Citation: Ning Duan, Xiaopeng Zhao. On global well-posedness to 3D Navier-Stokes-Landau-Lifshitz equations[J]. AIMS Mathematics, 2020, 5(6): 6457-6463. doi: 10.3934/math.2020416
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Consider the Cauchy problem of 3D Navier-Stokes-Landau-Lifshitz equation
{ut+(u⋅∇)u−νΔu+∇⋅(∇d⊙∇d)=0,dt+(u⋅∇)d=Δd+|∇d|2d+d×Δd,∇⋅u=0,|d|=1,u(x,0)=u0(x),d(x,0)=d0(x). | (1.1) |
where u(x,t) describes the velocity, p represent the pressure and d(x,t) stands for the magnetic moment respectively. The constant ν>0 means the shear viscosity coefficient of the fluid, and the symbol ∇d⊙∇d denotes a 3×3 matrix whose (i,j)th entry is given by ∂id⋅∂jd for 1≤i,j≤3. We note that if d=0, system (1.1) reduces to be the classical Navier-stokes equations [9,11,13,17,22,23], which have drawn much attention. Moreover, if u=0 in system (1.1), we obtain the Landau-Lifshitz system [1,2,3]. In this paper, for the sake of simplicity, we set the coefficient ν≡1. and the operator Λ2δ is defined through the Fourier transform (see [15]), namely
Λ2δf(x)=(−Δ)δf(x)=∫R3|x|2δˆf(ξ)e2πix⋅ξdξ, |
and ˆf is the Fourier transform of f.
For the study on the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations, we refer the reader to Wang and Guo [18,19] and Guo and Liu [8]. By using the Faedo-Galerkin approximation and weak compactness theory, the authors studied the existence and uniqueness of the weak solution to system (1.1) in two-dimension and three-dimension. There are also some papers related to the strong solutions to Navier-Stokes-Landau-Lifshitz equations. In [4], by using energy methods and delicate estimates from harmonic analysis, Fan, Gao and Guo obtained some regularity criteria for the strong solutions in Besov and multiplier spaces; Supposed that d0∈˙B3/22,1(R3) and u0=(uh0,u30)∈˙B1/22,1(R3) with |d0|=1 and ∇⋅u0=0, by using the Fourier frequency localization and Bony's paraproduct decomposition, Zhai, Li and Yan [21] proved that there exists a unique global solution (u,d) with
{u∈C([0,∞);˙B1/22,1)⋂˜L((0,∞);˙B1/22,1)⋂L1((0,∞);˙B5/22,1),d∈C([0,∞);˙B3/22,1)⋂˜L([0,∞);˙B3/22,1)⋂L1([0,∞);˙B7/22,1), | (1.2) |
provided that the initial data satisfies
C{ν(‖uh0‖˙B1/22,1+‖d0‖˙B3/22,1)+((‖uh0‖˙B1/22,1+‖d0‖˙B3/22,1)1/2(‖u30‖˙B1/22,1+ν)12)}≤ν2. |
Recently, with the help of an energy method, Wei, Li and Yao [20] established the global well-posedness of strong solutions for system (1.1) provided that ‖u0‖H1+‖d0‖H2 is sufficiently small. Moreover, by applying the Fourier splitting method, the authors also showed that the time decay rates of the higher-order spatial derivatives of the solutions.
REMARK 1.1. When the term d×Δd is omitted, the system (1.1) reduces to the liquid crystals equations, which have been studied by many researchers, see for instance, [5,6,7,12,14,16] and the reference therein.
In this paper, we first show the following local well-posedness result, which can be proved by using Banach fixed point theorem and the standard linearization argument. Since the proof is so standard, we omit it here.
LEMMA 1.2 (Local well-posedness). Let (u0,∇d0)∈H2(R3). Then, there exists a small time ˜T>0 and a unique strong solution (u(x,t),∇d(x,t)) to system (1.1) satisfying
{(u,∇d)∈C([0,˜T];H2)⋂L2(0,˜T;H3),(ut,∇dt)∈L∞(0,˜T;L2)⋂L2(0,˜T;H1). | (1.3) |
Now, we give the following theorem on the small initial data global well-posedness for system (1.1).
THEOREM 1.3 (Small initial data global well-posedness). Suppose (u0,d0)∈Hs(R3)×Hs+1(R3) with s>2 and divu0=0. There exists a sufficiently small constant K>0 and any ε>0 such that if ‖u0‖H12+‖∇d0‖H12+ε≤K, then there exists a unique global strong solution (u,d) and satisfy
(u,∇d)∈C(0,T;Hs(R3))∩L2(0,T;Hs+1(R3)). |
REMARK 1.4. It is worth pointing out that the constant ε in Theorem 1 can not reduced to 0 because of the Sobolev's embedding
‖d‖L∞≤‖d‖2ε1+2εL6‖Λ32+εd‖11+2εL2≤‖Λd‖2ε1+2εL2‖Λ32+εd‖11+2εL2≤‖Λd‖L2+‖Λ32+εd‖L2. |
REMARK 1.5. Since one only need ‖u0‖H12+‖∇d0‖H12+ε is sufficiently small, this paper can be seen as an improvement of Wei, Li and Yao [20].
Multiplying (1.1)1 and (1.1)2 by u and Δd+|∇d|2d, we obtain the fundamental energy estimate
ddt(‖u‖2L2+‖∇d‖2L2)+‖∇u‖2L2+‖Δd+|∇d|2d‖2L2=0,∀t≥0. | (2.1) |
Taking Λ12 to (1.1)1, taking Λ32 to (1.1)2, multiplying by Λ12u and Λ32d respectively, summing them up, we find that
12ddt(‖Λ12u‖2L2+‖Λ32d‖2L2)+‖Λ32u‖2L2+‖Λ52d‖2L2=−∫R3Λ12(u⋅∇u)⋅Λ12udx+∫R3Λ12[∇⋅(∇d⊙∇d)]⋅Λ12udx−∫R3Λ32(u⋅∇d)⋅Λ32ddx+∫R3Λ32(|∇d|2d)⋅Λ32ddx+∫R3Λ32(d×Δd)⋅Λ32ddx=I1+I2+I3+I4+I5. | (2.2) |
By using the Kato-Ponce inequality [10], it yields that
|I1|≤C‖Λ32u‖L2‖Λ12u‖L6‖u‖L3≤C‖Λ12u‖L2‖Λ32u‖2L2, | (2.3) |
|I2|≤C‖Λ12u‖L6‖Λ52d‖L2‖∇d‖L3≤C‖Λ32d‖L2(‖Λ32u‖2L2+‖Λ52d‖2L2), | (2.4) |
and
|I3|≤C‖Λ32d‖L6‖Λ32(u⋅∇d)‖L65≤C‖Λ52d‖L2(‖Λ32u‖L2‖∇d‖L3+‖u‖L3‖Λ32∇d‖L2)≤C(‖Λ32d‖L2+‖Λ12u‖L2)(‖Λ32u‖2L2+‖Λ52d‖2L2). | (2.5) |
Note that |d|=1, we have ‖d‖L∞≤C and d⋅d=1. Hence, Δ(d⋅d)=0, which implies that
∇⋅(d∇d)=|∇d|2+dΔd=0. |
Therefore, we have
d⋅Δd=−|∇d|2. |
Using the above equality, we easily obtain
|I4|≤C‖Λ52d‖L2‖Λ12(|∇d|2d)‖L2≤C‖Λ52d‖L2(‖Λ32d‖L6‖∇d‖L3‖d‖L∞+‖Λ12d‖L6‖d⋅Δd‖L3)=C‖Λ52d‖L2(‖Λ32d‖L6‖∇d‖L3‖d‖L∞+‖Λ12d‖L6‖d‖L∞‖Δd‖L3)≤C‖Λ32d‖L2‖Λ52d‖2L2. | (2.6) |
Moreover, since
‖d‖L∞≤‖d‖2ε1+2εL6‖Λ32+εd‖11+2εL2≤‖Λd‖dε1+dεL2‖Λ32+εd‖11+dεL2≤‖Λd‖L2+‖Λ32+εd‖L2, |
we easily obtain
|I5|≤C‖Λ52d‖L2‖Λ12(d×Δd)‖L2≤C‖Λ52d‖L2(‖Λ12d‖L6‖Δd‖L3+‖d‖L∞‖Λ52d‖L2)≤C‖Λ52d‖L2[‖Λ12d‖L6‖Δd‖L3+(‖Λd‖L2+‖Λ32+εd‖L2)‖Λ52d‖L2]≤C(‖Λd‖L2+‖Λ32d‖L2+‖Λ32+εd‖L2)‖Λ52d‖2L2. | (2.7) |
It then follows from (2.2)–(2.7) that
12ddt(‖Λ12u‖2L2+‖Λ32d‖2L2)+‖Λ32u‖2L2+‖Λ52d‖2L2≤C(‖Λ12u‖L2+‖Λd‖L2+‖Λ32d‖L2+‖Λ32+εd‖L2)(‖Λ32u‖2L2+‖Λ52d‖2L2). | (2.8) |
Taking Λ32+ε to (1.1)2, multiplying by Λ32+εd, we deduce that
12ddt‖Λ32+εd‖2L2+‖Λ52+εd‖2L2=−∫R3Λ32+ε(u⋅∇d)⋅Λ32+εddx+∫R3Λ32+ε(|∇d|2d)⋅Λ32+εddx+∫R3Λ32+ε(d×Δd)⋅Λ32+εddx=I6+I7+I8. | (2.9) |
Using the Kato-Ponce inequality again, we derive that
|I6|≤C‖Λ52+εd‖L2‖Λ12+ε(u⋅∇d)‖L2≤C‖Λ52+εd‖L2(‖Λ12+εu‖L62ε+1‖∇d‖L31−ε+‖u‖L3‖Λ32+εd‖L6)≤C‖Λ52+εd‖L2(‖Λ32u‖L2‖Λ32+εd‖L2+‖Λ12u‖L2‖Λ52+εd‖L2)≤C(‖Λ32+εd‖L2+‖Λ12u‖L2)(‖Λ52+εd‖2L2+‖Λ32u‖2L2), | (2.10) |
|I7|≤C‖Λ52+εd‖L2‖Λ12+ε(|∇d|2d)‖L2≤C‖Λ52+εd‖L2(‖Λ32+εd‖L6‖∇d‖L3‖d‖L∞+‖Λ12+εd‖L6‖d‖L∞‖Δd‖L3)≤C(Λ32d‖L2+‖Λ32+εd‖L2)(‖Λ52+εd‖2L2+‖Λ52d‖2L2), | (2.11) |
and
|I8|≤C‖Λ52+εd‖L2‖Λ12+ε(d×Δd)‖L2≤C‖Λ52+εd‖L2(‖Λ12+εd‖L6‖Δd‖L3+‖d‖L∞‖Λ52+εd‖L2)≤C‖Λ52+εd‖L2[‖Λ32+εd‖L2‖Λ52d‖L2+(‖Λd‖L2+‖Λ32+εd‖L2)‖Λ52+εd‖L2]≤C(‖Λ32+εd‖L2+‖Λd‖L2)(‖Λ52+εd‖2L2+‖Λ52d‖2L2), | (2.12) |
where we have used the facts that d⋅Δd=−|∇d|2 in (2.11) and ‖d‖L∞≤‖Λd‖L2+‖Λ32+εd‖L2 in (2.12), respectively. Combining (2.9)–(2.12) together gives
12ddt‖Λ32+εd‖2L2+‖Λ52+εd‖2L2≤C(‖Λ32+εd‖L2+‖Λd‖L2+‖Λ32d‖L2+‖Λ12u‖L2)(‖Λ52+εd‖2L2+‖Λ52d‖2L2+‖Λ32u‖2L2). | (2.13) |
Summing up (2.1), (2.8) and (2.13), we arrive at
12ddt(‖u‖2L2+‖∇d‖2L2+‖Λ12u‖2L2+‖Λ32d‖2L2+‖Λ32+εd‖2L2)+‖∇u‖2L2+‖Δd+|∇d|2d‖2L2+‖Λ32u‖2L2+‖Λ52d‖2L2+‖Λ52+εd‖2L2≤C(‖Λd‖L2+‖Λ12u‖L2+‖Λ32d‖L2+‖Λ32+εd‖L2)(‖Λ32u‖2L2+‖Λ52d‖2L2+‖Λ52+εd‖2L2). | (2.14) |
Taking K small enough such that
‖Λd‖L2+‖Λ12u‖L2+‖Λ32d‖L2+‖Λ32+εd‖L2<K<12C, | (2.15) |
then, ‖u‖2L2+‖∇d‖2L2+‖Λ12u‖2L2+‖Λ32d‖2L2+‖Λ32+εd‖2L2 is decreasing. So, for any 0<T<∞, we have
ddt(‖u‖2L2+‖∇d‖2L2+‖Λ12u‖2L2+‖Λ32d‖2L2+‖Λ32+εd‖2L2)+‖∇u‖2L2+‖Δd+|∇d|2d‖2L2+‖Λ32u‖2L2+‖Λ52d‖2L2+‖Λ52+εd‖2L2≤0, | (2.16) |
which means
{u∈L∞(0,T;H12)∩L2(0,T;H32),∇d∈L∞(0,T;H12+ε)∩L2(0,T;H32+ε). | (2.17) |
By Lemma and (2.17), we easily obtain the higher-order norm estimates for the solution, this complete the proof.
The three-dimensional incompressible Navier-Stokes-Landau-Lifshitz equations is an important hydrodynamics equations. The well-posedness and large time behavior of its solutions were studied by many authors. The latest result on the global well-posedness was studied by Wei, Li and Yao [20]. The author supposed that ‖u0‖H1+‖d0−ω0‖H2 is sufficiently small, obtained the small initial data global well-posedness. In this paper, we improve the global well-posedness result in [20], only assume that ‖u0‖H12+‖∇d0‖H12+ε (ε>0) is sufficiently small, prove the global well-posedness for 3D Navier-Stokes-Landau-Lifshitz equations.
This paper was supported by the Fundamental Research Funds for the Central Universities (grant No. N2005006, N2005031).
The authors declare that they have no competing interests.
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