Research article

On global well-posedness to 3D Navier-Stokes-Landau-Lifshitz equations

  • Received: 09 July 2020 Accepted: 13 August 2020 Published: 19 August 2020
  • MSC : 35Q35, 35D35, 76D05

  • In this paper, we prove the global well-posedness of solutions for the Cauchy problem of three-dimensional incompressible Navier-Stokes-Landau-Lifshitz equations under the condition that u0H12+d0H12+ε (ε>0) is sufficiently small. This result can be seen as an improvement of the previous paper [20].

    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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  • In this paper, we prove the global well-posedness of solutions for the Cauchy problem of three-dimensional incompressible Navier-Stokes-Landau-Lifshitz equations under the condition that u0H12+d0H12+ε (ε>0) is sufficiently small. This result can be seen as an improvement of the previous paper [20].


    Consider the Cauchy problem of 3D Navier-Stokes-Landau-Lifshitz equation

    {ut+(u)uνΔu+(dd)=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 dd denotes a 3×3 matrix whose (i,j)th entry is given by idjd for 1i,j3. 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 ν1. and the operator Λ2δ is defined through the Fourier transform (see [15]), namely

    Λ2δf(x)=(Δ)δf(x)=R3|x|2δˆf(ξ)e2πixξdξ,

    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

    {uC([0,);˙B1/22,1)˜L((0,);˙B1/22,1)L1((0,);˙B5/22,1),dC([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 u0H1+d0H2 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 u0H12+d0H12+εK, 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

    dLd2ε1+2εL6Λ32+εd11+2εL2Λd2ε1+2εL2Λ32+εd11+2εL2ΛdL2+Λ32+εdL2.

    REMARK 1.5. Since one only need u0H12+d0H12+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(u2L2+d2L2)+u2L2+Δd+|d|2d2L2=0,t0. (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(Λ12u2L2+Λ32d2L2)+Λ32u2L2+Λ52d2L2=R3Λ12(uu)Λ12udx+R3Λ12[(dd)]Λ12udxR3Λ32(ud)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|CΛ32uL2Λ12uL6uL3CΛ12uL2Λ32u2L2, (2.3)
    |I2|CΛ12uL6Λ52dL2dL3CΛ32dL2(Λ32u2L2+Λ52d2L2), (2.4)

    and

    |I3|CΛ32dL6Λ32(ud)L65CΛ52dL2(Λ32uL2dL3+uL3Λ32dL2)C(Λ32dL2+Λ12uL2)(Λ32u2L2+Λ52d2L2). (2.5)

    Note that |d|=1, we have dLC and dd=1. Hence, Δ(dd)=0, which implies that

    (dd)=|d|2+dΔd=0.

    Therefore, we have

    dΔd=|d|2.

    Using the above equality, we easily obtain

    |I4|CΛ52dL2Λ12(|d|2d)L2CΛ52dL2(Λ32dL6dL3dL+Λ12dL6dΔdL3)=CΛ52dL2(Λ32dL6dL3dL+Λ12dL6dLΔdL3)CΛ32dL2Λ52d2L2. (2.6)

    Moreover, since

    dLd2ε1+2εL6Λ32+εd11+2εL2Λddε1+dεL2Λ32+εd11+dεL2ΛdL2+Λ32+εdL2,

    we easily obtain

    |I5|CΛ52dL2Λ12(d×Δd)L2CΛ52dL2(Λ12dL6ΔdL3+dLΛ52dL2)CΛ52dL2[Λ12dL6ΔdL3+(ΛdL2+Λ32+εdL2)Λ52dL2]C(ΛdL2+Λ32dL2+Λ32+εdL2)Λ52d2L2. (2.7)

    It then follows from (2.2)–(2.7) that

    12ddt(Λ12u2L2+Λ32d2L2)+Λ32u2L2+Λ52d2L2C(Λ12uL2+ΛdL2+Λ32dL2+Λ32+εdL2)(Λ32u2L2+Λ52d2L2). (2.8)

    Taking Λ32+ε to (1.1)2, multiplying by Λ32+εd, we deduce that

    12ddtΛ32+εd2L2+Λ52+εd2L2=R3Λ32+ε(ud)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|CΛ52+εdL2Λ12+ε(ud)L2CΛ52+εdL2(Λ12+εuL62ε+1dL31ε+uL3Λ32+εdL6)CΛ52+εdL2(Λ32uL2Λ32+εdL2+Λ12uL2Λ52+εdL2)C(Λ32+εdL2+Λ12uL2)(Λ52+εd2L2+Λ32u2L2), (2.10)
    |I7|CΛ52+εdL2Λ12+ε(|d|2d)L2CΛ52+εdL2(Λ32+εdL6dL3dL+Λ12+εdL6dLΔdL3)C(Λ32dL2+Λ32+εdL2)(Λ52+εd2L2+Λ52d2L2), (2.11)

    and

    |I8|CΛ52+εdL2Λ12+ε(d×Δd)L2CΛ52+εdL2(Λ12+εdL6ΔdL3+dLΛ52+εdL2)CΛ52+εdL2[Λ32+εdL2Λ52dL2+(ΛdL2+Λ32+εdL2)Λ52+εdL2]C(Λ32+εdL2+ΛdL2)(Λ52+εd2L2+Λ52d2L2), (2.12)

    where we have used the facts that dΔd=|d|2 in (2.11) and dLΛdL2+Λ32+εdL2 in (2.12), respectively. Combining (2.9)–(2.12) together gives

    12ddtΛ32+εd2L2+Λ52+εd2L2C(Λ32+εdL2+ΛdL2+Λ32dL2+Λ12uL2)(Λ52+εd2L2+Λ52d2L2+Λ32u2L2). (2.13)

    Summing up (2.1), (2.8) and (2.13), we arrive at

    12ddt(u2L2+d2L2+Λ12u2L2+Λ32d2L2+Λ32+εd2L2)+u2L2+Δd+|d|2d2L2+Λ32u2L2+Λ52d2L2+Λ52+εd2L2C(ΛdL2+Λ12uL2+Λ32dL2+Λ32+εdL2)(Λ32u2L2+Λ52d2L2+Λ52+εd2L2). (2.14)

    Taking K small enough such that

    ΛdL2+Λ12uL2+Λ32dL2+Λ32+εdL2<K<12C, (2.15)

    then, u2L2+d2L2+Λ12u2L2+Λ32d2L2+Λ32+εd2L2 is decreasing. So, for any 0<T<, we have

    ddt(u2L2+d2L2+Λ12u2L2+Λ32d2L2+Λ32+εd2L2)+u2L2+Δd+|d|2d2L2+Λ32u2L2+Λ52d2L2+Λ52+εd2L20, (2.16)

    which means

    {uL(0,T;H12)L2(0,T;H32),dL(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 u0H1+d0ω0H2 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 u0H12+d0H12+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.



    [1] F. Alouges, A. Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal., 18 (1992), 1071-1084. doi: 10.1016/0362-546X(92)90196-L
    [2] B. Guo, M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Patial. Dif., 1 (1993), 311-334. doi: 10.1007/BF01191298
    [3] W. E, X. Wang, Numerical methods for the Landau-Lifshitz equation, SIAM J. Numer. Anal., 38 (2000), 1647-1665. doi: 10.1137/S0036142999352199
    [4] J. Fan, H. Gao, B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl. 363 (2010), 29-37. doi: 10.1016/j.jmaa.2009.07.047
    [5] J. Fan, Y. Zhou, Uniform local well-posedness for an Ericksen-Leslie's density-dependent parabolic-hyperbolic liquid crystals model, Appl. Math. Lett., 74 (2017), 79-84. doi: 10.1016/j.aml.2017.04.012
    [6] E. Feireisl, E. Rocca, G. Schimperna, et al. On a hyperbolic system arising in liquid crystals modeling, J. Hyperbol. Differ. Eq., 15 (2018), 15-35. doi: 10.1142/S0219891618500029
    [7] D. Golovaty, P. Sternberg, R. Venkatraman, A Ginzburg-Landau-type problem for highly anisotropic nematic liquid crystals, SIAM J. Math. Anal., 51 (2019), 276-320. doi: 10.1137/18M1178360
    [8] B. Guo, F. Liu, Weak and smooth solutions to incompressible Navier-Stokes-Landau-LifshitzMaxwell equations, Front. Math. China, 14 (2019), 1133-1161. doi: 10.1007/s11464-019-0800-x
    [9] T. Kato, Strong Lp-solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182
    [10] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704
    [11] H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22- 35.
    [12] F. Lin, J. Lin, C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x
    [13] F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible NavierStokes equations in R3, Ann. Inst. Henri Poincare, 13 (1996), 319-336.
    [14] M. Schonbek, Y. Shibata, Global well-posedness and decay for a Q tensor model of incompressible nematic liquid crystals in RN, J. Differ. Equations, 266 (2019), 3034-3065.
    [15] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, USA, 1970.
    [16] I. W. Stewart, T. R. Faulkner, The stability of nematic liquid crystals under crossed electric and magnetic fields, Appl. Math. Lett., 13 (2000), 23-28.
    [17] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
    [18] G. Wang, B. Guo, Existence and uniqueness of the weak solution to the incompressible NavierStokes-Landau-Lifshitz model in 2-dimension, Acta Math. Sci. Ser. B (Engl. Ed.), 37 (2017), 1361- 1372.
    [19] G. Wang, B. Guo, Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6141-6166.
    [20] R. Wei, Y. Li, Z. Yao, Decay rates of higher-order norms of solutions to the Navier-Stokes-LandauLifshitz system, Appl. Math. Mech. (English Ed.), 39 (2018), 1499-1528. doi: 10.1007/s10483-018-2380-8
    [21] X. Zhai, Y. Li, W. Yan, Global solutions to the Navier-Stokes-Landau-Lifshitz system, Math. Nachr., 289 (2016), 377-388. doi: 10.1002/mana.201400419
    [22] C. Zhao, Y. Li, Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach, Nonlinear Anal. Real World Appl., 53 (2020), 103077.
    [23] X. Zhao, Y. Zhou, Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations, Discrete Cont. Dyn. Syst. B, In press, doi: 10.3934/dcdsb.2020142.
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