1. Introduction
The Oldroyd-B model is a widely used constitutive model to describe the motion of viscoelastic fluids. One of the known derivations is that it can be derived as a macroscopic closure of Navier–Stokes–Fokker–Planck system which is a micro-macro model describing dilute polymeric fluids in dumbbell Hookean setting, see [Reference Barrett, Lu and Süli3, Reference Le Bris and Lelièvre7] for the incompressible case and the compressible case, respectively. The compressible Oldroyd-B model in the space-time cylinder $Q_T=\mathbb {R}^{3}\times (0,T]$
is stated as follows:
where the pressure $P(\rho )$
and the density $\rho (x,t)\geq 0$
of the fluid are supposed to be related by the typical power law relation for simplicity:
for some known constants $a>0$
, $\gamma >1$
; $u(x,t)\in \mathbb {R}^{3}$
denotes the velocity field of the fluid. $\mu$
and $\nu$
are viscosity coefficients satisfying $\mu \geq 0,\; 2\mu +3\nu \geq 0$
. The polymer number density $\eta (x,t)\geq 0$
represents the integral of the probability density function $\psi$
which is a microscopic variable in the modelling of dilute polymer chains, i.e.,
where $\psi$
is governed by the Fokker–Planck equation. The extra stress tensor $\mathbb {T}(x,t)= (\mathbb {T}_{i,j})(x,t)\in \mathbb {R}^{3\times 3},\ 1\leq i,j\leq 3$
is a positive definite symmetric matrix defined on $Q_T$
, and the notation $\mathrm {div} (u\mathbb {T})$
is understood as
The constant parameter $\varepsilon$
is the centre-of-mass diffusion coefficient and other parameters $k, L, \mathfrak {z}, A_0, \lambda$
are all positive and known numbers, whose meanings were explained in [Reference Barrett, Lu and Süli3]. (1.1) is known as diffusive Oldroyd-B model when the diffusion coefficient $\varepsilon >0$
. The corresponding micro-macro version of (1.1) can be referred for instance to [Reference Barrett and Süli2] and references therein.
Note that the centre-of-mass diffusion term is usually smaller than other effects ([Reference Bhave, Armstrong and Brown5]). For such a reason, in early mathematical studies of macroscopic Oldroyd-B model, the stress diffusion is omitted, see [Reference Oldroyd44]. In this context, [Reference Renardy46] established the local existence theory for Dirichlet problem. Guillopé and Saut [Reference Guillopé and Saut23] obtained the existence and uniqueness of global strong solution in the Sobolev space $H^{s}(\Omega )$
for bounded domains $\Omega \in \mathbb {R}^{3}$
with a small initial data. Some other related results can be referred to [Reference Fernández-Cara, Guillén and Ortega21, Reference Molinet and Talhouk43]. In exterior domains, Hieber, Naito and Shibata [Reference Hieber, Naito and Shibata24] obtained a global existence and uniqueness of the solution provided the initial data and the coupling constant are sufficiently small. Fang, Hieber and Zi [Reference Fang, Hieber and Zi18] extended the work [Reference Hieber, Naito and Shibata24] to the case without any smallness assumption on the coupling constant. The existence of a global-large-data weak solution was established by Lions and Masmoudi [Reference Lions and Masmoudi36] in the corotational derivative setting. For long-time behaviour of the solution, please refer to [Reference Hieber, Wen and Zi25, Reference Huang, Wang, Wen and Zi31]. There are also some interesting results on other macroscopic model of Oldroyd type concerning viscoelastic flow introduced by Lin, Liu and Zhang [Reference Lin, Liu and Zhang35], for example [Reference Hu and Lin26, Reference Hu and Wu29, Reference Lai, Lin and Wang33] and others.
However, the centre-of-mass diffusion can be physically justified to model the shear and vorticity banding phenomena [Reference Bhave, Menon, Armstrong and Brown6, Reference Cates and Fielding9, Reference Chupin and Martin10, Reference Dhont and Briels13, Reference El-Kareh and Leal32, Reference Liu and Fredrickson37, Reference Málek, Pruša, Skřivan and Süli42], although it is small. In this case, some interesting works have been achieved. More specifically, the global-in-time existence of large-data solutions in two dimensional setting was obtained by Barrett-Boyaval [Reference Barrett and Boyaval1] for weak solutions and by Constantin and Kliegl [Reference Constantin and Kliegl11] for strong solutions. In three-dimensional setting, Bathory, Bulíček and Málek [Reference Bathory, Bulíček and Málek4] proved the global existence of weak solutions for a generalized rate-type viscoelastic fluids in bounded domains. For the inviscid case, Elgindi and Rousset [Reference Elgindi and Rousset17] obtained the global existence and uniqueness of regular solutions in two dimensions with arbitrarily large initial data when $Q=Q(\nabla u,\mathbb {T})$
is omitted and with small initial data when $Q\neq 0$
. We refer to [Reference Elgindi and Liu16] for the three-dimensional case with small initial data. For the case of fractional Laplace, please refer to [Reference Constantin, Wu, Zhao and Zhu12]. Very recently, the second author, the third author and their collaborators [Reference Huang, Wang, Wen and Zi31] studied the long-time behaviour of the solutions and obtained some decay estimates. These results are concerned with homogeneous fluids, i.e., the density is constant.
For the compressible case, there are a lot of fundamental problems which are still open. We recall some mathematical results for compressible viscoelastic models, which have been the subject of related fields in recent years. The well posedness in local time and global well posedness near an equilibrium for macroscopic models of three-dimensional compressible viscoelastic fluids were considered in [Reference Fang and Zi19, Reference Hu and Wang27, Reference Hu and Wu28, Reference Lei34, Reference Qian and Zhang45] (see [Reference Bulíček, Feireisl and Málek8] for global existence of weak solutions). In particular, Fang and Zi [Reference Fang and Zi19] proved the local well posedness of strong solutions to a compressible Oldroyd-B model and established a blow-up criterion. Soon afterwards, the authors [Reference Zi53] obtained the global well posedness in critical spaces. Lei [Reference Lei34], Fang and Zi [Reference Fang and Zi20], and Guillopé, Salloum and Talhouk [Reference Guillopé, Salloum and Talhouk22] investigated the incompressible limit problem in torus, the whole space and bounded domain, respectively. Very recently, Zhou, Zhu and Zi [Reference Zhou, Zhu and Zi51] obtained some time-decay estimates of strong solutions. Zhu [Reference Zhu52] obtained the global well posedness of small classical solutions to a generalized inviscid compressible Oldroyd-B model in Sobolev space $H^{s}$
for $s\geq 5$
. In [Reference Barrett, Lu and Süli3], Barrett, Lu and Süli not only showed the derivation of the compressible viscous Oldroyd-B model with stress diffusion (1.1) via a macroscopic closure of a micro-macro model, but also proved the existence of global-in-time finite-energy weak solutions with arbitrarily large initial data in two dimensions. The global-in-time existence of solutions strong or weak with arbitrarily large initial data is unknown in three dimensions either with stress diffusion or not. In two and three dimensional setting, Lu and Zhang [Reference Lu and Zhang40] obtained the local-in-time well posedness of strong solutions together with a blow-up criterion and weak-strong uniqueness. Very recently, the second author and the third author [Reference Wang and Wen47] showed the global well posedness and optimal time-decay rates of strong solutions for Cauchy problem in three dimensions. In critical Besov spaces, one can refer to [Reference Zhai and Li50]. Less is known concerning the vanishing of centre-of-mass diffusion and the inviscid case in (1.1) either for global well posedness or for long time behaviour, until very recently the first author, the third author and their collaborator investigated the first case (i.e., $\varepsilon =0$
) in [Reference Liu, Lu and Wen38]. This work is devoted to the latter one which is more challenging.
More precisely, we consider the case that $\mu =\nu =0$
in (1.1), i.e.,
1.1 Reformulation of the problem
In this section, we give a reformulation of (1.2) to make the analysis more convenient right behind. In fact, when $\varepsilon =0$
and the viscosity coefficients are fixed, a similar reformulation was given in our previous work [Reference Liu, Lu and Wen38]. Thus this section is a slight modification of the corresponding part in [Reference Liu, Lu and Wen38]. More specifically, multiplying (1.2)$_3$
by $k\mathbb {I}_{ij}$
, we have
Then subtracting (1.3) from (1.2)$_4$
yields that
Further, denoting $\tau _{ij}=\mathbb {T}_{ij}-k\eta \mathbb {I}_{ij}$
, and conducting direct calculations, we can get
and
Putting (1.5) and (1.6) into (1.4) yields
which is
Next, the term of the right-hand side in (1.2)$_2$
can be transformed into the following form
which together with (1.2)$_2$
implies that
Finally, combining (1.2)$_1$
, (1.8), (1.2)$_3$
and (1.7) yields
which is equipped with the following initial condition:
Note that (1.9) is equivalent to the system (1.2) with the regularity of the solution in the present paper and that it seems more convenient to consider (1.9) in the proof. Therefore we will state the main results afterwards for the reformulated system (1.9) only.
1.2 Main results
Our main results are stated as follows.
Theorem 1.1 Let $L\geq 1,\,\mathfrak {z}\geq 0$
. Assume that $(\rho _0-\tilde {\rho }, u_0, \eta _0-\tilde {\eta }, \tau _0)\in H^{3}(\mathbb {R}^{3})$
for constants $\tilde {\rho }, \tilde {\eta }>0$
. Then there exists a positive constant $\theta$
sufficiently small such that if
the initial-value problem (1.9)–(1.10) admits a unique global strong solution $(\rho, u, \eta, \tau )$
which satisfies
Theorem 1.2 In addition to the conditions of theorem 1.1, we assume that $(\rho _0-\tilde {\rho }, u_0, \eta _0-\tilde {\eta }, \mathrm {div}\tau _0)\in L^{1}(\mathbb {R}^{3})$
. Then there exists a positive constant $C$
independent of $t$
such that the solution $(\rho,u,\eta,\tau )$
satisfies the following time-decay estimates:
for any $t\geq 0$
.
Remark 1.3 From (1.9)$_4$
and the conclusion in theorem 1.2, one can observe that the decay rate of $\|\nabla ^{l}\tau \|_{L^{2}(\mathbb {R}^{3})}$
is the same as that of $\|\nabla ^{l+1}u\|_{L^{2}(\mathbb {R}^{3})}$
where the maximum of $l$
is 2 according to the regularity of the solutions. Therefore the decay rate of $\|\nabla ^{3}\tau (t)\|_{L^{2}(\mathbb {R}^{3})}$
is not expected to be sharper.
We would like to introduce the main novelty of this work. Compared with [Reference Wang and Wen47] where the global well posedness of strong solution for (1.1) with positive shear viscosity $\mu$
is established subject to some smallness assumptions, the vanishing of viscosity in the present paper will bring new challenges such as the loss of regularity for the velocity. We introduce a good unknown $\tau _{ij}=\mathbb {T}_{ij}-k\eta \mathbb {I}_{ij}$
for $i,j=1,2,3$
inspired by [Reference Liu, Lu and Wen38, Reference Lu and Pokorný39] and derive some new dissipative estimates of velocity from the equation of $\tau _{ij}$
such that the smallness of the initial data does not depend on the viscosity. This demonstrates that the coupling yields new dissipative effect of the velocity satisfying the Euler equation only. Besides, the construction of the basic energy estimates in $H^{3}$
-norm relies strongly on the dissipative estimate of $\nabla ^{4}u$
due to the second term of the continuity equation and the pressure term of the momentum equation. It seems impossible to get the dissipative estimate of $\nabla ^{4}u$
in the non-viscous case. To handle the issue, inspired by [Reference Zhu52], we use the variation of the continuity equation $\mathrm {div}u=-\frac {\rho _t+\beta u\cdot \nabla \rho }{r_1+\beta \rho }$
and integration by parts to transfer the derivative to other term. Concerning the optimal time-decay estimates, the loss of dissipation of velocity due to the vanishing of viscosity is the main difficulty compared with the viscous case in [Reference Wang and Wen47]. Delicate energy method and low-high-frequency decomposition is the main tool in the proof. In this context, we observe that the reformulated equation of $\tau _{ij}$
can produce the dissipation mechanism of velocity such that the decay estimates do not rely on the viscosity, which is the key to obtain the optimal time-decay estimates of the solution except for its highest order. It is unusual that the optimal decay rate for the highest-order derivatives of the solution to some hyperbolic-parabolic systems even as (1.1) with viscosity (see [Reference Wang and Wen47]) could be obtained. To get the dissipative estimate for the hyperbolic quantities $\nabla ^{k}\rho$
and $\nabla ^{k}\tau$
where $k=3$
is the maximal one, the usual energy method is to construct the interaction energy functional between the second-order and the third-order. Therefore it implies the decay rate for the third-order will be the same as that for the second-order. Here we use the low-high-frequency decomposition and employ the high-frequency part of velocity at $2^{th}$
order as a test function of the equation of $\nabla ^{2}\mathrm {div}\tau _{ij}$
. The high-frequency quantity will make the damping term in the equation of $\nabla ^{2}\mathrm {div}\tau _{ij}$
keep the desired order, see § 4.4 for more details. This is different from our previous work [Reference Wang and Wen48] for compressible Navier–Stokes equations with reaction diffusion where a new observation for cancellation of a low-medium-frequency quantity was adopted to get the optimal time-decay estimate at the highest order, see also [Reference Wu, Zhang and Zhou49] for a two-phase fluid model. In addition, to get the decay estimates of the low-frequency part to the linearized system (A.1), we apply the Hodge decomposition to $u$
and $\tau$
, and transfer the linearized system into two systems (A.2) and (A.3). We introduce some corrected modes different from [Reference Wang and Wen47] to overcome the difficulties caused by the lack of dissipation of $u$
. With the help of these estimates, the decay properties for the low-frequency part of the solutions to the nonlinear system are obtained by using the Duhamel principle. Combining the delicate energy estimates with the decay estimates of low-frequency part, we obtain the same decay rates of the solution to (1.9) up to the second-order as those for viscous case, see [Reference Wang and Wen47]. Moreover the decay rate for the third-order in the present paper is sharper.
The rest of the paper is organized as follows. In § 2, we linearize the reformulated system which will make the following analysis more convenient. In § 3, the proof of the global well posedness of the solutions will be given by using delicate energy method combined with the continuity technique. In § 4, we establish some optimal time-decay estimates and finish the proof of theorem 1.2.
2. Linearization of the reformulated system
To simplify the proof of the main theorems, we linearize the reformulated problem (1.9)–(1.10) as follows. Taking change of variables by $(\rho, u, \eta, \tau )\rightarrow (\rho '+\tilde {\rho }, \beta u', \eta '+\tilde {\eta }, \tau )$
with $\beta >0$
to be determined, the initial-value problem (1.9)–(1.10) is written as below
where
with initial data
Denote the scaled parameters and constants by
and define the nonlinear functions of $\rho '$
by
Finally, (we remove all $'$
in the following system for brevity) we rewrite the system (1.9)–(1.10) with linearized part on the left as
and
with initial data
It is worth noticing that the proof of theorems 1.1 and 1.2 can be translated into that for the solution to (2.1)–(2.3).
3. Proof of theorem 1.1
In this section, we will prove theorem 1.1 via taking vanishing viscosity limit of (1.1). In fact, the global existence and uniqueness of solutions to the corresponding viscous case has been achieved by the second author and the third author in [Reference Wang and Wen47] where the smallness of initial data depends on the viscosity coefficients. Therefore the aim in this section is to derive some a priori estimates globally in time subject to some smallness of data independent of the viscosity coefficients. We assume that $\mu,\nu \leq 1$
in the section for simplicity.
After conducting a reformulation similar to (2.1), (1.1) can be converted to the following form.
where
and
We begin with a local existence and uniqueness result of the initial-value problem (3.1) and (2.3).
Proposition 3.1 Local existence and uniqueness
Assume that
Then, there exists a constant $T_0 > 0$
depending on $\mu$
, $\nu$
and $\|(\rho _0, u_0, \eta _0, \tau _0)\|_{H^{3}(\mathbb {R}^{3})}$
, such that the initial-value problem (3.1) and (2.3) has a unique strong solution $(\rho ^{\mu,\nu }, u^{\mu,\nu }, \eta ^{\mu,\nu }, \tau ^{\mu,\nu })$
over $\mathbb {R}^{3}\times [0,T_0]$
, which satisfies
where $Q_{T_0}=\mathbb {R}^{3}\times (0,T_0)$
.
Proof. The proof can be achieved by using some standard iteration arguments, please refer for instance to [Reference Fang and Zi19, Reference Huang, Ding and Wen30, Reference Lu and Zhang40]. We omit the details for brevity.
Proposition 3.2 (A priori estimate) Under the assumptions of theorem 1.1, there exists a positive constant $\delta$
independent of $\mu$
and $\nu$
and at least bigger than $\frac {3\theta }{2}$
$[$
determined by (3.55) for some $\theta$
given by (1.11) $]$
, such that if the strong solution of the initial-value problem (3.1) and (2.3) satisfies
for any $t\in [0,T]$
, where $0 < T^{*}\le +\infty$
is the maximum existence time for the solution and $T\in (0,T^{*})$
, then the following estimates
holds for any $t\in [0,T]$
.
Remark 3.3 A similar result has been obtained by the second author and the third author in [Reference Wang and Wen47] (proposition 3.2) when $\delta$
depends on $\mu$
and $\nu$
. Proposition 3.2 removes the dependence between $\delta$
and the viscosity coefficients, which gives the possibilities to take the vanishing viscosity limit.
Based on the propositions 3.1 and 3.2, the global existence of solutions to the initial-value problem (3.1) and (2.3) will be established with the help of the standard continuity arguments. Then, with the aid of the uniform estimates (3.4) and some compactness arguments, we conclude that a subsequence of solutions $(\rho ^{\mu,\nu }, u^{\mu,\nu }, \eta ^{\mu,\nu },\tau ^{\mu,\nu })$
converges to a limit $(\rho,u,\eta,\tau )$
(in some strong sense) which is a strong solution to the original problem (2.1)–(2.3). Therefore to prove theorem 1.1, it suffices to prove proposition 3.2 which will be achieved step by step in the following lemmas.
Throughout the rest of the paper, we denote $L^{p}:=L^{p}(\mathbb {R}^{3})$
and $\int f\,\mathrm {d}x:=\int _{\mathbb {R}^{3}} f\,\mathrm {d}x$
, and let $C \geq 1$
represent a generic positive constant that depends on some known constants but is independent of $\theta$
, $\delta$
, $\mu$
, $\nu$
, $t$
and $T^{*}$
.
Although the solutions usually depend on $\mu$
and $\nu$
, one can find that the following results and procedures are applicable to the case $\mu =\nu =0$
. For brevity, we omit the superscripts throughout lemmas 3.4–3.6.
Lemma 3.4 Under the same assumptions of theorem 1.1 and (3.3), the following estimate
holds for any $t\in [0,T]$
.
Proof. Applying derivatives $\nabla ^{\ell }(\ell =0,1,2,3)$
to the system (3.1), taking inner product with $\nabla ^{\ell }\rho$
, $\nabla ^{\ell } u$
, $\frac {r_2}{\beta \tilde {\eta }}\nabla ^{\ell } \eta$
and $\frac {r_3}{2\beta k\tilde {\eta }}\nabla ^{\ell } \tau$
respectively, and then adding the results, we can obtain
Before we estimate each term on the right-hand side of (3.6), it is worth noticing that the disappearance of the viscous terms in the momentum equation (3.1)$_2$
leads to partial loss of regularity of velocity $u$
. Hence, to derive some uniform estimates independent of $\mu$
and $\nu$
, those terms containing the fourth derivative of density $\rho$
or velocity $u$
can not be directly controlled. In the following proof, we will list them separately and deal with them in detail.
Firstly, for the first term on the right-hand side of (3.6), by noticing the definition of $S_1$
, we have
The reason why we discuss $\ell$
separately here is to make the proof more concise when proving decay estimates later in this article. We first deal with the lower derivative terms, $I_{11}$
and $I_{12}$
, using Hölder inequality, Sobolev inequality, Cauchy inequality and lemma A.5, it holds that
and
Then, for $I_{13}$
, which can be divided into the following five terms.
Next, we turn to deal with the terms. In the same way, we first deal with the lower derivative terms, $I_{13_{1}}$
, $I_{13_{3}}$
and $I_{13_{4}}$
, using Hölder inequality, Sobolev inequality, Cauchy inequality and lemma A.5, it holds that
Then, for the terms containing the fourth derivative, $I_{13_{2}}$
and $I_{13_{5}}$
. On one hand, for $I_{13_{2}}$
, by virtue of integration by parts, Hölder inequality and Sobolev inequality, we can directly deduce
On the other hand, for $I_{13_{5}}$
, by using (3.1)$_1$
: $\mathrm {div}u=-\frac {\rho _t+\beta u\cdot \nabla \rho }{r_1+\beta \rho }$
, we can get
Further, the second term and the last term on the right-hand side of (3.13) equal
where we use integration by parts.
(3.13), combined with (3.14), Hölder inequality, Sobolev inequality and Cauchy inequality, yields
Putting (3.11), (3.12) and (3.15) into (3.10) yields
Now, substituting (3.8), (3.9) and (3.16) into (3.7), it holds that
Secondly, the second term on the right-hand side of (3.6) equals
From (3.2), we first estimate $I_{21}$
and $I_{22}$
. Using Hölder inequality, Sobolev inequality and Cauchy inequality, it holds that
and
$I_{23}$
can be divided into the following five terms:
Then, we turn to deal with the terms $I_{23_{1}}$
–$I_{23_{5}}$
. For the terms $I_{23_{1}}$
–$I_{23_{3}}$
, using Hölder inequality, Sobolev inequality, Cauchy inequality and lemma A.5, we have
and
For $I_{23_{4}}$
, using integration by parts, we can easily get
where we have used Hölder inequality, Sobolev inequality and Cauchy inequality. Then, we are going to deal with $I_{23_{5}}$
which can be split into the following two terms.
For the last term in (3.26), using integration by parts and (3.1)$_1$
again, we have
where the observation that $\mathrm {div}u=\frac {\rho _t+\beta u\cdot \nabla \rho }{r_1+\beta \rho }$
is crucial due to the loss of regularity of velocity, which can be seen for instance in [Reference Zhu52].
Further, processing method similar to $I_{13_{5}}$
, the last term on the right-hand side of (3.27) can be dealt with like
Together with (3.26), (3.27) and (3.28), using Hölder inequality, Sobolev inequality and Cauchy inequality, we get
Hence, substituting (3.22), (3.23), (3.24), (3.25) and (3.29) into (3.21), it holds that
Now, putting (3.19), (3.20) and (3.30) into (3.18), we have
Next, for the third term on the right-hand side of (3.6), using (2.2)$_3$
, we have
For $I_{31}$
and $I_{32}$
, similar to $I_{11}$
and $I_{12}$
, it is not hard to get
and
Thanks to Hölder inequality, Sobolev inequality, Cauchy inequality and lemma A.5, $I_{33}$
can be controlled like
Note that we have used integration by parts in (3.34) to reduce the order of spatial derivative of velocity $u$
. Hence, combining with (3.32), (3.33) and (3.34) yields
Finally, for the last term on the right-hand side of (3.6), we have
Thanks to (2.2)$_4$
, and using Hölder inequality, Sobolev inequality and Cauchy inequality again, $I_{41}$
and $I_{42}$
can be controlled like
and
For $I_{43}$
, we will take advantage of the higher integrability of $\tau$
to deal with each term on the right-hand side through integration by parts. In other words, thanks to (2.2)$_4$
again, $I_{43}$
can be estimated as
where we have used Hölder inequality, Sobolev inequality, Cauchy inequality and lemma A.5.
Owing to (3.36), (3.37) and (3.38), we get
Finally, plugging (3.17), (3.31), (3.35) and (3.39) into (3.6), we then obtain the following inequality:
Choosing $\delta$
sufficiently small in the above inequality, (3.5) will be established. Thus, we complete the proof of this lemma.
In the following lemmas, we obtain some dissipation estimates of velocity $u$
and density $\rho$
which are independent of the viscosity coefficients.
Lemma 3.5 Under the same assumptions of theorem 1.1 and (3.3), we then have the following estimate:
Proof. Let $\ell = 0,1,2$
. Applying operator $\nabla ^{\ell }\mathrm {div}$
to the equation of (3.1)$_4$
and $\nabla ^{\ell }$
to the equation of (3.1)$_2$
, multiplying the results by $\nabla ^{\ell } u$
and $\nabla ^{\ell }\mathrm {div}\tau$
, respectively, summing them up and then integrating it over $\mathbb {R}^{3}$
, we get the equality:
First, by the definition of $S_4$
and $\tilde {S}_2$
, integration by parts, Hölder inequality, Sobolev inequality, Cauchy inequality and lemma A.5, it holds that
and
where the terms on the right-hand side of the above equality can be controlled as
and
For the remaining two terms on the second line of (3.41), by using integration by parts, Höder inequality and Cauchy inequality, it is straightforward to show that
Finally, the remaining terms in (3.41) can be estimated by using integration by parts, Hölder inequality and Cauchy inequality as follows
Plugging (3.42)–(3.47) into (3.41), then choosing $\delta$
sufficiently small yields
Lemma 3.6 Under the same assumptions of theorem 1.1 and (3.3), we then have the following estimate:
Proof. Let $\ell = 0,1,2$
. Applying operator $\nabla ^{\ell }$
to the equation of (3.1)$_2$
and (3.1)$_1$
, multiplying the results by $\nabla ^{\ell +1} \rho$
and $-\nabla ^{\ell }\mathrm {div}u$
respectively, then summing them up and integrating it over $\mathbb {R}^{3}$
, we get the equality:
Likewise for (3.43)–(3.45), from (3.2), Höder inequality, Sobolev inequality, Cauchy inequality and lemma A.5, the first line on the right-hand side of (3.49) can be controlled like
and
Similarly, for the last line in (3.49), recalling the definition of $S_1$
, using Höder inequality, Sobolev inequality, Cauchy inequality and lemma A.5 again, we have
Now, plugging (3.50)–(3.52) into (3.49), then choosing $\delta$
sufficiently small yields
In what follows, based on lemmas 3.4–3.6, we are ready to prove proposition 3.2.
Proof of proposition 3.2 Combined with (3.5), $\epsilon _2$
(3.40) and $\epsilon _1$
(3.48), it holds that
Firstly, choosing a fixed positive constant $\epsilon \leq \frac {\beta k \tilde {\eta }r_1}{64C^{2}}$
, and then taking
and
and finally choosing $\delta$
sufficiently small, we obtain
Next, integrating (3.53) over $(0,t)$
, we get
where
Since (3.3) and the smallness of $\delta$
, $\epsilon _1$
and $\epsilon _2$
, it is easy to check that $\mathcal {J}(t)$
is equivalent to
Moreover, by the virtue of (1.11) and (3.54), there exists a constant $C_1$
independent of $\mu,\nu,\theta,\delta,\epsilon,\epsilon _1,\epsilon _2,t$
and $T^{*}$
, such that
Letting
and then we can get (3.4) and complete the proof of proposition 3.2.
Due to the priori estimate stated in proposition 3.2 and the standard continuity arguments, the following estimate
holds. Next, we only need to show $T^{*}=\infty$
. In fact, owing to proposition 3.1 and the time-uniform estimates stated in (3.56), it concludes that $T^{*}=\infty$
. Thus, we get the global existence of solutions to the initial-value problem (3.1) and (2.3).
Thanks to (3.4), we deduce that $(\rho ^{\mu,\nu }, u^{\mu,\nu },\eta ^{\mu,\nu },\tau ^{\mu,\nu })$
, $(\rho ^{\mu,\nu }_t, u^{\mu,\nu }_t)$
and $(\eta ^{\mu,\nu }_t,\tau ^{\mu,\nu }_t)$
are uniformly bounded in $L^{\infty }([0,\infty );H^{3})$
, $L^{\infty }([0,\infty );H^{2})$
and $L^{\infty }([0,\infty );H^{1})$
, respectively. Moreover, $(\eta ^{\mu,\nu },\tau ^{\mu,\nu })$
is uniformly bounded in $L^{2}([0,\infty );H^{4})$
. Hence, there exists a subsequence $(\rho ^{\mu,\nu }, u^{\mu,\nu }, \eta ^{\mu,\nu },\tau ^{\mu,\nu })$
such that
The regularity of the limit is good enough to ensure that $(\rho, u, \eta,\tau )$
is a strong solution to the original problem (2.1)–(2.3). Therefore the proof for the first part of theorem 1.1, i.e., global existence, is complete. By the standard energy method, we can prove that the solution in theorem 1.1 is unique, provided that $\|(\rho, u, \eta,\tau )(t)\|_{H^{3}}$
is sufficiently small. Therefore, we finish the proof of theorem 1.1.
4. Proof of theorem 1.2
In this part, we are going to obtain the decay estimates of $(\rho,u,\eta,\tau )$
to the original problem (2.1)–(2.3). To do this, the strategy is to combine the energy method with the spectral analysis of the corresponding linear system. The connection between the two aspects is the Duhamel's principle.
Proposition 4.1 Under the assumptions of theorem 1.2, there is a constant $C$
independent of $t$
such that the solution $(\rho,u,\eta,\tau )$
of initial-value problem (2.1)–(2.3) satisfies the following estimates:
for any $t\geq 0$
.
4.1 Energy estimates
First of all, we are going to get the optimal time-decay estimate of $\|\nabla (\rho,u,\eta )(t)\|_{L^{2}}$
.
Lemma 4.2 Under the same assumptions of theorem 1.2 and (3.3), there exist two small positive constants $\epsilon _3$
and $\epsilon _4$
which will be determined in the proof of the lemma and lemma 4.3, such that
where
Proof. Following arguments similar to the proof of lemma 3.4 for the case $\ell =1,2,3$
and $\mu =\nu =0$
, one has,
In addition, for the case $\ell =1,2$
and $\mu =\nu =0$
in (3.40) and (3.48), we can get
and
Hence, $\epsilon _3$
(4.4) together with (4.2) and $\epsilon _4$
(4.3) yields
Firstly, choosing a fixed positive constant $\epsilon \leq \frac {\beta k \tilde {\eta }r_1}{64C^{2}}$
, and taking
and
and finally choosing $\delta$
sufficiently small, we get (4.1).
Moreover, with lemmas 4.2 and A.3, the following result holds.
Lemma 4.3 Under the same assumptions of theorem 1.2 and (3.3), we have
for some positive constant $C_2$
independent of $\delta$
.
Proof. By lemma A.3, we have
Thus, (4.1) leads to
By adding $\epsilon _3\frac {r_1c_0^{2}}{8}\|\nabla \rho ^{L}\|_{L^{2}}^{2}+ \epsilon _4\frac {\beta k\tilde {\eta }c_0^{2}}{8}\|\nabla u^{L}\|_{L^{2}}^{2}+\frac {r_2\varepsilon c_0^{2}}{8\beta \tilde {\eta }}\|\nabla \eta ^{L}\|_{L^{2}}^{2}$
to both sides of inequality (4.6), we have
where we let $c_0\in (0,1]$
.
Note that, by virtue of (3.3) and the smallness of $\delta$
, $\epsilon _3$
and $\epsilon _4$
, it is easy to check that $\mathcal {H}_1(t)$
is equivalent to
Then there exists a positive constant $C_2>0$
such that
By using Gronwall's inequality, we get (4.5).
In the same way, we show the following estimates of $\|\nabla ^{2}(\rho,u,\eta,\tau )(t)\|_{H^{1}}$
which are the basis for getting the optimal decay estimate of $\|\nabla ^{2}(\rho,u,\eta )(t)\|_{L^{2}}$
.
Lemma 4.4 Under the same assumptions of theorem 1.2 and (3.3), there exist two small positive constants $\epsilon _5$
and $\epsilon _6$
which will be determined in the proof of the lemma and lemma 4.5, such that
where
Proof. Following arguments similar to the proof of lemma 3.4 for the case $\ell =2,3$
and $\mu =\nu =0$
, one has,
In addition, for the case $\ell =2$
and $\mu =\nu =0$
in (3.40) and (3.48), we can deduce
and
Together with $\epsilon _5$
(4.10), (4.8) and $\epsilon _6$
(4.9) yields
Firstly, choosing a fixed positive constant $\epsilon \leq \frac {\beta k \tilde {\eta }r_1}{64C^{2}}$
, and taking
and
and finally choosing $\delta$
sufficiently small, we get (4.7).
Based on lemmas 4.4 and A.3, the following result holds.
Lemma 4.5 Under the same assumptions of theorem 1.2 and (3.3), we have
for some positive constant $\tilde {C_2}$
independent of $\delta$
.
Proof. By lemma A.3, we have
Thus, (4.7) leads to
By adding $\epsilon _5\frac {r_1c_0^{2}}{8}\|\nabla ^{2} \rho ^{L}\|_{L^{2}}^{2}+ \epsilon _6\frac {\beta k\tilde {\eta }c_0^{2}}{8}\|\nabla ^{2} u^{L}\|_{L^{2}}^{2}+\frac {r_2\varepsilon c_0^{2}}{8\beta \tilde {\eta }}\|\nabla ^{2}\eta ^{L}\|_{L^{2}}^{2}$
to both sides of inequality (4.12), and choosing $\delta$
sufficiently small, we have
where we let $c_0\in (0,1]$
.
Moreover, by virtue of (3.3) and the smallness of $\delta$
, $\epsilon _5$
and $\epsilon _6$
, it is easy to check that $\mathcal {H}_2(t)$
is equivalent to
Then there exists a positive constant $\tilde {C_2}>0$
such that
By using Gronwall's inequality again, we get (4.11) directly.
4.2 Decay estimates of the low-frequency parts
Next, with the help of lemmas 4.3 and 4.5, we will study the decay rates of solution $(\rho,u,\eta,\tau )$
. As it can be seen from (4.5) and (4.11), we only need to analyse the low-frequency part ($|\xi |\leq c_0$
) of $(\rho,u,\eta )$
.
Letting $\mathbb {A}$
be the following matrix of differential operators of the form
and setting
we obtain from the linearized problem of (A.1) as below:
Applying the Fourier transform to (4.13) with respect to the $x$
-variable and solving the ordinary differential equation with respect to $t$
, we have
where $\mathcal {A}(t)=e^{-t\mathbb {A}}(t\geq 0)$
is the semigroup generated by the linear operator $\mathbb {A}$
and $\mathcal {A}(t)f:=\mathcal {F}^{-1}(e^{-t\mathbb {A}_{\xi }}\hat {f}(\xi ))$
with
Then, from proposition A.2, we have the following result.
Lemma 4.6 For any integer $m\geq 0$
, the following time-decay estimates for the low-frequency part, i.e.,
and
hold for any $t\in (0,\infty )$
.
Remark 4.7 (4.14) and (4.15) are used to obtain the optimal time-decay estimates of $\|(\rho,u,\eta,\tau )(t)\|_{H^{2}}$
and $\|\nabla ^{3}(\rho,u,\eta,\tau )(t)\|_{L^{2}}$
, respectively.
In what follows, based on the estimates in lemma 4.6, we establish time-decay estimates for the low-frequency part of solutions to the nonlinear problem (2.1)–(2.3). Denoting
then from (2.2), we have
where
Using the Duhamel's principle, the solution of (4.16) can be stated as follows:
Lemma 4.8 Under the assumptions of theorem 1.2 and (3.3), for any integer $m\geq 1$
, there exists a positive constant $C$
such that
and
Remark 4.9 With the aid of (4.18) and (4.19), the optimal time-decay estimates of $\|(\rho,u,\eta,\tau )(t)\|_{H^{2}}$
and $\|\nabla ^{3}(\rho,u,\eta,\tau )(t)\|_{L^{2}}$
will be obtained, respectively.
Proof. From (4.17), using lemma 4.6, we have
where we have used the fact that
Hence, (4.18) is obtained.
Moreover, for (4.19), using (4.17) and lemma 4.6 again, we have
where
Thus, together with (4.21), (4.22) and (4.23), (4.19) can be obtained.
4.3 Optimal decay rates of $\|(\rho,u,\eta )(t)\|_{H^{2}}$
and $\|\tau (t)\|_{H^{1}}$
In this subsection, we will obtain the time-decay estimates of $(\rho, u,\eta,\tau )$
with the aid of lemmas 4.3, 4.5 and 4.8. Firstly, we consider the decay estimate of $\|(\nabla \rho, \nabla u,\nabla \eta,\nabla \tau )(t)\|_{H^{2}}$
.
Proposition 4.10 Under the same assumptions of theorem 1.2 and (3.3), we have
Proof. Owing to (4.5) and (4.18), we can obtain
where $\mathcal {I}(t)=\sup \limits _{0\leq s\leq t}(1+s)^{\frac {5}{2}}\mathcal {H}_1(s)$
.
Further, by virtue of the definition of $\mathcal {I}(t)$
and the smallness of $\delta$
, we can obtain
which is (4.24). We complete the proof of the proposition.
Then, based on proposition 4.10, we can obtain the next proposition.
Proposition 4.11 Under the same assumptions of theorem 1.2 and (3.3), we have
Proof. Firstly, for $\|\tau (t)\|_{L^{2}}$
, we multiply (2.1)$_4$
by $2\tau$
and then integrate the resulting equation over $\mathbb {R}^{3}$
to obtain
where we have used Cauchy inequality and the fact that
Choosing $\delta$
sufficiently small in (4.27), it holds that
Further, using Gronwall's inequality, the above inequality gives
where we have used (4.24). Then we obtain (4.25).
Next, for $\|(\rho, u, \eta )(t)\|_{L^{2}}$
, thanks to (4.18), let $m=0$
, we find that
where we have used (4.24).
In addition, by using lemma A.3 and (4.24) again, we have
Combining with (4.28) and (4.29), and owing to (A.32), we can get (4.26) directly.
Further, the optimal decay estimates of second order for $(\rho,u,\eta )$
and first order for $\tau$
in the sense of $L^{2}$
norm are obtained as below.
Proposition 4.12 Under the same assumptions of theorem 1.2 and (3.3), we have
Proof. Owing to (4.11), (4.20), (4.21) and propositions 4.10 and 4.11, we can obtain
which is (4.30).
Finally, for (4.31), multiplying $\nabla$
(2.1)4 by $2\nabla \tau$
and then integrating the result equation over $\mathbb {R}^{3}$
, similar to (4.25), we get
Further, using Gronwall's inequality, the above inequality gives
where we have used (4.30). Hence, we complete the proof of the proposition.
4.4 Optimal decay rates of $\|\nabla ^{3}(\rho,u,\eta )(t)\|_{L^{2}}$ and $\|\nabla ^{2}\tau (t)\|_{L^{2}}$
Inspired by [Reference Wang and Wen48, Reference Wu, Zhang and Zhou49], we are going to study the optimal decay estimates of $\|\nabla ^{3}(\rho,u,\eta )(t)\|_{L^{2}}$
and $\|\nabla ^{2}\tau (t)\|_{L^{2}}$
. In the process, we have made full use of the benefit of frequency decomposition.
Proposition 4.13 Under the same assumptions of theorem 1.2 and (3.3), we have
Lemma 4.15 Under the same assumptions of theorem 1.2 and (3.3), there exist two small positive constants $\epsilon _7$
and $\epsilon _8$
which will be determined in the proof of the lemma and lemma 4.16, such that
where
Proof. Following some arguments similar to the proof of lemma 3.4 for the case $\ell =3$
and $\mu =\nu =0$
, one has
where we have used the following inequality:
which is established by Hölder inequality, sobolev inequality and Gagliardo-Nirenberg inequality.
In addition, applying operator $\nabla ^{2}\mathrm {div}$
to the equation of (2.1)$_4$
and $\nabla ^{2}$
to the equation of (2.1)$_2^{h}$
, multiplying the results by $\nabla ^{2} u^{h}$
and $\nabla ^{2}\mathrm {div}\tau$
respectively, then summing them up and integrating it over $\mathbb {R}^{3}$
, we can get
where we have used the frequency decomposition (A.32). Then, similar to the case $\ell =2$
and $\mu =\nu =0$
in (3.40), we have
and
Finally, combined with lemma A.3, the rest term of (4.36) can be deal with like
Together with (4.37), (4.38) and (4.39), using Hölder inequality and Cauchy inequality, and choosing $\delta$
and $\epsilon$
small enough, we can deduce from (4.36)
where we have used the fact that
Similarly, applying operator $\nabla ^{2}$
to the equation of (2.1)$_2$
and (2.1)$_1^{h}$
, multiplying the results by $\nabla ^{3} \rho ^{h}$
and $-\nabla ^{2}\mathrm {div}u$
respectively, then summing them up and integrating it over $\mathbb {R}^{3}$
, we have
Further, from (4.41) and referring to the case $\ell =2$
and $\mu =\nu =0$
in (3.48), we can deduce the following inequality:
Hence, $\epsilon _7$
(4.42) together with (4.35) and $\epsilon _8$
(4.40) yields
Firstly, choosing a fixed positive constant $\epsilon \leq \frac {\beta k \tilde {\eta }r_1}{64C^{2}}$
, and taking
and
and finally choosing $\delta$
sufficiently small, we get (4.34).
Moreover, with lemmas 4.15 and A.3, the following result holds.
Lemma 4.16 Under the same assumptions of theorem 1.2 and (3.3), we have
for some positive constant $\tilde {C_3}$
independent of $\delta$
.
Proof. By lemma A.3, we have
Thus, (4.34) leads to
By adding $\epsilon _7\frac {r_1}{4}\|\nabla ^{3} \rho ^{L}\|_{L^{2}}^{2}+ \epsilon _8\frac {\beta k\tilde {\eta }}{4}\|\nabla ^{3} u^{L}\|_{L^{2}}^{2}+\frac {r_2\varepsilon c_0^{2}}{2\beta \tilde {\eta }}\|\nabla ^{3}\eta ^{L}\|_{L^{2}}^{2}$
to both sides of inequality (4.44), taking
and choosing $\delta$
sufficiently small, we have
Moreover, it follows from integration by parts, the Young inequality and lemma A.3 that
Hence, by virtue of (3.3) and the smallness of $\delta$
, $\epsilon _7$
and $\epsilon _8$
, it is easy to check that $\mathcal {H}_3(t)$
is equivalent to
Then there exists a positive constant $\tilde {C_3}>0$
such that
By using Gronwall's inequality, we get (4.43).
With the help of lemmas 4.15–4.16, we are ready to prove proposition 4.13.
Proof of proposition 4.13 Thanks to the case $m=3$
in (4.19), (4.43), propositions 4.10, 4.11 and 4.12, we can obtain
which is (4.32).
Then, for (4.33), multiplying $\nabla$
(2.1)4 by $2\nabla ^{2}\tau$
and then integrating the result equation over $\mathbb {R}^{3}$
, similar to (4.25) and (4.31), we get
Using Gronwall's inequality, the above inequality yields
where we have used (4.32). Hence, we complete the proof of the proposition.
Finally, based on propositions 4.10–4.13, the decay rates of the solution stated in proposition 4.1 are obtained. Thus, we finish the proof of theorem 1.2.
Acknowledgments
W. Wang was supported by the National Natural Science Foundation of China $\#$
11871341. H. Wen was supported by the National Natural Science Foundation of China $\#$
12071152 and by Guangdong Basic and Applied Basic Research Foundation #2020B1515310015.
Appendix A.1. Estimates on the linearized system
Let us consider the following linear system for $(\bar {\rho }, \bar {u}, \bar {\eta }, \mathrm {div}\bar {\tau })$
:
As it can be seen from (4.5), (4.11) and (4.43), to study the decay estimates of $(\rho,u,\eta,\tau )$
, we only need to analyse the low frequency part ($|\xi |\leq c_0$
) of $(\bar {\rho },\bar {u},\bar {\eta },\bar {\tau })$
.
If we adopt $\Lambda ^{s}:=(-\Delta )^{\frac{s}{2}}$
as the notation for the pseudo-differential operator defined by $\Lambda ^{s}f:=\mathcal {F}^{-1}(|\xi |^{s}\hat {f}(\xi ))$
, we only need to study $\bar {\rho }$
, $d:=\Lambda ^{-1}\mathrm {div}\bar {u}$
and $\mathbb {P}\bar {u}:=\Lambda ^{-1}\text {curl}\bar {u}$
, where $\text {curl}_i^{j}\bar {u}=\partial _j\bar {u}^{i}-\partial _i\bar {u}^{j}$
; $\bar {\eta }$
, $q:=\Lambda ^{-1}\mathrm {div}\mathrm {div}\bar {\tau }$
and $\mathbb {P}\mathrm {div}\bar {\tau }:=\Lambda ^{-1}\text {curl}\mathrm {div}\bar {\tau }$
. Indeed, by the definition of $\mathbb {P}$
, we have
We see that $(\bar {\rho },d, \bar {\eta },q)$
and $(\mathbb {P}\bar {u},\mathbb {P}\mathrm {div}\bar {\tau })$
satisfy
and
Applying Fourier transform to the linearized system (A.2) and (A.3), we arrive at
and
A.1.1 Estimates on $(\hat {\bar {\rho }},\hat {d},\hat {\bar {\eta }},\hat {q})$
We introduce the following corrected modes different from those in [Reference Wang and Wen47]:
Then the system (A.4) can be rewritten as
where the coefficient $A_1$
is defined by
From the corrected modes, it is not hard to find that the estimates of $(\hat {\mathbf {a}},\hat {\mathbf {o}},\hat {\mathbf {z}},\hat {\mathbf {q}})$
can be easily translated into the estimates of $(\hat {\bar {\rho }},\hat {d},\hat {\bar {\eta }},\hat {q})$
for small $|\xi |$
. Next, let us turn to study the estimates of $(\hat {\mathbf {a}},\hat {\mathbf {o}},\hat {\mathbf {z}},\hat {\mathbf {q}})$
. From (A.10), we easily obtain
Multiplying (A.10)$_1$
and (A.10)$_2$
by $\bar {\hat {\mathbf {o}}}$
and $\bar {\hat {\mathbf {a}}}$
, respectively, yields
Combined with (A.11) and $-\tilde {\epsilon }|\xi |\times$
(A.12), it holds that
It is natural to derive the estimates for those terms on the right-hand side of (A.13). First, the first two terms can be controlled by
where
Similarly, we can drive the bound of the last term.
Substituting (A.14) and (A.15) into (A.13) yields
Now, we move on and derive the estimates of $\hat {\mathbf {q}}$
as follows. Multiplying (A.10)$_4$
by $\bar {\hat {\mathbf {q}}}$
and using Cauchy inequality, we then obtain the following inequality:
which implies
Finally, combining with (A.16) and (A.17) yields
Taking $\tilde {\epsilon }:=\min \left \{\frac {r_1\varepsilon }{2r_2\beta \tilde {\eta }},\frac {\lambda r_3\beta k\tilde {\eta }}{A_0r_1}\right \}$
, and introducing the Lyapunov functional
It is clear that, for $|\xi |\leq \frac {1}{\tilde {\epsilon }}$
, we have $\mathcal {L}_{com}^{2}$
is equivalent to $|\hat {\mathbf {a}}|^{2}+|\hat {\mathbf {o}}|^{2} +\frac {r_2}{\beta \tilde {\eta }}|\hat {\mathbf {z}}|^{2}+|\hat {\mathbf {q}}|^{2}$
. Hence, from (A.18), using Cauchy inequality, there exists a small positive constant $c_1\leq \frac {1}{\tilde {\epsilon }}$
depending only on the parameters $\varepsilon, k, A_0, \beta, \tilde {\eta }, \lambda, \tilde {\epsilon }$
and $r_i\;(i=1,2,3)$
such that
Namely, we have
for some positive constant $C_3$
independent of $|\xi |$
. Then it follows form (A.19) that for $\xi \leq c_1$
Recalling the relation between $(\hat {\mathbf {a}},\hat {\mathbf {o}},\hat {\mathbf {z}},\hat {\mathbf {q}})$
with $(\hat {\bar {\rho }},\hat {d},\hat {\bar {\eta }},\hat {q})$
in (A.6)–(A.9), and thanks to (A.20), we can easily deduce the following inequality holds.
A.1.2 Estimates on $(\widehat {\mathbb {P}\bar {u}},\widehat {\mathbb {P}\mathrm {div}\bar {\tau }})$
We introduce the following corrected modes:
Then the system (A.5) can be rewritten as
Multiplying (A.24) with $\bar {\hat {\mathbf {v}}}$
and $\bar {\hat {\mathbf {w}}}$
respectively and use Cauchy inequality, we have
and
Together with (A.25) and (A.26), it holds that
By choosing $|\xi |\leq c_2$
, which makes $\frac {A_0}{2\lambda }+\left (\varepsilon -\frac {2\lambda }{A_0}r_3\beta k\tilde {\eta }-\frac {A_0}{2\lambda r_3}\beta k\tilde {\eta }-\frac {2\lambda }{A_0\beta k\tilde {\eta }}\right. \left.r_3(\varepsilon -\frac {2\lambda }{A_0}r_3\beta k\tilde {\eta })^{2}\right )|\xi |^{2}\geq \frac {A_0}{4\lambda }\geq \frac {A_0}{4\lambda }|\xi |^{2}$
true. Then, from (A.27), we have
Hence, there exists a positive constant $C_4$
independent of $|\xi |$
such that
Thanks to (A.28), we can deduce
Further, from the definitions (A.22) and (A.23), (A.29) implies that, for $|\xi |\leq c_2$
,
A.1.3 Decay estimates of the low frequency part
Taking $c_0=\min \{c_1, c_2\}$
, and combining with (A.21) and (A.30), we have the following proposition.
Proposition A.1 It holds that, for $|\xi |\leq c_0$
,
and
for some positive constant $C_5$
independent of $\xi$
.
For the low frequency part, we have the following decay estimates for solutions to the linearized problem (A.1), (2.3).
Proposition A.2 It holds that
and
for any $|m|>0$
.
Proof. By using Plancherel's theorem and proposition A.1, we have
Moreover, we can also deduce
By similar calculations, we can get
and
Then, we finish the proof of proposition A.2.
Appendix A.2. Some useful inequalities
Let $0\leq \varphi _0(\xi )\leq 1$
be a function in $\mathcal {C}_0^{\infty }(\mathbb {R}^{3})$
such that
where $c_0$
is a positive constant. Based on the Fourier transform, we can define a low and high frequency decomposition $(f^{L}(x), f^{h}(x))$
for a function $f(x)$
as follows
The following lemma can be obtained directly from the definition (A.32) and Plancherel's theorem.
Lemma A.3 [Reference Liu, Lu and Wen38]
If $f\in H^{m}(\mathbb {R}^{3})(m \geq 2)$
is divided into two parts $(f^{L}, f^{h})$
by the low and high frequency decomposition (A.32). It holds that
for any integers $m_1$
and $m_2$
with $m_2 \leq m_1\leq m$
.
Finally, the following elementary inequality will also be used.
Lemma A.4 [Reference Duan, Ukai, Yang and Zhao15]
If $a>1$
and $b\in [0,a]$
, then it holds that
Lemma A.5 [Reference Duan, Ruan and Zhu14, Reference Majda and Bertozzi41]
Let $m\geq 1$
be an integer, then we have
where $1\leq p, p_i\leq +\infty,\ (i=1,2,3,4)$
and
