Abstract
This paper is devoted to the stability and decay estimates of solutions to the two-dimensional magneto-micropolar fluid equations with partial dissipation. Firstly, focus on the 2D magneto-micropolar equation with only velocity dissipation and partial magnetic diffusion, we obtain the global existence of solutions with small initial in \(H^s({\mathbb {R}}^2)\) \((s>1)\), and by fully exploiting the special structure of the system and using the Fourier splitting methods, we establish the large time decay rates of solutions. Secondly, when the magnetic field has partial dissipation, we show the global existence of solutions with small initial data in \(\dot{B}^0_{2,1}({\mathbb {R}}^2)\). In addition, we explore the decay rates of these global solutions are correspondingly established in \(\dot{B}^m_{2,1}({\mathbb {R}}^2)\) with \(0 \le m \le s\), when the initial data belongs to the negative Sobolev space \(\dot{H}^{-l}({\mathbb {R}}^2)\) (for each \(0 \le l <1\)).
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1 Introduction
The magneto-micropolar equations were introduced in [1] to describe the motion of an incompressible, electrically conducting micropolar fluids in the presence of an arbitrary magnetic field. It belongs to a class of fluids with nonsymmetric stress tensor and includes, as special cases, the classical fluids modeled by the Navier-Stokes equation (see, e.g., [5, 31, 39]), magnetohydrodynamic (MHD) equations (see, e.g., [26]) and micropolar equations (see., e.g., [15, 16]). The 3D incompressible magneto-micropolar fluid equations can be written as:
where \(x = (x_1, x_2, x_3) \in {\mathbb {R}}^3\) and \(t\ge 0\), \(u(x, t), \omega (x, t), b(x, t)\) and \(\pi (x, t)\) denote the velocity of the fluid, microrotational velocity, the magnetic field and the hydrostatic pressure, respectively, \(\mu , \chi\) and \(\frac{1}{\nu }\) are, respectively, kinematic viscosity, vortex viscosity and magnetic Reynolds number. \(\gamma\) and \(\kappa\) are angular viscosities, and this is an isotropic system. The 3D magneto-micropolar equations reduce to the 2D micropolar equations when
More explicitly, the 2D incompressible magneto-micropolar fluid equations can be written as
where we have written \(u=(u_1, u_2)\), \(b=(b_1, b_2)\) and \(\omega\) for \(\omega _3\) for notational brevity. It is worth noting that, in the 2D case,
is a scalar function representing the vorticity, and \(\nabla \times \omega = (\partial _{2}\omega , -\partial _{1}\omega )\).
The magneto-micropolar equations play an important role in engineering and physics and have attracted considerable attention from the community of mathematical fluids (see, e.g., [20, 25, 28, 29]). When (2) has full dissipation (namely, \(\mu\), \(\chi\), \(\kappa\), \(\nu > 0\)), the global existence and uniqueness of solutions could be obtained easily (see, e.g., [20, 28]). However, for the inviscid case (namely, (2) with \(\mu > 0\), \(\chi >0\), \(\kappa = \nu = 0\) and \(\Delta u\) replaced by u), the global regularity problem is still a challenging open problem. Therefore, it is natural to study the intermediate cases, namely (2) with partial dissipation.
This paper aims at a system of the 2D magneto-micropolar equations that is closely related to (2),
Physically, the partial dissipation assumption is natural in the study of geophysical fluids. It turns out that, in certain regimes and under suitable scaling, certain dissipation can become small and be ignored. Anisotropic magnetic diffusion also arises in the modeling of reconnecting plasmas. When the resistivity of electrically conducting fluids such as certain plasmas and liquid metal is anisotropic and only in the mixed directions, the mixed magnetic diffusion may be relevant. In addition, mathematically, (3) allows us to explore the smoothing effect and the effect on large time behavior of the anisotropic magnetic diffusion. When the b with partial dissipation and zero angular viscosity, the global regularity problem for (3) can be quite difficult. However, many important progresses have recently been made on this direction (see, e.g., [6,7,8,9,10,11,12,13,14, 17, 23, 30, 36, 38, 42]). In [17, 23, 30], the global regularity of the 2D magneto-micropolar equations with various partial dissipation cases was obtained. Wang, Xu and Liu in [41] proved the uniqueness of global strong solution for the magneto-micropolar equations with zero angular viscosity in a smooth bounded domain. Yamazaki [43] obtained the global regularity of the Cauchy problem for the magneto-micropolar equations with zero angular viscosity
The magneto-micropolar equations share similarities with the Navier-Stokes equations, but they contain much richer structures than Navier–Stokes. It is well-known that the \(L^2\) decay problem of weak solutions to the 3D Navier–Stokes equations, i.e., (1) with \(\omega = 0\), \(b=0\) and \(\chi =0\), was proposed by the celebrated work of Leray [19]. By introducing the elegant method of Fourier splitting, the algebraic decay rate for weak solutions was first obtained by Schonbek [33]. Later, the result in [33] is sharpened and extended in [34], see also [35]. Recently, Niu and Shang [24] proved the \(L^2\)-decay estimates of weak solutions, and also proved the optimal decay rates of global solutions in \(\dot{H}^s({\mathbb {R}}^3)\)(\(s > \frac{3}{2}\)) and in \(\dot{B}^m_{2, 1}({\mathbb {R}}^3)\) with \(0 \le m \le \frac{1}{2}\). Shang and Gu [37] also proved the global existence of classical solutions for (3). Li [21] proved the \(L^2\)-decay estimates for global solutions of (8) and their derivate with initial data in \(L^1({\mathbb {R}}^2)\). In addition, Li [21] also shown the global stability of these solutions in \(H^s({\mathbb {R}}^2) (s>1)\) and the decay rates of global solutions and their higher derivates.
Motivated by the results of the magneto-micropolar equations [43] and the related fluid models [11, 18]. In this paper, the first theorem states that system (3) has a unique golbal solution when the initial data \((u_0, \omega _0, b_0)\) is sufficiently small in \(H^s({\mathbb {R}}^2)\), and obtain the upper bounds of time decay rates of the global solution to (3) in \(L^2 ({\mathbb {R}}^2)\), as stated in the following theorem.
Theorem 1
Let \(\mu > 0\), \(\chi >0\), \(\nu >0\) and \(\kappa >0\). Assume that \((u_0, \omega _0, b_0) \in H^s({\mathbb {R}}^2)\) with \(s>0\) and \(\nabla \cdot u_0 = \nabla \cdot b_0 = 0\). Then the following two statements hold:
(I) Let \(s>1\), then there exists a positive constant \(\epsilon _0\), such that for all \(0< \epsilon < \epsilon _0\), if
then system (3) has a unique global solution \((u, \omega , b)\) satisfying, for any \(t>0\),
where \(C>0\) is a constant independent of t.
(II) suppose that \((u_0, \omega _0,b_0)\in L^1({\mathbb {R}}^2)\), then the global solution \((u, \omega , b)\) has the following upper decay rates:
Moreover, when \(\mu < \sqrt{3} \chi\), then the global solution \((u, \omega , b)\) of the system (3) has the following upper decay rates
Finally, we consider the 2D magneto-micropolar equations with partial dissipation for the magnetic field, which can be written as
Motivated by the [21, 24], we establish the global existence results to system (8) in Besov spaces \(B^s_{2,1}({\mathbb {R}}^2)\). Furthermore, we study the large time decay rates of these global solutions in the Besove spaces \(B^s_{2,1}({\mathbb {R}}^2)\), as stated in the following theorem.
Theorem 2
Let \(\mu > 0\), \(\chi >0\), \(\nu >0\) and \(\kappa >0\). Assume that \((u_0, \omega _0, b_0) \in B^s_{2, 1}({\mathbb {R}}^2)\) with \(s>0\) and \(\nabla \cdot u_0 = \nabla \cdot b_0 = 0\). Then the following two statements hold:
(I) Let \(s\ge 1\), then there exists a positive constant \(\epsilon _0\), such that for all \(0< \epsilon < \epsilon _0\), if
then system (8) has a unique global solution \((u, \omega , b)\) satisfying, for any \(t>0\),
where \(C>0\) is a constant independent of t.
(II) Let \(s\ge 1\), suppose that \((u_0, \omega _0, b_0 ) \in \dot{H}^{-l}({\mathbb {R}}^2)\) with \(0 \le l < 1\). Then for all real numbers m with \(0 \le m \le s\), the global solution \((u, \omega , b)\) established in (I) satisfies the following decay estimates:
Remark 1
-
(i)
Since \(L^p({\mathbb {R}}^2) \hookrightarrow \dot{H}^{-l}({\mathbb {R}}^2)\) when \(l\in [0,1)\) and \(p \in (1,2]\), and \(L^p({\mathbb {R}}^2) \hookrightarrow \dot{B}^{-l}_{2, \infty }({\mathbb {R}}^2)\) when \(l\in (0,1]\) and \(p \in [1,2)\), thus Theorem 2 also hold for \((u_0, \omega _0, b_0) \in L^p({\mathbb {R}}^2)\) with \(p\in [1,2]\).
-
(ii)
Because of the divergence free condition \(\nabla \cdot b=0\), then \(\Vert \nabla b\Vert _{L^2({\mathbb {R}}^2)} = \Vert \nabla \times b\Vert _{L^2({\mathbb {R}}^2)}\), thus for the full dissipation 2D magneto-micropolar equations, we also have the same results as Theorem 1 and Theorem 2.
Remark 2
-
(i)
Compared to the classical magneto-micropolar equations (1), the full Laplacian operator is replaced by partial magnetic diffusion in systems (3) and (8). Theorem 1 and Theorem 2 indicate that the mixed partial magnetic diffusion has the same effect as the full Laplacian in deriving the large time behavior, in the sense that the decay rates in Theorem 1 and Theorem 2 coincide with the solutions of system (1).
-
(ii)
In Theorem 2, by assuming the initial data small in the critical Besov space \(\dot{B}_{2, 1}^{0}\), we can establish the global well-posedness to (8). However, due to the lack of micro-rotational velocity dissipation and the complex structure of the magneto-micropolar equations, it appears difficult to show the global well-posedness in critical Besov space to the solutions of (3).
To prove Theorem 2, we focus on the uniform bounds of \(\Vert (u, \omega , b)\Vert _{B^s_{2,1}}\). As preparation, we firstly show the global existence of solutions with small data in \(\dot{B}^0_{2,1}({\mathbb {R}}^2)\), then used the \(\Vert (u(t), \omega (t), b(t))\Vert _{\dot{B}^0_{2,1}} \le C \epsilon\), to obtain (10). The rest of this paper is divided into four sections. Sections 2 and 3 state the proofs of Theorem 1 and Theorem 2, respectively. An appendix containing the Littlewood-Paley decomposition, the definition of Besov spaces, and several useful calculus inequalities are also given for the convenience of the readers. To simplify the notation, we will write \(\partial _1\) for \(\partial _{x_1}\), \(\partial _2\) for \(\partial _{x_2}\), \(\int f\) for \(\int _{{\mathbb {R}}^2} f dx\), \(\Vert f\Vert _{L^p}\) for \(\Vert f\Vert _{L^p({\mathbb {R}}^2)}\), \(\Vert f\Vert _{\dot{H}^s}\) and \(\Vert f\Vert _{H^s}\) for \(\Vert f\Vert _{\dot{H}^s({\mathbb {R}}^2)}\) and \(\Vert f\Vert _{H^s( {\mathbb {R}}^2)}\) respectively, \(\Vert f\Vert _{\dot{B}^s_{p,r}}\) and \(\Vert f\Vert _{B^s_{p,r}}\) for \(\Vert f\Vert _{\dot{B}^s_{p,r}({\mathbb {R}}^2)}\) and \(\Vert f\Vert _{B^s_{p,r}({\mathbb {R}}^2)}\) respectively, and \(L_t^{q}(\dot{B}^s_{p,r})\) and \(\tilde{L}_t^{q}(\dot{B}^s_{p,r})\) for \(L_t^{q}(\dot{B}^s_{p,r}({\mathbb {R}}^2))\) and \(\tilde{L}_t^{q}(\dot{B}^s_{p,r}({\mathbb {R}}^2))\) respectively.
2 The Proof of Theorem 1
This section is devoted to the proof of Theorem 1. We first prove the global well-posedness part (I) of Theorem 1. As preparation, we give the following global a priori estimates.
Proposition 3
Let \((u_0, \omega _0, b_0) \in L^2\). Then for any \(t>0\), the solution \((u, \omega , b)\) of (3) satisfies
Proof
Taking the \(L^2\)-inner product to (3) with \((u, \omega , b_1, b_2)\) we have
where we used the facts that,
By Hölder’s inequality and the Young inequality, we have
Inserting (15) into (14), we obtain
Because of the divergence free condition \(\nabla \cdot b =0\), we have \(\Vert \nabla b\Vert _{L^2({\mathbb {R}}^2)}^2 = \Vert \nabla \times b\Vert _{L^2({\mathbb {R}}^2)}^2 \le 2 \left( \Vert \partial _2 b_1\Vert _{L^2({\mathbb {R}}^2)}^2 + \Vert \partial _1 b_2\Vert _{L^2({\mathbb {R}}^2)}^2 \right)\). Integrating (16) in [0, t], we can get
This completes the proof of Proposition 3\(\square\)
Next, we want to establish the global a priori \(H^s\) estimates. Applying \(\dot{\Delta }_j\) to (3), we have
where \([\dot{\Delta }_j, f \cdot \nabla ]g = \dot{\Delta }_j (f \cdot \nabla g) - f \cdot \dot{\Delta }_j (\nabla g)\) is commutator. Dotting (17) - (20) by \(\dot{\Delta }_j u\), \(\dot{\Delta }_j \omega\), \(\dot{\Delta }_j b_1\) and \(\dot{\Delta }_j b_2\) respectively, integrating the resulting equations in \({\mathbb {R}}^2\), and adding them together, we have
where we used the facts that
and
Due to the divergence free condition \(\nabla \cdot b = 0\), we have
Then, we can derive from the above inequalities
Due to
then, we have
Multiplying (22) by \(2^{2sj}\), taking the \(l^2_j\) over \(j \in {\mathbb {Z}}\), nothing that \(\dot{B}^s_{2,2} = \dot{H}^s\) and using Hölder’s inequality, we yield
where \(c_0 = \min \left\{\mu , \nu , \frac{8 \chi \mu }{\mu + 2 \chi } \right\}\). Adding the resulting inequality and (12) together, we have
Using commutator estimate (A5), and nothing that for \(s>1\),
one obviously derives
Similarly, we have
and
Taking advantage of the commutator estimate (A6), we imply that
Combining the above estimates together, we get
Then the Young inequality leads to
This inequality indicates that, if the initial data \((u_0, \omega _0, b_0)\) satisfy, for \(0< \epsilon < \epsilon _0 = \left(\frac{c_0}{2C} \right)^2\),
then the corresponding solution remains for all time. Namely,
In fact, if suppose (25) is not true and \(T_0\) is the first time such that (25) is violated, i.e.,
and (25) holds for any \(0 \le t < T_0\). We can deduce from (24) that for any \(0 \le t \le T_0\),
Therefore,
This is a contradiction. Thus, we get the uniform bound of (25). In addition,
Therefore, the proof of (I) of Theorem 1 is completed.
Next, we start to prove (II) of Theorem 1.
Proposition 4
Let \((u, \omega , b)\) be the global solutions of the system (3) with the initial data \((u_0, \omega _0, b_0) \in (L^1({\mathbb {R}}^2) \cap L^2({\mathbb {R}}^2) )^3\). Then \((u, \omega , b)\) satisfies the following inequality,
Proof of Proposition 4
Applying the Fourier transform to system (3), we obtain:
Multiplying the (30)\(_1\), (30)\(_2\), (30)\(_3\) and (30)\(_4\) by \(\bar{\hat{u}}\), \(\bar{\hat{\omega }}\), \(\bar{\hat{b}}_1\) and \(\bar{\hat{b}}_2\) respectively, and summing up, we have, noting that \(\vert \hat{u}\vert ^2 = \hat{u} \bar{\hat{u}}\)
For \(K_1\), taking divergence to the first equation of (3), one yields
And taking Fourier transformation obeys, nothing that \(\vert \hat{u}\vert = \vert \bar{\hat{u}} \vert\)
For \(K_2\),
Similarly, we obtain
Inserting \(K_1\) - \(K_{10}\) into (31), we derive that
which immediately yields
Integrating (32) in [0, t], we obtain
Thus the proof of Proposition 4 is completed. \(\square\)
Next, we obtain the result of Theorem 1 by using Proposition 4 and the generalized Fourier splitting method.
Let
where \(h(t) \in C^{\infty } [0, + \infty )\) is a positive function with respect to t and satisfies
where \(c_1 = \min \{ \mu , \nu , \frac{4 \chi \mu }{u + 2 \chi }\}\).
Multiplying both side of (12) by h(t), we have
By using the Plancherel Theorem for (34), we get
Applying (33), we can obtain
Combining the result of (35) and (36), we get
Employing (29), we have
Substituting (38) to (37), we have
Next, taking \(h(t) = [\ln (e + t)]^3\), then we have
and
Combining (39) - (41), we have
Now, taking \(h(t) = (1+t)^2\) and inserting it into (39), together with (42) and Hölder’s inequality, we have
where
From (43), we have
Taking \({\mathcal {N}}(t) = (1 + t) \left( \Vert u(t)\Vert _{L^2}^2 + \Vert \omega (t)\Vert _{L^2}^2 + \Vert b(t)\Vert _{L^2}^2 \right)\), then we have
Applying Gronwall’s inequality, we obtain
which implies the following decay
Therefore, the proof of (6) is completed.
Next, we will prove (7). The vorticity \(\Omega = \nabla \times u\), \(j = \nabla \times b\) satisfies
where
Due to the lack of angular viscosity for the system (3), it is crucial to deal with \(- 2 \chi \omega\) in (45) by introducing a new function \(Z = \Omega - \frac{2 \chi }{\mu + \chi } \omega\) in [11]. Subtracting \(\frac{2\chi }{\mu + \chi } \times\) (3)\(_2\) from (45), we have
Taking the \(L^2\)-inner products of (47), (3)\(_2\) and (46) with Z, \(\omega\) and j, respectively, we obtain
where
due to the divergence free condition \(\nabla \cdot b = \partial _1 b_1 + \partial _2 b_2\). By Hölder’s inequality
where we have used the fact that the Calderon-Zygmund operators are bounded on \(L^p (1< p < + \infty )\). It is easy to verify that
Indeed,
Then it follows from the above bounds and (50) that
Combining (48), (49) and (51), we have
Next, we consider \(I_1\) - \(I_6\), respectively. Applying the Young inequality
By using Hölder’s inequality, the Gagliardo-Nirenberg inequality, and the Young inequality,
and
Inserting \(I_1\) - \(I_6\) into (52), we have
Due to \(\mu < \sqrt{3} \chi\), then
Applying Gronwall’s inequality, we have
where we also used (13).
Multiplying (12) by \((1+ t)^n\), by (6), we have
where \(c_1 = \min \{\mu , \frac{4 \chi \mu }{\mu + 2 \chi }, \nu \}\). Integrating (55) in time, we have
Multiplying (53) by \((1+ r)^n\) and integrating with respect to r over \((\frac{t}{2}, t)\), we have
Nothing that
Combining the results of (57) and (48), for \(t \ge 1\), we get
then we have
Therefore, the proof of Theorem 1 is completed.
3 The Proof of Theorem 2
This section is devoted to the proof of Theorem 2. Firstly, we first prove the global stability part (I) of Theorem 2. As we know, the key step is to establish the global \(a \ priori \ B^s_{2, 1}({\mathbb {R}}^2)\) estimates of the solution.
Proof of (I) of Theorem 2
As preparation, in the following proposition, we state that system (8) has unique global solution when the initial data \((u_0, \omega _0, b_0)\) is sufficiently small in \(\dot{B}^0_{2,1}({\mathbb {R}}^2)\).
Proposition 5
Assume \((u_0, \omega _0, b_0)\) satisfies the conditions in (I) of Theorem 2 and (9), then system (8) has a unique global solution \((u, \omega , b)\) satisfying, for any \(t > 0\),
Proof
Now, we turn to establish the global \(a \ priori \ B^s_{2,1}\) estimates. Applying \(\dot{\Delta }_j\) to (8), we have
where \([\dot{\Delta }_j, f \cdot \nabla ]g = \dot{\Delta }_j (f \cdot \nabla g) - f \cdot \dot{\Delta }_j (\nabla g)\) is commutator. Dotting (60) - (63) by \(\dot{\Delta }_j u\), \(\dot{\Delta }_j \omega\), \(\dot{\Delta }_j b_1\) and \(\dot{\Delta }_j b_2\) respectively, integrating the resulting equations in \({\mathbb {R}}^2\), and adding them together, we have
where we used the facts that
and
Due to the divergence free condition \(\nabla \cdot b = 0\), we have
Then, we can derive from the above inequalities
Due to
then, we have
Applying Hölder’s inequality, the Young inequality, Bernstein’s inequality and the divergence free condition \(\nabla \cdot u = \nabla \cdot b = 0\) to (65), we obtain
Then, (66) implies
where \(c_2 = \min \{\mu , \nu , 2\kappa \}\). For (67), applying Bernstein’s inequality and integrating it in [0, t], we obtain
Taking the \(l^1_j\) over \(j \in {\mathbb {Z}}\), one yields
Denote \(c_3 = 2 c_1 \tilde{c}_2 c_2^{\star }\). Using Lemma 13, Lemma 14, and noting that \(\Vert f\Vert _{L^{\infty }} \le C \Vert f\Vert _{\dot{B}^1_{2.1}}\) and \(\Vert f\Vert _{\tilde{L}^1_t (\dot{B}^s_{2,1})} \approx \Vert f\Vert _{L^1_t(\dot{B}^s_{2,1})}\), we have
This inequality indicates that, for any \(0< \epsilon < \epsilon _0\),
then bootstrap argument yields
which completed the Proposition 5. \(\square\)
Next, we start to prove the decay estimates assertion of (II). As a tool, we first verify the following Proposition in the negative Sobolev space \(\dot{H}^{-l}\), with \(0< l < 1\).
Proposition 6
Let \(c_2 = \min \{\mu , 2\kappa , \nu \}\). Then for \(0< l <1\), we have
Proof of Proposition 6
Due to (64) and with the divergence free condition \(\nabla \cdot u = \nabla \cdot b = 0\), we derive that
Multiplying the above inequality by \(2^{-2lj}\) and taking the \(l_j^2\) over \(j \in {\mathbb {Z}}\), and noting that \(\dot{B}^{-l}_{2,2} = \dot{H}^{-l}\), we have
Applying Lemma 10, Hölder’s inequality and the Gagliardo-Nirenberg inequality, we obtain
and
Then
Similarly,
Combining (72) - (76), we obtain
where \(c_2 = \min \{\mu , 2\kappa , \nu \}\). Thus the proof of Proposition 6 is completed. \(\square\)
Next, we continue to prove (10). Multiplying (67) by \(2^{sj}\) and utilizing Bernstein’s inequality and integrating it in [0, t], finally, taking the \(l^1_j\) over \(j \in {\mathbb {Z}}\), we have
Since \((u_0, \omega _0, b_0) \in B^s_{2,1}({\mathbb {R}}^2)\), we have
By Lemma 13, Lemma 14 and noting that \(\Vert f\Vert _{L^1_t(\dot{B}^s_{2,1})} \approx \Vert f\Vert _{\tilde{L}^1_t(\dot{B}^s_{2,1})}\) and \(\Vert f\Vert _{L^{\infty }} \le C \Vert f\Vert _{\dot{B}^1_{2,1}}\), yield
For \(A_3\) - \(A_5\), by the similar method as \(A_2\), together with the Young inequality, we get
Similarly,
Inserting \(A_1\) - \(A_5\) into (77), we derive that
Combining (78) and (59) with \(\epsilon\) sufficiently small, we have
Because \(\dot{B}^0_{2,1} \hookrightarrow \dot{B}^0_{2,2}\) and \(\dot{B}^0_{2,2} \sim \dot{H}^0 \sim L^2\), thus (59) implies
Combining (79) and (80), we have
which completed the proof of (I) in Theorem 2. \(\square\)
We now turn to prove the decay part (II) of Theorem 2.
Proof of (II) of Theorem 2
Multiplying (67) by \(2^{mj}\), and taking the \(l^1_j\) over \(j\in {\mathbb {Z}}\), we obtain
where \(y(t) = \left\| 2^{mj} \sqrt{\Vert \dot{\Delta }_j u\Vert ^2 + \Vert \dot{\Delta }_j \omega \Vert ^2 +\Vert \dot{\Delta }_j b\Vert ^2 }\right\| _{l^1_j}\). Using Lemma 13 and Lemma 15, together with \(\Vert f\Vert _{L^{\infty }} \le C \Vert f\Vert _{\dot{B}^1_{2,1}}\), we have
Then this inequality, together with (59) with \(\epsilon\) sufficiently small, we have
Applying Lemma 15 and \(\dot{B}^s_{2,2} \hookrightarrow \dot{B}^s_{2, \infty }\), we infer that
and
Therefore, if
then we can obtain from (81) - (83), there exists a constant \(a_0 >0\) such that,
It follows from this that
which implies
which immediately yields (11).
Finally, to make the process more complete, we need to verify that (85) holds for \(0 \le l <1\). To this end, we divide the proof into two cases.
Case 1. (\(l=0\)) Using the fact that \(\dot{H}^0 = L^2\), by (13) we have
Then it yields (85).
Case 2. (\(0<l < 1\)) Assume that
Suppose that for all \(t \in [0,T]\)
If we can derive that for all \(t \in [0, T ]\),
then an application of the bootstrapping argument would imply that the solution \((u, \omega , b)\) of system (8) satisfies (90) for all \(t \in [0, T]\), which implies (85). With (86) and (87) at our disposal, we show that (90) holds.
With the help of (87) and Lemma 8, we know that
Similarly,
Integrating (71) in [0, t] with \(0<t\le T\), together with (80), and (91) - (94), one infers that
By choosing \(\epsilon\) in (80) sufficiently small, then the above inequality yields (90) for all \(t \in [0, t]\), which closes the proof. Then we have (85) and completed the proof of (11). \(\square\)
Data Availability
No data was used for the research described in the article.
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Appendix A: Functional Space
Appendix A: Functional Space
This appendix provides the definition of Littlewood-Paley decomposition and the definition of Besov space. Some related inequalities used in the previous sections are also included. Materials presented in this appendix can be found in several books and many papers (see, e.g., [3, 4, 22, 27, 40]).
We start with several notations. \({\mathcal {S}}\) denotes the usual Schwarz class and \(\mathcal {S'}\) its dual, the space of tempered distributions. To introduce the Littlewood-Paley decomposition, we write for each \(j\in {\mathbb {Z}}\)
The Littlewood-Paley decomposition asserts the existence of a sequence of functions \(\{\Phi _{j}\}_{j\in {\mathbb {Z}}} \subset {\mathcal {S}}\) such that
and
Therefore, for a general function \(\psi \in {\mathcal {S}}\), we have
We now choose \(\Psi \in {\mathcal {S}}\) such that
Then, for any \(\psi \in {\mathcal {S}}\),
and hence
in \(\mathcal {S'}\) for any \(f \in \mathcal {S'}\). To define the inhomogeneous Besov space, we set
To define the homogeneous Besov space, we set
Besides the Fourier localization operators \(\Delta _j\), the partial sum \(S_j\) is also a useful notation. For an integer j,
For any \(f \in \mathcal {S'}\), the Fourier transform of \(S_j f\) is supported on the ball of radius \(2^j\). It is clear from (A1) that \(S_j \rightarrow Id\) as \(j \rightarrow \infty\) in the distributional sense.
Definition 1
(see, e.g., [4, 22])The inhomogeneous and homogeneous Besov spaces \(B^s_{p,q}\) and \(\dot{B}^s_{p,q}\) with \(s \in {\mathbb {R}}\) and \(p,q \in [1, \infty ]\) consists of \(f \in \mathcal {S'}\) and \(f \in \mathcal {S'} {\setminus } {\mathcal {P}}\), respectively, satisfying
and
respectively, where \({\mathcal {P}}\) represents the set of polynomials.
Many frequently used function spaces are special cases of Besov spaces. The following lemma lists some useful equivalence and embedding relations.
Lemma 7
(see, e.g., [4, 22]) For any \(s\in R\),
For any \(s\in {\mathbb {R}}\) and \(1< q < \infty\),
For any non-integer \(s>0\), the Hölder space \(C^s\) is equivalent to \(B^s_{\infty , \infty }\).
In the following Lemmas, we stated a Sobolev-type embedding theorem for Besov space.
Lemma 8
(see [4]) Let \(1\le p_1 \le p_2 \le \infty\) and \(1 \le r_1 \le r_2 \le \infty\). Then, for any real number s, the space \(\dot{B}^s_{p_1, r_1}\) is continuously emdedded in \(\dot{B}^{s-d(\frac{1}{p_1} - \frac{1}{p_2})}_{p_2, r_2}\).
Lemma 9
(see [2]) For every \(s \in {\mathbb {R}}\), \(\epsilon > 0\), \(1< p < + \infty\) and \(1 \le q \le + \infty\), we have
Lemma 10
(see [4]) If p belongs to (1, 2], then \(L^{p}({\mathbb {R}}^d)\) embeds continuously in \(\dot{H}^s ({\mathbb {R}}^d)\) with \(s = \frac{d}{2} - \frac{d}{p}\).
We also used the space-time space defined below.
Definition 2
(see, e.g., [4, 22]) For \(t>0\), \(s\in {\mathbb {R}}\) and \(1 \le p,q,r \le \infty\), the inhomogenous and homogenous space-times spaces \(L^r_{t} B^s_{p,q}\), \(L^r_{t} \dot{B}^s_{p,q}\) and \(\tilde{L}^r_{t} B^s_{p,q}\), \(\tilde{L}^r_{t} \dot{B}^s_{p,q}\) are defined through the norms
and
respectively.
The inhomogeneous space-time space has the following properties.
As \(q=r\),
The homogeneous space-time space has similar properties.
Bernstein’s inequalities are useful tools in dealing with Fourier localized functions. These inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives. The upper bounds also hold when the fractional operators are replaced by partial derivatives.
Lemma 11
(see, e.g., [4, 22]) Let \(\alpha \ge 0\) and \(1 \le p \le q \le \infty\). 1) If f satisfies
for some integer j and a constant \(K>0\), then
2) If f satisfies
for some integer j and constants \(0< K_1 \le K_2\), then
where \(C_1\) and \(C_2\) are constants depending on \(\alpha , p\) and q.
Next, we give several useful calculus inequalities. We first give two lemmas regarding commutator estimates and product law. Lemma 12 and Lemma 13 below with \(p_1 = q_2 = \infty\) and \(q_1 = p_2\) have previously been obtained in [4, 22]. Here we state the following more general cases without detailed proofs since they can be proved by following the methods in [4, 22].
Lemma 12
Let \(s > -1\), \((p, r, p_1, p_2, q_1, q_2) \in [1, \infty ]\) with \(\frac{1}{p} = \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}\) and u be a smooth divergence free vector field. Then for \(j\in {\mathbb {Z}}\),
where \([\dot{\Delta }_j, u \cdot \nabla ]v = \dot{\Delta }_j (u \cdot \nabla v) - u \cdot \dot{\Delta }_j(\nabla v)\).
Lemma 13
Suppose that \(s>0\) and \((p, r, p_1, p_2, q_1, q_2) \in [1, \infty ]\) with \(\frac{1}{p} = \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}\). Then the following hold true
For inhomogeneous Besov space has the similar inequality.
Finally, we recall the following Besov space interpolation estimate and the inequality for homogeneous Besov space.
Lemma 14
(see [32]) Fixed \(m> l > k\), and \(1 \le p \le q \le r \le \infty\), we have
These parameters satisfy the following restrictions
Also \(1 \le p' \le q' \le r' \le \infty\) and solving we have \(\theta = \frac{m-l}{m-k} \in (0,1]\).
Lemma 15
(see [4, 22]) Let s, \(s_1\) and \(s_2\) be real numbers. Let \(s_1 < s_2\), \(0< \theta <1\), \(1 \le p \le \infty\) and \(1 \le r_1 \le r_2 \le \infty\). Then
For inhomogeneous Besov space has the similar inequality.
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Li, M., He, J. Large Time Behavior and Stability for Two-Dimensional Magneto-Micropolar Equations with Partial Dissipation. J Nonlinear Math Phys 30, 1567–1600 (2023). https://doi.org/10.1007/s44198-023-00144-2
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DOI: https://doi.org/10.1007/s44198-023-00144-2