The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien

Abstract

In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture $L^{2}$ -convergence result via auxiliary logarithmic time decay estimates with the initial data in $L^{p} (R^{2}) \cap L^{2} (R^{2})$ . Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

Keywords

Viscoelastic fluids sharp time decay rates global-in-time stability large solutions

1. Introduction

The global well-posedness of strong solutions with general initial data for the Magnetohydrodynamic (MHD) flows of isothermal incompressible polymer fluids in two space dimensions has been a longstanding open problem. The objective of this paper is to establish the sharp time decay rates and global-in-time stability of large solutions to the following two-dimensional incompressible linear Phan-Thien-Tanner (PTT) system with magnetic field: $\begin{array}{l} (1.1) & \{\begin{matrix} u_{t} + u \cdot \nabla u - μ △ u + \nabla p = μ_{1} div τ + H \cdot \nabla H - \nabla \frac{| H |^{2}}{2}, (t, x) \in R^{+} \times R^{2}, \\ τ_{t} + u \cdot \nabla τ + (a + b (tr τ)) τ + Q (τ, \nabla u) = μ_{2} D (u), (t, x) \in R^{+} \times R^{2}, \\ H_{t} + u \cdot \nabla H - H \cdot \nabla u - ν △ H = 0, (t, x) \in R^{+} \times R^{2}, \\ div u = 0, div H = 0, (t, x) \in R^{+} \times R^{2}, \end{matrix} \end{array}$ with the initial data $\begin{matrix} u |_{t = 0} = u_{0} (x), τ |_{t = 0} = τ_{0} (x), H |_{t = 0} = H_{0} (x) . \end{matrix}$ From a physical viewpoint, p, u, τ, and H denote, respectively, the isotropic pressure, the velocity of the fluid, the polymer contribution to the stress tensor, and the magnetic field. $Q (τ, \nabla u)$ is a given bilinear form $\begin{matrix} Q (τ, \nabla u) = τ Ω (u) - Ω (u) τ + λ (D (u) τ + τ D (u)), \end{matrix}$ with $D (u) : = \frac{\nabla u + {(\nabla u)}^{T}}{2}$ , $Ω (u) : = \frac{\nabla u - {(\nabla u)}^{T}}{2}$ . The constant $μ > 0$ is the viscosity coefficient, $μ_{1} > 0$ is the elastic coefficient, $ν > 0$ is the magnetic diffusion coefficient. $a > 0$ and $μ_{2} > 0$ imply the relation between the characteristic flow time and elastic time. $b ⩾ 0$ is related to the rates of creation and destruction for the polymeric network junctions. $λ \in [- 1, 1]$ is a physical parameter, the system is called co-rotational when $λ = 0$ .

In fact, what we call PTT system was firstly derived independently by Nhan Phan-Thien and Roger I. Tanner [39,40]. Let $f : R \to R$ be a smooth function, the constitutive equations are $\begin{array}{l} T = μ_{s} τ + 2 η D (u), \\ λ_{1} \frac{D τ}{D t} + (1 + \frac{ϵ λ_{1}}{η} f (tr τ)) τ = 2 η (1 - \frac{λ_{2}}{λ_{1}}) D (u), \end{array}$ the “objective derivative” $\frac{D τ}{D t}$ is denoted by $\begin{matrix} \frac{D τ}{D t} : = \frac{\partial τ}{\partial t} + u \cdot \nabla τ + τ Ω (u) - Ω (u) τ + λ (D (u) τ + τ D (u)), \end{matrix}$ where T represents the stress tensor, which relates to the rates of creation and destruction of network junctions, τ is an extra-stress tensor, and u is the velocity tensor. The constant $μ_{s} > 0$ is the elastic coefficient, $η > 0$ is the polymer viscosity coefficient, $λ_{1} > 0$ is the relaxation time of the fluid, $λ_{2} ⩾ 0$ is the retardation time of the fluid, ϵ and λ are physical parameters. For the sake of simplicity, we focus here upon the linear PTT system, that is, $f (x) = x$ .

In the absence of displacement current, the Maxwell equations of electromagnetism are $\begin{array}{l} \nabla \times H = J, \nabla \times E = - μ_{e} H_{t}, \\ J = σ (E + μ_{e} (u \times H)), div H = 0, div E = 0, \end{array}$ where H is the magnetic field, E is the electric field, and J is the electric current density vector field. The constant $μ_{e} > 0$ is the magnetic permeability, $σ > 0$ is the electrical conductivity.

This flow is governed by the continuity, momentum balance and induction equations: $\begin{array}{l} div u = 0, \\ u_{t} + u \cdot \nabla u + \nabla p = div T + μ_{e} (\nabla \times H) \times H, \\ H_{t} = \nabla \times (u \times H) + \frac{1}{μ_{e} σ} \nabla \times (\nabla \times H), div H = 0 . \end{array}$ Setting $a = \frac{1}{λ_{1}}$ , $b = \frac{ϵ}{η}$ , $μ = η$ , $μ_{1} = μ_{s}$ , $μ_{2} = \frac{2 η}{λ_{1}} (1 - \frac{λ_{2}}{λ_{1}})$ , $ν = \frac{1}{μ_{e} σ}$ , and $μ_{e} = 1$ , thus (1.1) was derived.

In the past and recent years, important progress has been made in the global well-posedness theory provided that the initial data is a small perturbation around equilibrium for the multi-dimensional incompressible Navier-Stokes equations. In the theory and applications of Nonlinear Partial Differential Equations, it is of considerable significance to investigate the global well-posedness and long-time asymptotic behavior of strong solutions with general initial data. Though, it poses a great challenge to extend the global well-posedness of classical solutions to the multi-dimensional Navier-Stokes equations with a small perturbation to general initial data, due to the strong nonlinearity. So the importance problem is the global existence of classical solutions to this system in a functional space with the initial data as large as possible. If we relax the smallness condition, there are a great number of results concerning the global well-posedness for the multi-dimensional incompressible Navier-Stokes equations (see for instance [1,17,37]). We should mention that Gui and Zhang [18], He et al. [21] proved that the global-in-time stability of large solutions to the incompressible and compressible Navier-Stokes equations.

Naturally, a new and interesting question was raised when studying the PTT system (see e.g. the works [7,9,10] concerning the global well-posedness of small solutions): whether the global-in-time stability of large solutions is still true for the Cauchy problem (1.1). Therefore, the aim of this paper is to prove the global-in-time stability of large solutions for (1.1). More precisely, we shall obtain the $L^{2}$ -convergence rate of solutions that gives the detailed behavior of the solutions of (1.1) with general initial data. The result obtained in the present work will help us to obtain the optimal upper and lower bounds of time decay rates of large solutions and their derivatives.

To our knowledge, there are a number of numerical results for (1.1), but many critical mathematical problems still remain to be answered. It is interesting to note that (1.1) reduces to Oldroyd-B system when $b = 0$ and $H \equiv 0$ . In recent years, there have been extensive research works on the study of the related incompressible Oldroyd-B type models for viscoelastic fluids. Local existence of solutions of incompressible viscoelastic fluids for arbitrary initial data and global existence for sufficiently small initial data in appropriate spaces were established by Guillopé and Saut [19,20]. Fernández-Cara et al. [16] proved the local existence in Sobolev spaces. A global existence result of weak solutions in the co-rotational case was developed by Lions and Masmoudi [34]. Chemin and Masmoudi [5] initiated the study of the well-posedness in critical Besov spaces. Some blow up criterions were studied in [15,31]. More recently, many results related to the global existence with initial data, which is a small perturbation around equilibrium, were shown in [6,11,30,32,46,48]. So far, the global existence of strong solutions with general initial data is still an open problem, even for incompressible viscoelastic fluids. For a class of large initial data, Fang et al. [14], Jiang et al. [28] established the global existence of strong solutions. We mention in passing that the well-posedness and other works for compressible flows were widely studied in [25,26,35,36,41,47]. A nice review of long time behavior of solutions may be found in [22,45].

The incompressible PTT system with magnetic field is the combination of the constitutive equations and incompressible MHD equations. For completeness, we present some related works on fluid system of MHD type, which has attracted considerable attention in the recent years. Duvaut and Lions [13], Sermange and Temam [44] constructed the well-known weak solutions of the incompressible MHD system. Later on, the global existence of smooth solutions in two dimensions was obtained in [29]. Abidi and Paicu [2] established the global existence of strong solutions for inhomogeneous MHD system in critical Besov spaces. In three dimensions, Ducomet and Feireisl [12] first proved the global existence of weak solutions to the compressible MHD system on a bounded spatial domain supplemented with conservative boundary conditions. Hu and Wang [23,24] proved that the compressible MHD system has global weak solutions with general initial data. In suitable Sobolev spaces, Cao [3,4] proved similar results for two-dimensional incompressible MHD system. More mathematical results concerning the MHD system, one may refer to [27,33,38,42].

In this paper, we are going to study the sharp time decay rates (for both upper and lower bounds) and global-in-time stability of large solutions for (1.1). As we have seen yet, the result obtained in this paper is the first one on the global-in-time stability of large solutions for two-dimensional PTT system of incompressible viscoelastic fluids.

The purpose of this paper is to study the global-in-time stability, for which obtaining the long time asymptotic behavior of large solutions is fundamental. The key point is to obtain the rates of $L^{2}$ -convergence of strong solutions for (1.1) without any small assumption, which is nontrivial to achieve in the two-dimensional critical case. It should be pointed out that the rates of $L^{2}$ -convergence of strong solutions for three-dimensional incompressible PTT system was recently obtained in [8]. However, the two-dimensional case can not be treated in the framework of the three-dimensional case. This is somewhat interesting owing to the difficulty mentioned above. In order to overcome this difficulty, we use the method introduced by Schonbek [43]. As shown in [43], a powerful tool in the study of the optimal time decay rates of solutions is the Fourier time-splitting method. The first step is establish the energy equality for large solutions due to the special structure of (1.1). Let us denote by $S (t) : = {ξ | | ξ | ⩽ g (t)}$ the time sphere, with $g (t)$ satisfying $g (t) ⩽ {(1 + t)}^{- \frac{1}{2}}$ . We shall construct a new function $g (t)$ with logarithmic-type, by means of an iterative process to obtain an auxiliary logarithmic time decay estimate, that is, $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}} ⩽ {ln}^{- 1} (e + t) . \end{matrix}$ We then construct $g (t)$ with polynomial-type, again take advantage of an iterative process to obtain the optimal time decay rate, that is, $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}} ⩽ {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1)}, \end{matrix}$ with the initial date in $L^{p} (R^{2}) \cap L^{2} (R^{2})$ , $1 ⩽ p < 2$ . The choice of $g (t)$ is the most crucial in the proof of the rates of $L^{2}$ -convergence of solutions.

Based on the convergence result of $(u, τ, H)$ in $L^{2}$ , it is thus enough to prove the rates of $L^{2}$ -convergence of its derivatives. We shall use the time-weighted energy estimates to obtain the rates of $H^{3}$ -convergence of strong solutions for (1.1). The choice of time weights may be suggested by looking at the linearized system to get hints on how the solution decays. Indeed, we may decompose the solution $(u, τ, H)$ into the linear term $(u_{l}, τ_{l}, H_{l})$ and the perturbed term $(u_{δ}, τ_{δ}, H_{δ})$ . Thanks to the spectral analysis of the semigroup for the linearized system, we may deduce the exact expression for the Fourier transform of Green’s function. Therefore, we obtain the precise lower bounds of time decay rates of $(u_{l}, τ_{l}, H_{l})$ for various order of derivatives. Together the upper bounds of time decay rates of $(u, τ, H)$ and the lower bounds of time decay rates of $(u_{l}, τ_{l}, H_{l})$ , our strategy for the nonlinear analysis is using the semigroup method and Duhamel principle to get the exact expression of $(u_{δ}, τ_{δ}, H_{δ})$ . Thus, we obtain the upper bounds of time decay rates of $(u_{δ}, τ_{δ}, H_{δ})$ , which is faster than $(u_{l}, τ_{l}, H_{l})$ due to the nonlinearity. Therefore, it suffices to prove that the lower bounds of time decay rates of $(u, τ, H)$ for lower order of derivatives. In addition, we get better convergence results concerning lower bounds of time decay rates of higher order derivatives of $(u, H)$ by using interpolation inequalities. These convergence results imply that the time decay rates of large solutions for (1.1) are sharp in two dimensions.

Finally, we preform standard energy estimates to obtain the global-in-time stability result. We show that the perturbed solutions will remain close to the reference solutions for a long time if their initial data are close to each other. Note that with these convergence results, if reference solutions are close to the equilibrium after a long time, then the perturbed solutions are also close to the equilibrium after a long time. This global-in-time stability result implies that the set of the smooth and bounded solutions for (1.1) is an open set.

Notation.
For the sake of simplicity, we restrict our attention here to the case where $λ = 0$ (similar results may be obtained for nonzero λ) and $μ = μ_{1} = μ_{2} = a = b = ν = 1$ . Unless otherwise indicated, we henceforth omit $R^{2}$ and set $H^{s} = H^{s} (R^{2})$ , $L^{p} = L^{p} (R^{2})$ . Throughout this paper, C denotes a generic positive constant independent of time t.

We now state the main results of this paper. We first prove the $L^{2}$ -decay for large solutions of (1.1).
Theorem 1.1.
For $λ = 0$ , let $(u, τ, H)$ be a global solution of ( 1.1 ) with the initial data $(u_{0}, τ_{0}, H_{0}) \in L^{p} (R^{2}) \cap L^{2} (R^{2})$ , $1 ⩽ p < 2$ , and assume that $tr τ_{0} ⩾ 0$ , $div u_{0} = 0$ , $div H_{0} = 0$ and ${(τ_{0})}_{i j} = {(τ_{0})}_{j i}$ . Then, there exists a positive constant C, depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ , such that for any $t ⩾ 0$ , $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1)} . \end{matrix}$
Remark 1.2.
The analysis in [8] relies heavily on the basic fact that, if $p = 1$ , the integral of ${(1 + t)}^{- \frac{3}{2}}$ is still a polynomial function in the three-dimensional case, we then get a better decay rate by means of an iterative scheme. In the two-dimensional case, the above proof fails because, one of the main new difficulties is the integral of ${(1 + t)}^{- 1}$ , which is an increasing logarithmic function. So we must make use of an auxiliary logarithmic time decay estimate.

Now, combining the uniform bounds for $(\nabla u, \nabla τ, \nabla H)$ in $H^{2} (R^{2})$ with the above result that $(u, τ, H)$ convergence in $L^{2} (R^{2})$ , and using the interpolation inequalities, we get the $H^{3}$ -decay for large solutions of (1.1).
Theorem 1.3.
Under the assumptions of Theorem 1.1 , assume further that for an arbitrary $C_{0} > 0$ , there exists a suitable large time $T_{0}$ , depending only on $C_{0}$ , such that $\begin{matrix} {‖ (\nabla u, \nabla τ, \nabla H) (T_{0}) ‖}_{H^{2} (R^{2})} ⩽ C_{0} . \end{matrix}$ Then, $(\nabla u, \nabla τ, \nabla H)$ belongs to $L^{\infty} ([T_{0}, + \infty); H^{2} (R^{2}))$ , and for any $t ⩾ T_{0}$ , $\begin{matrix} {‖ (\nabla^{k} u, \nabla^{k} τ, \nabla^{k} H) (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1) - \frac{k}{2}}, k = 1, 2, 3 . \end{matrix}$ Moreover, for any $t ⩾ T_{0}$ , $\begin{matrix} {‖ \nabla^{k} τ (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1) - \frac{k + 1}{2}}, k = 0, 1, 2, \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ and $C_{0}$ .

Next, we pay our attention to the lower bounds of time decay rates for large solutions of (1.1).
Theorem 1.4.
Under the assumptions of Theorem 1.1 with $p = 1$ , suppose further that there exists two nonzero constants $c_{0}$ and ${\tilde{c}}_{0}$ , such that $\begin{matrix} \int_{R^{2}} u_{0} (x) d x = c_{0}, \int_{R^{2}} H_{0} (x) d x = {\tilde{c}}_{0}, \end{matrix}$ and assume further that for an arbitrary $C_{1} > 0$ , there exists a suitable large time $T_{1}$ , depending only on $C_{1}$ , such that for any $t ⩽ T_{1}$ , $\begin{matrix} {‖ (\nabla u, \nabla τ, \nabla H) (t) ‖}_{H^{2} (R^{2})} ⩽ C_{1} . \end{matrix}$ Then, for large time t, $\begin{matrix} C^{- 1} {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}} ⩽ {‖ \nabla^{k} u (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}}, k = 0, 1, 2, 3, \\ C^{- 1} {(1 + t)}^{- 1} ⩽ {‖ τ (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- 1}, \\ C^{- 1} {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}} ⩽ {‖ \nabla^{k} H (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}}, k = 0, 1, 2, 3, \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ and $C_{1}$ .
Remark 1.5.
We might only require the boundedness of the norm $(\nabla u, \nabla τ, \nabla H)$ in $H^{2}$ , instead of the smallness of the norm $(\nabla u, \nabla τ, \nabla H)$ in $H^{2}$ . Our Theorems 1.3–1.4 not only relax the smallness conditions, but also give lower bounds of time decay rates for large solutions.
Remark 1.6.
Note that we use the equation ${(1.1)}_{2}$ to improve the decay rate of τ as well as $\nabla u$ . For the higher spatial derivatives of order k $(k = 1, 2, 3)$ of τ, however, we do not obtain their sharp time decay rates. The main difficulty is that Theorem 1.4 does not give the lower bounds of decay rates of $\nabla^{k} τ$ ( $k = 1, 2, 3$ ). Although the lower bound of $L^{2}$ -decay rate of the linear term $\nabla^{k} τ_{l}$ is ${(1 + t)}^{- 1 - \frac{k}{2}}$ , the $L^{2}$ -decay rate of the difference term $\nabla^{k} τ_{δ}$ obtained from Lemma 2.3 is equal to the $L^{2}$ -decay rate of $\nabla^{k} τ_{l}$ . This is the reason why we cannot derive the sharp time decay rates of the higher spatial derivatives of order k ( $k = 1, 2, 3$ ) of τ as well as u and H. One may refer to Section 5 for more details of the proof.

With the $H^{3}$ -decay for large solutions on hand, we keep our powder dry to investigate the global-in-time stability for large solutions of (1.1).
Theorem 1.7.
For $λ = 0$ , let $(\tilde{u}, \tilde{τ}, \tilde{H})$ be a global solution of ( 1.1 ) with the initial data $({\tilde{u}}_{0}, {\tilde{τ}}_{0}, {\tilde{H}}_{0})$ satisfying that for any $t ⩾ 0$ , $\begin{matrix} {‖ (\tilde{u}, \tilde{τ}, \tilde{H}) (t) ‖}_{H^{4} (R^{2})} ⩽ \tilde{C}, \end{matrix}$ and $({\tilde{u}}_{0}, {\tilde{τ}}_{0}, {\tilde{H}}_{0}) \in L^{p} (R^{2})$ , $1 ⩽ p < 2$ . Then there exists a positive constant $δ_{0}$ , depending on $\tilde{C}$ , such that for any $0 < δ ⩽ δ_{0}$ , if $\begin{matrix} {‖ (u_{0} - {\tilde{u}}_{0}, τ_{0} - {\tilde{τ}}_{0}, H_{0} - {\tilde{H}}_{0}) ‖}_{H^{3} (R^{2})} ⩽ δ, \end{matrix}$ then ( 1.1 ) admits a global unique solution $(u, τ, H)$ with initial data $(u_{0}, τ_{0}, H_{0})$ . Moreover, for any $t ⩾ 0$ , $\begin{matrix} {‖ (u - \tilde{u}, τ - \tilde{τ}, H - \tilde{H}) (t) ‖}_{H^{3} (R^{2})} ⩽ C min {{(1 + C_{2} ln | δ |)}^{- \frac{1}{p} + \frac{1}{2}}, δ + {(1 + t)}^{- \frac{1}{p} + \frac{1}{2}}}, \end{matrix}$ where $C_{2}$ is a positive constant independent of δ.
Remark 1.8.
The set of global large solutions is an open set, which is pointed out in Theorem 1.7. The question of whether or not the set of global large solutions is closed remains open. Besides, if we can prove that the set of global large solutions is a closed set, using the fact that an open and closed set is an empty set or a complete set, we may conclude immediately that the set of global large solutions is a complete set.

Finally, we investigate the general case where $λ \in [- 1, 1]$ .
Theorem 1.9.
For $λ \in [- 1, 1]$ , let $(u, τ, H)$ be a global solution for ( 1.1 ) in the space $C ((0, + \infty); H^{3} (R^{2}))$ with the initial data $(u_{0}, τ_{0}, H_{0}) \in L^{1} (R^{2})$ , $div u_{0} = 0$ , $div H_{0} = 0$ and ${(τ_{0})}_{i j} = {(τ_{0})}_{j i}$ . Moreover, we require that $τ \in L^{2} ((0, + \infty); L^{2} (R^{2}))$ . There then exists a suitable large time $T_{3}$ , such that for any $t ⩾ T_{3}$ , $\begin{matrix} {‖ (\nabla^{k} u, \nabla^{k} H) (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}}, k = 0, 1, 2, 3, \\ {‖ \nabla^{k} τ (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} - \frac{k + 1}{2}}, k = 0, 1, 2, {‖ \nabla^{3} τ (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- 2} . \end{matrix}$ Suppose further that $\int_{R^{2}} u_{0} (x) d x$ and $\int_{R^{2}} H_{0} (x) d x$ are nonzero, then for large time t, $\begin{matrix} {‖ (\nabla^{k} u, \nabla^{k} H) (t) ‖}_{L^{2} (R^{2})} ⩾ C^{- 1} {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}}, k = 0, 1, 2, 3, {‖ τ (t) ‖}_{L^{2} (R^{2})} ⩾ C^{- 1} {(1 + t)}^{- 1}, \end{matrix}$ where C is a positive constant independent of t.
Remark 1.10.
The difference between the proofs for Theorem 1.9 and Theorems 1.1–1.4 is the $L^{2}$ -decay estimates of large solutions for (1.1) (see Proposition 3.1 below for $λ = 0$ ). Actually, we cannot obtain the energy equality (3.3) of Proposition 3.1 directly in the case $λ \in [- 1, 1]$ . However, it is easy to check that for $λ \in [- 1, 1]$ , $\begin{matrix} (1.2) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2} (R^{2})}^{2} + {‖ (\nabla u, τ, \nabla H) ‖}_{L^{2} (R^{2})}^{2} \\ = \int_{R^{2}} (tr τ) | τ |^{2} d x + \int_{R^{2}} | \nabla u | | τ |^{2} d x \\ ⩽ C ‖ τ ‖_{L^{\infty} (R^{2})} (‖ τ ‖_{L^{2} (R^{2})}^{2} + ‖ \nabla u ‖_{L^{2} (R^{2})}^{2}) \\ ⩽ C ‖ τ ‖_{L^{2} (R^{2})}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2} (R^{2})}^{\frac{1}{2}} (‖ τ ‖_{L^{2} (R^{2})}^{2} + ‖ \nabla u ‖_{L^{2} (R^{2})}^{2}) . \end{matrix} \end{matrix}$ Since $τ \in C ((0, + \infty); H^{3} (R^{2})) \cap L^{2} ((0, + \infty); L^{2} (R^{2}))$ , it is easy to verify that ${lim}_{t \to + \infty} ‖ τ (t) ‖_{L^{2} (R^{2})} = 0$ . Thus, for any small constant $δ^{'} > 0$ , there exists a large time $T^{'}$ such that $‖ τ (t) ‖_{L^{2} (R^{2})} < δ^{'}$ for any $t ⩾ T^{'}$ , which, together with (1.2), one obtains that for any $t ⩾ T^{'}$ , $\begin{matrix} (1.3) & \begin{matrix} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2} (R^{2})}^{2} + {‖ (\nabla u, τ, \nabla H) ‖}_{L^{2} (R^{2})}^{2} ⩽ 0, \end{matrix} \end{matrix}$ provided that $δ^{'}$ is suitably small. With the help of the energy inequality (1.3), it then follows as a similar way to the proof of Proposition 3.1 that $‖ (u, τ, H) (t) ‖_{L^{2} (R^{2})} ⩽ C^{'} {(1 + t)}^{- \frac{1}{2}}$ for any $t ⩾ T^{'}$ , where $C^{'}$ is a positive constant depending on the initial data. Based on this $L^{2}$ -decay estimate, we then can obtain the other decay estimates with higher spatial derivatives for large solutions via the similar methods as the proof of the case where $λ = 0$ . We omit details here.
Remark 1.11.
In contrast with the case $λ = 0$ , we draw the conclusion without the assumption $tr τ_{0} ⩾ 0$ , the proofs are often simpler if we assume that $(u, τ, H)$ in $C ((0, + \infty); H^{3} (R^{2}))$ . Obviously, Theorem 1.7 is still true in a much more general case.
Remark 1.12.
Theorems 1.1–1.4 are valid for corresponding Oldroyd-B model when $b = 0$ .
Remark 1.13.
The special case which we consider in this paper is $f (x) = x$ . Under appropriate assumptions on f, we shall see that Theorems 1.1–1.4 hold for an arbitrary function f.

The other parts of this paper is organized as follows. We recall some useful estimates in Section 2. In Section 3, we apply ourselves to prove Theorem 1.1. The proof of Theorem 1.3 is given in Section 4. We use spectral analysis to prove Theorem 1.4 in Section 5. Section 6 is denoted to give the proof of Theorem 1.7.
2. Preliminaries

Firstly, we recall the well-known Gagliardo-Nirenberg inequalities, which will be useful in the following sections.

Lemma 2.1.
Let $p ⩾ 2$ , $q > 1$ and $r > 2$ , there exists a constant $C > 0$ , depending only on p, q and r, such that for any functions $f \in H^{1} (R^{2})$ , $g \in L^{q} (R^{2})$ and $\nabla g \in L^{r} (R^{2})$ , we get $\begin{matrix} ‖ f ‖_{L^{p} (R^{2})}^{p} ⩽ C ‖ f ‖_{L^{2} (R^{2})}^{2} ‖ \nabla f ‖_{L^{2} (R^{2})}^{p - 2}, \\ ‖ g ‖_{L^{\infty} (R^{2})} ⩽ C ‖ g ‖_{L^{q} (R^{2})}^{\frac{q (r - 2)}{2 r + q (r - 2)}} ‖ \nabla g ‖_{L^{r} (R^{2})}^{\frac{2 r}{2 r + q (r - 2)}} . \end{matrix}$

We now introduce the general Gagliardo-Nirenberg inequalities.
Lemma 2.2.
Let $m \in N_{+}$ , $k \in N_{+}$ , $p ⩾ 2$ , and satisfying $\begin{matrix} m > k, \frac{1}{2} - \frac{m - k}{2} ⩽ \frac{1}{p} ⩽ \frac{1}{2} . \end{matrix}$ Then, if $f \in L^{2} (R^{2})$ and $\nabla^{m} f \in L^{2} (R^{2})$ , we have $\nabla^{k} f \in L^{p} (R^{2})$ , and there exists a constant C, depending on m, k, p, such that $\begin{matrix} {‖ \nabla^{k} f ‖}_{L^{p} (R^{2})} ⩽ C ‖ f ‖_{L^{2} (R^{2})}^{θ} {‖ \nabla^{m} f ‖}_{L^{2} (R^{2})}^{1 - θ}, \end{matrix}$ where $\begin{matrix} \frac{1}{p} = θ (\frac{1}{2} + \frac{k}{2}) + (1 - θ) (\frac{1}{2} - \frac{m - k}{2}) . \end{matrix}$

The following lemma will be of frequent use in Section 5. Lemma 2.3.
For any $1 ⩽ r_{1} < r_{2} < + \infty$ , it holds that $\begin{matrix} \int_{T_{0}}^{t} {(1 + t - s)}^{- r_{1}} {(1 + s)}^{- r_{2}} d s ⩽ C {(1 + t)}^{- r_{1}}, \forall t ⩾ T_{0} . \end{matrix}$ In particular, we have $\begin{matrix} \int_{T_{0}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- 1} d s ⩽ C {(1 + t)}^{- 1} ln (1 + t), \forall t ⩾ T_{0}, \end{matrix}$ where C is a positive constant independent on t.
Proof.
For any $1 ⩽ r_{1} < r_{2} < + \infty$ , it is not difficult to check that $\begin{matrix} \int_{T_{0}}^{t} {(1 + t - s)}^{- r_{1}} {(1 + s)}^{- r_{2}} d s \\ = {(1 + \frac{t - T_{0}}{2})}^{- r_{1}} \int_{T_{0}}^{\frac{T_{0} + t}{2}} {(1 + s)}^{- r_{2}} d s + {(1 + \frac{T_{0} + t}{2})}^{- r_{2}} \int_{\frac{T_{0} + t}{2}}^{t} {(1 + t - s)}^{- r_{1}} d s \\ ⩽ C {(1 + t)}^{- r_{1}} . \end{matrix}$ Note that in the special case $r_{1} = r_{2} = 1$ , we have $\begin{matrix} \int_{T_{0}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- 1} d s \\ = {(1 + \frac{t - T_{0}}{2})}^{- 1} \int_{T_{0}}^{\frac{T_{0} + t}{2}} {(1 + s)}^{- 1} d s + {(1 + \frac{T_{0} + t}{2})}^{- 1} \int_{\frac{T_{0} + t}{2}}^{t} {(1 + t - s)}^{- 1} d s \\ ⩽ C {(1 + t)}^{- 1} ln (1 + t) . \end{matrix}$ The proof is therefore complete. □

3. Upper bounds of $L^{2}$ -decay for large solutions

The key idea of this section is first to prove the energy equality for large solutions of (1.1), and then using Fourier time-splitting method to show the convergence of large solutions in $L^{2} (R^{2})$ . We seek to find some special functions $g (t)$ for obtaining optimal time decay rates of large solutions by means of an iterative scheme. To the best our knowledge, the optimal time decay rates of heat equation in $L^{2} (R^{2})$ is ${(1 + t)}^{- \frac{1}{2}}$ , and $\int_{0}^{t} {(1 + s)}^{- 1} d s = ln (1 + t)$ is an increasing function, so we directly apply iterative scheme clearly fails in two dimensions. In order to overcome the difficulty, we may use the logarithmic decay estimates.

Proposition 3.1.
Let $(u, τ, H)$ be a global smooth solution of ( 1.1 ) with the initial data $(u_{0}, τ_{0}, H_{0}) \in L^{p} (R^{2}) \cap L^{2} (R^{2})$ , $1 ⩽ p < 2$ . Then we have $\begin{matrix} (3.1) & {‖ (u, τ, H) (t) ‖}_{L^{2} (R^{2})}^{2} ⩽ C {(1 + t)}^{- (\frac{2}{p} - 1)}, \forall t ⩾ 0 . \end{matrix}$ Moreover, we have $\begin{matrix} (3.2) & {(1 + t)}^{- \frac{1}{2}} \int_{0}^{t} {(1 + s)}^{\frac{2}{p} - \frac{1}{2}} {‖ (\nabla u, τ, \nabla H) (s) ‖}_{L^{2} (R^{2})}^{2} d s ⩽ C, \forall t ⩾ 0, \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ .
Proof.
We multiply equation ${(1.1)}_{1}$ by u, equation ${(1.1)}_{2}$ by τ, equation ${(1.1)}_{3}$ by H, respectively, and integrate by parts to discover $\begin{matrix} (3.3) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + {‖ (\nabla u, τ, \nabla H) ‖}_{L^{2}}^{2} + \int (tr τ) | τ |^{2} d x = 0 . \end{matrix} \end{matrix}$ Let $X (t, x)$ be the position of x at time t such that $\begin{matrix} X_{t} (t, x) = u (t, X (t, x)), (t, x) \in R^{+} \times R^{2}, X (0, x) = x, x \in R^{2} . \end{matrix}$ Applying the operator tr to ${(1.1)}_{2}$ yields $\begin{matrix} \frac{d}{d t} tr τ (t, X (t, x)) + a tr τ (t, X (t, x)) + b {(tr τ)}^{2} (t, X (t, x)) = 0, \end{matrix}$ then we get $\begin{matrix} tr τ (t, X (t, x)) = \frac{a tr τ_{0} (x)}{b tr τ_{0} (x) (e^{a t} - 1) + a e^{a t}} ⩾ 0, \end{matrix}$ owing to $a > 0$ , $b ⩾ 0$ and $tr τ_{0} (x) ⩾ 0$ . Hence, we integrate from 0 to t to find $\begin{matrix} (3.4) & \begin{matrix} sup_{0 ⩽ s ⩽ t} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} + \int_{0}^{t} {‖ (\nabla u, τ, \nabla H) (s) ‖}_{L^{2}}^{2} d s ⩽ C, \forall t ⩾ 0, \end{matrix} \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ .

Now, setting $S (t) : = {ξ | | ξ | ⩽ C g (t)}$ , with $g (t)$ satisfying $g (t) ⩽ C {(1 + t)}^{- \frac{1}{2}}$ . We take the Fourier transform of (1.1), multiply the first equation by $\overline{\hat{u}}$ , the second by $\overline{\hat{τ}}$ , and the third by $\overline{\hat{H}}$ , respectively, to discover $\begin{matrix} (3.5) & \begin{matrix} \frac{1}{2} \int_{S (t)} ({| \hat{u} (t, ξ) |}^{2} + {| \hat{τ} (t, ξ) |}^{2} + {| \hat{H} (t, ξ) |}^{2}) d ξ \\ + \int_{0}^{t} \int_{S (t)} (| ξ |^{2} {| \hat{u} (s, ξ) |}^{2} + {| \hat{τ} (s, ξ) |}^{2} + | ξ |^{2} {| \hat{H} (s, ξ) |}^{2}) d ξ d s \\ = \frac{1}{2} \int_{S (t)} ({| \hat{u} (0, ξ) |}^{2} + {| \hat{τ} (0, ξ) |}^{2} + {| \hat{H} (0, ξ) |}^{2}) d ξ \\ - Re \int_{0}^{t} \int_{S (t)} \hat{(u \cdot \nabla u + H \cdot \nabla H - \nabla \frac{| H |^{2}}{2})} \cdot \overline{\hat{u}} (s) d ξ d s \\ - Re \int_{0}^{t} \int_{S (t)} \hat{(u \cdot \nabla τ + (tr τ) τ + Q (τ, \nabla u))} \cdot \overline{\hat{τ}} (s) d ξ d s \\ + Re \int_{0}^{t} \int_{S (t)} \hat{(H \cdot \nabla u - u \cdot \nabla H)} \cdot \overline{\hat{H}} (s) d ξ d s \\ - Re \int_{0}^{t} \int_{S (t)} (\hat{div τ} \cdot \overline{\hat{u}} + \hat{D (u)} \cdot \overline{\hat{τ}}) (s) d ξ d s - Re \int_{0}^{t} \int_{S (t)} \hat{\nabla p} \cdot \overline{\hat{u}} (s) d ξ d s . \end{matrix} \end{matrix}$ We find that $\begin{matrix} Re (\hat{div τ} \cdot \overline{\hat{u}} + \hat{D (u)} \cdot \overline{\hat{τ}}) = Re (i ξ_{i} {\hat{τ}}_{i j} {\overline{\hat{u}}}_{j} + i ξ_{i} {\hat{u}}_{j} {\overline{\hat{τ}}}_{i j}) = 0, \end{matrix}$ since $τ_{i j} = τ_{j i}$ . And the imcompressible condition ensures that $\begin{matrix} \hat{\nabla p} \cdot \overline{\hat{u}} = i \hat{p} ξ \cdot \overline{\hat{u}} = \hat{p} \overline{\hat{div u}} = 0 . \end{matrix}$ Because the initial data $(u_{0}, τ_{0}, H_{0}) \in L^{p}$ , invoking Hölder’s inequality, we then compute $\begin{matrix} \frac{1}{2} \int_{S (t)} ({| \hat{u} (0, ξ) |}^{2} + {| \hat{τ} (0, ξ) |}^{2} + {| \hat{H} (0, ξ) |}^{2}) d ξ ⩽ C g^{2 (\frac{2}{p} - 1)} (t) (‖ u_{0} ‖_{L^{p}}^{2} + ‖ τ_{0} ‖_{L^{p}}^{2} + ‖ H_{0} ‖_{L^{p}}^{2}), \end{matrix}$ with $1 ⩽ p < 2$ . Cauchy’s inequality implies that $\begin{array}{l} Re \int_{0}^{t} \int_{S (t)} \hat{(u \cdot \nabla u + H \cdot \nabla H - \nabla \frac{| H |^{2}}{2})} \cdot \overline{\hat{u}} (s) d ξ d s \\ ⩽ \int_{0}^{t} \int_{S (t)} | ξ | | \hat{(u \otimes u + H \otimes H)} | | \hat{u} | d ξ d s \\ ⩽ ϵ \int_{0}^{t} \int_{S (t)} | ξ |^{2} | \hat{u} |^{2} d ξ d s + C_{ϵ} \int_{0}^{t} \int_{S (t)} {| \hat{(u \otimes u + H \otimes H)} |}^{2} d ξ d s \\ ⩽ ϵ \int_{0}^{t} \int_{S (t)} | ξ |^{2} | \hat{u} |^{2} d ξ d s + C_{ϵ} g^{2} (t) \int_{0}^{t} (‖ u ‖_{L^{2}}^{4} + ‖ H ‖_{L^{2}}^{4}) d s . \end{array}$ Similar to the above estimate, we have $\begin{array}{l} Re \int_{0}^{t} \int_{S (t)} \hat{(H \cdot \nabla u - u \cdot \nabla H)} \cdot \overline{\hat{H}} (s) d ξ d s \\ ⩽ ϵ \int_{0}^{t} \int_{S (t)} | ξ |^{2} | \hat{H} |^{2} d ξ d s + C_{ϵ} \int_{0}^{t} \int_{S (t)} | \hat{u \otimes H} |^{2} d ξ d s \\ ⩽ ϵ \int_{0}^{t} \int_{S (t)} | ξ |^{2} | \hat{H} |^{2} d ξ d s + C_{ϵ} g^{2} (t) \int_{0}^{t} (‖ u ‖_{L^{2}}^{4} + ‖ H ‖_{L^{2}}^{4}) d s, \\ Re \int_{0}^{t} \int_{S (t)} \hat{(u \cdot \nabla τ + (tr τ) τ + Q (τ, \nabla u))} \cdot \overline{\hat{τ}} (s) d ξ d s \\ ⩽ ϵ \int_{0}^{t} \int_{S (t)} | \hat{τ} |^{2} d ξ d s + C_{ϵ} \int_{0}^{t} \int_{S (t)} (| ξ |^{2} | \hat{u \otimes τ} |^{2} + {| \hat{(tr τ) τ} |}^{2} + {| \hat{Q (τ, \nabla u)} |}^{2} d ξ) d s \\ ⩽ ϵ \int_{0}^{t} \int_{S (t)} | \hat{τ} |^{2} d ξ d s + C_{ϵ} g^{4} (t) \int_{0}^{t} ‖ u ‖_{L^{2}}^{2} ‖ τ ‖_{L^{2}}^{2} d s + C_{ϵ} g^{2} (t) \int_{0}^{t} ‖ τ ‖_{L^{2}}^{4} d s \\ + C_{ϵ} g^{2} (t) \int_{0}^{t} ‖ \nabla u ‖_{L^{2}}^{2} ‖ τ ‖_{L^{2}}^{2} d s . \end{array}$ Returning to (3.5), we employ the inequalities above and choose ϵ small enough to find $\begin{matrix} (3.6) & \begin{matrix} \frac{1}{2} \int_{S (t)} ({| \hat{u} (t, ξ) |}^{2} + {| \hat{τ} (t, ξ) |}^{2} + {| \hat{H} (t, ξ) |}^{2}) d ξ \\ + \int_{0}^{t} \int_{S (t)} (| ξ |^{2} {| \hat{u} (s, ξ) |}^{2} + {| \hat{τ} (s, ξ) |}^{2} + | ξ |^{2} {| \hat{H} (s, ξ) |}^{2}) d ξ d s \\ ⩽ C g^{2 (\frac{2}{p} - 1)} (t) + C g^{2} (t) \int_{0}^{t} ({‖ (u, τ, H) ‖}_{L^{2}}^{4} + ‖ \nabla u ‖_{L^{2}}^{2} ‖ τ ‖_{L^{2}}^{2}) (s) d s, 1 ⩽ p < 2 . \end{matrix} \end{matrix}$ Consequently, (3.4) yields the inequality $\begin{matrix} \int_{0}^{t} ({‖ (u, τ, H) ‖}_{L^{2}}^{4} + ‖ \nabla u ‖_{L^{2}}^{2} ‖ τ ‖_{L^{2}}^{2}) (s) d s ⩽ C (e + t) . \end{matrix}$ Substituting the above inequality into (3.6) yields $\begin{array}{l} \frac{1}{2} \int_{S (t)} ({| \hat{u} (t, ξ) |}^{2} + {| \hat{τ} (t, ξ) |}^{2} + {| \hat{H} (t, ξ) |}^{2}) d ξ \\ + \int_{0}^{t} \int_{S (t)} (| ξ |^{2} {| \hat{u} (s, ξ) |}^{2} + {| \hat{τ} (s, ξ) |}^{2} + | ξ |^{2} {| \hat{H} (s, ξ) |}^{2}) d ξ d s \\ ⩽ C g^{2 (\frac{2}{p} - 1)} (t) + C g^{2} (t) (e + t) . \end{array}$ Returning once more to inequalities (3.3), we obtain $\begin{matrix} (3.7) & \begin{matrix} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + g^{2} (t) {‖ (u, τ, H) ‖}_{L^{2}}^{2} \\ ⩽ g^{2} (t) \int_{S (t)} ({| \hat{u} (t, ξ) |}^{2} + {| \hat{τ} (t, ξ) |}^{2} + {| \hat{H} (t, ξ) |}^{2}) d ξ \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) (e + t) . \end{matrix} \end{matrix}$ Now, taking $g^{2} (t) = \frac{α}{(e + t) ln (e + t)}$ (α will be determined later), we claim that $\begin{matrix} (3.8) & g^{\frac{4}{p}} (t) ⩽ (e + t) g^{4} (t) {ln}^{- β} (e + t), \forall β \in R . \end{matrix}$ Indeed, $\begin{matrix} g^{\frac{4}{p}} (t) = & α^{\frac{2}{p}} {(e + t)}^{- \frac{2}{p}} {ln}^{- \frac{2}{p}} (e + t) \\ = & {(e + t)}^{1 - \frac{2}{p}} {ln}^{2 + β - \frac{2}{p}} (e + t) α^{\frac{2}{p} - 2} (e + t) g^{4} (t) {ln}^{- β} (e + t) \\ ⩽ & C α^{\frac{2}{p} - 2} (e + t) g^{4} (t) {ln}^{- β} (e + t), \end{matrix}$ owing to $1 ⩽ p < 2$ . Choosing $α = 2$ , (3.7) and (3.8) then say $\begin{matrix} \frac{d}{d t} ({ln}^{2} (e + t) {‖ (u, τ, H) ‖}_{L^{2}}^{2}) ⩽ C g^{4} (t) (e + t) {ln}^{2} (e + t) ⩽ \frac{C}{e + t}, \end{matrix}$ that is, $\begin{matrix} (3.9) & \begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ C {ln}^{- 1} (e + t), \forall t ⩾ 0 . \end{matrix} \end{matrix}$ Substituting (3.9) into (3.6), and following the similar argument used in the previous proof, for any $0 < α < 1$ , we see that $\begin{array}{l} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + g^{2} (t) {‖ (u, τ, H) ‖}_{L^{2}}^{2} \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) (\int_{0}^{t} {ln}^{- 2} (e + s) d s + 1) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) ({(e + t)}^{α} \int_{0}^{t} {(e + t)}^{- α} {ln}^{- 2} (e + s) d s + 1) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) ({(e + t)}^{α} \int_{0}^{t} {(e + s)}^{- α} {ln}^{- 2} (e + s) ((1 - α) ln (e + s) - 1) d s + 1) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) {(e + t)}^{α} ({(e + s)}^{1 - α} {ln}^{- 1} (e + s) |_{e^{2}}^{t} + 1) \\ (3.10) & ⩽ C (e + t) g^{4} (t) {ln}^{- 1} (e + t), \end{array}$ since (3.8).

Choosing $α = 3$ , and multiply (3.10) by ${ln}^{3} (e + t)$ , we then find $\begin{matrix} \frac{d}{d t} ({ln}^{3} (e + t) {‖ (u, τ, H) ‖}_{L^{2}}^{2}) ⩽ \frac{C}{e + t}, \end{matrix}$ which implies that $\begin{matrix} (3.11) & {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ C {ln}^{- 2} (e + t), \forall t ⩾ 0 . \end{matrix}$ Inserting (3.11) into (3.6), and repeating the same process in the above again, for any $0 < α < \frac{2}{p} - 1$ , we readily check $\begin{matrix} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + g^{2} (t) {‖ (u, τ, H) ‖}_{L^{2}}^{2} \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) sup_{0 ⩽ s ⩽ t} {(e + s)}^{α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} \\ \times (\int_{0}^{t} {(e + s)}^{- α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} d s + \int_{0}^{t} {‖ \nabla u (s) ‖}_{L^{2}}^{2} d s) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) sup_{0 ⩽ s ⩽ t} {(e + s)}^{α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} (\int_{0}^{t} {(e + s)}^{- α} {ln}^{- 2} (e + s) d s + 1) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) sup_{0 ⩽ s ⩽ t} {(e + s)}^{α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} \\ \times (\int_{0}^{t} {(e + s)}^{- α} {ln}^{- 2} (e + s) ((1 - α) {ln}^{α} (e + s) - (2 - α) {ln}^{- 1 + α} (e + s)) d s + 1) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) sup_{0 ⩽ s ⩽ t} {(e + s)}^{α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} ({(e + s)}^{1 - α} {ln}^{- 2 + α} (e + s) |_{0}^{t} + 1) \\ ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) {(e + t)}^{1 - α} {ln}^{- 2 + α} (e + t) sup_{0 ⩽ s ⩽ t} {(e + s)}^{α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} . \end{matrix}$ Taking $g^{2} (t) = \frac{α}{e + t}$ , we see that $\begin{matrix} (3.12) & \begin{matrix} \frac{d}{d t} ({(e + t)}^{α} {‖ (u, τ, H) ‖}_{L^{2}}^{2}) \\ ⩽ C {(e + t)}^{α - \frac{2}{p}} + C {(e + t)}^{- 1} {ln}^{- 2 + α} (e + t) sup_{0 ⩽ s ⩽ t} {(e + s)}^{α} {‖ (u, τ, H) (s) ‖}_{L^{2}}^{2} . \end{matrix} \end{matrix}$ Let $X (t) : = {(e + t)}^{α} ‖ (u, τ, H) (t) ‖_{L^{2}}^{2}$ , $Y (t) : = {sup}_{0 ⩽ s ⩽ t} X (s)$ , thus (3.12) implies $\begin{matrix} X (t) ⩽ C (1 + \int_{0}^{t} {(e + s)}^{- 1} {ln}^{- 2 + α} (e + s) Y (s) d s), \end{matrix}$ since $α < \frac{2}{p} - 1$ . By virtue of the integral form of Gronwall’s inequality, we get $\begin{matrix} Y (t) ⩽ C, \forall t ⩾ 0 . \end{matrix}$ Hence, $\begin{matrix} (3.13) & \begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ C {(e + t)}^{- α}, \forall t ⩾ 0 . \end{matrix} \end{matrix}$ We first investigate the case $1 < p < 2$ , choosing $α = \frac{1}{p} - \frac{1}{2} \in (0, \frac{2}{p} - 1)$ , we then get $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ C {(e + t)}^{- \frac{1}{p} + \frac{1}{2}}, \forall t ⩾ 0 . \end{matrix}$ Consequently, $\begin{matrix} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + g^{2} (t) {‖ (u, τ, H) ‖}_{L^{2}}^{2} ⩽ C g^{\frac{4}{p}} (t) + C g^{4} (t) {(e + t)}^{- \frac{2}{p} + 2} . \end{matrix}$ Taking $g^{2} (t) = \frac{\frac{2}{p} - \frac{1}{2}}{e + t}$ yields the estimate $\begin{matrix} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + \frac{\frac{2}{p} - \frac{1}{2}}{e + t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} ⩽ C {(e + t)}^{- \frac{2}{p}} . \end{matrix}$ Multiplying the above inequality by ${(e + t)}^{\frac{2}{p} - \frac{1}{2}}$ , we deduce $\begin{matrix} \frac{d}{d t} ({(e + t)}^{\frac{2}{p} - \frac{1}{2}} {‖ (u, τ, H) ‖}_{L^{2}}^{2}) ⩽ C {(e + t)}^{- \frac{1}{2}}, \end{matrix}$ it follows that $\begin{matrix} (3.14) & \begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ C {(e + t)}^{- \frac{2}{p} + 1}, \forall t ⩾ 0 . \end{matrix} \end{matrix}$ Now consider the case $p = 1$ , choosing $α = \frac{2}{3} \in (0, 1)$ , we then have $\begin{matrix} (3.15) & \begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ C {(e + t)}^{- \frac{2}{3}}, \forall t ⩾ 0 . \end{matrix} \end{matrix}$ Therefore, $\begin{matrix} \frac{d}{d t} {‖ (u, τ, H) ‖}_{L^{2}}^{2} + g^{2} (t) {‖ (u, τ, H) ‖}_{L^{2}}^{2} ⩽ C g^{4} (t) . \end{matrix}$ Taking $g^{2} (t) = \frac{\frac{3}{2}}{e + t}$ in this case, we conclude the estimate $\begin{matrix} \frac{d}{d t} ({(e + t)}^{\frac{3}{2}} {‖ (u, τ, H) ‖}_{L^{2}}^{2}) ⩽ C {(1 + t)}^{- \frac{1}{2}}, \end{matrix}$ it follows that $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{L^{2}}^{2} ⩽ {(e + t)}^{- 1}, \forall t ⩾ 0 . \end{matrix}$ This inequality and (3.14) imply (3.1), it is then easy to deduce (3.2). This completes the proof of the proposition. □

4. Upper bounds of $H^{3}$ -decay for large solutions

According to Proposition 3.1, it suffices to prove that the time decay rates for high order derivatives of $(u, τ, H)$ . Moreover, we will show that the rates is optimal in this section.

Proposition 4.1.
Let $(u, τ, H)$ be a global smooth solution of ( 1.1 ) with the initial data $(u_{0}, τ_{0}, H_{0}) \in L^{p} (R^{2}) \cap L^{2} (R^{2})$ , $1 ⩽ p < 2$ . Furthermore, suppose that for an arbitrary $C_{0} > 0$ , there exists a suitable large time $T_{0}$ , depending on $C_{0}$ , such that $\begin{matrix} {‖ (\nabla u, \nabla τ, \nabla H) (T_{0}) ‖}_{H^{2} (R^{2})} ⩽ C_{0} . \end{matrix}$ Then $(\nabla u, \nabla τ, \nabla H) \in L^{\infty} ([T_{0}, + \infty); H^{2} (R^{2}))$ , we have $\begin{matrix} (4.1) & {‖ (\nabla^{k} u, \nabla^{k} τ, \nabla^{k} H) (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1) - \frac{k}{2}}, \forall t ⩾ T_{0}, k = 1, 2, 3, \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ and $C_{0}$ .
Proof.
At this stage, we use a bootstrap argument to prove this proposition. Now, we assume that following estimate holds: $\begin{matrix} (4.2) & \begin{matrix} {‖ (\nabla u, \nabla τ, \nabla H) (t) ‖}_{H^{2}} ⩽ C_{0}, \forall t ⩾ T_{0} . \end{matrix} \end{matrix}$ Applying ∇ to (1.1), multiplying the first equation by $\nabla u$ , the second by $\nabla τ$ , the third by $\nabla H$ , and integrating over $R^{2}$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ (\nabla u, \nabla τ, \nabla H) ‖}_{L^{2}}^{2} + {‖ (\nabla^{2} u, \nabla τ, \nabla^{2} H) ‖}_{L^{2}}^{2} \\ = \int (\nabla div τ \cdot \nabla u + \nabla D (u) \cdot \nabla τ) d x - \int \nabla (u \cdot \nabla u) \cdot \nabla u d x - \int \nabla ((tr τ) τ) \nabla τ d x \\ - \int \nabla ((u \cdot \nabla τ) + Q (τ, \nabla u)) \cdot \nabla τ d x \\ + \int (\nabla (H \cdot \nabla H) \cdot \nabla u + \nabla (H \cdot \nabla u) \cdot \nabla H) d x \\ (4.3) & - \int (\nabla (u \cdot \nabla H) \cdot \nabla H + \nabla^{2} \frac{| H |^{2}}{2} \cdot \nabla u) d x . \end{array}$ The fact $τ_{i j} = τ_{j i}$ ensures that $\begin{matrix} \int (\nabla div τ \cdot \nabla u + \nabla D (u) \cdot \nabla τ) d x = 0 . \end{matrix}$ And $div u = 0$ leads to the following identity $\begin{matrix} \int \nabla (u \cdot \nabla u) \cdot \nabla u d x = 0 . \end{matrix}$ Using integration by parts and the Gagliardo-Nirenberg inequality, we get $\begin{matrix} \int \nabla ((tr τ) τ) \nabla τ d x ⩽ C ‖ τ ‖_{L^{\infty}} ‖ \nabla τ ‖_{L^{2}}^{2} ⩽ C ‖ τ ‖_{L^{4}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{4}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{2} ⩽ C ‖ τ ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{5}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{4}} . \end{matrix}$ By using the Gagliardo-Nirenberg inequality, Sobolev interpolation inequality and the divergence-free condition, the fourth term of the right-hand side of (4.3) can be controlled as follows: $\begin{matrix} \int \nabla ((u \cdot \nabla τ) + Q (τ, \nabla u)) \cdot \nabla τ d x \\ ⩽ C ‖ \nabla u ‖_{L^{4}} ‖ \nabla τ ‖_{L^{4}} ‖ \nabla τ ‖_{L^{2}} + C {‖ \nabla^{2} u ‖}_{L^{2}} ‖ τ ‖_{L^{\infty}} ‖ \nabla τ ‖_{L^{2}} \\ ⩽ C ‖ u ‖_{L^{2}}^{\frac{1}{4}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{3}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{3}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{2}} + C ‖ τ ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{4}} ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + ‖ \nabla τ ‖_{L^{2}}^{2}) \\ ⩽ ϵ ‖ \nabla τ ‖_{L^{2}}^{2} + C_{ϵ} ‖ u ‖_{L^{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{2} {‖ \nabla^{2} u ‖}_{L^{2}}^{3} + C ‖ τ ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{4}} ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + ‖ \nabla τ ‖_{L^{2}}^{2}) . \end{matrix}$ The last two of the right-hand side of (4.3) can be computed by using $div u = div H = 0$ and Lemma 2.2 as follows: $\begin{matrix} \int (\nabla (H \cdot \nabla H) \cdot \nabla u + \nabla (H \cdot \nabla u) \cdot \nabla H) d x - \int (\nabla (u \cdot \nabla H) \cdot \nabla H + \nabla^{2} \frac{| H |^{2}}{2} \cdot \nabla u) d x \\ = \int (\partial_{i} H^{j} \partial_{j} H^{k} \partial_{i} u^{k} + \partial_{i} H^{j} \partial_{i} H^{k} \partial_{j} u^{k} + \partial_{i} H^{j} \partial_{k} H^{j} \partial_{i} u^{k}) d x \\ ⩽ C ‖ \nabla u ‖_{L^{2}} ‖ \nabla H ‖_{L^{4}}^{2} \\ ⩽ C ‖ u ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ H ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} H ‖}_{L^{2}}^{\frac{3}{2}} \\ ⩽ C ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ H ‖_{L^{2}}^{\frac{1}{2}} ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + {‖ \nabla^{2} H ‖}_{L^{2}}^{2}) . \end{matrix}$ Utilizing the above calculations in equality (4.3), and recalling the inequality (3.1) and (4.2), we then obtain that for $t ⩾ T_{0}$ , $\begin{matrix} (4.4) & \begin{matrix} \frac{d}{d t} {‖ (\nabla u, \nabla τ, \nabla H) ‖}_{L^{2}}^{2} + {‖ (\nabla^{2} u, \nabla τ, \nabla^{2} H) ‖}_{L^{2}}^{2} ⩽ 0, \end{matrix} \end{matrix}$ if ϵ is small enough and $T_{0}$ is large enough. Setting $S (t) : = {ξ | | ξ | ⩽ C {(1 + t)}^{- \frac{1}{2}}}$ , we compute $\begin{matrix} {‖ (\nabla^{2} u, \nabla^{2} H) ‖}_{L^{2}}^{2} ⩾ & \int_{R^{2} / S (t)} | ξ |^{4} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ \\ ⩾ & \frac{C}{1 + t} \int_{R^{2}} | ξ |^{2} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ - \frac{C}{{(1 + t)}^{2}} \int_{S (t)} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ . \end{matrix}$ Hence, for any $t ⩾ T_{0}$ , $\begin{matrix} \frac{d}{d t} {‖ (\nabla u, \nabla τ, \nabla H) ‖}_{L^{2}}^{2} + \frac{C}{1 + t} {‖ (\nabla u, \nabla τ, \nabla H) ‖}_{L^{2}}^{2} \\ ⩽ \frac{C}{{(1 + t)}^{2}} \int_{S (t)} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ ⩽ C {(1 + t)}^{- \frac{2}{p} - 1} . \end{matrix}$ Consequently, this estimate implies finally for any $t ⩾ T_{0}$ , $\begin{matrix} (4.5) & \begin{matrix} {‖ (\nabla u, \nabla τ, \nabla H) (t) ‖}_{L^{2}}^{2} \\ ⩽ C {(1 + t)}^{- \frac{2}{p} - \frac{1}{2}} ({(1 + T_{0})}^{\frac{2}{p} + \frac{1}{2}} {‖ (\nabla u, \nabla τ, \nabla H) (T_{0}) ‖}_{L^{2}}^{2} - {(1 + T_{0})}^{\frac{1}{2}} + {(1 + t)}^{\frac{1}{2}}) \\ ⩽ C {(1 + t)}^{- \frac{2}{p}}, \end{matrix} \end{matrix}$ and $\begin{matrix} (4.6) & \begin{matrix} \int_{T_{0}}^{t} {(1 + s)}^{\frac{2}{p} + \frac{1}{2}} {‖ (\nabla^{2} u, \nabla τ, \nabla^{2} H) (s) ‖}_{L^{2}}^{2} d s ⩽ C {(1 + t)}^{\frac{1}{2}} . \end{matrix} \end{matrix}$ From similar arguments, we observe $\begin{matrix} (4.7) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ (\nabla^{2} u, \nabla^{2} τ, \nabla^{2} H) ‖}_{L^{2}}^{2} + {‖ (\nabla^{3} u, \nabla^{2} τ, \nabla^{3} H) ‖}_{L^{2}}^{2} \\ = \int (\nabla^{2} div τ \cdot \nabla^{2} u + \nabla^{2} D (u) \cdot \nabla^{2} τ) d x - \int \nabla^{2} (u \cdot \nabla u) \cdot \nabla^{2} u d x \\ - \int \nabla^{2} ((tr τ) τ) \nabla^{2} τ d x - \int \nabla^{2} ((u \cdot \nabla τ) + Q (τ, \nabla u)) \cdot \nabla^{2} τ d x \\ + \int (\nabla^{2} (H \cdot \nabla H) \cdot \nabla^{2} u + \nabla^{2} (H \cdot \nabla u) \cdot \nabla^{2} H) d x \\ - \int (\nabla^{2} (u \cdot \nabla H) \cdot \nabla^{2} H + \nabla^{3} \frac{| H^{2} |}{2} \cdot \nabla^{2} u) d x : = \sum_{i = 1}^{6} I_{i} . \end{matrix} \end{matrix}$ The fact that $τ_{i j} = τ_{j i}$ implies $I_{1} = 0$ . By integration by parts and Sobolev inequality, we have $\begin{array}{l} I_{2} ⩽ C ‖ \nabla u ‖_{L^{2}} {‖ \nabla^{2} u ‖}_{L^{4}}^{2} ⩽ C ‖ \nabla u ‖_{L^{2}} {‖ \nabla^{2} u ‖}_{L^{2}} {‖ \nabla^{3} u ‖}_{L^{2}} ⩽ ϵ {‖ \nabla^{3} u ‖}_{L^{2}}^{2} + C_{ϵ} ‖ \nabla u ‖_{L^{2}}^{2} {‖ \nabla^{2} u ‖}_{L^{2}}^{2}, \\ I_{3} ⩽ C (‖ τ ‖_{L^{\infty}} {‖ \nabla^{2} τ ‖}_{L^{2}} + ‖ \nabla τ ‖_{L^{4}}^{2}) {‖ \nabla^{2} τ ‖}_{L^{2}} ⩽ C (‖ τ ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{4}} + ‖ \nabla τ ‖_{L^{2}}) {‖ \nabla^{2} τ ‖}_{L^{2}}^{2}, \\ I_{4} ⩽ C (‖ \nabla u ‖_{L^{\infty}} {‖ \nabla^{2} τ ‖}_{L^{2}} + {‖ \nabla^{2} u ‖}_{L^{4}} ‖ \nabla τ ‖_{L^{4}} + {‖ \nabla^{3} u ‖}_{L^{2}} ‖ τ ‖_{L^{\infty}}) {‖ \nabla^{2} τ ‖}_{L^{2}} \\ ⩽ C (‖ \nabla u ‖_{L^{2}}^{\frac{1}{4}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{3} u ‖}_{L^{2}}^{\frac{1}{4}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{2} + {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{3} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{3}{2}} \\ + {‖ \nabla^{3} u ‖}_{L^{2}} ‖ τ ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{5}{4}}) \\ ⩽ C ({‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} + ‖ τ ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{4}} + ‖ \nabla u ‖_{L^{2}}^{\frac{1}{4}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{3} u ‖}_{L^{2}}^{\frac{1}{4}}) \\ \times ({‖ \nabla^{2} τ ‖}_{L^{2}}^{2} + {‖ \nabla^{3} u ‖}_{L^{2}}^{2}) \\ I_{5} + I_{6} ⩽ C (‖ \nabla u ‖_{L^{2}} {‖ \nabla^{2} H ‖}_{L^{4}}^{2} + ‖ \nabla H ‖_{L^{2}} {‖ \nabla^{2} u ‖}_{L^{4}} {‖ \nabla^{2} H ‖}_{L^{4}}) \\ ⩽ C (‖ \nabla u ‖_{L^{2}} {‖ \nabla^{2} H ‖}_{L^{2}} {‖ \nabla^{3} H ‖}_{L^{2}} + ‖ \nabla H ‖_{L^{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{3} u ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} H ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{3} H ‖}_{L^{2}}^{\frac{1}{2}}) \\ ⩽ ϵ ({‖ \nabla^{3} u ‖}_{L^{2}}^{2} + {‖ \nabla^{3} H ‖}_{L^{2}}^{2}) + C_{ϵ} (‖ \nabla u ‖_{L^{2}}^{2} {‖ \nabla^{2} H ‖}_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2} {‖ \nabla^{2} u ‖}_{L^{2}} {‖ \nabla^{2} H ‖}_{L^{2}}) . \end{array}$ Substituting the above estimates into (4.7), choosing ϵ small enough and take advantage of (4.2), (3.1) and (4.5), we conclude that for $t ⩾ T_{0}$ , $\begin{matrix} (4.8) & \begin{matrix} \frac{d}{d t} {‖ (\nabla^{2} u, \nabla^{2} τ, \nabla^{2} H) ‖}_{L^{2}}^{2} + {‖ (\nabla^{3} u, \nabla^{2} τ, \nabla^{3} H) ‖}_{L^{2}}^{2} \\ ⩽ C (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2}) ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + {‖ \nabla^{2} H ‖}_{L^{2}}^{2}) . \end{matrix} \end{matrix}$ Setting $S (t) : = {ξ | | ξ | ⩽ C {(1 + t)}^{- \frac{1}{2}}}$ , we also compute $\begin{matrix} {‖ (\nabla^{3} u, \nabla^{3} H) ‖}_{L^{2}}^{2} ⩾ & \int_{R^{2} / S (t)} | ξ |^{6} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ \\ ⩾ & \frac{C}{1 + t} \int_{R^{2}} | ξ |^{4} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ - \frac{C}{{(1 + t)}^{3}} \int_{S (t)} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ . \end{matrix}$ Hence, for any $t ⩾ T_{0}$ , $\begin{matrix} (4.9) & \begin{matrix} \frac{d}{d t} {‖ (\nabla^{2} u, \nabla^{2} τ, \nabla^{2} H) ‖}_{L^{2}}^{2} + \frac{C}{1 + t} {‖ (\nabla^{2} u, \nabla^{2} τ, \nabla^{2} H) ‖}_{L^{2}}^{2} \\ ⩽ \frac{C}{{(1 + t)}^{3}} \int_{S (t)} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ + C (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2}) ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + {‖ \nabla^{2} H ‖}_{L^{2}}^{2}) \\ ⩽ C {(1 + t)}^{- \frac{2}{p} - 2} + C (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2}) ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + {‖ \nabla^{2} H ‖}_{L^{2}}^{2}) . \end{matrix} \end{matrix}$ Multiplying this inequality by ${(1 + t)}^{\frac{2}{p} + \frac{3}{2}}$ , and applying (4.5), we deduce that for any $t ⩾ T_{0}$ , $\begin{matrix} (4.10) & \begin{matrix} {‖ (\nabla^{2} u, \nabla^{2} τ, \nabla^{2} H) (t) ‖}_{L^{2}}^{2} \\ ⩽ C {(1 + t)}^{- \frac{2}{p} - \frac{3}{2}} ({(1 + T_{0})}^{\frac{2}{p} + \frac{3}{2}} {‖ (\nabla^{2} u, \nabla^{2} τ, \nabla^{2} H) (T_{0}) ‖}_{L^{2}}^{2} - {(1 + T_{0})}^{\frac{1}{2}} + {(1 + t)}^{\frac{1}{2}} \\ + sup_{T_{0} ⩽ s ⩽ t} (1 + s) {‖ (\nabla u, \nabla H) (s) ‖}_{L^{2}}^{2} \int_{T_{0}}^{t} {(1 + s)}^{\frac{2}{p} + \frac{1}{2}} {‖ (\nabla^{2} u, \nabla^{2} H) (s) ‖}_{L^{2}}^{2} d s) \\ ⩽ C {(1 + t)}^{- \frac{2}{p} - 1}, \end{matrix} \end{matrix}$ and $\begin{matrix} \int_{T_{0}}^{t} {(1 + s)}^{\frac{2}{p} + \frac{3}{2}} {‖ (\nabla^{3} u, \nabla^{2} τ, \nabla^{3} H) (s) ‖}_{L^{2}}^{2} d s ⩽ C {(1 + t)}^{\frac{1}{2}} . \end{matrix}$ Finally, we estimate $‖ (\nabla^{3} u, \nabla^{3} τ, \nabla^{3} H) ‖_{L^{2}}$ , the proof is similar to the one above. We obviously have $\begin{matrix} \frac{d}{d t} {‖ (\nabla^{3} u, \nabla^{3} τ, \nabla^{3} H) ‖}_{L^{2}}^{2} + {‖ (\nabla^{4} u, \nabla^{3} τ, \nabla^{4} H) ‖}_{L^{2}}^{2} \\ ⩽ C (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2} + {‖ \nabla^{2} u ‖}_{L^{2}} + {‖ \nabla^{2} H ‖}_{L^{2}}) ({‖ \nabla^{3} u ‖}_{L^{2}}^{2} + {‖ \nabla^{3} H ‖}_{L^{2}}^{2}) \\ + C ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + ‖ \nabla τ ‖_{L^{2}}^{2}) {‖ \nabla^{2} τ ‖}_{L^{2}}^{2} {‖ \nabla^{3} u ‖}_{L^{2}}^{2} + C ‖ \nabla τ ‖_{L^{2}}^{2} {‖ \nabla^{2} τ ‖}_{L^{2}}^{4} . \end{matrix}$ Consequently, for any $t ⩾ T_{0}$ , $\begin{matrix} \frac{d}{d t} {‖ (\nabla^{3} u, \nabla^{3} τ, \nabla^{3} H) ‖}_{L^{2}}^{2} + \frac{C}{1 + t} {‖ (\nabla^{3} u, \nabla^{3} τ, \nabla^{3} H) ‖}_{L^{2}}^{2} \\ ⩽ \frac{C}{{(1 + t)}^{4}} \int_{S (t)} (| \hat{u} |^{2} + | \hat{H} |^{2}) d ξ + C (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2} + {‖ \nabla^{2} u ‖}_{L^{2}} + {‖ \nabla^{2} H ‖}_{L^{2}}) ({‖ \nabla^{3} u ‖}_{L^{2}}^{2} \\ + {‖ \nabla^{3} H ‖}_{L^{2}}^{2}) + C ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + ‖ \nabla τ ‖_{L^{2}}^{2}) {‖ \nabla^{2} τ ‖}_{L^{2}}^{2} {‖ \nabla^{3} u ‖}_{L^{2}}^{2} + C ‖ \nabla τ ‖_{L^{2}}^{2} {‖ \nabla^{2} τ ‖}_{L^{2}}^{4} \\ ⩽ C {(1 + t)}^{- \frac{2}{p} - 3} + C (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla H ‖_{L^{2}}^{2} + {‖ \nabla^{2} u ‖}_{L^{2}} + {‖ \nabla^{2} H ‖}_{L^{2}}) ({‖ \nabla^{3} u ‖}_{L^{2}}^{2} + {‖ \nabla^{3} H ‖}_{L^{2}}^{2}) \\ + C ({‖ \nabla^{2} u ‖}_{L^{2}}^{2} + ‖ \nabla τ ‖_{L^{2}}^{2}) {‖ \nabla^{2} τ ‖}_{L^{2}}^{2} {‖ \nabla^{3} u ‖}_{L^{2}}^{2} + C ‖ \nabla τ ‖_{L^{2}}^{2} {‖ \nabla^{2} τ ‖}_{L^{2}}^{4}, \end{matrix}$ from which it follows that $\begin{array}{l} {‖ (\nabla^{3} u, \nabla^{3} τ, \nabla^{3} H) (t) ‖}_{L^{2}}^{2} \\ ⩽ C {(1 + t)}^{- \frac{2}{p} - \frac{5}{2}} ({(1 + T_{0})}^{\frac{2}{p} + \frac{5}{2}} {‖ (\nabla^{3} u, \nabla^{3} τ, \nabla^{3} H) (T_{0}) ‖}_{L^{2}}^{2} - {(1 + T_{0})}^{\frac{1}{2}} + {(1 + t)}^{\frac{1}{2}} \\ + C sup_{T_{0} ⩽ s ⩽ t} (1 + s) ({‖ \nabla u (s) ‖}_{L^{2}}^{2} + {‖ \nabla H (s) ‖}_{L^{2}}^{2} + {‖ \nabla^{2} u (s) ‖}_{L^{2}} + {‖ \nabla^{2} H (s) ‖}_{L^{2}}) \\ \times \int_{T_{0}}^{t} {(1 + s)}^{\frac{2}{p} + \frac{3}{2}} ({‖ \nabla^{3} u (s) ‖}_{L^{2}}^{2} + {‖ \nabla^{3} H (s) ‖}_{L^{2}}^{2}) d s \\ + sup_{T_{0} ⩽ s ⩽ t} (1 + s) ({‖ \nabla^{2} u (s) ‖}_{L^{2}}^{2} + {‖ \nabla τ (s) ‖}_{L^{2}}^{2}) {‖ \nabla^{2} τ (s) ‖}_{L^{2}}^{2} \int_{T_{0}}^{t} {(1 + s)}^{\frac{2}{p} + \frac{3}{2}} {‖ \nabla^{3} u (s) ‖}_{L^{2}}^{2} d s \\ + sup_{T_{0} ⩽ s ⩽ t} (1 + s) {‖ \nabla τ (s) ‖}_{L^{2}}^{2} {‖ \nabla^{2} τ (s) ‖}_{L^{2}}^{2} \int_{T_{0}}^{t} {(1 + s)}^{\frac{2}{p} + \frac{3}{2}} {‖ \nabla^{2} τ (s) ‖}_{L^{2}}^{2} d s) \\ (4.11) & ⩽ C {(1 + t)}^{- \frac{2}{p} - 2} . \end{array}$ Combining the above three estimates (4.5), (4.10) and (4.11), we see that $\begin{matrix} {‖ (\nabla^{k} u, \nabla^{k} τ, \nabla^{k} H) (t) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- \frac{1}{2} (\frac{2}{p} - 1) - \frac{k}{2}}, \forall t ⩾ T_{0}, k = 1, 2, 3, \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ and $C_{0}$ . A positive $C_{0}$ being given, we can choose $T_{0}$ sufficiently large, then it is now easy to complete the proof by means of a bootstrap argument. □

In what follows, we want to improve the estimate of τ by using the structure of the equation ${(1.1)}_{2}$ . Proposition 4.2.
Under the assumptions of Proposition 4.1 , we have $\begin{matrix} (4.12) & {‖ \nabla^{k} τ (t) ‖}_{L^{2} (R^{2})} ⩽ C {(1 + t)}^{- \frac{1}{p} - \frac{k}{2}}, \forall t ⩾ T_{0}, k = 0, 1, 2, \end{matrix}$ where C is a positive constant depending only on the initial data and $C_{0}$ .
Proof.
We apply $\nabla^{k} (k = 0, 1, 2)$ to ${(1.1)}_{2}$ , multiply the result equation by $\nabla^{k} τ$ , and integrate over $R^{2}$ , $\begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \nabla^{k} τ ‖}_{L^{2}}^{2} + {‖ \nabla^{k} τ ‖}_{L^{2}}^{2} \\ ⩽ \int \nabla^{k} D (u) \cdot \nabla^{k} τ d x + \int \nabla^{k} ((tr τ) τ) \cdot \nabla^{k} τ d x + \int \nabla^{k} (u \cdot \nabla τ) \cdot \nabla^{k} τ d x . \end{matrix}$ By virtue of Proposition 4.1, this gives $\begin{matrix} \frac{d}{d t} {‖ \nabla^{k} τ ‖}_{L^{2}}^{2} + {‖ \nabla^{k} τ ‖}_{L^{2}}^{2} ⩽ C {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2}, k = 0, 1, 2 . \end{matrix}$ Therefore, for any $t ⩾ T_{0}$ , we have $\begin{matrix} {‖ \nabla^{k} τ (t) ‖}_{L^{2}}^{2} ⩽ & e^{T_{0} - t} {‖ \nabla^{k} τ (T_{0}) ‖}_{L^{2}}^{2} + \int_{T_{0}}^{t} e^{s - t} {‖ \nabla^{k + 1} u (s) ‖}_{L^{2}}^{2} d s \\ ⩽ & e^{T_{0} - t} {‖ \nabla^{k} τ (T_{0}) ‖}_{L^{2}}^{2} + \int_{T_{0}}^{t} e^{s - t} {(1 + s)}^{- \frac{2}{p} - k} d s \\ ⩽ & C {(1 + t)}^{- \frac{2}{p} - k} . \end{matrix}$ This completes the proof of the proposition. □

5. Lower bounds of decay for large solutions

In this section, we are concerned with the precise large time behavior of large solutions of the lower bound in $H^{1} (R^{2})$ . To do so, we shall use semigroup method and Duhamel principle. Here, we will derive lower bounds of time decay rates for $(u, τ, H)$ if the average of the initial data $\int_{R^{2}} u_{0} (x) d s$ and $\int_{R^{2}} H_{0} (x) d s$ are nonzero. This result, Proposition 4.1 and Proposition 4.2 imply that the time decay rates of large solutions are sharp in two dimensions.

Firstly, we consider the following linearized system: $\begin{array}{l} (5.1) & \{\begin{matrix} \partial_{t} u_{l} - △ u_{l} - P div τ_{l} = 0, \\ \partial_{t} τ_{l} + τ_{l} - D u_{l} = 0, \\ \partial_{t} H_{l} - △ H_{l} = 0, \end{matrix} \end{array}$ with the initial data $(u_{l}, τ_{l}, H_{l}) |_{t = 0} = (u_{0}, τ_{0}, H_{0})$ . Thanks to the spectral analysis of the semigroup for the first two equations of the linearized system (5.1), we can deduce the exact expression for the Fourier transform of Green’s function (see [48] for more details). On the other hand, the third equation of the linearized system is the standard heat equation, which is well understood. This is illustrated by the following proposition.

Proposition 5.1.
For any $t > 0$ , $ξ \in R^{2}$ , we define $\begin{matrix} M : = \frac{e^{λ_{1} t} - e^{λ_{2} t}}{λ_{1} - λ_{2}}, N : = \frac{(λ_{1} + 1) e^{λ_{1} t} - (λ_{2} + 1) e^{λ_{2} t}}{λ_{1} - λ_{2}}, \\ L : = \frac{(λ_{2} + 1) e^{λ_{1} t} - (λ_{1} + 1) e^{λ_{2} t}}{λ_{1} - λ_{2}}, \end{matrix}$ where $\begin{matrix} λ_{1} = \frac{- (1 + | ξ |^{2}) + \sqrt{{(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2}}}{2}, λ_{2} = \frac{- (1 + | ξ |^{2}) - \sqrt{{(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2}}}{2}, \end{matrix}$ then the solutions of ( 5.1 ) can be expressed as follows: $\begin{array}{l} {\hat{u}}_{l}^{j} (t, ξ) = N (t, ξ) {\hat{u}}_{0}^{j} (ξ) + M (t, ξ) i ξ_{n} (δ_{j, k} - \frac{ξ_{j} ξ_{k}}{| ξ |^{2}}) {\hat{τ}}_{0}^{n, k} (ξ), \\ {\hat{τ}}_{l}^{j, k} (t, ξ) = e^{- t} {\hat{τ}}_{0}^{j, k} (ξ) - (L (t, ξ) + e^{- t}) (\frac{ξ_{j} ξ_{n}}{| ξ |^{2}} (δ_{k, m} - \frac{ξ_{k} ξ_{m}}{| ξ |^{2}}) {\hat{τ}}_{0}^{n, m} + \frac{ξ_{k} ξ_{n}}{| ξ |^{2}} (δ_{j, m} - \frac{ξ_{j} ξ_{m}}{| ξ |^{2}}) {\hat{τ}}_{0}^{n, m}) \\ + i M (t, ξ) (ξ_{j} {\hat{u}}_{0}^{k} (ξ) + ξ_{k} {\hat{u}}_{0}^{j} (ξ)), \\ {\hat{H}}_{l}^{j} (t, ξ) = e^{- | ξ |^{2} t} {\hat{H}}_{0}^{j} (t, ξ), \end{array}$ for any $t ⩾ 0$ .

Defining the difference $u_{δ} : = u - u_{l}$ , $τ_{δ} : = τ - τ_{l}$ , $H_{δ} : = H - H_{l}$ , we then have $\begin{array}{l} (5.2) & \{\begin{matrix} \partial_{t} u_{δ} - △ u_{δ} - P div τ_{δ} = - P div (u \otimes u + H \otimes H), \\ \partial_{t} τ_{δ} + τ_{δ} - D (u_{δ}) = - div (u \otimes τ) - Q (\nabla u, τ) - (tr τ) τ, \\ \partial_{t} H_{δ} - △ H_{δ} = - div (u \otimes H + H \otimes u), \end{matrix} \end{array}$ with the initial data $(u_{δ}, τ_{δ}, H_{δ}) |_{t = 0} = (0, 0, 0)$ . By means of the semigroup method and Duhamel principle, the solutions of (5.2) can be expressed as follows: $\begin{matrix} (5.3) & \begin{matrix} {\hat{u}}_{δ}^{j} (t, ξ) \\ = - \int_{0}^{t} N (t - s, ξ) i ξ_{n} (δ_{j, k} - \frac{ξ_{j} ξ_{k}}{| ξ |^{2}}) (\hat{u^{n} u^{k}} + \hat{H^{n} H^{k}}) (s) d s \\ - \int_{0}^{t} M (t - s, ξ) i ξ_{n} (δ_{j, k} - \frac{ξ_{j} ξ_{k}}{| ξ |^{2}}) {(\hat{div (u \otimes τ)} + \hat{Q (\nabla u, τ)} + \hat{(tr τ) τ})}^{n, k} (s) d s \\ : = J_{1} + J_{2}, \\ {\hat{τ}}_{δ}^{j, k} (t, ξ) \\ = - \int_{0}^{t} M (t - s, ξ) (ξ_{n} ξ_{j} (δ_{k, m} - \frac{ξ_{k} ξ_{m}}{| ξ |^{2}}) (\hat{u^{n} u^{m}} + \hat{H^{n} H^{m}}) + ξ_{k} ξ_{n} (δ_{j, k} - \frac{ξ_{j} ξ_{k}}{| ξ |^{2}}) (\hat{u^{n} u^{k}} \\ + \hat{H^{n} H^{k}})) (s) d s - \int_{0}^{t} L (t - s, ξ) {(\hat{div (u \otimes τ)} + \hat{Q (\nabla u, τ)} + \hat{(tr τ) τ})}^{j, k} (s) d s \\ : = K_{1} + K_{2}, \\ {\hat{H}}_{δ}^{j} (t, ξ) = - \int_{T_{0}}^{t} e^{- | ξ |^{2} (t - s)} {\hat{div (u \otimes H + H \otimes u)}}^{j} (s) d s, \end{matrix} \end{matrix}$ for any $t ⩾ 0$ .

More precisely, we have the following proposition. Proposition 5.2.
Let $(u, τ, H)$ be a global smooth solution of ( 1.1 ) with the initial data $(u_{0}, τ_{0}, H_{0}) \in L^{1} (R^{2}) \cap L^{2} (R^{2})$ , $\int_{R^{2}} u_{0} (x) d x = c_{0} \neq 0$ and $\int_{R^{2}} H_{0} (x) d x = {\tilde{c}}_{0} \neq 0$ . Suppose further that there exists a time $T_{1} ⩾ T_{0}$ such that $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{H^{3} (R^{2})} ⩽ C_{1}, \forall t ⩽ T_{1} . \end{matrix}$ Then for large time t, we have $\begin{matrix} {‖ (\nabla^{k} u, \nabla^{k} H) (t) ‖}_{L^{2} (R^{2})} ⩾ C^{- 1} {(1 + t)}^{- \frac{1}{2} - \frac{k}{2}}, k = 0, 1, 2, 3, \end{matrix}$ and $\begin{matrix} {‖ \nabla^{k} τ (t) ‖}_{L^{2} (R^{2})} ⩾ C^{- 1} {(1 + t)}^{- 1 - \frac{k}{2}}, \end{matrix}$ where C is a positive constant depending only on the initial data $(u_{0}, τ_{0}, H_{0})$ , $C_{0}$ and $C_{1}$ .
Proof.
In order to derive the lower bounds of time decay rates of large solutions, we need to verify the approximation of the Green’s function in Fourier space. From Proposition 5.1, it is easy to check that the eigenvalues $\begin{matrix} λ_{1} = - | ξ |^{2}, λ_{2} = - 1 for | ξ | ⩽ 1, \\ λ_{1} = - 1, λ_{2} = - | ξ |^{2} for | ξ | ⩾ 1 . \end{matrix}$ We thus obtain $\begin{matrix} M = \frac{e^{- | ξ |^{2} t} - e^{- t}}{1 - | ξ |^{2}}, N = e^{- | ξ |^{2} t}, L = - e^{- t}, \end{matrix}$ then the expression of ${\hat{τ}}_{l}$ can be simplified to $\begin{matrix} {\hat{τ}}_{l}^{j, k} (t, ξ) = e^{- t} {\hat{τ}}_{0}^{j, k} (ξ) + i M (t, ξ) (ξ_{j} {\hat{u}}_{0}^{k} (ξ) + ξ_{k} {\hat{u}}_{0}^{j} (ξ)) . \end{matrix}$ For any nonnegative integer k, we readily get $\begin{matrix} (5.4) & \begin{matrix} {‖ {\hat{\nabla^{k} u}}_{l} (t) ‖}_{L^{2}}^{2} ⩾ & \int_{| ξ | ⩽ η} | ξ |^{2 k} | {\hat{u}}_{l} |^{2} d ξ \\ ⩾ & \frac{1}{2} \int_{| ξ | ⩽ η} | ξ |^{2 k} \sum_{j} {| N {\hat{u}}_{0}^{j} |}^{2} d ξ - \int_{| ξ | ⩽ η} | ξ |^{2 k} \sum_{j} {| M i ξ_{n} (δ_{j, k} - \frac{ξ_{j} ξ_{k}}{| ξ |^{2}}) {\hat{τ}}_{0}^{n, k} |}^{2} d ξ \\ ⩾ & \frac{1}{2} \int_{| ξ | ⩽ η} | ξ |^{2 k} e^{- 2 | ξ |^{2} t} | {\hat{u}}_{0} |^{2} d ξ - C \int_{| ξ | ⩽ η} | ξ |^{2 (k + 1)} (e^{- 2 | ξ |^{2} t} + e^{- 2 t}) | {\hat{τ}}_{0} |^{2} d ξ . \end{matrix} \end{matrix}$ Indeed, since $u_{0} \in L^{1} (R^{2})$ , we must have ${\hat{u}}_{0} \in C (R^{2})$ . Hence also $| {\hat{u}}_{0} (ξ) | ⩾ \frac{c_{0}}{2}$ when $0 ⩽ | ξ | ⩽ η$ , if η is sufficiently small. We then observe $\begin{matrix} \frac{1}{2} \int_{| ξ | ⩽ η} | ξ |^{2 k} e^{- 2 | ξ |^{2} t} | {\hat{u}}_{0} |^{2} d ξ ⩾ \frac{c_{0}^{2}}{8} \int_{| ξ | ⩽ η} | ξ |^{2 k} e^{- 2 | ξ |^{2} t} d ξ ⩾ C^{- 1} {(1 + t)}^{- k - 1} . \end{matrix}$ In view of $τ_{0} \in L^{1} (R^{2})$ , we have $\begin{matrix} \int_{| ξ | ⩽ η} | ξ |^{2 (k + 1)} (e^{- 2 | ξ |^{2} t} + e^{- 2 t}) | {\hat{τ}}_{0} |^{2} d ξ ⩽ C ‖ τ_{0} ‖_{L^{1}}^{2} \int_{| ξ | ⩽ η} | ξ |^{2 (k + 1)} e^{- 2 | ξ |^{2} t} d ξ ⩽ C {(1 + t)}^{- k - 2} . \end{matrix}$ Combining these estimates with (5.4), for large time t, we deduce that $\begin{matrix} (5.5) & \begin{matrix} {‖ \nabla^{k} u_{l} (t) ‖}_{L^{2}}^{2} = {‖ \hat{\nabla^{k} u_{l}} (t) ‖}_{L^{2}}^{2} ⩾ C^{- 1} {(1 + t)}^{- k - 1} - C {(1 + t)}^{- k - 2} ⩾ C^{- 1} {(1 + t)}^{- k - 1} . \end{matrix} \end{matrix}$ We also have $\begin{matrix} {‖ {\hat{\nabla^{k} τ}}_{l} (t) ‖}_{L^{2}}^{2} ⩾ & \int_{| ξ | ⩽ η} | ξ |^{2 k} | {\hat{τ}}_{l} |^{2} d ξ \\ ⩾ & \frac{1}{2} \int_{| ξ | ⩽ η} | ξ |^{2 k} \sum_{j, k} | i M (t, ξ) (ξ_{j} {\hat{u}}_{0}^{k} + ξ_{k} {\hat{u}}_{0}^{j}) |^{2} d ξ - \int_{| ξ | ⩽ η} | ξ |^{2 k} \sum_{j, k} e^{- 2 t} {| {\hat{τ}}_{0}^{j, k} |}^{2} d ξ \\ ⩾ & \frac{1}{2} \int_{| ξ | ⩽ η} | ξ |^{2 (k + 1)} e^{- 2 | ξ |^{2} t} | {\hat{u}}_{0} |^{2} d ξ - C \int_{| ξ | ⩽ η} | ξ |^{2 (k + 1)} e^{- 2 t} | {\hat{u}}_{0} |^{2} d ξ \\ - C \int_{| ξ | ⩽ η} | ξ |^{2 k} e^{- 2 t} | {\hat{τ}}_{0} |^{2} d ξ . \\ ⩾ & C^{- 1} {(1 + t)}^{- k - 2} - C e^{- 2 t}, \end{matrix}$ owing to $(u_{0}, τ_{0}) \in L^{1} (R^{2}) \cap L^{2} (R^{2})$ . Thus, for large time t, we deduce that $\begin{matrix} {‖ \nabla^{k} τ_{l} (t) ‖}_{L^{2}}^{2} = {‖ \hat{\nabla^{k} τ_{l}} (t) ‖}_{L^{2}}^{2} ⩾ C^{- 1} {(1 + t)}^{- k - 2} - C e^{- 2 t} ⩾ C^{- 1} {(1 + t)}^{- k - 2} . \end{matrix}$ Using the properties of the heat semigroup, we yield a rather explicit lower bounds of time decay rates for $\nabla^{k} H_{l}$ , for large time t, we then have $\begin{matrix} {‖ {\nabla^{k} H}_{l} (t) ‖}_{L^{2}}^{2} & = {‖ {\hat{\nabla^{k} H}}_{l} (t) ‖}_{L^{2}}^{2} ⩾ \int_{| ξ | ⩽ η} | ξ |^{2 k} | {\hat{H}}_{l} |^{2} d ξ \\ ⩾ \int_{| ξ | ⩽ η} | ξ |^{2 k} e^{- 2 | ξ |^{2} t} | {\hat{H}}_{0} |^{2} d ξ ⩾ C^{- 1} {(1 + t)}^{- k - 1}, \end{matrix}$ owing to $H_{0} \in L^{1} (R^{2})$ and $| {\hat{H}}_{0} (ξ) | ⩾ \frac{{\tilde{c}}_{0}}{2}$ when $0 ⩽ | ξ | ⩽ η$ .

Next, we shall establish the upper bounds of time decay rates for $(u_{δ}, τ_{δ}, H_{δ})$ . In fact, combining this with the lower bounds of time decay rates for $(u_{l}, τ_{l}, H_{l})$ , we can get the lower bounds of time decay rates for $(u, τ, H)$ . By virtue of Proposition 4.1, we have $‖ (\nabla^{k} u, \nabla^{k} H) (t) ‖_{L^{2}}^{2} ⩽ C {(1 + t)}^{- k - 1}$ , $\forall t ⩾ T_{0}$ , $k = 1, 2, 3$ . Note that Minkowshi’s inequality and Lemma 2.3 imply that $\begin{matrix} ‖ J_{1} ‖_{L^{2}} ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- 1} (‖ u \otimes u ‖_{L^{1}} + ‖ u \otimes u ‖_{L^{2}} + ‖ H \otimes H ‖_{L^{1}} + ‖ H \otimes H ‖_{L^{2}}) d s \\ ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- 1} (‖ u ‖_{L^{2}} ‖ u ‖_{L^{2}} + ‖ u ‖_{L^{2}} ‖ \nabla u ‖_{L^{2}} \\ + ‖ H ‖_{L^{2}} ‖ H ‖_{L^{2}} + ‖ H ‖_{L^{2}} ‖ \nabla H ‖_{L^{2}}) d s \\ ⩽ & C (\int_{0}^{T_{1}} {(1 + t - s)}^{- 1} d s + \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- 1} d s), \end{matrix}$ we get $\begin{matrix} ‖ J_{1} ‖_{L^{2}} ⩽ C T_{1} {(1 + t - T_{1})}^{- 1} + C {(1 + t)}^{- 1} ln (1 + t) . \end{matrix}$ In light of Proposition 4.1 and Proposition 4.2, we have $‖ \nabla^{k} τ (t) ‖_{L^{2}}^{2} ⩽ C {(1 + t)}^{- k - 2}$ , $‖ \nabla^{3} τ (t) ‖_{L^{2}}^{2} ⩽ C {(1 + t)}^{- 4}$ , $\forall t ⩾ T_{0}$ , $k = 0, 1, 2$ . Thus, Minkowshi’s inequality, Lemma 2.1 and Lemma 2.3 then lead to $\begin{matrix} ‖ J_{2} ‖_{L^{2}} ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- \frac{3}{2}} (‖ u \otimes τ ‖_{L^{1}} + ‖ u \otimes τ ‖_{L^{2}}) d s \\ + \int_{0}^{t} {(1 + t - s)}^{- 1} (‖ \nabla u \otimes τ ‖_{L^{1}} + ‖ \nabla u \otimes τ ‖_{L^{2}} + ‖ tr τ \otimes τ ‖_{L^{1}} + ‖ tr τ \otimes τ ‖_{L^{2}}) d s \\ ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- \frac{3}{2}} (‖ u ‖_{L^{2}} ‖ τ ‖_{L^{2}} + ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} ‖ τ ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}}) d s \\ + \int_{0}^{t} {(1 + t - s)}^{- 1} (‖ \nabla u ‖_{L^{2}} ‖ τ ‖_{L^{2}} + ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ τ ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} \\ + ‖ τ ‖_{L^{2}}^{2} + ‖ τ ‖_{L^{2}} ‖ \nabla τ ‖_{L^{2}}) d s \\ ⩽ & C (\int_{0}^{T_{1}} {(1 + t - s)}^{- 1} d s + \int_{T_{1}}^{t} {(1 + t - s)}^{- \frac{3}{2}} {(1 + s)}^{- \frac{3}{2}} d s \\ + \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- 2} d s) \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- 1} + C {(1 + t)}^{- 1} . \end{matrix}$ Utilizing these calculations in ${(5.3)}_{1}$ , we obtain the estimate $\begin{matrix} {‖ u_{δ} (t) ‖}_{L^{2}} ⩽ C T_{1} {(1 + t - T_{1})}^{- 1} + C {(1 + t)}^{- 1} ln (1 + t), \forall t ⩾ T_{1}, \end{matrix}$ we thereupon conclude from (5.5) that for large time t, $\begin{matrix} (5.6) & \begin{matrix} {‖ u (t) ‖}_{L^{2}} & ⩾ {‖ u_{l} (t) ‖}_{L^{2}} - {‖ u_{δ} (t) ‖}_{L^{2}} \\ ⩾ C^{- 1} {(1 + t)}^{- \frac{1}{2}} - C T_{1} {(1 + t - T_{1})}^{- 1} - C {(1 + t)}^{- 1} ln (1 + t) ⩾ C^{- 1} {(1 + t)}^{- \frac{1}{2}} . \end{matrix} \end{matrix}$ Similarly, we have $\begin{array}{rcl} {‖ | ξ | J_{1} ‖}_{L^{2}} & ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} (‖ u \otimes u ‖_{L^{1}} + ‖ u \otimes u ‖_{L^{2}} + ‖ H \otimes H ‖_{L^{1}} + ‖ H \otimes H ‖_{L^{2}}) d s \\ + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} ({‖ div (u \otimes u) ‖}_{L^{1}} + {‖ div (u \otimes u) ‖}_{L^{2}} + {‖ div (H \otimes H) ‖}_{L^{1}} \\ + {‖ div (H \otimes H) ‖}_{L^{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} (‖ u \otimes u ‖_{L^{1}} + ‖ u \otimes u ‖_{L^{2}} + ‖ H \otimes H ‖_{L^{1}} + ‖ H \otimes H ‖_{L^{2}}) d s \\ + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} (‖ u ‖_{L^{2}} ‖ \nabla u ‖_{L^{2}} + ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} + ‖ H ‖_{L^{2}} ‖ \nabla H ‖_{L^{2}} \\ + ‖ H ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla H ‖_{L^{2}} {‖ \nabla^{2} H ‖}_{L^{2}}^{\frac{1}{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} d s + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- \frac{1}{2}} {(1 + s)}^{- 1} d s \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} + C {(1 + T_{1})}^{- \frac{1}{4}} {(1 + t)}^{- 1}, \end{array}$ and $\begin{matrix} {‖ | ξ | J_{2} ‖}_{L^{2}} ⩽ & \int_{0}^{t} {(1 + t - s)}^{- \frac{3}{2}} ({‖ div (u \otimes τ) ‖}_{L^{1}} + {‖ div (u \otimes τ) ‖}_{L^{2}} \\ + {‖ Q (\nabla u, τ) ‖}_{L^{1}} + {‖ Q (\nabla u, τ) ‖}_{L^{2}} \\ + {‖ (tr τ) τ ‖}_{L^{1}} + {‖ (tr τ) τ ‖}_{L^{2}}) d s \\ ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- \frac{3}{2}} (‖ \nabla u ‖_{L^{2}} ‖ τ ‖_{L^{2}} + ‖ u ‖_{L^{2}} ‖ \nabla τ ‖_{L^{2}} \\ + ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ τ ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} \\ + ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{2}} + ‖ τ ‖_{L^{2}}^{2} + ‖ τ ‖_{L^{2}} ‖ \nabla τ ‖_{L^{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} d s + C \int_{T_{1}}^{t} {(1 + t - s)}^{- \frac{3}{2}} {(1 + s)}^{- 2} d s \\ + C \int_{T_{1}}^{t} {(1 + t - s)}^{- \frac{3}{2}} {(1 + s)}^{- \frac{5}{2}} d s \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} + C {(1 + t)}^{- \frac{3}{2}} . \end{matrix}$ Substituting these two estimates into (5.3), we find that $\begin{array}{rcl} {‖ \nabla u_{δ} (t) ‖}_{L^{2}} & ⩽ & {‖ | ξ | \hat{u_{δ}} (t) ‖}_{L^{2}} \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} + C {(1 + T_{1})}^{- \frac{1}{4}} {(1 + t)}^{- 1} + C {(1 + t)}^{- \frac{3}{2}}, \forall t ⩾ T_{1} . \end{array}$ Thus, for large time t, $\begin{matrix} (5.7) & \begin{matrix} {‖ \nabla u (t) ‖}_{L^{2}} ⩾ & {‖ \nabla u_{l} (t) ‖}_{L^{2}} - {‖ \nabla u_{δ} (t) ‖}_{L^{2}} \\ ⩾ & C^{- 1} {(1 + t)}^{- 1} - C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} - C {(1 + T_{1})}^{- \frac{1}{4}} {(1 + t)}^{- 1} - C {(1 + t)}^{- \frac{3}{2}} \\ ⩾ & C^{- 1} {(1 + t)}^{- 1} . \end{matrix} \end{matrix}$ Consequently, from (3.1) and (5.7), Lemma 2.2 leads us to get that $\begin{matrix} (5.8) & {‖ \nabla^{k} u (t) ‖}_{L^{2}} ⩾ C {‖ \nabla u (t) ‖}_{L^{2}}^{k} {‖ u (t) ‖}_{L^{2}}^{1 - k} ⩾ C^{- 1} {(1 + t)}^{- \frac{k + 1}{2}}, k = 2, 3 . \end{matrix}$ We return now to the estimate of $τ_{δ}$ by using similar computations. Combining Minkowshi’s inequality, Lemma 2.1 and Lemma 2.3, we infer that $\begin{matrix} ‖ K_{1} ‖_{L^{2}} ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} (‖ u \otimes u ‖_{L^{1}} + ‖ u \otimes u ‖_{L^{2}} + ‖ H \otimes H ‖_{L^{1}} + ‖ H \otimes H ‖_{L^{2}}) d s \\ + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} ({‖ div (u \otimes u) ‖}_{L^{1}} + {‖ div (u \otimes u) ‖}_{L^{2}} + {‖ div (H \otimes H) ‖}_{L^{1}} \\ + {‖ div (H \otimes H) ‖}_{L^{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} d s \\ + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} (‖ u ‖_{L^{2}} ‖ \nabla u ‖_{L^{2}} + ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} \\ + ‖ H ‖_{L^{2}} ‖ \nabla H ‖_{L^{2}} + ‖ H ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla H ‖_{L^{2}} {‖ \nabla^{2} H ‖}_{L^{2}}^{\frac{1}{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- \frac{3}{2}} d s + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- \frac{3}{2}} d s \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} + C {(1 + T_{1})}^{- \frac{1}{4}} {(1 + t)}^{- 1}, \end{matrix}$ and $\begin{matrix} ‖ K_{2} ‖_{L^{2}} ⩽ & \int_{0}^{t} e^{- (t - s)} ({‖ div (u \otimes τ) ‖}_{L^{2}} + {‖ Q (\nabla u, τ) ‖}_{L^{2}} + {‖ (tr τ) τ ‖}_{L^{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} e^{- (t - s)} d s + C \int_{T_{1}}^{t} e^{- (t - s)} (‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} τ ‖}_{L^{2}}^{\frac{1}{2}} \\ + ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ τ ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla τ ‖_{L^{2}}^{\frac{1}{2}} + ‖ τ ‖_{L^{2}} ‖ \nabla τ ‖_{L^{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} e^{- (t - s)} d s + C \int_{T_{1}}^{t} e^{- (t - s)} {(1 + s)}^{- \frac{5}{2}} d s \\ ⩽ & C T_{1} e^{T_{1}} e^{- t} + C {(1 + t)}^{- \frac{5}{2}} . \end{matrix}$ These estimates and ${(5.3)}_{2}$ imply that $\begin{array}{rcl} {‖ τ_{δ} (t) ‖}_{L^{2}} & ⩽ & C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} + C {(1 + T_{1})}^{- \frac{1}{4}} {(1 + t)}^{- 1} \\ + C T_{1} e^{T_{1}} e^{- t} + C {(1 + t)}^{- \frac{5}{2}}, \forall t ⩾ T_{1} . \end{array}$ Finally, we find that for large time t, $\begin{array}{rcl} {‖ τ (t) ‖}_{L^{2}} & ⩾ & {‖ τ_{l} (t) ‖}_{L^{2}} - {‖ τ_{δ} (t) ‖}_{L^{2}} \\ ⩾ & C^{- 1} {(1 + t)}^{- 1} - C T_{1} {(1 + t - T_{1})}^{- \frac{3}{2}} - C {(1 + T_{1})}^{- \frac{1}{4}} {(1 + t)}^{- 1} \\ - C T_{1} e^{T_{1}} e^{- t} - C {(1 + t)}^{- \frac{5}{2}} \\ (5.9) & ⩾ & C^{- 1} {(1 + t)}^{- 1} . \end{array}$ A similar estimate holds for $H_{δ}$ , then $\begin{matrix} ‖ H_{δ} {(t) ‖}_{L^{2}} ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- 1} (‖ u \otimes H ‖_{L^{1}} + ‖ H \otimes u ‖_{L^{2}}) d s \\ ⩽ & C \int_{0}^{t} {(1 + t - s)}^{- 1} (‖ u ‖_{L^{2}} ‖ H ‖_{L^{2}} + ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} ‖ H ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla H ‖_{L^{2}}^{\frac{1}{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- 1} d s + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- 1} d s \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- 1} + C {(1 + t)}^{- 1} ln (1 + t), \forall t ⩾ T_{1} . \end{matrix}$ Hence, we find that for large enough time t, $\begin{matrix} (5.10) & \begin{matrix} {‖ H (t) ‖}_{L^{2}} & ⩾ {‖ H_{l} (t) ‖}_{L^{2}} - {‖ H_{δ} (t) ‖}_{L^{2}} \\ ⩾ C^{- 1} {(1 + t)}^{- \frac{1}{2}} - C T_{1} {(1 + t - T_{1})}^{- 1} - C {(1 + t)}^{- 1} ln (1 + t) \\ ⩾ C^{- 1} {(1 + t)}^{- \frac{1}{2}} . \end{matrix} \end{matrix}$ We also obtain that $\begin{matrix} {‖ \nabla H_{δ} (t) ‖}_{L^{2}} ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- 2} (‖ u \otimes H ‖_{L^{1}} + ‖ H \otimes u ‖_{L^{2}}) d s \\ + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} ({‖ div (u \otimes H) ‖}_{L^{1}} + {‖ div (H \otimes u) ‖}_{L^{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- 2} d s + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} (‖ \nabla u ‖_{L^{2}} ‖ H ‖_{L^{2}} + ‖ u ‖_{L^{2}} ‖ \nabla H ‖_{L^{2}} \\ + ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} u ‖}_{L^{2}}^{\frac{1}{2}} ‖ H ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla H ‖_{L^{2}}^{\frac{1}{2}} + ‖ u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla u ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla H ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} H ‖}_{L^{2}}^{\frac{1}{2}}) d s \\ ⩽ & C \int_{0}^{T_{1}} {(1 + t - s)}^{- 2} d s + C \int_{T_{1}}^{t} {(1 + t - s)}^{- 1} {(1 + s)}^{- 1} {(1 + s)}^{- \frac{1}{2}} d s \\ ⩽ & C T_{1} {(1 + t - T_{1})}^{- 2} + C {(1 + T_{0})}^{- \frac{1}{4}} {(1 + t)}^{- 1}, \forall t ⩾ T_{1}, \end{matrix}$ it is obvious that if time t is large enough, then $\begin{matrix} (5.11) & \begin{matrix} {‖ \nabla H (t) ‖}_{L^{2}} ⩾ & {‖ \nabla H_{l} (t) ‖}_{L^{2}} - {‖ \nabla H_{δ} (t) ‖}_{L^{2}} \\ ⩾ & C^{- 1} {(1 + t)}^{- 1} - C T_{1} {(1 + t - T_{1})}^{- 2} - C {(1 + T_{0})}^{- \frac{1}{4}} {(1 + t)}^{- 1} \\ ⩾ & C^{- 1} {(1 + t)}^{- 1} . \end{matrix} \end{matrix}$ Consequently, we deduce from Lemma 2.2, (3.1) and (5.11) that $\begin{matrix} {‖ \nabla^{k} H (t) ‖}_{L^{2}} ⩾ C {‖ \nabla H (t) ‖}_{L^{2}}^{k} {‖ H (t) ‖}_{L^{2}}^{1 - k} ⩾ C^{- 1} {(1 + t)}^{- \frac{k + 1}{2}}, k = 2, 3 . \end{matrix}$ This bound, together with (5.6)–(5.11), completes the proof of the proposition. □

6. Global-in-time stability for large solutions

In fact, we shall show that the perturbed solutions will remain close to the reference solutions for a long time if their initial data are close to each other. The above convergence results will help us to prove that the reference solutions are close to the equilibrium after a long time, then the perturbed solutions are also close to the equilibrium after a long time.

More precisely, let $(u, τ, H)$ and $(\tilde{u}, \tilde{τ}, \tilde{H})$ be two global solutions of (1.1) with the initial data $(u_{0}, τ_{0}, H_{0})$ and $({\tilde{u}}_{0}, {\tilde{τ}}_{0}, {\tilde{H}}_{0})$ , respectively. We denote $v : = u - \tilde{u}$ , $σ : = τ - \tilde{τ}$ , and $B : = H - \tilde{H}$ , it would follow that the equations: $\begin{array}{l} (6.1) & \{\begin{matrix} v_{t} + v \cdot \nabla v - △ v + \nabla (p - \tilde{p}) \\ = div σ - v \cdot \nabla \tilde{u} - \tilde{u} \cdot \nabla v + B \cdot \nabla B + B \cdot \nabla \tilde{H} + \tilde{H} \cdot \nabla B \\ - \nabla \frac{| B |^{2}}{2} - B^{j} \partial_{i} {\tilde{H}}^{j} - {\tilde{H}}^{j} \partial_{i} B^{j}, \\ σ_{t} + v \cdot \nabla σ + σ \\ = D (v) - (tr σ) σ - (tr σ) \tilde{τ} - (tr \tilde{τ}) σ - v \cdot \nabla \tilde{τ} - \tilde{u} \cdot \nabla σ \\ - Q (σ, \nabla v) - Q (σ, \nabla \tilde{u}) - Q (\tilde{τ}, \nabla v), \\ B_{t} + v \cdot \nabla B - △ B = - v \cdot \nabla \tilde{H} - \tilde{u} \cdot \nabla B + B \cdot \nabla v + \tilde{H} \cdot \nabla v + B \cdot \nabla \tilde{u}, \end{matrix} \end{array}$ with the initial data $(v, σ, B) |_{t = 0} = (v_{0}, σ_{0}, B_{0}) = (u_{0} - {\tilde{u}}_{0}, τ_{0} - {\tilde{τ}}_{0}, H_{0} - {\tilde{H}}_{0})$ .

Proposition 6.1.

Let $(\tilde{u}, \tilde{τ}, \tilde{H})$ be a global solution of ( 1.1 ) satisfying the assumptions of Theorem 1.7 . Given a positive constant $δ > 0$ , if the initial data $(v_{0}, σ_{0}, B_{0})$ satisfying that $\begin{matrix} {‖ (v_{0}, σ_{0}, B_{0}) ‖}_{H^{3} (R^{2})} ⩽ δ, \end{matrix}$ then there exists a constant $C_{2}$ independent of δ, such that for any $t \in [0, C_{2} | ln δ |]$ , it holds that $\begin{matrix} (6.2) & {‖ (v, σ, B) (t) ‖}_{H^{3} (R^{2})} ⩽ δ^{\frac{1}{2}} . \end{matrix}$

Proof.

We use the continuity argument to prove the desired results. Denoting $\tilde{T}$ as follows: $\begin{matrix} (6.3) & \tilde{T} : = sup {t | (6.2) holds for any s \in [0, t]} \end{matrix}$ The existence of $\tilde{T}$ can be obtained by the local well-posedness result, then we need to prove that $T_{2} ⩽ C_{2} | ln δ |$ , where $C_{2}$ is a constant independent of δ. For $0 ⩽ k ⩽ 3$ , applying $\nabla^{k}$ to (6.1), multiplying the first equation by $\nabla^{k} v$ , the second by $\nabla^{k} σ$ , the third by $\nabla^{k} B$ , respectively, and integrating over $R^{2}$ , we get $\begin{array}{l} \frac{d}{d t} {‖ (v, σ, B) ‖}_{H^{3}} + {‖ (\nabla v, σ, \nabla B) ‖}_{H^{3}} \\ = \sum_{k = 0}^{3} \int (\nabla^{k} div σ \cdot \nabla^{k} v + \nabla^{k} D (v) \cdot \nabla^{k} σ) d x \\ - \sum_{k = 0}^{3} \int \nabla^{k} ((tr σ) σ + (tr σ) τ_{2} + (tr τ_{2}) σ) \cdot \nabla^{k} σ d x \\ - \sum_{k = 0}^{3} \int (\nabla^{k} (v \cdot \nabla v - B \cdot \nabla B) \cdot \nabla^{k} v + \nabla^{k} (v \cdot \nabla σ) \cdot \nabla^{k} σ \\ + \nabla^{k} (v \cdot \nabla B - B \cdot \nabla v) \cdot \nabla^{k} B) d x \\ - \sum_{k = 0}^{3} \int \nabla^{k} \frac{| B |^{2}}{2} \nabla^{k} v d x - \sum_{k = 0}^{3} \int (\nabla^{k} (v \cdot \nabla u_{2} + u_{2} \cdot \nabla v - B \cdot \nabla H_{2} + B^{j} \partial_{i} H_{2}^{j}) \cdot \nabla^{k} v \\ + \nabla^{k} (v \cdot \nabla τ_{2} + u_{2} \cdot \nabla σ) \cdot \nabla^{k} σ + \nabla^{k} (v \cdot \nabla H_{2} + u_{2} \cdot \nabla B + B \cdot \nabla u_{2}) \cdot \nabla^{k} B) d x \\ + \sum_{k = 0}^{3} \int (\nabla^{k} (H_{2} \cdot \nabla B - H_{2}^{j} \partial_{i} B^{j}) \cdot \nabla^{k} v + \nabla^{k} (H_{2} \cdot \nabla v) \cdot \nabla^{k} B) d x \\ - \sum_{k = 0}^{3} \int \nabla^{k} (Q (σ, \nabla v)) \cdot \nabla^{k} σ d x - \sum_{k = 0}^{3} \int \nabla^{k} (Q (σ, \nabla u_{2}) + Q (τ_{2}, \nabla v)) \cdot \nabla^{k} σ d x \\ : = \sum_{i = 1}^{8} L_{i} . \end{array}$ Using integrating by parts, we get $L_{1} = 0$ . Applying Sobolev inequalities, Young inequality and (6.3), we calculate that $\begin{array}{l} L_{2} ⩽ C (‖ \nabla σ ‖_{H^{1}} + ‖ τ_{2} ‖_{H^{3}}) ‖ σ ‖_{H^{3}}^{2} ⩽ C δ^{\frac{1}{2}} ‖ σ ‖_{H^{3}}^{2} + C ‖ σ ‖_{H^{3}}^{2}, \\ L_{3} ⩽ C ‖ \nabla v ‖_{H^{2}} (‖ \nabla B ‖_{H^{2}}^{2} + ‖ \nabla v ‖_{H^{2}}^{2} + ‖ \nabla σ ‖_{H^{2}}^{2}) ⩽ C δ^{\frac{1}{2}} ({‖ (\nabla v, \nabla σ, \nabla B) ‖}_{H^{2}}^{2}), \\ L_{4} ⩽ C ‖ B ‖_{H^{2}} ‖ v ‖_{H^{3}} ‖ \nabla B ‖_{H^{2}} + C ‖ B ‖_{H^{2}} {‖ \nabla^{3} v ‖}_{L^{2}} {‖ \nabla^{4} B ‖}_{L^{2}} \\ ⩽ ϵ {‖ \nabla^{4} B ‖}_{L^{2}}^{2} + C_{ϵ} δ ‖ B ‖_{H^{2}}^{2} + C δ^{\frac{1}{2}} ‖ B ‖_{H^{3}}^{2}, \\ L_{5} ⩽ C {‖ (\nabla u_{2}, \nabla τ_{2}, \nabla H_{2}) ‖}_{H^{3}} {‖ (v, σ, B) ‖}_{H^{3}} ({‖ (\nabla v, \nabla B) ‖}_{H^{2}} + ‖ σ ‖_{H^{3}}) \\ ⩽ ϵ ({‖ (\nabla v, \nabla B) ‖}_{H^{2}}^{2} + ‖ σ ‖_{H^{3}}^{2}) + C_{ϵ} {‖ (v, σ, B) ‖}_{H^{3}}^{2}, \\ L_{6} ⩽ C ‖ H_{2} ‖_{H^{3}} {‖ (v, B) ‖}_{H^{3}} {‖ (\nabla v, \nabla B) ‖}_{H^{3}} ⩽ ϵ {‖ (\nabla v, \nabla B) ‖}_{H^{3}}^{2} + C_{ϵ} {‖ (v, B) ‖}_{H^{3}}^{2}, \\ L_{7} ⩽ C ‖ \nabla v ‖_{H^{3}} ‖ σ ‖_{H^{3}} ‖ \nabla σ ‖_{H^{2}} ⩽ C δ^{\frac{1}{2}} ‖ σ ‖_{H^{3}}^{2}, \\ L_{8} ⩽ C ‖ \nabla u_{2} ‖_{H^{3}} ‖ σ ‖_{H^{3}}^{2} + ‖ τ_{2} ‖_{H^{3}} ‖ \nabla v ‖_{H^{3}} ‖ σ ‖_{H^{3}} ⩽ ϵ ‖ \nabla v ‖_{H^{3}}^{2} + C_{ϵ} ‖ σ ‖_{H^{3}}^{2} . \end{array}$ Combining the above estimates and choosing ϵ small enough, we easily get $\begin{matrix} \frac{d}{d t} {‖ (v, σ, H) ‖}_{H^{3}} ⩽ C {‖ (v, σ, H) ‖}_{H^{3}} . \end{matrix}$ that is, $\begin{matrix} (6.4) & \begin{matrix} {‖ (v, σ, H) (t) ‖}_{H^{3}} ⩽ C e^{C t}, \forall t ⩾ 0, \end{matrix} \end{matrix}$ which implies that $t ⩾ C_{2} | ln δ |$ , thus completes the proof of the proposition. □

It is thus enough to prove Theorem 1.7.

Proof of Theorem 1.7.

Thanks to Proposition 6.1, we choose $T_{2} = \frac{1}{2} (1 + C_{2} | ln δ |)$ , then we observe $\begin{matrix} {‖ (\tilde{u}, \tilde{τ}, \tilde{H}) (T_{2}) ‖}_{H^{3}} ⩽ C {(1 + C_{2} | ln δ |)}^{- \frac{1}{p} + \frac{1}{2}} . \end{matrix}$ Then at time $T_{2}$ , (1.1) is close to equilibrium regime. Hence we obtain the global existence for $(u, τ, H)$ . Moreover, owing to the definition of $\tilde{T}$ , we have for any $t ⩾ 0$ , $\begin{matrix} {‖ (u, τ, H) (t) ‖}_{H^{3}} ⩽ C min {{(1 + C_{2} | ln δ |)}^{- \frac{1}{p} + \frac{1}{2}}, δ^{\frac{1}{2}} + {(1 + t)}^{- \frac{1}{p} + \frac{1}{2}}} . \end{matrix}$ This completes the proof of Theorem 1.7. □

Footnotes

Acknowledgements

Yuhui Chen is supported by Guangdong Basic and Applied Basic Research Foundation grant No. 2019A1515110733. Minling Li is supported by NNSF of China grant No. 11971497. Qinghe Yao is supported by NNSF of China grant No. 11972384 and National Key R&D Program of International Collaboration grant No. 2018YFE9103900. Zheng-an Yao is supported by NNSF of China grant No. 11971496 and National Key R&D Program of China grant No. 2020YFA0712500.

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The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

Abstract

Keywords

1. Introduction

Footnotes

Acknowledgements

References