Processing math: 60%
Case report

Laparoscopic uterosacral nerve block: A fertility preserving option in chronic pelvic pain

  • Received: 31 July 2019 Accepted: 17 September 2019 Published: 26 September 2019
  • Chronic pelvic pain (CPP) can cause extreme physical distress in women and has widespread socio-economic consequences. Nerve root blocks have become a safe and effective treatment modality in multiple specialties in both the diagnosis and treatment of pain. We describe a novel technique of a laparoscopic uterosacral nerve block (USNB) and demonstrate its effectiveness in the treatment of a complex case of CPP. USNB has potential diagnostic, prognostic and therapeutic implications. It should therefore be considered as part of the multi-disciplinary management of women with CPP of suspected uterine origin such as adenomyosis, degenerating fibroids or following myomectomy.

    Citation: Benjamin P Jones, Srdjan Saso, Timothy Bracewell-Milnes, Jen Barcroft, Jane Borley, Teodor Goroszeniuk, Kostas Lathouras, Joseph Yazbek, J Richard Smith. Laparoscopic uterosacral nerve block: A fertility preserving option in chronic pelvic pain[J]. AIMS Medical Science, 2019, 6(4): 260-267. doi: 10.3934/medsci.2019.4.260

    Related Papers:

    [1] Yuhui Chen, Ronghua Pan, Leilei Tong . The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $. Electronic Research Archive, 2021, 29(2): 1945-1967. doi: 10.3934/era.2020099
    [2] Guochun Wu, Han Wang, Yinghui Zhang . Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $. Electronic Research Archive, 2021, 29(6): 3889-3908. doi: 10.3934/era.2021067
    [3] Jingjing Zhang, Ting Zhang . Local well-posedness of perturbed Navier-Stokes system around Landau solutions. Electronic Research Archive, 2021, 29(4): 2719-2739. doi: 10.3934/era.2021010
    [4] Yue Cao . Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Electronic Research Archive, 2020, 28(1): 27-46. doi: 10.3934/era.2020003
    [5] Jun Zhou . Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28(1): 67-90. doi: 10.3934/era.2020005
    [6] Jiayi Han, Changchun Liu . Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian. Electronic Research Archive, 2021, 29(5): 3509-3533. doi: 10.3934/era.2021050
    [7] Ting Liu, Guo-Bao Zhang . Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, 2021, 29(4): 2599-2618. doi: 10.3934/era.2021003
    [8] Xiu Ye, Shangyou Zhang . A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh. Electronic Research Archive, 2021, 29(6): 3609-3627. doi: 10.3934/era.2021053
    [9] Huafei Di, Yadong Shang, Jiali Yu . Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28(1): 221-261. doi: 10.3934/era.2020015
    [10] Chungen Liu, Huabo Zhang . Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, 2021, 29(5): 3281-3295. doi: 10.3934/era.2021038
  • Chronic pelvic pain (CPP) can cause extreme physical distress in women and has widespread socio-economic consequences. Nerve root blocks have become a safe and effective treatment modality in multiple specialties in both the diagnosis and treatment of pain. We describe a novel technique of a laparoscopic uterosacral nerve block (USNB) and demonstrate its effectiveness in the treatment of a complex case of CPP. USNB has potential diagnostic, prognostic and therapeutic implications. It should therefore be considered as part of the multi-disciplinary management of women with CPP of suspected uterine origin such as adenomyosis, degenerating fibroids or following myomectomy.


    In this paper, we are concerned with the sharp decay rates of solutions to the Cauchy problem for the isentropic Navier-Stokes equations:

    $ {tρ+div(ρu)=0,(t,x)R+×R3,t(ρu)+div(ρuu)+p(ρ)=divT,(t,x)R+×R3,lim|x|ρ=ˉρ,lim|x|u=0,tR+,(ρ,u)|t=0=(ρ0,u0),xR3, $ (1.1)

    which governs the motion of a isentropic compressible viscous fluid. The unknown functions $ \rho $ and $ u $ represent the density and velocity of the fluid respectively. The pressure $ p = p(\rho) $ is a smooth function in a neighborhood of a positive constant $ \bar \rho $ s.t. $ p'(\bar \rho)>0 $. $ T $ is the viscosity stress tensor given by $ T = \mu({\nabla} u+({\nabla} u)^t)+\nu({\mathop{{\rm{div}}}\nolimits} u)I $ with $ I $ the identity matrix. We assume that the constant viscosity coefficients $ \mu>0 $ and $ \nu $ satisfy $ \nu+\frac23 \mu>0 $. Throughout this article, by optimal time decay rate, we refer to the best possible decay rate in upper bound as many literatures, and the sharp time decay rate includes the best possible upper and lower bounds.

    Using the classical spectral method, the optimal time decay rate (upper bound) of the linearized equations of the isentropic Navier-Stokes equations are well known. One may then expect that the small solution of the nonlinear equations (1.1) have the same decay rate as the linear one. Our work is devoted to proving the sharp time decay rate (for both upper and lower bound) for the nonlinear system.

    In the case of one space dimension, Zeng [24] and Liu-Zeng [15] offered a detailed analysis of the solution to a class of hyperbolic-parabolic system through point-wise estimate, including the isentropic Navier-Stokes system. For multi-dimensional Navier-Stokes equations (and/or Navier-Stokes-Fourier system), the $ H^s $ global existence and time-decay rate of strong solutions with the initial perturbation small in $ H^s \cap L^1 $ are obtained in whole space first by A. Matsumura and T. Nishida [17], [18]. When the small initial perturbation belongs to $ H^3 $ only, using a weighted energy method, A. Matsumura [16] showed the time-decay rate $ (1+t)^{-\frac34} $ of upper bound in $ L^\infty $-norm. Since then, there are concrete development on the upper bound time-decay estimates: the optimal $ L^p $ (with $ 2\leq p \leq \infty $) upper bound decay rate was proved by G. Ponce [19], combining the spectral analysis on linearized system and the energy method for small initial perturbation in $ L^1 $. For the isentropic Navier-Stokes equations with artificial viscosity, D. Hoff and K. Zumbrun [6], [7] studied the Green's function and derived the $ L^p $ ($ 1 \leq p\leq\infty $) upper bound time decay rate of diffusive waves for the small initial perturbation belongs to $ H^m \cap L^1 $ with $ m\geq4 $. Liu and Wang [14] studied the point-wise estimates of the Green function of the linearized isentropic Navier-Stokes system in 3D and then analyzed the coupling of nonlinear diffusion waves, obtained the optimal (upper bound) decay rate. These results were further extended to the exterior problem [12], [11], or the half space problem [9], [10], [8]. Recently, Guo and Wang in [5] developed a new general energy method for proving the optimal (upper bound) time decay rates of the solutions to the dissipative equations in the whole space, using a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.

    When additional external force is taken into account, the external force does affect the long time behavior of solutions. The upper bound of time decay rates were studied intensively, see for instance [1] and [2] on unbounded domain, [22], [23] on the convergence of the non-stationary flow to the corresponding steady flow when the initial date are small in $ H^3 \cap L^{\frac65} $, and [4], [3], on the optimal $ L^p-L^q $ upper bound decay rates for potential forces.

    The main goal of current paper is to establish the sharp decay rate, on both upper and lower bounds, to the solutions of (1.1) using relatively simple energy method. We remark that similar results had been pursued by M. Schonbek [20], [21] for incompressible Navier-Stokes equations, and by Li, Matsumura-Zhang [13] for isentropic Navier-Stokes-Poisson system. Although they share the same spirit in obtaining the lower bound decay rates, the feature of the spectrum near zero exhibits quite different behaviors, leading to different analysis. For instance, we explored the elegant structure of the higher order nonlinear terms of Navier-Stokes, when choosing conservative variables: density and momentum. The conservative form of the sharp equations provided a natural derivative structure in these terms, leading to the possibility of a faster decay rate estimate. We will make a more detailed comparison later in this paper.

    Define $ n = \rho- \bar \rho $, and let $ m = \rho u = (n+ \bar \rho)u $ be the momentum. We rewrite (1.1) as

    $ {tn+divm=0,(t,x)R+×R3,tm+c2nˉμm(ˉμ+ˉν)divm=F,(t,x)R+×R3,lim|x|n=0,lim|x|m=0,tR+,(n,m)|t=0=(ρ0ˉρ,ρ0u0),xR3, $ (1.2)

    where $ \bar\mu = \frac{\mu}{\bar\rho} $, $ \bar\nu = \frac{\nu}{\bar\rho} $, $ c = \sqrt{p'(\bar \rho)}>0 $ is the sound speed, and

    $ F=div{mmn+ˉρ+ˉμ(nmn+ˉρ)}{(ˉμ+ˉν)div(nmn+ˉρ)+(p(n+ˉρ)p(ˉρ)c2n)}. $

    It is this structure of $ F $ that plays an important role in our analysis.

    Our aim is to obtain a clear picture of the large time behavior of $ U = (n, m) $ in $ L^2({\mathop{\mathbb R\kern 0pt}\nolimits}^3) $ when $ U_0 = (\rho_0-\bar \rho,\rho_0 u_0) $ is sufficiently smooth and small. We introduce the following initial value problem of the linearized Navier-Stokes system corresponding to (1.2):

    $ {t˜n+div˜m=0,(t,x)R+×R3,t˜m+c2˜nˉμ˜m(ˉμ+ˉν)div˜m=0,(t,x)R+×R3,lim|x|˜n=0,lim|x|˜m=0,tR+,(˜n,˜m)|t=0=(ρ0ˉρ,ρ0u0),xR3, $ (1.3)

    where $ \bar\mu = \frac{\mu}{\bar\rho} $, $ \bar\nu = \frac{\nu}{\bar\rho} $, $ c = \sqrt{p'(\bar \rho)} $. It is known that the $ L^2 $-norm of $ \widetilde U = (\widetilde{n} , \widetilde{m} ) $ decays at the optimal upper bound rate $ (1+t)^{-\frac34} $ for generic small initial data, see for instance [18]. A detailed proof on the optimal lower and upper bound rate will be given in the section 3 of this paper. In section 4, we prove that $ \|(U-\widetilde U)(\cdot, t)\|_{L^2} $ decays at a faster rate than $ \|\widetilde U(\cdot, t)\|_{L^2} $, under some reasonable conditions on the initial data. Therefore, $ \|U(\cdot, t)\|_{L^2} $ shares the sharp decay rate of $ (1+t)^{-\frac34} $.

    Notation. For $ a \lesssim b $, we mean that there is a uniform constant $ C $, which may be different on different lines, such that $ a \leq Cb $. And $ a \approx b $ stands for $ a \lesssim b $ and $ b\lesssim a $.

    We now state our main result.

    Theorem 1.1. Assume that $ (n_0, m_0)\in L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^3)\cap H^3({\mathop{\mathbb R\kern 0pt}\nolimits}^3) $, $ \delta_0 = : \|(n_0,m_0)\|_{L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^3)\cap H^3({\mathop{\mathbb R\kern 0pt}\nolimits}^3)} $ is sufficiently small, and

    $ R3(n0,m0)dx0, $ (1.4)

    then there is a unique global classical solution $ \widetilde U = (\widetilde{n}, \widetilde{m})\in \mathcal C([0,\infty); H^3({\mathop{\mathbb R\kern 0pt}\nolimits}^3)) $ of the linearized system (1.3) satisfying for some positive constant $ C $

    $ C1(1+t)34k2k˜n(t)L2(R3)C(1+t)34k2,k=0,1,2,3,C1(1+t)34k2k˜m(t)L2(R3)C(1+t)34k2,k=0,1,2,3, $

    and the initial value problem (1.2) has a unique solution $ U = (n, m)\in \mathcal C([0,\infty); H^3({\mathop{\mathbb R\kern 0pt}\nolimits}^3)) $. Moreover, let $ n_h = n-\widetilde{n} $ and $ m_h = m-\widetilde{m} $, then it holds that

    $ k(nh,mh)(t)L2(R3)δ20(1+t)54k2,k=0,1,2,3mh(t)L2(R3)δ20(1+t)114,3nh(t)L2(R3)δ0(1+t)74. $

    As a consequence, there exists a positive constant $ C_1 $ such that

    $ C11(1+t)34k2kn(t)L2(R3)C1(1+t)34k2,k=0,1,2,C11(1+t)34k2km(t)L2(R3)C1(1+t)34k2,k=0,1,2,3. $

    Remark 1.1. We remark that this theorem is valid under the condition (1.4) which is important in the lower bound estimate to the linearized problem. When (1.4) fails, the decay rate of the linearized system (1.3) depends on the order of the degeneracy of moments. Assume $ (n_0, m_0)\in L^1\cap H^3 $ and belong to certain appropriate weighted $ L^p $ spaces, similar situation happened also in the incompressible Navier-Stokes equations, c.f. [20], [21]. We also note that our condition (1.4) is weaker than those in most of previous results where the differentiability of Fourier transform of initial disturbance is required in general.

    Remark 1.2. In [13], Li, Matsumura-Zhang proved the lower bound decay rate of the linearized isentropic Navier-Stokes-Poisson system, they only require $ |\widehat n_0(\xi)|>c_0>0 $ for $ |\xi|\ll 1 $ with $ c_0 $ a constant due to the special structure of the spectrum from the help of the Poisson term. This condition is proposed in Fourier space, similar to (1.4) in some sense. In our case, the spectrum is different and the different structure leads to different sharp decay rates.

    In what follows, we will set $ n = \rho-\bar \rho $, $ u = u-0 $. We rewrite (1.1) in the perturbation form as

    $ {tn+ˉρdivu=ndivuun,tu+γˉρnˉμu(ˉμ+ˉν)divu=uuˉμf(n)u(ˉμ+ˉν)f(n)divug(n)n,lim|x|n=0,lim|x|u=0,(n,u)|t=0=(ρ0ˉρ,u0), $ (2.1)

    where $ \bar \mu = \frac{\mu}{ \bar \rho } $, $ \bar \nu = \frac{\nu}{ \bar \rho } $, $ \gamma = \frac{p'(\bar \rho)}{\bar \rho^2} $, and the nonlinear functions $ f $ and $ g $ are defined by

    $ f(n):=nn+ˉρ,g(n):=p(n+ˉρ)n+ˉρp(ˉρ)ˉρ. $ (2.2)

    We assume that there exist a time of existence $ T>0 $ and sufficiently small $ \delta>0 $, such that a priori estimate

    $ n(t)H3+u(t)H3δ, $ (2.3)

    holds for any $ t\in[0,T] $. First of all, by (2.3) and Sobolev's inequality, we obtain that

    $ ˉρ2n+ˉρ2ˉρ. $

    Hence, we immediately have

    $ |f(n)|,|g(n)|C|n|,|kf(n)|,|kg(n)|CkN+, $ (2.4)

    where $ f(n) $ and $ g(n) $ are nonlinear functions of $ n $ defined by (2.2).

    Next, we begin with the energy estimates including $ n $ and $ u $ themselves. The following results is essentially due to A. Matsumura and T. Nishida [17], [18].

    Theorem 2.1. Assume that $ (n_0, u_0)\in H^3({\mathop{\mathbb R\kern 0pt}\nolimits}^3) $, then there exists a constant $ \delta_0>0 $ such that if

    $ n0H3+u0H3δ0, $

    then the problem (2.1) admits a unique global solution $ (n(t), u(t)) $ satisfying that for all $ t\geq0 $,

    $ n(t)2H3+u(t)2H3+t0(n(τ)2H2+u(τ)2H3)dτC(n02H3+u02H3), $

    where $ C $ is a positive constant independent of time.

    The proof of this theorem is divided into several subsections.

    For $ k = 0 $, multiplying the first equation in (2.1) by $ \gamma n $ and the second equation in (2.1) by $ u $, summing up and then integrating the result over $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $ by parts. By virtue of Hölder's inequality, Sobolev's inequality and the fact (2.4), we obtain that

    $ 12ddtR3(γ|n|2+|u|2)dx+R3(ˉμ|u|2+(ˉμ+ˉν)|divu|2)dx=R3γ(ndivuun)n(uu+ˉμf(n)u+(ˉμ+ˉν)f(n)divu+g(n)n)udxnL3uL2nL6+(uL3uL2+nL3nL2)uL6+(uLnL2+nLuL2)uL2(nL3+uL3+nL+uL)(n2L2+u2L2). $ (2.5)

    Now for $ 1\leq k \leq 3 $, applying $ {\nabla}^k $ to (2.1) and then multiplying the first equation by $ \gamma{\nabla}^k n $ and the second equation by $ {\nabla}^k u $, summing up and integrating over $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $. For $ k = 1 $ we have

    $ 12ddtR3(γ|n|2+|u|2)dx+R3(ˉμ|2u|2+(ˉμ+ˉν)|divu|2)dx(nL+uL+nL+uL)(n2L2+u2L2+2u2L2). $ (2.6)

    For $ k = 2 $ we have

    $ 12ddtR3(γ|2n|2+|2u|2)dx+R3(ˉμ|3u|2+(ˉμ+ˉν)|2divu|2)dx(nL+uL+nL+uL)(2n2L2+2u2L2+3u2L2). $ (2.7)

    For $ k = 3 $ we have

    $ 12ddtR3(γ|3n|2+|3u|2)dx+R3(ˉμ|4u|2+(ˉμ+ˉν)|3divu|2)dx(nL+uL+nL+uL)(3n2L2+3u2L2+4u2L2)+nL34uL22nL6+uL34uL22uL6+2nL3(3nL2+4uL2)2uL6. $ (2.8)

    Summing up the above estimates, noting that $ \delta>0 $ is small, we obtain that

    $ ddt0k3(γkn2L2+ku2L2)+C11k4ku2L2C2δ1k3kn2L2. $ (2.9)

    For $ 0\leq k \leq 2 $, applying $ {\nabla} ^k $ to the second equation in (2.1) and then multiplying by $ {\nabla}^{k+1} n $. The key idea is to integrate by parts in the $ t $-variable and to use the continuity equation. Thus integrating the results by parts for both the $ t $- and $ x $-variables, we obtain for $ k = 0 $ that

    $ ddtR3undx+γˉρR3|n|2dxu2L2+nL22uL2+(nL+uL)(n2L2+u2L2), $ (2.10)

    for $ k = 1 $, we get

    $ ddtR3u2ndx+γˉρR3|2n|2dx2u2L2+2nL23uL2+((n,u)L+(n,u)L)×(n2L2+2n2L2+2u2L2), $ (2.11)

    and for $ k = 2 $ we have

    $ ddtR32u3ndx+γˉρR3|3n|2dx3u2L2+3nL24uL2+((n,u)L+(n,u)L)×(2n2L2+2u2L2+3n2L2+3u2L2). $ (2.12)

    Plugging the above estimates, using the smallness of $ \delta>0 $, we obtain that

    $ ddt0k2R3kuk+1ndx+C31k3kn2L2C41k4ku2L2. $ (2.13)

    Proof of Theorem 2.1. Multiplying (2.13) by $ \frac{2C_2\delta}{C_3} $, adding it with (2.9), with the help of smallness of $ \delta>0 $, we deduce that there exists a constant $ C_5>0 $ such that

    $ ddt{0k3(γkn2L2+ku2L2)+2C2δC30k2R3kuk+1ndx}+C5{1k3kn2L2+1k4ku2L2}0. $ (2.14)

    Next, we define $ \mathcal E(t) $ to be $ C_5^{-1} $ times the expression under the time derivative in (2.14). Then we may write (2.14) as

    $ ddtE(t)+n(t)2H2+u(t)2H30. $ (2.15)

    Observe that since $ \delta $ is small, then there exists a constant $ C_6>0 $ such that

    $ C16(n(t)2H3+u(t)2H3)E(t)C6(n(t)2H3+u(t)2H3). $

    Then integrating (2.15) directly in time, we get

    $ sup0tT(n(t)2H3+u(t)2H3)+C6T0(n(τ)2H2+u(τ)2H3)dτC26(n02H3+u02H3). $

    Using a standard continuity argument along with classical local wellposedness theory, this closes the a priori assumption (2.3) if we assume $ \|n_0\|_{H^3}+\|u_0\|_{H^3}\leq\delta_0 $ is sufficiently small. We can then extend the solution globally in time and complete the proof of Theorem 2.1.

    In this section, we consider the initial value problem for the linearized Navier-Stokes system

    $ {t˜n+div˜m=0,(t,x)R+×R3,t˜m+c2˜nˉμ˜m(ˉμ+ˉν)div˜m=0,(t,x)R+×R3,lim|x|˜n=0,lim|x|˜m=0,tR+,(˜n,˜m)|t=0=(ρ0ˉρ,ρ0u0),xR3, $ (3.1)

    where $ \bar\mu = \frac{\mu}{\bar\rho} $, $ \bar\nu = \frac{\nu}{\bar\rho} $, $ c = \sqrt{p'(\bar \rho)} $.

    In terms of the semigroup theory for evolutionary equations, the solution $ (\widetilde{n}, \widetilde{m}) $ of the linearized Navier-Stokes problem (3.1) can be expressed for $ \widetilde U = (\widetilde{n}, \widetilde{m})^ t $ as

    $ ˜Ut=B˜U,t0,˜U(0)=˜U0, $

    which gives rise to

    $ ˜U(t)=S(t)˜U0=etB˜U0,t0, $

    where $ B $ is defined as

    $ B = {\left( 0divc2ˉμangle+(ˉμ+ˉν)div \right).} $

    What left is to analyze the differential operator $ B $ in terms of its Fourier expression $ A(\xi) $ and show the long time properties of the semigroup $ S(t) $. Applying the Fourier transform to system (3.1), we have

    $ tˆ˜U(t,ξ)=A(ξ)ˆ˜U(t,ξ),t0,ˆ˜U(0,ξ)=ˆ˜U0(ξ), $

    where $ \xi = (\xi_1,\xi_2,\xi_3)^t $, and $ A(\xi) $ is defined as

    $ A(ξ)=(0iξtc2iξˉμ|ξ|2I3×3(ˉμ+ˉν)ξξ). $

    The eigenvalues of the matrix $ A $ can be computed by

    $ det(A(ξ)λI)=(λ+ˉμ|ξ|2)2(λ2+(2ˉμ+ˉν)|ξ|2λ+c2|ξ|2)=0, $

    which implies

    $ λ0=ˉμ|ξ|2(double),λ1=λ1(|ξ|),λ2=λ2(|ξ|). $

    The semigroup $ e^{tA} $ is expressed as

    $ etA=eλ0tP0+eλ1tP1+eλ2tP2, $

    where the project operators $ P_i $ can be computed as

    $ Pi=ijA(ξ)λjIλiλj. $

    By a direct computation, we can verify the exact expression for the Fourier transform $ \widehat G(t,\xi) $ of Green's function $ G(t,x) = e^{tB} $ as

    $ ˆG(t,ξ)=etA=(λ1eλ2tλ2eλ1tλ1λ2iξt(eλ1teλ2t)λ1λ2c2iξ(eλ1teλ2t)λ1λ2eλ0t(Iξξ|ξ|2)+ξξ|ξ|2λ1eλ1tλ2eλ2tλ1λ2)=(ˆNˆM). $

    Indeed, we can make the following decomposition for $ (\widetilde n, \widetilde m) = G \ast \widetilde U_0 $ as

    $ ˆ˜n=ˆNˆ˜U0=(ˆN+ˆN)ˆ˜U0,ˆ˜m=ˆMˆ˜U0=(ˆM+ˆM)ˆ˜U0, $

    where

    $ ˆN=(λ1eλ2tλ2eλ1tλ1λ20),ˆN=(0iξt(eλ1teλ2t)λ1λ2),ˆM=(c2iξ(eλ1teλ2t)λ1λ20),ˆM=(0eλ0t(Iξξ|ξ|2)+ξξ|ξ|2λ1eλ1tλ2eλ2tλ1λ2). $

    We further decompose the Fourier transform $ \widehat N $, $ \widehat M $ into low frequency term and high frequency term below.

    Define

    $ ˆN=ˆN1+ˆN2,ˆN=ˆN1+ˆN2,ˆM=ˆM1+ˆM2,ˆM=ˆM1+ˆM2, $

    where $ (\cdot)_1 = \chi(\xi)(\cdot) $, $ (\cdot)_2 = (1-\chi(\xi))(\cdot) $, and $ \chi(\xi) $ is a smooth cut off function such that

    $ χ(ξ)={1,|ξ|R,0,|ξ|R+1. $

    Then we have the following decomposition for $ (\widetilde n, \widetilde m) = G \ast \widetilde U_0 $ as

    $ ˆ˜n=ˆNˆ˜U0=ˆN1ˆ˜U0+ˆN2ˆ˜U0=(ˆN1+ˆN1)ˆ˜U0+(ˆN2+ˆN2)ˆ˜U0,ˆ˜m=ˆMˆ˜U0=ˆM1ˆU0+ˆM2ˆ˜U0=(ˆM1+ˆM1)ˆ˜U0+(ˆM2+ˆM2)ˆ˜U0. $ (3.2)

    To derive the long time decay rate of solution, we need to use accurate approximation to the Fourier transform $ \widehat G(t,x) $ of Green's function for both lower frequency and high frequency. In terms of the definition of the eigenvalues, we are able to obtain that it holds for $ |\xi|\leq\eta $ for some small positive constant $ \eta $ that

    $ λ1=2ˉμ+ˉν2|ξ|2+i24c2|ξ|2(2ˉμ+ˉν)2|ξ|4=a+bi,λ2=2ˉμ+ˉν2|ξ|2i24c2|ξ|2(2ˉμ+ˉν)2|ξ|4=abi, $ (3.3)

    and we have

    $ λ1eλ2tλ2eλ1tλ1λ2=e12(2ˉμ+ˉν)|ξ|2t[cos(bt)+12(2ˉμ+ˉν)|ξ|2sin(bt)b]O(1)e12(2ˉμ+ˉν)|ξ|2t,|ξ|η, $
    $ λ1eλ1tλ2eλ2tλ1λ2=e12(2ˉμ+ˉν)|ξ|2t[cos(bt)12(2ˉμ+ˉν)|ξ|2sin(bt)b]O(1)e12(2ˉμ+ˉν)|ξ|2t,|ξ|η, $
    $ eλ1teλ2tλ1λ2=e12(2ˉμ+ˉν)|ξ|2tsin(bt)bO(1)1|ξ|e12(2ˉμ+ˉν)|ξ|2t,|ξ|η, $

    where

    $ b=124c2|ξ|2(2ˉμ+ˉν)2|ξ|4c|ξ|+O(|ξ|3),|ξ|η. $

    For the high frequency $ |\xi|\geq\eta $, we are also able to obtain that it holds for $ |\xi|\geq\eta $ that

    $ λ1=2ˉμ+ˉν2|ξ|212(2ˉμ+ˉν)2|ξ|44c2|ξ|2=ab,λ2=2ˉμ+ˉν2|ξ|2+12(2ˉμ+ˉν)2|ξ|44c2|ξ|2=a+b, $ (3.4)

    and we have

    $ λ1eλ2tλ2eλ1tλ1λ2=12e(a+b)t[1+e2bt]a2be(a+b)t[1e2bt]O(1)eR0t,|ξ|η, $
    $ λ1eλ1tλ2eλ2tλ1λ2=a+b2be(a+b)t[1e2bt]+e(ab)tO(1)eR0t,|ξ|η, $
    $ eλ1teλ2tλ1λ2=12be(a+b)t[1e2bt]O(1)1|ξ|2eR0t,|ξ|η, $

    where

    $ b=12(2ˉμ+ˉν)2|ξ|44c2|ξ|212(2ˉμ+ˉν)|ξ|22c22ˉμ+ˉν+O(|ξ|2),|ξ|η. $

    Here $ R_0 $, $ \eta $ are some fixed positive constants.

    In this section, we apply the spectral analysis to the semigroup for the linearized Navier-Stokes system. We will establish the $ L^2 $ and $ L^p $ ($ 2\leq p \leq \infty $) time decay rate of the global solutions for the linearized Navier-Stokes system.

    With the help of the formula for Green's function in Fourier space and the asymptotic analysis on its elements, we are able to establish the $ L^2 $ time decay rate. Indeed, we have the $ L^2 $-time decay rate of the global strong solution to the problem for the linearized Navier-Stokes system as follows.

    Proposition 4.1. Let $ U_0 = (n_0, m_0)\in L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^3)\cap H^l({\mathop{\mathbb R\kern 0pt}\nolimits}^3) $ with $ l\geq3 $, then $ (\widetilde n,\widetilde m) $ solves the linearized Navier-Stokes system (3.1) and satisfies for $ 0\leq k\leq l $ that

    $ k(˜n,˜m)(t)L2(R3)C(1+t)34k2(U0L1(R3)+kU0L2(R3)), $

    where $ C $ is a positive constant independent of time.

    Proof. A straightforward computation together with the formula of the Green's function $ \widehat G(t,\xi) $ gives

    $ ˆ˜n(t,ξ)=λ1eλ2tλ2eλ1tλ1λ2ˆn0iξˆm0(eλ1teλ2t)λ1λ2{O(1)e12(2ˉμ+ˉν)|ξ|2t(|ˆn0|+|ˆm0|),|ξ|η,O(1)eR0t(|ˆn0|+|ˆm0|),|ξ|η,ˆ˜m(t,ξ)=c2iξ(eλ1teλ2t)λ1λ2ˆn0+eλ0tˆm0+(λ1eλ1tλ2eλ2tλ1λ2eλ0t)ξ(ξˆm0)|ξ|2{O(1)eˉμ|ξ|2t(|ˆn0|+|ˆm0|),|ξ|η,O(1)eR0t(|ˆn0|+|ˆm0|),|ξ|η, $

    here and below, $ R_0 $, $ \eta $ are some fixed positive constants. Therefore, we have the $ L^2 $-decay rate for $ (\widetilde n, \widetilde m) $ as

    $ (ˆ˜n,ˆ˜m)(t)2L2(R3)=|ξ|η|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ+|ξ|η|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ|ξ|ηe2ˉμ|ξ|2t(|ˆn0|2+|ˆm0|2)dξ+|ξ|ηe2R0t(|ˆn0|2+|ˆm0|2)dξ(1+t)32(n0,m0)2L1(R3)L2(R3). $

    And the $ L^2 $-decay rate on the derivatives of $ (\widetilde n,\widetilde m) $ as

    $ (^k˜n,^k˜m)(t)2L2(R3)=|ξ|η|ξ|2k|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ+|ξ|η|ξ|2k|(ˆ˜n,ˆ˜m)(t,ξ)|2dξ|ξ|ηe2ˉμ|ξ|2t|ξ|2k(|ˆn0|2+|ˆm0|2)dξ+|ξ|ηe2R0t|ξ|2k(|ˆn0|2+|ˆm0|2)dξ(1+t)32k((n0,m0)2L1(R3)+(kn0,km0)2L2(R3)). $

    The proof of the Proposition 4.1 is completed.

    It should be noted that the $ L^2 $-time decay rates derived above are optimal.

    Proposition 4.2. Let $ U_0 = ( n_0, m_0)\in L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^3)\cap H^l({\mathop{\mathbb R\kern 0pt}\nolimits}^3) $ with $ l\geq3 $, assume that $ M_n = \int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3}n_0(x) d x $ and $ M_m = \int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} m_0(x) dx $ satisfies that $ M_n $, $ M_m $ are at least not all zeros, then the solution $ (\widetilde n,\widetilde m) $ of the linearized Navier-Stokes system (3.1) given by Proposition 4.1 satisfies for $ 0\leq k\leq l $

    $ C1(1+t)34k2k˜n(t)L2(R3)C(1+t)34k2,C1(1+t)34k2k˜m(t)L2(R3)C(1+t)34k2, $

    where $ C $ is a positive constant independent of time.

    Proof. We only show the case of $ k = 0 $ for simplicity, the argument applies to the other orders of derivatives. From the formula of the Green's function $ \widehat G(t,\xi) $, we deduce that

    $ ˆ˜n(t,ξ)=λ1eλ2tλ2eλ1tλ1λ2ˆn0iξˆm0(eλ1teλ2t)λ1λ2=e12(2ˉμ+ˉν)|ξ|2t[cos(bt)ˆn0iξˆm0sin(bt)b]+e12(2ˉμ+ˉν)|ξ|2t[12(2ˉμ+ˉν)|ξ|2sin(bt)bˆn0]=T1+T2,for|ξ|η, $
    $ ˆ˜m(t,ξ)=c2iξ(eλ1teλ2t)λ1λ2ˆn0+eλ0tˆm0+(λ1eλ1tλ2eλ2tλ1λ2eλ0t)ξ(ξˆm0)|ξ|2=[e12(2ˉμ+ˉν)|ξ|2t[cos(bt)ξ(ξˆm0)|ξ|2c2iξsin(bt)bˆn0]+eˉμ|ξ|2t[ˆm0ξ(ξˆm0)|ξ|2]]e12(2ˉμ+ˉν)|ξ|2t[12(2ˉμ+ˉν)|ξ|2sin(bt)bξ(ξˆm0)|ξ|2]=S1+S2,for|ξ|η, $

    here and below, $ \eta $ is a sufficiently small but fixed constant.

    It is easy to check that

    $ ˆ˜n(t,ξ)2L2=|ξ|η|ˆ˜n(t,ξ)|2dξ+|ξ|η|ˆ˜n(t,ξ)|2dξ|ξ|η|T1+T2|2dξ|ξ|η12|T1|2|T2|2dξ. $ (4.1)

    We then calculate that

    $ |ξ|η|T2|2dξˆn02L|ξ|ηe(2ˉμ+ˉν)|ξ|2t|ξ|4(sin(bt)b)2dξˆn02L|ξ|ηe(2ˉμ+ˉν)|ξ|2t|ξ|2dξ(1+t)52n02L1. $ (4.2)

    Since $ n_0(x) \in L^1 $ implies $ \widehat{n}_0(\xi) \in C({{\mathop{\mathbb R\kern 0pt}\nolimits}^3}) $. If $ \widehat{n}_0(0) = \int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3}n_0(x) d x\neq 0 $, we deduce that $ \widehat{n}_0(\xi)\neq 0 $ for $ |\xi|\leq\eta $ when $ \eta $ is sufficiently small. One finds that, when $ M_n\neq 0 $,

    $ |ˆn0(ξ)|21C|R3n0(x)dx|2M2nC,for|ξ|η. $

    For $ \widehat{m}_0 $, a similar argument yields that, when $ M_m\neq 0 $, we have

    $ |ξˆm0(ξ)|2|ξ|2|ξMm|2C|ξ|2,for|ξ|η. $

    When $ M_n\neq 0 $, $ M_m\neq 0 $, with the help of the above analysis, using $ b\sim c|\xi|+O(|\xi|^3) $ for $ |\xi|\leq \eta $, we obtain that

    $ |ξ|η|T1|2dξM2nC|ξ|ηe(2ˉμ+ˉν)|ξ|2tcos2(bt)dξ+1C|ξ|η|ξMm|2b2e(2ˉμ+ˉν)|ξ|2tsin2(bt)dξ $
    $ min{M2n,M2m3c2}C|ξ|ηe(2ˉμ+ˉν)|ξ|2t(cos2(bt)+sin2(bt))dξC1|ξ|ηe(2ˉμ+ˉν)|ξ|2tdξC1(1+t)32. $ (4.3)

    If $ M_n\neq 0 $, $ M_m = 0 $, and by the conituinity of $ \widehat{m}_0 $ near $ \xi = 0 $, there exists a small enough constant $ \epsilon $ such that $ \epsilon\to0 $ as $ \xi\to 0 $, and

    $ |ˆm0(ξ)|2<ϵ,for|ξ|η. $

    We thus use the help of spherical coordinates and the change of variables $ r = |\xi|\sqrt{t} $ to obtain that

    $ |ξ|η|T1|2dξM2nC|ξ|ηe(2ˉμ+ˉν)|ξ|2tcos2(bt)dξϵCc2|ξ|ηe(2ˉμ+ˉν)|ξ|2tsin2(bt)dξM2nCt32ηt0e(2ˉμ+ˉν)r2cos2(crt)r2drϵCc2t32ηt0e(2ˉμ+ˉν)r2sin2(crt)r2drM2nCt32[cηtπ]1k=0kπ+π4ctkπcte(2ˉμ+ˉν)r2cos2(crt)r2drϵCc2(1+t)32M2n2Ct32[cηtπ]1k=0kπ+π4ctkπcte(2ˉμ+ˉν)r2r2drϵCc2(1+t)32C11(1+t)32C12ϵ(1+t)32.C1(1+t)32 $ (4.4)

    In the case of $ M_n = 0 $, $ M_m\neq0 $, we can use a similar argument to obtain that

    $ |ξ|η|T1|2dξϵC|ξ|ηe(2ˉμ+ˉν)|ξ|2tcos2(bt)dξ+M2m3Cc2|ξ|ηe(2ˉμ+ˉν)|ξ|2tsin2(bt)dξC1(1+t)32. $ (4.5)

    Combining the above estimates (4.1), (4.2), (4.3), (4.4) and (4.5), we obtain the lower bound of the time decay rate for $ {\widetilde n}(t,x) $ as

    $ ˜n(t,x)2L2=ˆ˜n(t,ξ)2L2C1(1+t)32. $

    The lower bound of the time decay rate for $ {\widetilde m}(t,x) $ can be shown in a similar fashion. It is not difficult to derive that

    $ ˆ˜m(t,ξ)2L2|ξ|η12|S1|2|S2|2dξ, $ (4.6)

    then we find that

    $ |ξ|η|S2|2dξ(1+t)52m02L1. $ (4.7)

    We then calculate that

    $ |ξ|η|S1|2dξ{c4M2nC|ξ|η|ξ|2b2e(2ˉμ+ˉν)|ξ|2tsin2(bt)dξ+1C|ξ|η|ξMm|2|ξ|2e(2ˉμ+ˉν)|ξ|2tcos2(bt)dξ}+{|ξ|ηe12(4ˉμ+ˉν)|ξ|2tcos(bt)ξ(ξˆm0)|ξ|2(ˆm0ξ(ξˆm0)|ξ|2)dξ}=J1+J2. $

    A direct computation gives rise to

    $ J1C1(1+t)32,J2=0. $ (4.8)

    Combining the above estimates (4.6), (4.7) and (4.8), we obtain the lower bound of the time decay rate for $ {\widetilde m}(t,x) $ as

    $ ˜m(t,x)2L2=ˆ˜m(t,ξ)2L2C1(1+t)32. $

    Then the proof of Proposition 4.2 is completed.

    In this subsection, we establish the following $ L^p $-time decay rate of the global strong solution to the linearized Navier-Stokes system with $ p \in [2,+\infty] $.

    Proposition 4.3. Let $ U_0 = (n_0,m_0)\in L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^3)\cap W^{l,p}({\mathop{\mathbb R\kern 0pt}\nolimits}^3) $ with $ l\geq 3 $, then $ (\widetilde n,\widetilde m) $ solves the linearized Navier-Stokes system (3.1) and satisfies for $ 0\leq k\leq l $ and $ p \in [2,+\infty] $ that

    $ k(˜n,˜m)(t)Lp(R3)C(1+t)32(11p)k2(U0L1(R3)+kU0Lp(R3)), $

    where $ C $ is a positive constant independent of time.

    To prove Proposition 4.3, the following two lemmas in [6] are helpful.

    Lemma 4.1. Let $ n\geq 1 $ and assume that $ \hat f(\xi) \in L^\infty \cap C^{n+1}({\mathop{\mathbb R\kern 0pt}\nolimits}^n/\{0\}) $, with

    $ |αξˆf(ξ)|C{|ξ||α|+σ1,|ξ|R,|α|=n,|ξ||α|σ2,|ξ|R,|α|=n1,n,n+1, $

    where $ \sigma_1, \sigma_2>0 $ and $ n>2-2\sigma_2 $. Then $ \hat f(\xi) $ is continuous at $ 0 $ and $ \infty $, and

    $ f=m1+m2δ, $

    where $ m_1\in L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^n) $ satisfies $ \|m_1\|_{L^1({\mathop{\mathbb R\kern 0pt}\nolimits}^n)} \leq C(C') $, $ m_2 $ is the constant

    $ m2=(2π)n2lim|ξ|ˆf(ξ), $

    and $ \delta $ is the Dirac distribution. In particular, $ \hat f(\xi) $ is a strong $ L^p $ multiplier, $ 1\leq p \leq \infty $, in the sense that, for any $ g\in L^p $,

    $ fgLpCgLp,1p, $

    where $ C $ depends only on $ |m_2|\leq \| \hat f\|_{L^\infty} $ and the constant $ C' $ above.

    Lemma 4.2. Let $ \hat g(t,\xi) = \hat K(t,\xi)\hat f(\xi) $, where $ \hat K(t,\xi) = e^{-\vartheta|\xi|^2 t} $, $ \hat f(\xi)\in L^\infty\cap C^{n+1}({\mathop{\mathbb R\kern 0pt}\nolimits}^n) $, and

    $ |βξˆf(ξ)|C|ξ||β|,|β|n+1. $

    Then $ {\nabla}_x ^\alpha g(t,\cdot)\in L^p $ for $ t>0 $, and for all $ \alpha $, $ 1\leq p \leq \infty $, we have

    $ αxg(t,)LpC(|α|)tn2(11p)|α|2. $

    In particular, $ \widehat{{\nabla}_x ^\alpha g(t,x)} = (i\xi)^\alpha \hat g(t,\xi) $ is a strong $ L^p $ multiplier, with norm bounded by $ C(|\alpha|,\vartheta)C't^{-\frac{|\alpha|}2} $, where the constant $ C(|\alpha|,\vartheta) $ depends only on $ |\alpha| $ and $ \vartheta $.

    Now let us turn to the proof of Proposition 4.3.

    Proof of Proposition 4.3. We first analyze above higher frequency terms denoted by $ \widehat {(\cdot)}_2 $. Recall that

    $ λ1=(2ˉμ+ˉν)|ξ|2+2c22ˉμ+ˉν+O(|ξ|2),λ2=2c22ˉμ+ˉν+O(|ξ|2),|ξ|η. $

    We shall prove that the higher frequency terms are $ L^p $ Fourier multipliers with an exponential time decay coefficient $ C e^{-c_1t} $ for some constants $ c_1>0 $. For simplicity, we only show that $ \widehat {\mathcal N}_2 $ is an $ L^p $ Fourier multiplier at higher frequency as follows. It holds

    $ λ1eλ2tλ2eλ1tλ1λ2=eλ2t+λ2eλ2tλ1λ2λ2eλ1tλ1λ2. $

    By a direct computation, it is easy to verify

    $ \begin{eqnarray*} |{\nabla}_\xi^k \lambda_2|\lesssim|\xi|^{-2-k},\quad |\xi|\geq\eta, \end{eqnarray*} $

    which gives rise to

    $ \begin{eqnarray*} \begin{split} \bigg|{\nabla}_\xi^k \Big[(1-\chi(\cdot))e^{\lambda_2 t}\Big]\bigg|, \left|{\nabla}_\xi^k \Big[(1-\chi(\cdot))\frac{\lambda_2 e^{\lambda_2 t}}{\lambda_1-\lambda_2}\Big]\right|\lesssim{ \left\{\begin{array}{l} 0, \quad |\xi|\leq R,\\ e^{-c_1t}|\xi|^{-2-k},\quad |\xi|\geq R, \end{array}\right.} \end{split} \end{eqnarray*} $

    here and below, $ R>0 $ is a given constant. Thus, from Lemma 4.1 it follows that the inverse Fourier transform of the term $ (1-\chi(\cdot))\left(e^{\lambda_2 t}+\frac{\lambda_2 e^{\lambda_2 t}}{\lambda_1-\lambda_2}\right) $ is an $ L^p $ multiplier with the coefficient $ Ce^{-c_1t} $. The other part of $ \widehat {\mathcal N}_2 $ at higher frequency can be written as

    $ \begin{eqnarray*} (1-\chi(\cdot))\frac{\lambda_2 e^{\lambda_1 t}}{\lambda_1-\lambda_2} \sim e^{-\frac12(2\bar\mu+\bar\nu) |\xi|^2 t}\Big[(1-\chi(\cdot))\frac{e^{(-\lambda_2-\frac12(2\bar\mu+\bar\nu) |\xi|^2)t}}{\lambda_1-\lambda_2}\Big]. \end{eqnarray*} $

    We can regard $ e^{-\frac12 (2\bar\mu+\bar\nu) |\xi|^2 t} $ as the function $ K(t,\xi) $ of Lemma 4.2, and the rest term satisfies the condition. Thus, the inverse Fourier transform of $ (1-\chi(\cdot))\frac{\lambda_2 e^{\lambda_1 t}}{\lambda_1-\lambda_2} $ is also an $ L^p $ multiplier with the coefficient $ Ce^{-c_1t} $. These facts imply that $ \widehat {\mathcal N}_2 $ at higher frequency is an $ L^p $ multiplier with the coefficient $ Ce^{-c_1t} $. Applying the similar analysis to the terms $ \widehat {\mathfrak N}_2 $, $ \widehat {\mathcal M}_2 $, and $ \widehat {\mathfrak M}_2 $, we can show that their inverse Fourier transform are all $ L^p $ multiplier with the constant coefficient $ Ce^{-c_1t} $. Then

    $ \begin{equation} \|({\nabla}_x^k({\mathcal N}_2 \ast f),{\nabla}_x^k({\mathfrak N}_2 \ast f),{\nabla}_x^k( {\mathcal M}_2 \ast f),{\nabla}_x^k( {\mathfrak M}_2 \ast f))(t)\|_{L^p} \leq Ce^{-c_1t}\|{\nabla}_x^k f\|_{L^p}, \end{equation} $ (4.9)

    for all integer $ k\geq 0 $, and $ p\in[2,\infty] $.

    We also need to deal with the corresponding lower frequency terms denoted by $ \widehat {(\cdot)}_1 $. Recall that

    $ \begin{eqnarray*} \begin{split} &\frac{\lambda_1e^{\lambda_2 t}-\lambda_2e^{\lambda_1 t}}{\lambda_1-\lambda_2}, \frac{\lambda_1e^{\lambda_1 t}-\lambda_2e^{\lambda_2 t}}{\lambda_1-\lambda_2},\frac{|\xi|(e^{\lambda_1 t}-e^{\lambda_2 t})}{\lambda_1-\lambda_2}\sim O(1)e^{-\frac12 (2\bar\mu+\bar\nu)|\xi|^2t},\quad |\xi|\leq\eta, \end{split} \end{eqnarray*} $

    which imply that for $ |\xi|\leq\eta $ that

    $ \begin{eqnarray*} |\widehat{\mathcal N}_1|\sim O(1)e^{-c_2|\xi|^2t},\quad |\widehat{\mathfrak N}_1|\sim O(1)e^{-c_2|\xi|^2t},\\ |\widehat{\mathcal M}_1|\sim O(1)e^{-c_2|\xi|^2t},\quad |\widehat{\mathfrak M}_1|\sim O(1)e^{-c_2|\xi|^2t}, \end{eqnarray*} $

    for some constants $ c_2>0 $. Thus, by Hausdroff-Young's inequality with $ p\in[2, +\infty] $, we can obtain

    $ \begin{equation} \begin{split} \|({\nabla}^k{\mathcal N}_1,{\nabla}^k{\mathfrak N}_1,{\nabla}^k {\mathcal M}_1,{\nabla}^k {\mathfrak M}_1)(t)\|_{L^p} \leq& C\left(\int_{|\xi|\leq \eta}\big||\xi|^k e^{-c_2|\xi|^2t}\big|^q d\xi\right)^{\frac 1q}\\ \leq& C(1+t)^{-\frac32(1-\frac1p)-\frac k2}. \end{split} \end{equation} $ (4.10)

    Combining (4.9) and (4.10), we finally have for $ t>0 $ that

    $ \begin{eqnarray*} \begin{split} \|({\nabla}^k(N \ast f),{\nabla}^k(M \ast f))(t)\|_{L^p}& = \|({\nabla}^k((N_1+N_2) \ast f),{\nabla}^k((M_1+M_2) \ast f))(t)\|_{L^p}\\ &\leq C(1+t)^{-\frac32(1-\frac1p)-\frac k2}\|f\|_{L^1}+Ce^{-c_1t}\|{\nabla}^k f\|_{L^p}\\ &\leq C(1+t)^{-\frac32(1-\frac1p)-\frac k2}(\|f\|_{L^1}+\|{\nabla}^k f\|_{L^p}). \end{split} \end{eqnarray*} $

    The proof of Proposition 4.3 is completed.

    We are ready to prove Theorem 1.1 on the sharp time decay rate of the global solution to the initial value problem for the nonlinear Navier-Stokes system.

    In what follows, we will set $ n_h = n-\widetilde n $ and $ m_h = m-\widetilde m $, then we have

    $ \begin{equation} \left\{\begin{array}{l} \partial_t n_h + {\mathop{{\rm{div}}}\nolimits} m_h = 0,\qquad (t,x)\in{\mathop{\mathbb R\kern 0pt}\nolimits}^+\times{\mathop{\mathbb R\kern 0pt}\nolimits}^3,\\ \partial_t m_h + c^2 {\nabla} n_h- \bar\mu △ m_h- (\bar\mu+\bar\nu){\nabla}{\mathop{{\rm{div}}}\nolimits} m_h = F,\qquad (t,x)\in{\mathop{\mathbb R\kern 0pt}\nolimits}^+\times{\mathop{\mathbb R\kern 0pt}\nolimits}^3,\\ \lim\limits_{|x|\to\infty}n_h = 0, \quad\lim\limits_{|x|\to\infty} m_h = 0,\qquad t\in{\mathop{\mathbb R\kern 0pt}\nolimits}^+,\\ (n_h,m_h)\big|_{t = 0} = (0,0),\qquad x\in{\mathop{\mathbb R\kern 0pt}\nolimits}^3, \end{array}\right. \end{equation} $ (5.1)

    where $ \bar\mu = \frac{\mu}{\bar\rho} $, $ \bar\nu = \frac{\nu}{\bar\rho} $, $ c = \sqrt{p'(\bar \rho)} $, and

    $ \begin{eqnarray*} \begin{split} F = &-{\mathop{{\rm{div}}}\nolimits}\Big\{ \frac{(m_h+\widetilde m)\otimes (m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}+\bar\mu{\nabla}\big(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}\big)\Big\}\\ & - {\nabla}\Big\{(\bar\mu+\bar\nu){\mathop{{\rm{div}}}\nolimits}(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho})+\big(p(n_h+\widetilde n+\bar\rho)-p(\bar\rho)-c^2(n_h+\widetilde n)\big)\Big\}. \end{split} \end{eqnarray*} $

    Denote $ U_h = (n_h, m_h)^t $, we have the equivalent form of system (5.1) in vector form

    $ \begin{eqnarray*} \partial_t U_h = BU_h+H,\quad t\geq0,\qquad U_h(0) = 0, \end{eqnarray*} $

    where the nonlinear term $ H(\widetilde U, U_h) = (0, F(\widetilde U, U_h))^t $. Thus, we can represent the solution in term of the semigroup

    $ \begin{eqnarray*} U_h(t) = S(t)\ast U_{h}(0)+\int_0^t S(t-\tau)\ast H(\widetilde U, U_h)(\tau)d\tau, \end{eqnarray*} $

    which $ (n_h, m_h) $ can be decomposed as

    $ \begin{equation} n_h = N\ast U_{h}(0)+\int_0^t \mathfrak N (t-\tau)\ast H(\tau)d\tau, \end{equation} $ (5.2)
    $ \begin{equation} m_h = M\ast U_{h}(0)+\int_0^t \mathfrak M (t-\tau)\ast H(\tau)d\tau. \end{equation} $ (5.3)

    Furthermore, in view of the above definition for $ \widehat{\mathfrak N}(\xi) $ and $ \widehat{\mathfrak M}(\xi) $, it is easy to verify for some constants $ c_3>0 $, $ c_4>0 $, $ R_0>0 $, we discover that

    $ \begin{eqnarray*} |\widehat{\mathfrak N}(\xi)|\sim O(1)e^{-c_3|\xi|^2t}, \quad |\widehat{\mathfrak M}(\xi)|\sim O(1)e^{-c_3|\xi|^2t}, \quad|\xi|\leq\eta, \end{eqnarray*} $
    $ \begin{eqnarray*} |\widehat{\mathfrak N}(\xi)|\sim O(1)\frac1{|\xi|}e^{-R_0t}, \quad |\widehat{\mathfrak M}(\xi)|\sim O(1)\frac1{|\xi|^2}e^{-R_0t}+O(1)e^{-c_4|\xi|^2t}, \quad |\xi|\geq\eta. \end{eqnarray*} $

    Thus, applying a similar argument as in the proof of Proposition 4.1, we have

    $ \begin{equation} \|({\nabla}^k {\mathfrak N}\ast H, {\nabla}^k {\mathfrak M}\ast H)(t)\|_{L^2} \leq C(1+t)^{-\frac32(\frac1q-\frac12)-\frac12-\frac k 2}\big(\|Q\|_{L^q}+\|{\nabla}^{k+1} Q\|_{L^2}\big),\quad q = 1,2, \end{equation} $ (5.4)
    $ \begin{equation} \|({\nabla}^k {\mathfrak N}\ast H, {\nabla}^k {\mathfrak M}\ast H)(t)\|_{L^2} \leq C(1+t)^{-\frac32(\frac1q-\frac12)-\frac12-\frac k 2}\big(\|Q\|_{L^q}+\|{\nabla}^{k} Q\|_{L^2}\big),\quad q = 1,2, \end{equation} $ (5.5)
    $ \begin{equation} \|{\nabla}^k {\mathfrak M}\ast H(t)\|_{L^2} \leq C(1+t)^{-\frac32(\frac1q-\frac12)-\frac12-\frac k 2}\big(\|Q\|_{L^q}+\|{\nabla}^{k-1} Q\|_{L^2}\big),\quad q = 1,2, \end{equation} $ (5.6)

    for any non-negative integer $ k $ and

    $ \begin{equation} \begin{split} Q = &\Big|\frac{(m_h+\widetilde m)\otimes (m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}+\bar\mu{\nabla}\big(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho}\big)\Big|\\ &+\Big|(\bar\mu+\bar\nu){\mathop{{\rm{div}}}\nolimits}(\frac{(n_h+\widetilde n)(m_h+\widetilde m)}{n_h+\widetilde n+\bar\rho})+\big(p(n_h+\widetilde n+\bar\rho)-p(\bar\rho)-c^2(n_h+\widetilde n)\big)\Big|. \end{split} \end{equation} $ (5.7)

    For readers' convenience, we show how to estimate $ \|{\nabla}^k {\mathfrak M}\ast H(t)\|_{L^2} $ as an example. The other two estimates can be obtained by the similar argument. Indeed,

    $ \begin{eqnarray*} \begin{split} &\quad\|{\nabla}^k {\mathfrak M}\ast H(t)\|_{L^2}^2\\ &\lesssim\int_{|\xi|\leq \eta}e^{-2c_3|\xi|^2t}|\xi|^{2k}|\widehat H|^2d\xi +\int_{|\xi|\geq \eta}e^{-2R_0 t}|\xi|^{2k-4}|\widehat H|^2d\xi\\ &\quad+\int_{|\xi|\geq \eta}e^{-2c_4|\xi|^2 t}|\xi|^{2k}|\widehat H|^2d\xi\\ &\lesssim\int_{|\xi|\leq \eta}e^{-2c_3|\xi|^2t}|\xi|^{2k+2}|\widehat Q|^2d\xi +\int_{|\xi|\geq \eta}e^{-2R_0 t}|\xi|^{2k-2}|\widehat Q|^2d\xi\\ &\quad+\int_{|\xi|\geq \eta}e^{-2c_4|\xi|^2t}|\xi|^{2k+2}|\widehat Q|^2d\xi\\ &\lesssim(1+t)^{-3(\frac1q-\frac12)-1-k}\big(\|Q\|^2_{L^q({\mathop{\mathbb R\kern 0pt}\nolimits}^3)}+\|{\nabla}^{\tilde k} Q\|^2_{L^2({\mathop{\mathbb R\kern 0pt}\nolimits}^3)}\big),\quad q = 1,2,\quad k-1\leq\tilde k\in{\mathop{\mathbb N\kern 0pt}\nolimits}^+. \end{split} \end{eqnarray*} $

    In this subsection, we establish the faster decay rate for $ (n_h, m_h) $. We will start with an a priori assumption on a carefully chosen quantity $ \Lambda(t) $ defined in (5.8), and then later prove a better estimate with the help of the smallness of initial data.

    We begin with following Lemma.

    Lemma 5.1. Let $ r_1, r_2>0 $ be real, one has

    $ \begin{eqnarray*} \begin{split} \int_0^{\frac t 2}(1+t-\tau)^{-r_1}(1+\tau)^{-r_2} d\tau = &\int_0^{\frac t 2}(1+\frac t 2+\tau)^{-r_1}(1+\frac t 2-\tau)^{-r_2} d\tau\\ \lesssim&{ \left\{\begin{array}{l} (1+t)^{-r_1}, \quad \mathit{\text{for}} \quad r_2 > 1,\\ (1+t)^{-(r_1-\epsilon)},\quad \mathit{\text{for}} \quad r_2 = 1,\\ (1+t)^{-(r_1+r_2-1)},\quad \mathit{\text{for}} \quad r_2 < 1, \end{array}\right.} \end{split} \end{eqnarray*} $

    and

    $ \begin{eqnarray*} \begin{split} \int_{\frac t 2}^t(1+t-\tau)^{-r_1}(1+\tau)^{-r_2} d\tau = &\int_0^{\frac t 2}(1+t-\tau)^{-r_2}(1+\tau)^{-r_1} d\tau\\ \lesssim&{ \left\{\begin{array}{l} (1+t)^{-r_2}, \quad \mathit{\text{for}} \quad r_1 > 1,\\ (1+t)^{-(r_2-\epsilon)},\quad \mathit{\text{for}} \quad r_1 = 1,\\ (1+t)^{-(r_1+r_2-1)},\quad \mathit{\text{for}} \quad r_1 < 1, \end{array}\right.} \end{split} \end{eqnarray*} $

    where $ \epsilon>0 $ is a small but fixed constant.

    Proposition 5.1. Under the assumptions of Theorem 1.1, the solution $ (n_h, m_h) $ of the nonlinear system (5.1) satisfies for $ k = 0,1,2 $ that

    $ \begin{eqnarray*} \begin{split} &\|({\nabla} ^k n_h,{\nabla}^{k} m_h)\|_{L^2}\leq C\delta_0^2(1+t)^{-\frac54-\frac {k} 2},\\ &\|{\nabla} ^3 m_h\|_{L^2}\leq C\delta_0^2(1+t)^{-\frac{11}4},\quad \|{\nabla} ^3 n_h\|_{L^2}\leq C\delta_0(1+t)^{-\frac74}, \end{split} \end{eqnarray*} $

    where $ C $ is a positive constant independent of time.

    From (5.7), we deduce

    $ \begin{eqnarray*} Q(\widetilde U, U_h) = Q_1+Q_2+Q_3+Q_4, \end{eqnarray*} $

    which implies for a smooth solution $ (n,m) $ satisfying $ \|(n,m)\|_{H^3}<\infty $ that

    $ \begin{eqnarray*} \begin{split} &Q_1 = Q_1(\widetilde U, U_h)\sim O(1)\left(n_h^2+m_h\otimes m_h+\widetilde n^2+\widetilde m\otimes\widetilde m \right),\\ &Q_2 = Q_2(\widetilde U, U_h)\sim O(1)\left(\widetilde n n_h+\widetilde m\otimes m_h\right),\\ &Q_3 = Q_3(\widetilde U, U_h)\sim O(1)\left({\nabla}(n_h\cdot m_h)+{\nabla}(\widetilde n\cdot\widetilde m)\right),\\ &Q_4 = Q_4(\widetilde U, U_h)\sim O(1)\left({\nabla}(\widetilde n\cdot m_h)+{\nabla}( n_h\cdot\widetilde m) \right). \end{split} \end{eqnarray*} $

    Define

    $ \begin{equation} \begin{split} \Lambda(t) = :&\sup\limits_{0\leq s \leq t}\bigg\{\sum\limits_{k = 0}^2(1+s)^{\frac54+\frac k 2}{\delta_0}^{-\frac34}\|({\nabla} ^k n_h,{\nabla}^{k}m_h)(s)\|_{L^2}\\ &\quad+(1+s)^{\frac74}\|({\nabla}^3 n_h, {\nabla}^3 m_h)(s)\|_{L^2}\bigg \}. \end{split} \end{equation} $ (5.8)

    Proposition 5.2. Under the assumptions of Theorem 1.1, if for some $ T>0 $, $ \Lambda(t) \leq \delta_0^{\frac12} $ for any $ t\in[0,T] $, then it holds that

    $ \begin{eqnarray*} \begin{split} \Lambda(t)\leq C\delta_0^{\frac34},\quad t\in[0,T], \end{split} \end{eqnarray*} $

    where $ C $ is a positive constant independent of time.

    The proof of this Proposition 5.2 consists of following three steps.

    Starting with (5.4), (5.5), (5.6) and (5.8), we have after a complicate but straightforward computation that

    $ \begin{equation} \begin{split} \|(n_h, m_h)\|_{L^2}&\lesssim\int_0^t \|(\mathfrak N (t-\tau)\ast H(\tau), \mathfrak M (t-\tau)\ast H(\tau))\|_{L^2}d\tau\\ &\lesssim\int_0^{t} (1+t-\tau)^{-\frac54}\big(\|Q(\tau)\|_{L^1}+\| Q(\tau)\|_{L^2}\big)d\tau\\ &\lesssim\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right)\int_0^{t} (1+t-\tau)^{-\frac54}(1+\tau)^{-\frac32}d\tau\\ &\lesssim(1+t)^{-\frac54}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{equation} $ (5.9)

    It is easy to verify that

    $ \begin{eqnarray*} \begin{split} \|Q(t)\|_{L^1}\lesssim&\|Q_1\|_{L^1}+\|Q_2\|_{L^1}+\|Q_3\|_{L^1}+\|Q_4\|_{L^1}\\ \lesssim &\|(\widetilde n,\widetilde m)\|_{L^2}^2+\|( n_h, m_h)\|_{L^2}^2+ \|( n_h, m_h)\|_{L^2}\big(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}\\ &+\|({\nabla} n_h,{\nabla} m_h)\|_{L^2}\big)+\|(\widetilde n,\widetilde m)\|_{L^2}\left(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2} +\| ({\nabla} n_h,{\nabla} m_h)\|_{L^2}\right)\\ \lesssim & (1+t)^{-\frac32}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{eqnarray*} $

    Indeed, by virtue of Hölder's inequality and Gagliardo-Nirenberg's inequality, we obtain that

    $ \begin{eqnarray*} \|u\|_{L^\infty}\lesssim \|{\nabla} u\|_{L^2}^{\frac12}\|{\nabla} ^2 u\|_{L^2}^{\frac12}, \end{eqnarray*} $

    which implies that

    $ \begin{eqnarray*} \begin{split} &\|Q(t)\|_{L^2}\\ \lesssim &\|(\widetilde n,\widetilde m)\|_{L^\infty}\big(\|(\widetilde n,\widetilde m)\|_{L^2}+\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}+\|(n_h,m_h)\|_{L^2}\\ &+\|({\nabla} n_h,{\nabla} m_h)\|_{L^2}\big)+\|( n_h,m_h)\|_{L^\infty}\left(\|( n_h,m_h)\|_{L^2}+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\right)\\ &+\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^\infty}\|(n_h,m_h)\|_{L^2}\\ \lesssim &(1+t)^{-\frac94}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{eqnarray*} $

    Furthermore, exactly as in the estimate of the high order derivatives, we have

    $ \begin{equation} \begin{split} &\|({\nabla} n_h, {\nabla} m_h)\|_{L^2}\\ \lesssim &\int_0^{\frac t 2} \|({\nabla}\mathfrak N , {\nabla}\mathfrak M)(t-\tau)\ast H(\tau)\|_{L^2}d\tau+\int_{\frac t 2}^t \|(\mathfrak N, \mathfrak M )(t-\tau)\ast {\nabla} H(\tau)\|_{L^2}d\tau\\ \lesssim &\int_0^{\frac t 2} (1+t-\tau)^{-\frac74}\big(\| Q(\tau)\|_{L^1}+\|{\nabla} Q(\tau)\|_{L^2}\big)d\tau +\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}\|{\nabla} Q(\tau)\|_{L^2}d\tau\\ \lesssim&\left(\delta_0^2+\delta_0^{\frac98}\Lambda^2(t)\right)\Bigg(\int_0^{\frac t 2} (1+t-\tau)^{-\frac74}(1+\tau)^{-\frac32}d\tau+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}(1+\tau)^{-\frac{11}4}d\tau\Big)\\ \lesssim&(1+t)^{-\frac74}\left(\delta_0^2+\delta_0^{\frac98}\Lambda^2(t)\right), \end{split} \end{equation} $ (5.10)

    Similarly, it holds that

    $ \begin{eqnarray*} \begin{split} &\|{\nabla} Q(t)\|_{L^2}\nonumber\\ \lesssim &\|(\widetilde n,\widetilde m)\|_{L^\infty}\big(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}+\|({\nabla}^2\widetilde n,{\nabla}^2\widetilde m)\|_{L^2}+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\nonumber \end{split} \end{eqnarray*} $
    $ \begin{eqnarray*} \begin{split}&\quad+\|( {\nabla}^2 n_h,{\nabla}^2 m_h)\|_{L^2}\big)+\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^\infty}\big(\|({\nabla}\widetilde n,{\nabla}\widetilde m)\|_{L^2}+\|(n_h,m_h)\|_{L^2}\\ &\quad+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\big)+\|( n_h,m_h)\|_{L^\infty}\big(\|( {\nabla}^2\widetilde n,{\nabla}^2\widetilde m)\|_{L^2}+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\\ &\quad+\|({\nabla}^2 n_h,{\nabla}^2 m_h)\|_{L^2}\big)+\|( {\nabla} n_h,{\nabla} m_h)\|_{L^\infty}\|( {\nabla} n_h,{\nabla} m_h)\|_{L^2}\\ \lesssim & (1+t)^{-\frac{11}4}\left(\delta_0^2+\delta_0^{\frac98}\Lambda^2(t)\right). \end{split} \end{eqnarray*} $

    Thus, we also get that

    $ \begin{equation} \begin{split} &\|({\nabla}^2 n_h, {\nabla}^2 m_h)(t)\|_{L^2}\\ \lesssim& \int_0^{\frac t 2} \|({\nabla}^2 \mathfrak N, {\nabla}^2 \mathfrak M) (t-\tau)\ast H(\tau)\|_{L^2}d\tau\\ &\quad+\int_{\frac t 2}^t \|(\mathfrak N, \mathfrak M) (t-\tau)\ast {\nabla}^2 H(\tau)\|_{L^2}d\tau\\ \lesssim &\int_0^{\frac t 2}(1+t-\tau)^{-\frac94}\big(\|Q(\tau)\|_{L^1}+\|{\nabla}^2 Q(\tau)\|_{L^2}\big)d\tau\\ &\quad+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}\|{\nabla} ^2Q(\tau)\|_{L^2}d\tau\\ \lesssim& \left(\delta_0^2+\delta_0\Lambda(t)+\delta_0^{\frac34}\Lambda^2(t)\right)\bigg(\int_0^{\frac t 2} (1+t-\tau)^{-\frac94}(1+\tau)^{-\frac32}d\tau\\ &\quad+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}(1+\tau)^{-\frac{13}4}d\tau\bigg)\\ \lesssim&(1+t)^{-\frac94}\left(\delta_0^2+\delta_0\Lambda(t)+\delta_0^{\frac34}\Lambda^2(t)\right). \end{split} \end{equation} $ (5.11)

    Finally, we have

    $ \begin{eqnarray*} \begin{split} &\|{\nabla}^2 Q(t)\|_{L^2}\\ \lesssim &(\|(\widetilde n,\widetilde m)\|_{L^\infty}+\|(n_h,m_h)\|_{L^\infty})(\|({\nabla}^3\widetilde n,{\nabla}^3 \widetilde m)\|_{L^2}+\|({\nabla}^3n_h,{\nabla}^3 m_h)\|_{L^2})\\ &\quad+(\|({\nabla} \widetilde n,{\nabla}\widetilde m)\|_{L^\infty}+\|({\nabla} n_h,{\nabla} m_h)\|_{L^\infty}) (\|({\nabla}\widetilde n,{\nabla} \widetilde m)\|_{L^2}+\|({\nabla} n_h,{\nabla} m_h)\|_{L^2})\\ &\quad+(\|(\widetilde n,\widetilde m)\|_{L^\infty}+\|(n_h,m_h)\|_{L^\infty}+\|({\nabla} \widetilde n,{\nabla}\widetilde m)\|_{L^\infty}+\|({\nabla} n_h,{\nabla} m_h)\|_{L^\infty})\\ &\quad\times(\|({\nabla}^2\widetilde n,{\nabla}^2 \widetilde m)\|_{L^2}+\|({\nabla}^2 n_h,{\nabla}^2 m_h)\|_{L^2})\\ \lesssim&(1+t)^{-\frac{13}4}\left(\delta_0^2+\delta_0\Lambda(t)+\delta_0^{\frac34}\Lambda^2(t)\right). \end{split} \end{eqnarray*} $

    In this subsection, we will close the a priori estimates and complete the proof of Proposition 5.2. For this purpose, we need to derive the time decay rate of higher order derivatives of $ (n_h,m_h) $. We will establish the following lemma.

    Lemma 5.2. Under the assumption of Theorem 1.1, one has

    $ \begin{eqnarray*} \|{\nabla}^2 n(t)\|_{H^1}+\|{\nabla}^2 u(t)\|_{H^1}\lesssim (1+t)^{-\frac74}\left(\delta_0+\delta_0^{\frac34}\Lambda(t)\right). \end{eqnarray*} $

    In particular, it holds that

    $ \begin{eqnarray*} \|{\nabla}^3 (n_h, m_h)(t)\|_{L^2}\lesssim (1+t)^{-\frac74}\left(\delta_0+\delta_0^{\frac34}\Lambda(t)\right). \end{eqnarray*} $

    Proof. First of all, in view of (2.12), recovering the dissipation estimate for $ n $, we see that

    $ \begin{equation} \begin{split} &\frac{d}{dt}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^3 n dx +C_1\|{\nabla}^3 n\|_{L^2}^2 dx\\ \leq &C_2\left(\|{\nabla}^3 u\|_{L^2}^2+\|{\nabla}^4 u\|_{L^2}^2\right)+C(1+t)^{-\frac{3}2}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\\ &\quad\times\left(\|{\nabla}^2 n\|_{L^2}^2+\|{\nabla}^2 u\|_{L^2}^2+\|{\nabla}^3 u\|_{L^2}^2\right). \end{split} \end{equation} $ (5.12)

    Summing up (2.7) and (2.8) in the energy estimate for $ (n,u) $, we can directly derive

    $ \begin{equation} \begin{split} &\frac{d}{dt}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} \left(\gamma |{\nabla}^2 n|^2+|{\nabla}^2 u|^2 + \gamma|{\nabla}^3 n|^2 +|{\nabla}^3 u|^2 \right)dx + C_3\left(\|{\nabla}^3 u|^2 _{L^2}+\|{\nabla}^4 u\|^2 _{L^2}\right) \\ \leq &C(1+t)^{-\frac{3}2}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\|{\nabla}^2 n\|_{L^2}^2+\|{\nabla}^2 u\|_{L^2}^2+\|{\nabla}^3 n\|_{L^2}^2\right). \end{split} \end{equation} $ (5.13)

    Multiplying (5.12) by $ \epsilon_1\frac{C_3}{C_2} $ with $ \epsilon_1>0 $ a small but fixed constant, adding it with (5.13), we deduce that there exists a constant $ C_4>0 $ such that

    $ \begin{eqnarray*} \begin{split} &\frac{d}{dt}\bigg\{\sum\limits_{2\leq k\leq3}\left(\gamma \|{\nabla}^k n\|^2_{L^2}+\|{\nabla}^k u\|^2_{L^2} \right)+\epsilon_1\frac{C_3}{C_2}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^{3} n dx \bigg\}\\ &\quad+ C_4\Big(\|{\nabla}^{3} n\|_{L^2}^2+\sum\limits_{3\leq k\leq4}\|{\nabla}^{k} u\|^2_{L^2}\Big)\\ \leq &C(1+t)^{-\frac{3}2}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\|{\nabla}^2 n\|_{L^2}^2+\|{\nabla}^2 u\|_{L^2}^2\right). \end{split} \end{eqnarray*} $

    Next, we define

    $ \mathcal E_1(t) = \bigg\{\sum\limits_{2\leq k\leq3}\left(\gamma \|{\nabla}^k n\|^2_{L^2}+\|{\nabla}^k u\|^2_{L^2} \right)+\epsilon_1\frac{C_3}{C_2}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^{3} n dx \bigg\}. $

    Observe that since $ \epsilon_1\frac{C_3}{C_2} $ is small, then there exists a constant $ C_5>0 $ such that

    $ \begin{eqnarray*} C_5^{-1}\left(\|{\nabla}^2 n(t)\|^2_{H^1}+\|{\nabla}^2 u(t)\|^2_{H^1}\right) \leq\mathcal E_1(t)\leq C_5\left(\|{\nabla}^2 n(t)\|^2_{H^1}+\|{\nabla}^2 u(t)\|^2_{H^1}\right). \end{eqnarray*} $

    Then we arrive at

    $ \begin{eqnarray*} \frac{d}{dt}\mathcal E_1(t)+C_4\Big(\|{\nabla}^{3} n(t)\|_{L^2}^2+\|{\nabla}^3 u(t)\|^2_{H^1}\Big) \leq C(1+t)^{-5}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{eqnarray*} $

    Denote $ S(t) = \Big\{\xi\big| |\xi| \leq \sqrt{\frac{3(1+\gamma)}{C_4}}(1+t)^{-\frac12}\Big\} $ the time-dependent $ n $-dimensional sphere. This decomposition allows us to estimate $ L^2 $ time decay depend on $ (\widehat {n}, \widehat {u}) $ for frequency values $ \xi \in S(t) $, then we obtain that

    $ \begin{eqnarray*} \begin{split} &\frac{C_4}{3}\|{\nabla}^{3} (n, u)(x)\|_{L^2}^2 \geq\frac{C_4}{3}\int_{S(t)^c} |\xi|^6|(\widehat{n}, \widehat{u})(\xi)|^2d\xi\\ \geq&(1+\gamma)(1+t)^{-1}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} |\xi|^4|(\widehat{n}, \widehat{u})(\xi)|^2d\xi-(1+\gamma)(1+t)^{-1}\int_{S(t)} |\xi|^4|(\widehat{n}, \widehat{u})(\xi)|^2d\xi. \end{split} \end{eqnarray*} $

    Hence we have

    $ \begin{eqnarray*} \begin{split} &\frac{d}{dt}\mathcal E_1(t)+(1+t)^{-1}\mathcal E_1(t)+\|{\nabla}^{3} n\|_{L^2}^2+\|{\nabla}^3 u\|^2_{H^1}\\ \lesssim&(1+t)^{-5}\left(\delta_0+\delta_0^{\frac38}\Lambda(t)\right)\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right)+(1+t)^{-1}\int_{S(t)} |\xi|^4|(\widehat{n}, \widehat{u})(\xi)|^2d\xi\\ &\quad+(1+t)^{-1}\int_{{\mathop{\mathbb R\kern 0pt}\nolimits}^3} {\nabla}^2 u\cdot {\nabla}^{3} n dx. \end{split} \end{eqnarray*} $

    Multiplying the above equation by $ (1+t)^5 $, we obtain that

    $ \begin{eqnarray*} \begin{split} &\frac{d}{dt}\Big\{(1+t)^5\mathcal E_1(t)\Big\}+(1+t)^5\Big(\|{\nabla}^{3} n\|_{L^2}^2+\|{\nabla}^3 u\|^2_{H^1}\Big) \lesssim(1+t)^{\frac12}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{split} \end{eqnarray*} $

    Integrating it with respect to time from $ 0 $ to $ T $, then we have

    $ \begin{eqnarray*} \begin{split} &(1+t)^5\mathcal E_1(t)+\int_0^T(1+t)^5\Big(\|{\nabla}^{3} n\|_{L^2}^2+\|{\nabla}^3 u\|^2_{H^1}\Big)dt\\ \lesssim& \mathcal E_1(0)+(1+t)^{\frac32}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right), \end{split} \end{eqnarray*} $

    which implies that

    $ \begin{eqnarray*} \|{\nabla}^3 n\|^2_{L^2}+\|{\nabla}^3 u\|^2_{L^2}\lesssim\mathcal E_1(t)\lesssim (1+t)^{-5}\delta_0^2+(1+t)^{-\frac72}\left(\delta_0^2+\delta_0^{\frac32}\Lambda^2(t)\right). \end{eqnarray*} $

    Finally, we have

    $ \begin{eqnarray*} \|{\nabla}^3 n_h\|_{L^2}+\|{\nabla}^3 m_h\|_{L^2}\lesssim (1+t)^{-\frac74}\left(\delta_0+\delta_0^{\frac34}\Lambda(t)\right). \end{eqnarray*} $

    This completes the proof of this Lemma.

    In this subsection, we first combine the above a priori estimates of (5.8), (5.9), (5.10), (5.11) and Lemma 5.2 together to give the proof of the Proposition 5.2. In deed, for any $ t\in[0,T] $, we have shown that

    $ \begin{equation} \Lambda(t)\leq C\left(\delta_0+\delta_0^{\frac14}\Lambda(t)+\Lambda^2(t)\right) \leq C\delta_0^{\frac34}. \end{equation} $ (5.14)

    With the help of standard continuity argument, Proposition 5.2 and the smallness of $ \delta_0>0 $, implies that $ \Lambda(t)\leq C\delta_0^{\frac34} $ for any $ t>0 $. Moreover, we deduce the time decay estimate for $ (n_h, m_h) $ from (5.9), (5.10), (5.11), Lemma 5.2 and (5.14) that

    $ \begin{eqnarray*} \begin{split} &\|({\nabla}^k n_h, {\nabla}^k m_h)\|_{L^2}\lesssim \delta_0^2(1+t)^{-\frac54-\frac k2},\quad k = 0,1,\\ &\|{\nabla}^2 (n_h, m_h)\|_{L^2}\lesssim\delta_0^{\frac74}(1+t)^{-\frac94},\quad \|{\nabla}^3 (n_h, m_h)\|_{L^2}\lesssim\delta_0(1+t)^{-\frac74}. \end{split} \end{eqnarray*} $

    Consequently, for any $ t\in[0,T] $ we have

    $ \begin{equation} \Lambda(t)\leq C\delta_0. \end{equation} $ (5.15)

    From (5.11) and (5.15), thus we also get that

    $ \begin{eqnarray*} \|{\nabla}^2 (n_h, m_h)\|_{L^2}\lesssim\delta_0^2(1+t)^{-\frac94}. \end{eqnarray*} $

    For $ {\nabla}^3 m_h $, in view of the (5.6), we see that

    $ \begin{eqnarray*} \begin{split} &\|{\nabla}^3 m_h(t)\|_{L^2} \\\lesssim &\int_0^{\frac t 2}(1+t-\tau)^{-\frac{11}4}\big(\|Q(\tau)\|_{L^1}+\|{\nabla}^2 Q(\tau)\|_{L^2}\big)d\tau\\ &\quad+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}\|{\nabla} ^2Q(\tau)\|_{L^2}d\tau\\ \lesssim& \delta_0^2\bigg(\int_0^{\frac t 2} (1+t-\tau)^{-\frac{11}4}(1+\tau)^{-\frac32}d\tau+\int_{\frac t 2}^t (1+t-\tau)^{-\frac12}(1+\tau)^{-\frac{13}4}d\tau\bigg)\\ \lesssim&\delta_0^2(1+t)^{-\frac{11}4}. \end{split} \end{eqnarray*} $

    Hence, we finish the proof of the Proposition 5.1. Theorem 1.1 follows.

    Y. Chen is partially supported by the China Postdoctoral Science Foundation under grant 2019M663198, Guangdong Basic and Applied Basic Research Foundation under grant 2019A1515110733, NNSF of China under grants 11801586, 11971496 and China Scholarship Council. The research of R. Pan is partially supported by National Science Foundation under grants DMS-1516415 and DMS-1813603, and by National Natural Science Foundation of China under grant 11628103. L. Tong's research is partially supported by China Scholarship Council.


    Acknowledgments



    The authors would like to thank Dee Mclean for providing the artwork for Figure 3.

    Conflict of interest



    All authors declare no conflicts of interest in this paper.

    [1] Donaldson L (2009) 150 years of the Annual Report of the Chief Medical Officer: On the state of public health 2008. London: Department of Health.
    [2] Ahangari A (2014) Prevalence of chronic pelvic pain among women: An updated review. Pain Physician 17: E141–147.
    [3] van Wilgen CP, Keizer D (2012) The sensitization model to explain how chronic pain exists without tissue damage. Pain Manag Nurs 13: 60–65. doi: 10.1016/j.pmn.2010.03.001
    [4] Lamvu G (2011) Role of hysterectomy in the treatment of chronic pelvic pain. Obstet Gynecol 117: 1175–1178. doi: 10.1097/AOG.0b013e31821646e1
    [5] Mathias SD, Kuppermann M, Liberman RF, et al. (1996) Chronic pelvic pain: prevalence, health-related quality of life, and economic correlates. Obstet Gynecol 87: 321–327. doi: 10.1016/0029-7844(95)00458-0
    [6] Royal Collge of Obstetricians & Gynaecologists (RCOG) (2015) Scientific Impact Paper No. 46. Therapies Targeting the Nervous System for Chronic Pelvic Pain Relief. RCOG: London
    [7] Lee TT, Yang LC (2008) Pelvic denervation procedures: A current reappraisal. Int J Gynaecol Obstet 101: 304–308. doi: 10.1016/j.ijgo.2008.02.010
    [8] Huber SA, Northington GM, Karp DR (2015) Bowel and bladder dysfunction following surgery within the presacral space: an overview of neuroanatomy, function, and dysfunction. Int Urogynecol J 26: 941–946. doi: 10.1007/s00192-014-2572-x
    [9] Chen FP, Soong YK (1997) The efficacy and complications of laparoscopic presacral neurectomy in pelvic pain. Obstet Gynecol 90: 974–977. doi: 10.1016/S0029-7844(97)00484-5
    [10] Lichten EM, Bombard J (1987) Surgical treatment of primary dysmenorrhea with laparoscopic uterine nerve ablation. J Reprod Med 32: 37–41.
    [11] Daniels JP, Middleton L, Xiong T, et al. (2010) International LUNA IPD Meta-analysis Collaborative Group. Individual patient data meta-analysis of randomized evidence to assess the effectiveness of laparoscopic uterosacral nerve ablation in chronic pelvic pain. Hum Reprod Update 16: 568–576.
    [12] El-Din Shawki H (2011) The efficacy of laparoscopic uterosacral nerve ablation (LUNA) in the treatment of unexplained chronic pelvic pain: a randomized controlled trial. Gynecol Surg 8: 31–39. doi: 10.1007/s10397-010-0612-1
    [13] Daniels J, Gray R, Hills RK, et al. (2009) LUNA Trial Collaboration. Laparoscopic uterosacral nerve ablation for alleviating chronic pelvic pain: A randomized controlled trial. JAMA 302: 955–961.
    [14] Jedrzejczak P, Sokalska A, Spaczynski RZ, et al. (2009) Effects of presacral neurectomy on pelvic pain in women with and without endometriosis. Ginekol Pol 80: 172–178.
    [15] Rouholamin S, Jabalameli M, Mostafa A (2015) The effect of preemptive pudendal nerve block on pain after anterior and posterior vaginal repair. Adv Biomed Res 27: 153. doi: 10.4103/2277-9175.161580
    [16] Chanrachakul B, Likittanasombut P, O-Prasertsawat P, et al. (2001) Lidocaine versus plain saline for pain relief in fractional curettage: A randomized controlled trial. Obstet Gynecol 98: 592–595.
    [17] Naghshineh E, Shiari S, Jabalameli M (2015) Preventive effect of ilioinguinal nerve block on postoperative pain after cesarean section. Adv Biomed Res 4: 229. doi: 10.4103/2277-9175.166652
    [18] Binkert CA, Hirzel FC, Gutzeit A, et al. (2015) Superior hypogastric nerve block to reduce pain after uterine artery embolization: Advanced technique and comparison to epidural anesthesia. Cardiovasc Intervent Radiol 38: 1157–1161. doi: 10.1007/s00270-015-1118-z
    [19] Rapp H, Ledin Eriksson S, Smith P (2017) Superior hypogastric plexus block as a new method of pain relief after abdominal hysterectomy: Double-blind, randomised clinical trial of efficacy. BJOG 124: 270–276. doi: 10.1111/1471-0528.14119
    [20] Fujii M, Sagae S, Sato T, et al. (2002) Investigation of the localization of nerves in the uterosacral ligament: Determination of the optimal site for uterosacral nerve ablation. Gynecol Obstet Invest 54: discussion 16–7. doi: 10.1159/000066289
    [21] Matalliotakis IM, Katsikis IK, Panidis DK (2005) Adenomyosis: What is the impact on fertility? Curr Opin Obstet Gynecol 17: 261–264. doi: 10.1097/01.gco.0000169103.85128.c0
    [22] Desrosiers JA, Faucher GL (1964) Uterosacral block: A new diagnostic procedure. Obstet Gynecol 23: 671–677.
    [23] Rana MV, Candido KD, Raja O, et al. (2014) Celiac plexus block in the management of chronic abdominal pain. Curr Pain Headache Rep 18: 394. doi: 10.1007/s11916-013-0394-z
    [24] Soysal ME, Soysal S, Gurses E, et al. (2003) Laparoscopic presacral neurolysis for endometriosis-related pelvic pain. Hum Reprod 18: 588–592. doi: 10.1093/humrep/deg127
    [25] Byrd D, Mackey S (2008) Pulsed radiofrequency for chronic pain. Curr Pain Headache Rep 12: 37–41. doi: 10.1007/s11916-008-0008-3
  • This article has been cited by:

    1. Yuhui Chen, Minling Li, Qinghe Yao, Zheng-an Yao, The sharp time-decay rates for one-dimensional compressible isentropic Navier-Stokes and magnetohydrodynamic flows, 2022, 1674-7283, 10.1007/s11425-021-1937-9
    2. Jincheng Gao, Minling Li, Zheng-an Yao, Optimal decay of compressible Navier-Stokes equations with or without potential force, 2023, 342, 00220396, 63, 10.1016/j.jde.2022.09.030
    3. Yuhui Chen, Minling Li, Qinghe Yao, Zheng-an Yao, The sharp time decay rates for the incompressible Phan-Thien–Tanner system with magnetic field in R2, 2022, 129, 08939659, 107965, 10.1016/j.aml.2022.107965
    4. Xiuli Xu, Xueke Pu, Xiaoyu Xi, Optimal decay of the magnetohydrodynamic model for quantum plasmas with potential force, 2025, 0, 1531-3492, 0, 10.3934/dcdsb.2025054
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4958) PDF downloads(614) Cited by(0)

Figures and Tables

Figures(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog