uλ,dλ is a Cauchy sequence in L∞ℝ+;X−1×X0∩L1ℝ+;X1×X2 and the convergence in (51) is a strong one. In fact, (56) also yields the uniqueness of solutions in the space L∞ℝ+;X−1×X0∩L1ℝ+;X1×X2 under the assumption (1.1).Step 4.
Time continuity
To get the further time regularity ofut,x and dt,x, we come back to the Equation (39). We claim that ∂tuλ,∂tdλ are uniformly bounded in L1ℝ+;X−1×X0. In fact, form
(57)∂tuλ^+∇Πλ^−μΔuλ^=−∇·uλ⊗uλ^−∇·∇dλ⊙∇dλ^,∂tdλ^−νΔdλ^=−uλ·∇dλ^,ξ·uλ^=0,we have
(58)∫0∞∂tuλX−1dt≤μ∫0∞ΔuλX−1dt+∫0∞∇ΠλX−1dt+∫0∞∇·uλ⊗uλX−1dt+∫0∞∇·∇dλ⊙∇dλX−1dt≤C∫0∞uλX1dt+∫0∞uλ⊗uλX0dt+∫0∞∇dλ⊙∇dλX0dt≤C∫0∞uλX1dt+suptuλX−1∫0∞uλX1dt+suptdλX0∫0∞dλX2dt,
and(59)∫0∞∂tdλX0dt≤ν∫0∞dλX2dt+∫0∞uλ·∇dλX0dt≤C∫0∞dλX2dt+suptuλX−1∫0∞dλX2dt+suptdλX0∫0∞uλX1dt,which implies that
(60)∂tuλ,∂tdλ∈L1ℝ+;X−1×X0.(60) allows us to improve (52) and to finally conclude
(61)u,d∈Cℝ+;X−1×X0∩L1ℝ+;X1×X2,∂tu,∂td∈L1ℝ+;X−1×X0.


## 4. The Proof of Theorem 2

In this section, we show the proof of Theorem 2. Here, we only show the global uniform estimates foru,d in E, where the norm of E is defined by
(62)u,dE=uE1+dE2with
(63)uE1=uL~∞X−1+μuL~1X1,dE2=dL~∞X0+νdL~1X2.According tou0+d0≤η, for some sufficiently small η, there exists a positive constant M which will be chosen later such that
(64)u,dE≤Mη.Employing Lemmas8-10 and the boundness of the Fourier multiplier ℙ, we conclude that
(65)uL~∞X−1+μuL~1X1≤Cu0X−1+ℙ∇·u⊗u+∇d⊙∇dL~1X−1≤Cu0X−1+u⊗u+∇d⊙∇dL~1X0≤Cu0X−1+uL~∞X−1uL~1X1+∇dL~∞X−1∇dL~1X1≤Cu0X−1+uL~∞X−1uL~1X1+dL~∞X0dL~1X2≤Cη+M2η2,(66)dL~∞X0+νdL~1X2≤Cd0X0+u·∇dL~1X0+∇d2dL~1X0≤Cd0X0+uL~∞X−1∇dL~1X1+uL~1X1∇dL~∞X−1+dL~∞X0∇d2L~1X0≤Cd0X0+uL~∞X−1dL~1X2+uL~1X1dL~∞X0+dL~∞X0∇dL~∞X−1∇dL~1X1≤Cd0X0+uL~∞X−1dL~1X2+uL~1X1dL~∞X0+dL~∞X0dL~∞X0dL~1X2≤Cη+M2η2+M3η3.Combining (65) with (66), we deduce that
(67)u,dE≤Cη+M2η2+M3η3.TakingM=4C and then choosing η small enough such that
(68)M2η≤12,M3η2≤12,then we infer that
(69)u,dE≤12Mη,which implies that u,d∈L~∞X−1×L~∞X0∩L~1X1×L~1X2.At last, we consider the following iteration scheme:(70)uk+1t,x=u1−∫0te−μΔt−sℙuk·∇uk+∇·∇dke∇dkds,dk+1t,x=d1−∫0te−νΔt−suk·∇dk−∇dk2dkds,u1=e−μΔtu0,d1=e−νΔtd0.Based on (64), we can also get the global uniform estimates for uk,dk in E. Due to the similarity to the Steps 3–4, here, we omit the remaining part of the proof.


## 5. The Proof of Theorem 3

In this section, we show the proof of Theorem 3. Here, we only present a priori estimate (8) and skip the iteration step. Introducing the following function spaces
(71)E≝L~∞etΛX−1×L~∞etΛX0∩L~1etΛX1×L~1etΛX2.The norm ofE is defined by
(72)u,dE=uE1+dE2,with
(73)uE1=uL~∞etΛX−1+μuL~1etΛX1,dE2=dL~∞etΛX0+νdL~1etΛX2.Due to the small initial conditionsu0+d0≤η, there exists a positive constant M which will be chosen later such that
(74)u,dE≤Mη.Taking the same procedures (65)-(66), and then employing Lemmas 9-11 and the boundness of the Fourier multiplier ℙ, we have
(75)u,dE≤12Mη.



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*Source: 3317007-2022-06-02.xml*