Myosin is an actin-based motor protein that widely exists in muscle tissue and non-muscle tissue, and myosin of a diverse subfamily has obvious differences in structure and cell function. Many eukaryotes and even some unicellular organisms possess a variety of myosins. They have been well characterized in human, fungi and other organisms. However, the myosin gene family in Bemisia tabaci MEAM1 (Middle East-Asia Minor1 species) is poorly studied. In the study, we identified 15 myosin genes in B. tabaci MEAM1 based on a genome database. Myosin genes can be divided into ten classes, including subfamilies I, II, III, V, VI, VII, IX, XV, XVIII, XX in B. tabaci MEAM1. The amounts of myosin in Class I are the largest of the isoforms. Expression profiling of myosins by quantitative real-time PCR revealed that their expression differed among developmental stages and different tissues of B. tabaci MEAM1. The diversely may be related to the development characteristics of B. tabaci MEAM1. The BtaMyo-IIIb-like X1 was highly expressed in nymphs 4 instar which may be related to the development process before metamorphosis. Our outcome contributes to the basis for further research on myosin gene function in B. tabaci MEAM1 and homologous myosins in other biology.
Citation: Kui Wang, Zhifang Yang, Xiaohui Chen, Shunxiao Liu, Xiang Li, Liuhao Wang, Hao Yu, Hongwei Zhang. Characterization and analysis of myosin gene family in the whitefly (Bemisia tabaci)[J]. AIMS Molecular Science, 2022, 9(2): 91-106. doi: 10.3934/molsci.2022006
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Myosin is an actin-based motor protein that widely exists in muscle tissue and non-muscle tissue, and myosin of a diverse subfamily has obvious differences in structure and cell function. Many eukaryotes and even some unicellular organisms possess a variety of myosins. They have been well characterized in human, fungi and other organisms. However, the myosin gene family in Bemisia tabaci MEAM1 (Middle East-Asia Minor1 species) is poorly studied. In the study, we identified 15 myosin genes in B. tabaci MEAM1 based on a genome database. Myosin genes can be divided into ten classes, including subfamilies I, II, III, V, VI, VII, IX, XV, XVIII, XX in B. tabaci MEAM1. The amounts of myosin in Class I are the largest of the isoforms. Expression profiling of myosins by quantitative real-time PCR revealed that their expression differed among developmental stages and different tissues of B. tabaci MEAM1. The diversely may be related to the development characteristics of B. tabaci MEAM1. The BtaMyo-IIIb-like X1 was highly expressed in nymphs 4 instar which may be related to the development process before metamorphosis. Our outcome contributes to the basis for further research on myosin gene function in B. tabaci MEAM1 and homologous myosins in other biology.
Nematic liquid crystals are aggregates of molecules which possess same orientational order and are made of elongated, rod-like molecules. Hence, in the study of nematic liquid crystals, one approach is to consider the behavior of the director field d in the absence of the velocity fields. Unfortunately, the flow velocity does disturb the alignment of the molecules. More importantly, the converse is also true, that is, a change in the alignment will induce velocity. This velocity will in turn affect the time evolution of the director field. In this process, we cannot assume that the velocity field will remain small even when we start with zero velocity field.
In the 1960's, Ericksen [3,4] and Leslie [10,11] developed the hydrodynamic theory of liquid crystals. The Ericksen-Leslie system consists of the following equations [12]:
{ρt+∇⋅(ρu)=0,(ρu)t=ρFi+σji,j,ρ1(ωi)t=ρ1Gi+gi+πji,j, | (1.1) |
where (1.1)1, (1.1)2 and (1.1)3, represent the conservation of mass, linear momentum and angular momentum respectively. Besides, ρ denotes the fluid density, u=(u1,u2,u3) is the velocity vector and d=(d1,d2,d3) the direction vector,
{σji=−Pδij−ρ∂F∂dk,j+ˆσ′ji,πji=βjdi+ρ∂F∂di,j,gi=γdi−βjdi,j−ρ∂F∂di+ˆg′i, | (1.2) |
where Fi is the external body force, Gi denotes the external director body force and β, γ come from the restriction of the direction vector |d|=1. The following relations also hold:
{2ρF=k22di,jdi,j+(k11−k22−k24)di,idj,j+(k33−k22)didjdk,idkj+k24di,jdj,i,ˆσ′ji=μ1dkdpAkpdidj+μ2djNi+μ3diNj+μ4Aij+μ5djdkAki+μ6didkAkj,ˆgi=λ1Ni+λ2djAji, | (1.3) |
and
{ωi=˙di=∂di∂t+u⋅∇di,Ni=ωi+ωkidk,Nij=ωi,j+ωkidk,j, | (1.4) |
where
2Aij=ui,j+uj,i,2ωi,j=ui,j−uj,i. |
On the basis of the second law of thermodynamics and Onsager reciprocal relation, one obtain
λ1=μ2−μ3,λ2=μ5−μ6=−(μ2+μ3). |
The nonlinear constraint |d|=1 can also be relaxed by using the Ginzburg-Landau approximation, that is, instead of the restriction |d|=1, we add the term 1ε2(|d|2−1)2 in ρF. In addition, to further simplify the calculation, one take ρ1=0, βj=0, γ=0, Fi=0 and ρF=|∇d|2+1ε2(|d|2−1)2, choose the domain Ω=R3, obtain the simplified model of nematic liquid crystals:
{ρt+∇⋅(ρu)=0,(ρu)t+∇⋅(ρu⊗u)−μΔu−(μ+λ)∇∇⋅u+∇p(ρ)=−Δd⋅∇d,dt+u⋅∇d=Δd−f(d), | (1.5) |
with the following initial conditions
ρ(x,0)=ρ0(x),u(x,0)=u0(x),d(x,0)=d0(x),|d0(x)|=1, | (1.6) |
and
ρ0−ˉρ∈HN(R3),u0∈HN(R3),d0−ω0∈HN(R3), | (1.7) |
for any integer N≥3 with a fixed vector ω0∈S2, that is, |ω0|=1. In this paper, we assume that f(d)=1ε2(|d|2−1)d (ε>0) is the Ginzburg-Landau approximation and the pressure p=p(ρ) is a smooth function in a neighborhood of ˉρ with p′(ˉρ)>0 for ˉρ>0. Moreover, μ and λ are the shear viscosity and the bulk viscosity coefficients of the fluid, respectively. As usual, the following inequalities hold:
μ>0,3λ+2μ≥0. |
The study of liquid crystals can be traced back to Ericksen [3,4] and Leslie [10,11] in the 1960s. Since then, there is a huge amount of literature on this topic. For the incompressible case, we refer the author to [2,6,12,13,20] and the reference therein. There are also many papers related to the compressible case, see for instance, [1,5,7,8,17,19] and the reference cited therein.
In [19], the authors rewrote system (1.5) in the perturbation form as
{ϱt+ˉρ∇⋅u=−ϱ∇⋅u−u⋅∇ϱ,ut−ˉμΔu−(ˉμ+ˉλ)∇∇⋅u+γˉρ∇ϱ=−u⋅∇u−h(ϱ)(ˉμΔu+(ˉμ+ˉλ)∇∇⋅u)−g(ϱ)∇ϱ−ϕ(Δd⋅∇d),dt+u⋅∇d=Δd−f(d), | (1.8) |
where ϱ=ρ−ˉρ, ˉμ=μˉρ, ˉλ=λˉρ, γ=p′(ˉρ)¯ρ2 and the nonlinear functions of ϱ are defined by
h(ϱ)=ϱϱ+ˉρ,g(ϱ)=p′(ϱ+ˉρ)ϱ+ˉρ−p′(ˉρ)ˉρ,ϕ(ϱ)=1ρ+ˉρ. |
We remark that the functions h(ϱ), g(ϱ) and ϕ(ϱ) satisfy (see [19])
|h(ϱ)|,|g(ϱ)|≤C|ϱ|,|ϕ(l)(ϱ)|,|h(k)(ϱ)|,|g(k)(ϱ)|≤Cfor anyl≥0,k≥1. | (1.9) |
Wei, Li and Yao [19] obtained the small initial data global well-posedness provided that ‖ϱ0‖H3+‖u0‖H3+‖d0−ω0‖H4 is sufficiently small. Moreover, the authors also showed the optimal decay rates of higher order spatial derivatives of of strong solutions provided that (ϱ0,u0,∇d0)∈˙H−s for some s∈[0,12].
Next, we introduce the main results in [19]:
Lemma 1.1. (Small initial data global well-posedness [19]) Assume that N≥3 and (ϱ0,u0,d0−ω0)∈HN(R3)×HN(R3)×HN+1(R3). Then for a unit vector ω0, there exists a positive constant δ0 such that if
‖ϱ0‖H3+‖u0‖H3+‖d0−ω0‖H4≤δ0, | (1.10) |
then problem (1.8) has a unique global solution (ϱ(t),u(t),d(t)) satisfying that for all t≥0,
ddt(‖ϱ‖2HN+‖u‖2HN+‖∇d‖2HN)+C0(‖∇ϱ‖2HN−1+‖∇u‖2HN+‖∇∇d‖2HN)≤0. | (1.11) |
Lemma 1.2. (Decay estimates [19]) Assume that all the assumptions of Lemma 1.1 hold. Then, if (ϱ0,u0,∇d0)∈˙H−s for some s∈[0,12], we have
‖Λ−sϱ(t)‖2L2+‖Λ−su(t)‖2L2+‖Λ−s∇d(t)‖2L2≤C1,∀t≥0, | (1.12) |
and
‖∇lϱ‖HN−l+‖∇lu‖HN−l+‖∇l+1d‖HN−l≤C2(1+t)−l+s2,∀t≥0andl=0,1,⋯,N−1. | (1.13) |
The main purpose of this paper is to improve the decay results in [19]. First, we give a remark on the symbol stipulations of this paper.
Remark 1.3. In this paper, we use Hk(R3) (k∈R), to denote the usual Sobolev spaces with norm ‖⋅‖Hs, and Lp(R3) (1≤p≤∞) to denote the usual Lp spaces with norm ‖⋅‖Lp. We also introduce the homogeneous negative index Sobolev space ˙H−s(R3):
˙H−s(R3):={f∈L2(R3):‖|ξ|−sˆf(ξ)‖L2<∞} |
endowed with the norm ‖f‖˙H−s:=‖|ξ|−sˆf(ξ)‖L2. The symbol ∇l with an integer l≥0 stands for the usual spatial derivatives of order l. For instance, we define
∇lz={∂αxzi||α|=l,i=1,2,3},z=(z1,z2,z3). |
If l<0 or l is not a positive integer, ∇l stands for Λl defined by
Λsf(x)=∫R3|ξ|sˆf(ξ)e2πix⋅ξdξ, |
where ˆf is the Fourier transform of f. Besides, C and Ci (i=0,1,2,⋯) will represent generic positive constants that may change from line to line even if in the same inequality. The notation A≲B means that A≤CB for a universal constant C>0 that only depends on the parameters coming from the problem.
It is worth pointing out that in [19], the authors consider problem (1.5) in 3D case, the negative Sobolev norms were shown to be preserved along time evolution and enhance the decay rates. However, because the Ginzburg-Landau approximation term is difficulty to control, only s∈[0,12] were considered in [19]. In this paper, we ovcome the difficult caused by Ginzburg-Landau approximation, assume that s∈[0,32), obtain the optimal decay rates of higher order spatial derivatives of strong solutions for problem (1.5). Our main results are stated in the following theorem.
Theorem 1.4. Assume that all the assumptions of Lemma 1.1 hold. Then, if (ϱ0,u0,∇d0)∈˙H−s for some s∈[0,32), we have
‖Λ−sϱ(t)‖2L2+‖Λ−su(t)‖2L2+‖Λ−s∇d(t)‖2L2≤C0,∀t≥0, | (1.14) |
and
‖∇lϱ‖HN−l+‖∇lu‖HN−l+‖∇l+1d‖HN−l≤C(1+t)−l+s2,forl=0,1,⋯,N−1,∀t≥0. | (1.15) |
Note that the Hardy-Littlewood-Sobolev theorem implies that for p∈(1,2], Lp(R3)⊂˙H−s(R3) with s=3(1p−12)∈[0,32). Then, on the basis of Lemma 1.2 and Theorem 1.4, we obtain the optimal decay estimates for system (1.8).
Corollary 1.5. Under the assumptions of Lemma 1.2 and Theorem 1.4, if we replace the ˙H−s(R3) assumption by
(ϱ,u0,∇d0)∈Lp(R3),1<p≤2, |
then for l=0,1,⋯,N−1, , the following decay estimate holds:
‖∇lϱ‖HN−l+‖∇lu‖HN−l+‖∇l+1d‖HN−l≤C(1+t)−[32(1p−12)+l2],∀t≥0. | (1.16) |
Remark 1.6. Lemma 1.1 shows the global well-posedness of strong solutions for system (1.5) provided that the smallness assumption (1.10) holds. One can use the energy method to obtain the higher order energy estimates for the solution to prove this lemma (see [19]). We remark that the negative Sobolev norm estimates did not appear in the proving process of Lemma 1.1, it is only used in the decay estimates. Hence, the value of s in Lemma 1.2 and Theorem 1.4 do not affect the energy estimates (1.10) and (1.11). And those two estimates hold for both Lemma 1.2 and Theorem 1.4.
Remark 1.7. The main purpose of this paper is to prove Theorem 1.4 and Corollary 1.5 on the asymptotic behavior of strong solutions for a compressible Ericksen-Leslie system. We remark that the global well-posedness and asymptotic behavior of solutions are important for the study of nematic liquid crystals system. Thanks to the above properties of solutions, one can understand the model more profoundly. Our results maybe useful for the study of nematic liquid crystals.
The structure of this paper is organized as follows. In Section 2, we introduce some preliminary results. The proof of Theorem 1.4 is postponed in Section 3.
We first show a useful Sobolev embedding theorem in the following Lemma 2.1:
Lemma 2.1. ([15]) If 0≤s<32, one have
‖u‖L63−2s(R3)≲‖u‖˙Hs(R3)forallu∈˙Hs(R3). | (2.1) |
In [14], the author proved the following Gagliardo-Nirenberg inequality:
Lemma 2.2. ([14]) Let 0≤m,α≤l, then we have
‖∇αf‖Lp(R3)≲‖∇mf‖1−θLq(R3)‖∇lf‖θLr(R3), | (2.2) |
where θ∈[0,1] and α satisfies
α3−1p=(m3−1q)(1−θ)+(l3−1r)θ. | (2.3) |
Here, when p=∞, we require that 0<θ<1.
One also introduce the Kato-Ponce inequality which is of great importance in our paper.
Lemma 2.3. ([9]) Let 1<p<∞, s>0. There exists a positive constant C such that
‖∇s(fg)‖Lp(R3)≲‖f‖Lp1(R3)‖∇sg‖Lp2(R3)+‖∇sf‖Lq1(R3)‖g‖Lq2(R3), | (2.4) |
where p2,q2∈(1,∞) satisfying 1p=1p1+1p2=1q1+1q2.
The Hardy-Littlewood-Sobolev theorem implies the following Lp type inequality:
Lemma 2.4. ([16]) Let 0≤s<32, 1<p≤2 and 12+s3=1p, then
‖f‖˙H−s(R3)≲‖f‖Lp(R3). | (2.5) |
In the end, we introduce the special Sobolev interpolation lemma, which will be used in the proof of Theorem 1.4.
Lemma 2.5. ([18]) Let s≥0 and l≥0, then
‖∇lf‖L2(R3)≤‖∇l+1f‖1−θL2(R3)‖f‖θ˙H−s(R3),withθ=1l+1+s. | (2.6) |
Equation (1.8)3 can be rewritten as
(d−ω0)t−Δ(d−ω0)=−u⋅∇(d−ω0)−[f(d)−f(ω0)]. | (3.1) |
In [19], the authors proved the L2-norm estimate of d−ω0 provided that the assumptions of Lemma 1.1 hold.
Lemma 3.1. ([19]) Assume that all the assumptions of Lemma 1.1 hold. Then, the solution of (3.1) satisfies
‖d−ω0‖2L2+∫t0‖∇(d−ω0)‖2L2≤‖d0−ω0‖2L2. | (3.2) |
In the following, we prove the decay estimates of strong solutions for system (1.8). The case s∈[0,12] was shown in Lemma 1.2, one only need to consider the case s∈(12,32). We first derive the evolution of the negative Sobolev norms of the solution.
Lemma 3.2. Under the assumptions of Lemma 1.1, if s∈(12,32), we have
ddt∫R3(γ|Λ−sϱ|2+|Λ−su|2+|Λ−s∇d|2)dx+C∫R3(|∇Λ−s∇u|2+|Λ−s∇2d|2)dx≤C‖∇d‖2H1(‖Λ−su‖L2+‖Λ−sϱ‖L2+‖Λ−s∇d‖L2)+(‖ϱ‖L2+‖u‖L2+‖∇d‖L2)s−12(‖∇ϱ‖H1+‖∇u‖H1+‖∇2d‖H1)52−s×(‖Λ−su‖L2+‖Λ−sϱ‖L2+‖Λ−s∇d‖L2). | (3.3) |
Proof. Applying Λ−s to (1.8)1, (1.8)2, Λ−s∇ to (1.8)3, multiplying the resulting identities by γΛ−sϱ, Λ−su and Λ−s∇d respectively, summing up and integrating over R3 by parts, we arrive at
12ddt∫R3(γ|Λ−sϱ|2+|Λ−su|2+|Λ−s∇d|2)dx+∫R3(ˉμ|∇Λ−su|2+(ˉμ|∇Λ−su|2+(ˉμ+ˉλ)|∇⋅Λ−su|2+|Λ−s∇2d|2)dx=∫R3γΛ−s(−ϱ∇⋅u−u⋅∇ϱ)⋅Λ−sϱ−Λ−s[u⋅∇u+h(ϱ)(ˉμΔu+(ˉμ+ˉλ)∇∇⋅u)+g(ϱ)∇ϱ+ϕ(Δd⋅∇d)]⋅Λ−su−Λ−s∇(u⋅∇d+f(d))⋅Λ−s∇ddx=K1+K2+K3+K4+K5+K6+K7+K8. | (3.4) |
Note that s∈(12,32), it is easy to see that 12+s3<1 and 3s∈(2,6). For the terms K1, by using Lemmas 2.2 and 2.4, Hölder's inequality, Young's inequality together with the estimates established in Lemma 1.1, we deduce that
K1=−∫R3γΛ−s(ϱ∇⋅u)⋅Λ−sϱdx≤C‖Λ−s(ϱ∇⋅u)‖L2‖Λ−sϱ‖L2≤C‖ϱ∇⋅u‖L112+s3‖Λ−sϱ‖L2≤C‖ϱ‖L3s‖∇u‖L2‖Λ−sϱ‖L2≤C‖ϱ‖s−12L2‖∇ϱ‖32−sL2‖∇u‖L2‖Λ−sϱ‖L2, | (3.5) |
similarly, for K2–K6, we have
K2=−∫R3γΛ−s(u⋅∇ϱ)⋅Λ−sϱdx≤C‖Λ−s(u⋅∇ϱ)‖L2‖Λ−sϱ‖L2≤C‖u⋅∇ϱ‖L112+s3‖Λ−sϱ‖L2≤C‖u‖L3s‖∇ϱ‖L2‖Λ−sϱ‖L2≤C‖u‖s−12L2‖∇u‖32−sL2‖∇ϱ‖L2‖Λ−sϱ‖L2, | (3.6) |
K3=−∫R3γΛ−s(u⋅∇u)⋅Λ−sϱdx≤C‖Λ−s(u⋅∇u)‖L2‖Λ−sϱ‖L2≤C‖u⋅∇u‖L112+s3‖Λ−sϱ‖L2≤C‖u‖L3s‖∇u‖L2‖Λ−sϱ‖L2≤C‖u‖s−12L2‖∇u‖32−sL2‖∇u‖L2‖Λ−sϱ‖L2, | (3.7) |
K4=−∫R3Λ−s[h(ϱ)(ˉμΔu+(ˉμ+ˉλ)∇∇⋅u)]⋅Λ−sϱdx≤‖Λ−s[h(ϱ)(ˉμΔu+(ˉμ+ˉλ)∇∇⋅u)]‖L2‖Λ−sϱ‖L2≤C‖h(ϱ)(ˉμΔu+(ˉμ+ˉλ)∇∇⋅u)‖L112+s3‖Λ−sϱ‖L2≤C‖h(ϱ)‖L3s‖∇2u‖L2‖Λ−sϱ‖L2≤C‖ϱ‖L3s‖∇2u‖L2‖Λ−sϱ‖L2≤C‖ϱ‖s−12L2‖∇ϱ‖32−sL2‖∇2u‖L2‖Λ−sϱ‖L2, | (3.8) |
K5=−∫R3Λ−s[g(ϱ)∇ϱ]⋅Λ−sϱdx≤‖Λ−s[g(ϱ)∇ϱ]‖L2‖Λ−sϱ‖L2≤C‖g(ϱ)∇ϱ‖L112+s3‖Λ−sϱ‖L2≤C‖g(ϱ)‖L3s‖∇ϱ‖L2‖Λ−sϱ‖L2≤C‖ϱ‖L3s‖∇ϱ‖L2‖Λ−sϱ‖L2≤C‖ϱ‖s−12L2‖∇ϱ‖32−sL2‖∇ϱ‖L2‖Λ−sϱ‖L2, | (3.9) |
and
K6=−∫R3Λ−s(ϕ(ϱ)∇d⋅Δd)⋅Λ−sudx≤C‖Λ−s(ϕ(ϱ)∇d⋅Δd)‖L2‖Λ−su‖L2≤C‖ϕ(ϱ)∇d⋅Δd‖L112+s3‖Λ−su‖L2≤C‖ϕ(ϱ)‖L∞‖∇d⋅Δd‖L112+s3‖Λ−su‖L2≤C‖∇d⋅Δd‖L112+s3‖Λ−su‖L2≤C‖∇d‖L3s‖Δd‖L2‖Λ−su‖L2≤C‖∇d‖s−12L2‖Δd‖32−sL2‖Δd‖L2‖Λ−su‖L2, | (3.10) |
where we have used the fact (1.9) in (3.8)–(3.10). Next, by using Lemmas 2.2–2.4, Hölder's inequality, Young's inequality together with Lemma 1.1 on the energy estimates of the solutions, it yields that
K7=−∫R3Λ−s∇(u⋅∇d)⋅Λ−s∇ddx≤C‖Λ−s(∇u⋅∇d+u⋅∇2d)‖L2‖Λ−s∇d‖L2≤C(‖∇u⋅∇d‖L112+s3+‖u⋅∇2d‖L112+s3)‖Λ−s∇d‖L2≤C(‖∇d‖L3s‖∇u‖L2+‖u‖L3s‖∇2d‖L2)‖Λ−s∇d‖L2≤C(‖∇d‖s−12L2‖∇2d‖32−sL2‖∇u‖L2+‖u‖s−12L2‖∇u‖32−sL2‖∇2d‖L2)‖Λ−s∇d‖L2. | (3.11) |
For K8, we first consider s∈(12,1). Thanks to Lemma 2.2, one easily obtain
‖Λ2−sd‖L2+‖Λ2−s(d−ω0)‖L2+‖Λ2−s(d+ω0)‖L2≤C‖∇d‖sL2‖∇2d‖1−sL2≤C(‖∇d‖L2+‖∇2d‖L2). |
Then, by Hölder's inequality, Young's inequality, the facts |d|<1, |ω0|=1 together with Lemmas 1.1, 2.2 and 2.3, we derive that
K8=−∫R3Λ−s∇⋅[(d+ω0)(d−ω0)d]⋅Λ−s∇ddx≤C‖Λ−s∇⋅[(d+ω0)(d−ω0)d]‖L2‖Λ−s∇d‖L2≤C[‖d+ω0‖L6‖d−ω0‖L6‖Λ1−sd‖L6+‖d+ω0‖L6‖d‖L6‖Λ1−s(d−ω0)‖L6+‖d−ω0‖L6‖d‖L6‖Λ1−s(d+ω0)‖L6]‖Λ−s∇d‖L2≤C‖∇d‖2L2(‖Λ2−sd‖L2+‖Λ2−s(d−ω0)‖L2+‖Λ2−s(d+ω0)‖L2)‖Λ−s∇d‖L2≤C‖∇d‖L2(‖∇d‖L2+‖∇2d‖L2)‖Λ−s∇d‖L2≤C(‖∇d‖2L2+‖∇2d‖2L2)‖Λ−s∇d‖L2. | (3.12) |
Moreover, if s∈(1,32), the following inequality holds:
K8≤C‖Λ−s+1[(d+ω0)(d−ω0)d]‖L2‖Λ−s∇d‖L2≤C‖(d+ω0)(d−ω0)d‖L112+s−13‖Λ−s∇d‖L2≤C‖d+ω0‖L∞‖d−ω0‖L2‖d‖L3s−1‖Λ−s∇d‖L2≤C‖∇d‖12L2‖∇2d‖12L2‖d−ω0‖L2‖∇d‖(s−1)+12L2‖∇2d‖12−(s−1)L2‖Λ−s∇d‖L2≤C‖∇d‖sL2‖∇2d‖2−sL2‖Λ−s∇d‖L2≤C(‖∇d‖2L2+‖∇2d‖2L2)‖Λ−s∇d‖L2. | (3.13) |
Combining (3.4)–(3.12) together, we obtain (3.3) and complete the proof.
Now, we give the proof of our main results.
Proof of Theorem 1.4. First of all, the sketch of proof for the decay estimate with s∈[0,12] will be derived in the following. Note that this part follows more or less the lines of [19], so that we do note claim originality here. Then, by using this proved estimate, one can obtain the decay results for s∈(12,32).
Now, consider the decay for s∈[0,12]. We first establish the negative Sobolev norm estimates for the strong solutions, obtain one important inequality:
ddt(γ‖Λ−sϱ‖2L2+‖Λ−su‖2L2+‖Λ−s∇d‖2L2)+C(‖Λ−s∇u‖2L2+‖Λ−s∇2d‖2L2)≲(‖∇(ϱ,u)‖2H1+‖∇d‖2H2)(‖Λ−sϱ‖2L2+‖Λ−su‖2L2+‖Λ−s∇d‖2L2). |
Then, define
E−s(t)=‖Λ−sϱ(t)‖2L2+‖Λ−su(t)‖2L2+‖Λ−s∇d(t)‖2L2, |
we deduce from (3.2) and (3.3) that for s∈[0,12],
E−s(t)≤E−s(0)+C∫t0(‖∇(ϱ,u)‖2H1+‖∇d‖2H2)√E−s(t)dτ≤C(1+sup0≤τ≤t√E−s(t)), |
which implies (1.12) for s∈[0,12], i.e.,
‖Λ−sϱ(t)‖2L2+‖Λ−su(t)‖2L2+‖Λ−s∇d(t)‖2L2≤C0. | (3.14) |
Moreover, if l=1,2,⋯,N−1, we may use Lemma 2.4 to have
‖∇l+1f‖L2≥C‖Λ−sf‖−1l+sL2‖∇lf‖1+1l+sL2. | (3.15) |
Then, by (3.14) and (3.15), it yields that
‖∇l(∇ϱ,∇u,∇2d)‖2L2≥C(‖∇l(ϱ,u,∇d)‖2L2)1+1l+s. |
Hence, for l=1,2,⋯,N−1,
‖∇l(∇ϱ,∇u,∇2d)‖2HN−l−1≥C(‖∇l(ϱ,u,∇d)‖2HN−l)1+1l+s. | (3.16) |
Thus, we deduce from (1.11) the following inequality
ddt(‖∇lϱ‖2HN−l+‖∇lu‖2HN−l+‖∇l+1d‖2HN−l)+C0(‖∇lϱ‖2HN−l+‖∇lu‖2HN−l+‖∇l+1d‖2HN−l)1+1l+s≤0,forl=1,⋯,N−1. |
Solving this inequality directly gives
‖∇lϱ‖HN−l+‖∇lu‖HN−l+‖∇l+1d‖HN−l≤C(1+t)−l+s2,forl=1,⋯,N−1. | (3.17) |
Then, by (3.14), (3.17) and the interpolation, we obtain the following inequality holds for s∈[0,12]:
‖∇lϱ‖HN−l+‖∇lu‖HN−l+‖∇l+1d‖HN−l≤C(1+t)−l+s2,forl=0,1,⋯,N−1. | (3.18) |
Second, we consider the decay estimate for s∈(12,32). Notice that the arguments for s∈[0,12] can not be applied to this case. However, observing that we have ϱ0,u0,∇d0∈˙H−12 hold since ˙H−s⋂L2⊂˙H−s′ for any s′∈[0,s], we can deduce from (3.18) for (1.10) and (1.11) with s=12 that the following estimate holds:
‖∇lϱ‖2HN−l+‖∇lu‖2HN−l+‖∇l∇d‖2HN−l≤C0(1+t)−12−l,forl=0,1,⋯,N−1. | (3.19) |
Therefore, we deduce from (3.3) and (3.2) that for s∈(12,32),
E−s(t)≤E−s(0)+C∫t0‖∇d‖2H1√E−s(τ)dτ+C∫t0(‖ϱ‖L2+‖u‖L2+‖∇d‖L2)s−12(‖∇ϱ‖H1+‖∇u‖H1+‖∇2d‖H1)52−s√E−s(τ)dτ≤C0+Csupτ∈[0,t]√E−s(τ)+C∫t0(1+τ)−74+s2dτsupτ∈[0,t]√E−s(τ)≤C0+Csupτ∈[0,t]√E−s(τ), | (3.20) |
which implies that (1.12) holds for s∈(12,32), i.e.,
‖Λ−sϱ(t)‖2L2+‖Λ−su(t)‖2L2+‖Λ−s∇d(t)‖2L2≤C0. | (3.21) |
Moreover, thanks to (1.11) and (3.16), we can also obtain the following inequality for s∈(12,32):
ddt(‖∇lϱ‖2HN−l+‖∇lu‖2HN−l+‖∇l+1d‖2HN−l)+C0(‖∇lϱ‖2HN−l+‖∇lu‖2HN−l+‖∇l+1d‖2HN−l)1+1l+s≤0,forl=1,⋯,N−1, |
which implies
‖∇lϱ‖HN−l+‖∇lu‖HN−l+‖∇l+1d‖HN−l≤C(1+t)−l+s2,forl=1,⋯,N−1. | (3.22) |
Next, using (3.21), (3.22), and Lemma 2.5, we easily obtain
‖(ϱ,u,∇d)‖L2≤C(‖∇(ϱ,u,∇d)‖L2)s1+s(‖Λ−s(ϱ,u,∇d)‖11+sL2≤C(‖∇(ϱ,u,∇d)‖L2)s1+s≤C[(1+t)−1+s2]s1+s=C(1+t)−s2. | (3.23) |
It then follows from (3.22) and (3.23) that
‖ϱ‖HN−l+‖u‖HN−l+‖∇d‖HN−l≤C(1+t)−s2. |
Hence, we obtain (1.15) for s∈(12,32) and complete the proof.
In this paper, we consider the optimal decay estimates for the higher order derivatives of strong solutions for three-dimensional nematic liquid crystal system. We use the pure energy method, negative Sobolev norm estimates together with the classical Kato-Ponce inequality, Gagliardo-Nirenberg inequality, overcome the difficulties caused by the Ginzburg-Landau approximation and the coupling between the compressible Navier-Stokes equations and the direction equations, obtain the decay estimates. Since the result (1.16) is same to the decay of the heat equation, it is optimal. We remark that our results may attract the attentions of the researchers in the nematic liquid crystals filed.
The author would like to thank the anonymous referees and Dr. Xiaopeng Zhao for their helpful suggestions. This paper was supported by the Fundamental Research Funds of Heilongjiang Province (grant No. 145109131).
The author declares no conflict of interest.
[1] |
Ouyang Z, Zhao S, Yao S, et al. (2021) Multifaceted function of myosin-18, an unconventional class of the myosin superfamily. Frontiers in Cell and Developmental Biology 9: 632445. https://doi.org/10.3389/fcell.2021.632445 ![]() |
[2] |
Sebé-Pedrós A, Grau-Bové X, Richards TA, et al. (2014) Evolution and classification of myosins, a paneukaryotic whole-genome approach. Genome biology and evolution 6: 290-305. https://doi.org/10.1093/gbe/evu013 ![]() |
[3] |
Hartman MA, Spudich JA (2012) The myosin superfamily at a glance. Journal of cell science 125: 1627-1632. https://doi.org/10.1242/jcs.094300 ![]() |
[4] | Coluccio LM (2007) Myosins: a superfamily of molecular motors. Springer . https://doi.org/10.1007/978-1-4020-6519-4 |
[5] | Chen C-L, Wang Y, Sesaki H, et al. (2012) Myosin I links PIP3 signaling to remodeling of the actin cytoskeleton in chemotaxis. Science signaling 5: ra10-ra10. https://doi.org/10.1126/scisignal.2002446 |
[6] |
Dai J, Ting-Beall HP, Hochmuth RM, et al. (1999) Myosin I contributes to the generation of resting cortical tension. Biophysical journal 77: 1168-1176. https://doi.org/10.1016/S0006-3495(99)76968-7 ![]() |
[7] |
Falk DL, Wessels D, Jenkins L, et al. (2003) Shared, unique and redundant functions of three members of the class I myosins (MyoA, MyoB and MyoF) in motility and chemotaxis in Dictyostelium. Journal of cell science 116: 3985-3999. https://doi.org/10.1242/jcs.00696 ![]() |
[8] |
Haraguchi T, Honda K, Wanikawa Y, et al. (2013) Function of the head–tail junction in the activity of myosin II. Biochemical and biophysical research communications 440: 490-494. https://doi.org/10.1016/j.bbrc.2013.09.038 ![]() |
[9] |
Beach JR, Hammer JA (2015) Myosin II isoform co-assembly and differential regulation in mammalian systems. Experimental cell research 334: 2-9. https://doi.org/10.1016/j.yexcr.2015.01.012 ![]() |
[10] |
Singh K, Kim AB, Morgan KG (2021) Non-muscle myosin II regulates aortic stiffness through effects on specific focal adhesion proteins and the non-muscle cortical cytoskeleton. Journal of Cellular and Molecular Medicine 25: 2471-2483. https://doi.org/10.1111/jcmm.16170 ![]() |
[11] |
Betapudi V (2014) Life without double-headed non-muscle myosin II motor proteins. Frontiers in chemistry 2: 45. https://doi.org/10.3389/fchem.2014.00045 ![]() |
[12] | Newell-Litwa KA, Horwitz R, Lamers ML (2015) Non-muscle myosin II in disease: mechanisms and therapeutic opportunities. Disease models & mechanisms 8: 1495-1515. https://doi.org/10.1242/dmm.022103 |
[13] |
Wei Z, Liu X, Yu C, et al. (2013) Structural basis of cargo recognitions for class V myosins. Proceedings of the National Academy of Sciences 110: 11314-11319. https://doi.org/10.1073/pnas.1306768110 ![]() |
[14] | Renshaw H, Vargas-Muñiz JM, Juvvadi PR, et al. (2018) The tail domain of the Aspergillus fumigatus class V myosin MyoE orchestrates septal localization and hyphal growth. Journal of cell science 131: jcs205955. https://doi.org/10.1242/jcs.205955 |
[15] |
Bähler M, Elfrink K, Hanley P, et al. (2011) Cellular functions of class IX myosins in epithelia and immune cells. Biochemical Society Transactions 39: 1166-1168. https://doi.org/10.1042/BST0391166 ![]() |
[16] | Naalden D, van Kleeff PJ, Dangol S, et al. (2021) Spotlight on the roles of whitefly effectors in insect–plant interactions. Frontiers in plant science 2011: 1243. https://doi.org/10.3389/fpls.2021.661141 |
[17] |
Barbosa LdF, Marubayashi JM, De Marchi BR, et al. (2014) Indigenous American species of the Bemisia tabaci complex are still widespread in the Americas. Pest Management Science 70: 1440-1445. https://doi.org/10.1002/ps.3731 ![]() |
[18] |
Costa H, Brown J, Sivasupramaniam S, et al. (1993) Regional distribution, insecticide resistance, and reciprocal crosses between the A and B biotypes of Bemisia tabaci. International Journal of Tropical Insect Science 14: 255-266. https://doi.org/10.1017/S1742758400014703 ![]() |
[19] |
Ueda S, Kitamura T, Kijima K, et al. (2009) Distribution and molecular characterization of distinct Asian populations of Bemisia tabaci (Hemiptera: Aleyrodidae) in Japan. Journal of applied entomology 133: 355-366. https://doi.org/10.1111/j.1439-0418.2008.01379.x ![]() |
[20] |
Derrien S, Quinton P (2010) Hardware acceleration of HMMER on FPGAs. Journal of Signal Processing Systems 58: 53-67. https://doi.org/10.1007/s11265-008-0262-y ![]() |
[21] |
Li W, Godzik A (2006) Cd-hit: a fast program for clustering and comparing large sets of protein or nucleotide sequences. Bioinformatics 22: 1658-1659. https://doi.org/10.1093/bioinformatics/btl158 ![]() |
[22] | Chen Q, Wan Y, Lei Y, et al. (2016) Evaluation of CD-HIT for constructing non-redundant databases. IEEE BIBM 2016: 703-706. |
[23] |
Letunic I, Doerks T, Bork P (2015) SMART: recent updates, new developments and status in 2015. Nucleic acids research 43: D257-D260. https://doi.org/10.1093/nar/gku949 ![]() |
[24] |
Newman L, Duffus AL, Lee C (2016) Using the free program MEGA to build phylogenetic trees from molecular data. The American Biology Teacher 78: 608-612. https://doi.org/10.1525/abt.2016.78.7.608 ![]() |
[25] | Marchal E, Hult EF, Huang J, et al. (2013) Sequencing and validation of housekeeping genes for quantitative real-time PCR during the gonadotrophic cycle of Diploptera punctata. BMC research notes 6: 1-10. https://doi.org/10.1186/1756-0500-6-237 |
[26] |
Livak KJ, Schmittgen TD (2001) Analysis of relative gene expression data using real-time quantitative PCR and the 2−ΔΔCT method. methods 25: 402-408. https://doi.org/10.1006/meth.2001.1262 ![]() |
[27] |
Nambiar R, McConnell RE, Tyska MJ (2010) Myosin motor function: the ins and outs of actin-based membrane protrusions. Cellular and molecular life sciences 67: 1239-1254. https://doi.org/10.1007/s00018-009-0254-5 ![]() |
[28] |
Crow KD, Wagner GP (2005) What is the role of genome duplication in the evolution of complexity and diversity?. Molecular biology and evolution 23: 887-892. https://doi.org/10.1093/molbev/msj083 ![]() |
[29] |
Xie T, Zeng L, Chen X, et al. (2020) Genome-wide analysis of the lateral organ boundaries domain gene family in Brassica napus. Genes 11: 280. https://doi.org/10.3390/genes11030280 ![]() |
[30] |
Cheney RE, Riley MA, Mooseker MS (1993) Phylogenetic analysis of the myosin superfamily. Cell motility and the cytoskeleton 24: 215-223. https://doi.org/10.1002/cm.970240402 ![]() |
[31] |
Berg JS, Powell BC, Cheney RE (2001) A millennial myosin census. Molecular biology of the cell 12: 780-794. https://doi.org/10.1091/mbc.12.4.780 ![]() |
[32] |
He C, Liu S, Liang J, et al. (2020) Genome-wide identification and analysis of nuclear receptors genes for lethal screening against Bemisia tabaci Q. Pest management science 76: 2040-2048. https://doi.org/10.1002/ps.5738 ![]() |
[33] |
Karut K, Castle SJ, Karut ŞT, et al. (2020) Secondary endosymbiont diversity of Bemisia tabaci and its parasitoids. Infection, Genetics and Evolution 78: 104104. https://doi.org/10.1016/j.meegid.2019.104104 ![]() |
[34] |
Ghosh S, Ghanim M (2021) Factors Determining Transmission of Persistent Viruses by Bemisia tabaci and Emergence of New Virus–Vector Relationships. Viruses 13: 1808. https://doi.org/10.3390/v13091808 ![]() |
[35] |
Tan D, Hu H, Tong X, et al. (2019) Genome-wide identification and characterization of myosin genes in the silkworm, Bombyx mori. Gene 691: 45-55. https://doi.org/10.1016/j.gene.2018.12.011 ![]() |
[36] | Wagner SA, Beli P, Weinert BT, et al. (2012) Proteomic analyses reveal divergent ubiquitylation site patterns in murine tissues. Molecular & Cellular Proteomics 11: 1578-1585. https://doi.org/10.1074/mcp.M112.017905 |
[37] |
Christensen B, Lundby C, Jessen N, et al. (2012) Evaluation of functional erythropoietin receptor status in skeletal muscle in vivo: acute and prolonged studies in healthy human subjects. PloS one 7: e31857. https://doi.org/10.1371/journal.pone.0031857 ![]() |
[38] |
Koshida R, Tome S, Takei Y (2018) Myosin Id localizes in dendritic spines through the tail homology 1 domain. Experimental cell research 367: 65-72. https://doi.org/10.1016/j.yexcr.2018.03.021 ![]() |
[39] |
Girón-Pérez DA, Piedra-Quintero ZL, Santos-Argumedo L (2019) Class I myosins: Highly versatile proteins with specific functions in the immune system. Journal of Leukocyte Biology 105: 973-981. https://doi.org/10.1002/JLB.1MR0918-350RRR ![]() |
[40] |
Olety B, Wälte M, Honnert U, et al. (2010) Myosin 1G (Myo1G) is a haematopoietic specific myosin that localises to the plasma membrane and regulates cell elasticity. FEBS letters 584: 493-499. https://doi.org/10.1016/j.febslet.2009.11.096 ![]() |
[41] |
Okumura T, Sasamura T, Inatomi M, et al. (2015) Class I myosins have overlapping and specialized functions in left-right asymmetric development in Drosophila. Genetics 199: 1183-1199. https://doi.org/10.1534/genetics.115.174698 ![]() |
[42] |
Spitznagel D, O'Rourke JF, Leddy N, et al. (2010) Identification and characterization of an unusual class I myosin involved in vesicle traffic in Trypanosoma brucei. PloS one 5: e12282. https://doi.org/10.1371/journal.pone.0012282 ![]() |
[43] |
Maravillas-Montero JL, López-Ortega O, Patiño-López G, et al. (2014) Myosin 1g regulates cytoskeleton plasticity, cell migration, exocytosis, and endocytosis in B lymphocytes. European journal of immunology 44: 877-886. https://doi.org/10.1002/eji.201343873 ![]() |
[44] |
Vicente-Manzanares M, Ma X, Adelstein RS, et al. (2009) Non-muscle myosin II takes centre stage in cell adhesion and migration. Nature reviews Molecular cell biology 10: 778-790. https://doi.org/10.1038/nrm2786 ![]() |
[45] |
Rozbicki E, Chuai M, Karjalainen AI, et al. (2015) Myosin-II-mediated cell shape changes and cell intercalation contribute to primitive streak formation. Nature cell biology 17: 397-408. https://doi.org/10.1038/ncb3138 ![]() |
[46] |
Seabrooke S, Qiu X, Stewart BA (2010) Nonmuscle Myosin II helps regulate synaptic vesicle mobility at the Drosophilaneuromuscular junction. BMC neuroscience 11: 1-10. https://doi.org/10.1186/1471-2202-11-37 ![]() |
[47] |
Rubio MD, Johnson R, Miller CA, et al. (2011) Regulation of synapse structure and function by distinct myosin II motors. Journal of Neuroscience 31: 1448-1460. https://doi.org/10.1523/JNEUROSCI.3294-10.2011 ![]() |
[48] | Wang K, Okada H, Bi E (2020) Comparative analysis of the roles of non-muscle myosin-IIs in cytokinesis in budding yeast, fission yeast, and mammalian cells. Frontiers in Cell and Developmental Biology 2020: 1397. https://doi.org/10.3389/fcell.2020.593400 |
[49] |
Wang C, Xu Y, Wang X, et al. (2018) GEsture: an online hand-drawing tool for gene expression pattern search. Peer J 6: e4927. https://doi.org/10.7717/peerj.4927 ![]() |
[50] |
An BC, Sakai T, Komaba S, et al. (2014) Phosphorylation of the kinase domain regulates autophosphorylation of myosin IIIA and its translocation in microvilli. Biochemistry 53: 7835-7845. https://doi.org/10.1021/bi501247z ![]() |
[51] | Calábria LK, Vieira da Costa A, da Silva Oliveira RJ, et al. (2013) Myosins are differentially expressed under oxidative stress in chronic streptozotocin-induced diabetic rat brains. International Scholarly Research Notices 2013. https://doi.org/10.1155/2013/423931 |
[52] | Hilbrant M, Horn T, Koelzer S, et al. (2016) The beetle amnion and serosa functionally interact as apposed epithelia. E life 5: e13834. https://doi.org/10.7554/eLife.13834 |
[53] |
Warmke J, Yamakawa M, Molloy J, et al. (1992) Myosin light chain-2 mutation affects flight, wing beat frequency, and indirect flight muscle contraction kinetics in Drosophila. The Journal of cell biology 119: 1523-1539. https://doi.org/10.1083/jcb.119.6.1523 ![]() |
[54] |
Akbariazar E, Vahabi A, Abdi Rad I (2019) Report of a novel splicing mutation in the MYO15A gene in a patient with sensorineural hearing loss and spectrum of the MYO15A mutations. Clinical Medicine Insights: Case Reports 12: 1179547619871907. ![]() |
[55] |
Komaba S, Watanabe S, Umeki N, et al. (2010) Effect of phosphorylation in the motor domain of human myosin IIIA on its ATP hydrolysis cycle. Biochemistry 49: 3695-3702. https://doi.org/10.1021/bi902211w ![]() |
[56] |
Chibalina MV, Puri C, Kendrick-Jones J, et al. (2009) Potential roles of myosin VI in cell motility. Biochemical Society Transactions 37: 966-970. https://doi.org/10.1042/BST0370966 ![]() |
[57] |
Wei T, Huang T-S, McNeil J, et al. (2010) Sequential recruitment of the endoplasmic reticulum and chloroplasts for plant potyvirus replication. Journal of virology 84: 799-809. https://doi.org/10.1128/JVI.01824-09 ![]() |
[58] |
Agbeci M, Grangeon R, Nelson RS, et al. (2013) Contribution of host intracellular transport machineries to intercellular movement of turnip mosaic virus. PLoS Pathogens 9: e1003683. https://doi.org/10.1371/journal.ppat.1003683 ![]() |
1. | 玉芳 王, Multiplicity of Positive Solutions forNonlinear Elliptic Problems inExterior Domain, 2024, 14, 2160-7583, 605, 10.12677/PM.2024.145214 |