The sharp time decay rates for the incompressible Phan-Thien–Tanner system with magnetic field in ${\mathbb{R}}^2$

Abstract

Our focus in this paper is to investigate the sharp time decay rates (upper and lower bounds) for the higher order spatial derivatives of large solutions to the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid. In dimension two, the sharp time decay rates of large solution $(u,H)$ and all its derivatives have been shown by Chen et al. (2022). The $L^2$-decay rate of the stress tensor $\tau$ was improved to ${(1+t)}^{-1}$, however, the sharp time decay rates of any spatial derivatives of $\tau$ cannot be achieved. Based on Fourier time-splitting and semigroup methods, we shall obtain the optimal upper and lower bounds of time decay rates for the higher spatial derivatives of order $k$ ($k=1,2$) of $\tau$. More precisely, the $k$th order spatial derivatives of $\tau$ decay at ${(1+t)}^{-1-\frac{k}{2}}$-rate, which will be shown to be sharp. In addition, our result is valid for the incompressible Oldroyd-B system.

Introduction

In this paper, we would like to build the sharp time decay rates of large solutions to the following two-dimensional incompressible Phan-Thien–Tanner (PTT) system with magnetic field:

$$\left\{\begin{array}{c}{u}_t+u\cdot \nabla u-\mu △u+\nabla p={\mu }_{1}div\tau +H\cdot \nabla H-\nabla \frac{{\left|H\right|}^2}{2},(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^2,\hfill \\ {\tau }_t+u\cdot \nabla \tau +(a+bf(tr\tau ))\tau +Q(\nabla u,\tau )={\mu }_{2}D\left(u\right),(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^2,\hfill \\ H_t+u\cdot \nabla H-H\cdot \nabla u-\nu △H=0,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^2,\hfill \\ divu=0,divH=0,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^2,\hfill \end{array}\right.$$

with the initial data

$$u{|}_{t=0}=u_0\left(x\right),\tau {|}_{t=0}={\tau }_0\left(x\right),H{|}_{t=0}=H_0\left(x\right).$$

The unknown functions $u=(u_1,u_2)$, $\tau$, $H$ and $p$ stand for the velocity of the fluid, the polymer contribution to the stress tensor, the magnetic field and the pressure, respectively. $Q(\nabla u,\tau )$ is given by the bilinear form $Q(\nabla u,\tau )≔\tau \Omega \left(u\right)-\Omega \left(u\right)\tau +\lambda (D\left(u\right)\tau +\tau D\left(u\right))$, with $\Omega \left(u\right)≔\frac{1}{2}(\nabla u-{(\nabla u)}^T)$ and $D\left(u\right)≔\frac{1}{2}(\nabla u+{(\nabla u)}^T)$. The stress function $f$ is a given smooth function. The constant viscosity coefficient $\mu >0$, the elastic coefficient ${\mu }_1>0$, the magnetic diffusion coefficient $\nu >0$, the coefficients $a>0$ and ${\mu }_2>0$ imply the relation between the characteristic flow time and elastic time, and the coefficient $b\ge 0$ is related to the rates of creation and destruction for the polymeric network junctions. $\lambda$ is a constant that is typically in $[-1,1]$, the system is called co-rotational when $\lambda =0$. One may check [1], [2] for the detailed derivation of this system.

Note that if $b=0$ and $H\equiv 0$, then system (1.1) reduces to the incompressible Oldroyd-B system, which is well studied. In recent years, there are a number of references concerning the study of the related Oldroyd-B type model for viscoelastic fluids. Local existence result was first stated by Guillopé and Saut [3], [4], then local well-posedness result in Sobolev spaces was established by Fernández-Cara et al. [5]. Global existence of weak solutions with general initial data in the co-rotational case was proven by Lions and Masmoudi [6]. Chemin and Masmoudi [7] proved the well-posedness of local solutions and global small solutions in the critical Besov spaces. Later, blow up criteria were shown in [8], [9]. For most of the results concerning the global well-posedness for other types of Oldroyd model, one may refer to [10], [11], [12], [13], [14], [15]. We mention in passing that the well-posedness results for compressible viscoelastic fluids were widely studied in [16], [17], [18], [19], [20], [21]. More results on the optimal time decay rates of global small solutions may be found in [22], [23], [24]. To our knowledge, there is little mathematical result for the incompressible PTT system until that Masmoudi [25] firstly proved the global existence of weak solutions. Recently, Chen et al. [26] showed the global existence of strong small solutions and the blow-up phenomenon for the periodic incompressible linear PTT system without damping. We also note out that in [27], [28], Chen et al. showed the global well-posedness for the incompressible linear and generalized PTT systems with small initial data in the critical Besov spaces.

It is worth noting that almost all results on the global well-posedness and large-time behavior of strong solutions are restricted to the regime which is near the equilibrium. In the theory and applications of Nonlinear Partial Differential Equations, it is of considerable significance to investigate the global well-posedness and large-time behavior of strong solutions with general initial data. For Oldroyd-B system, Fang et al. [29], Jiang et al. [30] established the global existence of strong solutions with a class of large initial data. Recently, Chen et al. [1] investigated the global-in-time stability of large solutions for the system (1.1), and also proved the sharp time decay rates (for both upper and lower bounds) of the large solution $(u,\tau ,H)$ and all its derivatives except $\tau$. A natural question to ask is the following: can we obtain the sharp time decay rates (for both upper and lower bounds) of the higher spatial derivatives of $\tau$?

This paper is devoted to investigating the sharp time decay rates of the large solution $\tau$ for the system (1.1) and all its derivatives. To begin, we recall the time decay result of [1] as follows:

Proposition 1.1

For $f\left(x\right)=x$ and $\lambda =0$, let $(u,\tau ,H)$ be a global solution of the system (1.1) with the initial data $(u_0,{\tau }_0,H_0)\in L^1\left({\mathbb{R}}^2\right)\cap L^2\left({\mathbb{R}}^2\right)$, and assume that $tr{\tau }_0\ge 0$, $div{u}_0=0$, $div{H}_0=0$ and ${\left({\tau }_0\right)}_{ij}={\left({\tau }_0\right)}_{ji}$. Suppose that there exist two nonzero constants $c_0$ and ${\tilde{c}}_0$, such that

$${\int }_{{\mathbb{R}}^2}{u}_0\left(x\right)dx=c_0,{\int }_{{\mathbb{R}}^2}{H}_0\left(x\right)dx={\tilde{c}}_0,$$

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 $t\le T_1$,

$${\Vert (\nabla u,\nabla \tau ,\nabla H)\left(t\right)\Vert }_{H^2\left({\mathbb{R}}^2\right)}\le C_1.$$

Then, for large time $t$,

$$C^{-1}{(1+t)}^{-\frac{1}{2}-\frac{k}{2}}\le {\Vert {\nabla }^{k}u\left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-\frac{1}{2}-\frac{k}{2}},k=0,1,2,3,$$$$C^{-1}{(1+t)}^{-\frac{1}{2}-\frac{k}{2}}\le {\Vert {\nabla }^{k}H\left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-\frac{1}{2}-\frac{k}{2}},k=0,1,2,3,$$$$C^{-1}{(1+t)}^{-1}\le {\Vert \tau \left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-1},{\Vert {\nabla }^k\tau \left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-1-\frac{k}{2}},k=1,2,{\Vert {\nabla }^3\tau \left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-2},$$

where $C$ is a positive constant depending only on the initial data and $C_1$.

We can now state our main result.

Theorem 1.2

Under the assumptions of Proposition 1.1, then, for large time $t$,

$$C^{-1}{(1+t)}^{-1-\frac{k}{2}}\le {\Vert {\nabla }^k\tau \left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-1-\frac{k}{2}},k=0,1,2,$$$$C^{-1}{(1+t)}^{-\frac{5}{2}}\le {\Vert {\nabla }^3\tau \left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-2},$$

where $C$ is a positive constant depending only on the initial data and $C_1$.

Remark 1.3

In contrast with the decay estimate (1.2), we obtain the lower bound of $L^2$-decay rate of ${\nabla }^k\tau$ $(k=1,2,3)$. In fact, we note that Eq. (1.1)₂ was used for improving the upper bound of $L^2$-decay rate of ${\nabla }^k\tau$, which is equal to that of ${\nabla }^{k+1}u$ in [1]. For the highest spatial derivatives of order $3$ of $\tau$, it is impossible to derive any upper bound of $L^2$-decay rate of ${\nabla }^{4}u$. That is the reason why we cannot obtain the sharp time decay rate of ${\nabla }^3\tau$.

Remark 1.4

Theorem 1.2 is still valid for the corresponding Oldroyd-B system (i.e., $b=0$).

Remark 1.5

The special case which we consider in this paper is $f\left(x\right)=x$. Under appropriate assumptions on $f$, we shall see that Theorem 1.2 also holds true for an arbitrary function $f$, one can refer to [27].

Remark 1.6

To simplify the presentation, we only consider the case where $\lambda =0$. Similar results may be obtained for the general case where $\lambda \in [-1,1]$ under the additional conditions that large solution $(u,\tau ,H)\in C((0,+\infty );H^3\left({\mathbb{R}}^2\right))$. In contrast with the case $\lambda =0$, we draw the conclusion without the assumption $tr{\tau }_0\ge 0$, see [1] for more details.

Notation:For simplicity, we restrict our attention here to $\mu ={\mu }_1={\mu }_2=a=b=\nu =1$. Unless otherwise indicated, we henceforth omit ${\mathbb{R}}^2$ and set $H^s=H^s\left({\mathbb{R}}^2\right)$, $L^p=L^p\left({\mathbb{R}}^2\right)$. Throughout this paper, $C$ denotes a generic positive constant independent of time $t$.