Abstract
This paper is devoted to the large time decay of solutions of a three-dimensional Stokes-Magneto equations. It is shown that, when initial data belong to \(L^2\), weak solutions of the equations decay to zero in \(L^{3/2,\infty }\times L^2\) without a uniform rate, and this decay estimate is optimal. Furthermore, the optimal temporal decay estimates for weak solutions are established when initial data belongs to \(L^1\cap L^2\).
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1 Introduction
In this paper, we study the following equations
Here u is the velocity field, b is the magnetic field, \(p_*=p+\frac{1}{2}|b|^2\) is the total pressure, p is the pressure, \(\nu >0\) is the viscosity coefficient and \(\eta >0\) is the magnetic resistivity coefficient.
Equations (1.1)–(1.3) is obtained by removing the advective terms \((\partial _t+u\cdot \nabla )u\) from the u equation of the magnetohydrodynamics (MHD) equations. It is well-known that MHD equations, which was first derived by Alfvén, govern the motion of the electrically conducting fluids arising from plasmas, liquid metals, and electrolytes, etc (see [12]). It is also known that MHD equations are one of the most important equations in the study of phenomena arising from geophysics, astrophysics, cosmology and engineering (see, e.g., [2, 5]).
Equation (1.1)–(1.3) is closely connected with the method of magnetic relaxation (see [14]). When \(\eta =0\), Moffatt [16] argued that (1.1)–(1.3) on a smooth bounded domain \(\Omega \) should produce a magnetostatic equilibrium \(b^E(x)\) that satisfies
if the topology of the magnetic field is non-trivial. Note that (1.5) almost shares the same form with the steady Euler equation:
if one “identifies” \(b^E\) with velocity field \(u^E\), here \(\nabla h^E=\nabla (p_e+\frac{1}{2}|u^E|^2)\) and \(p_e\) denotes the pressure of the Euler equation. This indicates that the study of (1.1)–(1.3) might be helpful to understand the unstable Euler flows. Moffatt also argued that the steady state of some non-resistive MHD equations should also obey (1.5) (see [15]). However, there is no rigorous proof that the magnetic relaxation will yield a steady Euler flow. One of the reasons is that the global well-posedness of 3D MHD equations remains open (see [14] and references therein).
From a limiting state point of view, the dynamical model used to obtain the above steady state is not particularly important (see [8, 14, 16]). In fact, it was argued by Moffatt that dropping the acceleration terms from the u equation and working with a “Stokes” model might prove more mathematically amenable (see [8, 14, 16]). In recent years, the well-posedness of (1.1)–(1.4) and related models have attracted great attention. McCormick et al. proved the existence of weak solutions of the equations in [14], where the uniqueness of weak solutions for two-dimensional case is also shown. Furthermore, they proved that weak solutions of the 2D equations become regular if \(b_0\) is smooth (see [14]). We refer readers to [4, 8] for the local-in-time existence of regular solutions of 3D non-resistive MHD equations. Recently, we established an optimal regularity criterion for (1.1)–(1.4), and studied the global-in-time existence of strong solutions when initial data is small in critical Sobolev spaces or critical Besov spaces (see [22]). We also established global-in-time existence of strong solutions of the equations with arbitrary initial data when \(-\Delta \) in (1.2) is replaced by \((-\Delta )^\alpha \) with \(\alpha \ge 3/2\) (see [10]).
The purpose of this paper is to investigate the decay of weak solutions of (1.1)–(1.4). The analysis of decay of solutions of fluid flow motions originally goes back to Leray [13], in which he asked whether or not weak solutions of 3D Navier–Stokes equations decay to zero in \(L^2\) as time tends to infinity. Since then, this kind of problem has been extensively studied, see [3, 17,18,19,20] for Navier–Stokes equations and [1, 6, 7, 21] for MHD equations. In this paper, motivated by the work of [1, 20], we show that the \(L^{3/2,\infty }\) norm of velocity u and \(L^2\) norm of magnetic field b are decay to zero without a uniform rate when initial data belong to \(L^2\). It stated as follows:
Theorem 1.1
Let \(b_0\in L^2\) with \(\nabla \cdot b_0=0\). Assume that (u, b) a weak solution of the initial value problem (1.1)–(1.4). Then (u, b) satisfies
For the proof, it should be pointed out that in contrast with Navier–Stokes equations or MHD, there is no a priori bound for u in \(L^\infty (\mathbb {R}_+,L^2(\mathbb {R}^3))\) because of the absence of \(\partial _tu\) in (1.1). Note that this estimate for u is crucial to study the corresponding part for Navier–Stokes or MHD (see [1, 17, 20]). However, by the energy estimate of solutions, we would overcome this difficulty by some proper interpolation inequalities, see Sect. 3 below.
Furthermore, we show that the decay result obtained in Theorem 1.1 is optimal in the sense that for any sphere with radius \(\alpha \) in \(L^2(\mathbb {R}^3)\), there exists a \(b_0\) on the sphere such that the corresponding solutions should decay arbitrarily slow. The result is stated as follows:
Theorem 1.2
For any \(T>0\), \(\alpha >0\) and \(0<\epsilon <1\), there exists \(b_0\in L^2\) with \(\nabla \cdot b_0=0\) and \(\Vert b_0\Vert _{L^2}=\alpha \), such that if (u, b) is a weak solution of (1.1)–(1.4) corresponding to the initial data \(b_0\), then \(\frac{\Vert b(T)\Vert _{L^2}}{\Vert b_0\Vert _{L^2}}\ge 1-\epsilon \).
Motivated by [1, 20], we prove this result by choosing a suitable scaling transform \(\delta _\lambda \) on \(L^2\) that preserves \(L^2\)-norm. Since this scaling does not preserve the semi-norm in \(\dot{H}^1\), we assume without loss of generality that \(b_0\) belongs to a more regular space, say \(H^1\). Taking \(\delta _\lambda b_0\) as the initial data, we establish a global-in-time bound of solutions of (1.1)–(1.4) when \(\lambda \) is small, then we show Theorem 1.2, see Sect. 4.
The non-uniform decay of weak solutions derived in Theorem 1.1 can be improved if initial data satisfies some additional assumptions. More precisely, it is shown that when \(b_0\) belongs to \(L^1\cap L^2\), the \(L^2\) norm of b(t) will decay like \(O(t^{-3/4})\) as \(t\rightarrow \infty \). This indicates that, on the one hand, the temporal decay of \(\Vert b(t)\Vert _{L^2}\) can be uniformly dominated by \(t^{-3/4}\) (in the sense of ‘\(\lesssim \)’). On the other hand, this decay rate is optimal in the sense that the lower bound for rate of decay is proportional to \(t^{-3/4}\). The two results are stated as follows:
Theorem 1.3
Let \(b_0\in L^1\cap L^2\) with \(\nabla \cdot b_0=0\). Then there exists a Leray-Hopf weak solution of the initial value problem (1.1)–(1.4), which satisfies
Theorem 1.4
Let \(R_\beta =\{f\in L^1:\inf _{|\xi |\le \beta }|\hat{f}(\xi )|\ge \beta \}\) for \(\beta >0\) and let \(b_0\in L^1\cap L^2\cap R_\beta \) with \(\nabla \cdot b_0=0\). Then there exists a Leray-Hopf weak solution of (1.1)–(1.4) such that
Remark 1.5
The space \(R_\beta \) plays a crucial role in the proof of the optimal decay rate of solutions. When \(b_0\in L^2\cap R_\beta \), the weak solution of the heat equation
decays at most as \(\Vert b'(t)\Vert _{L^2}\ge c(\eta ,\beta )(1+t)^{-3/4}\) (see Lemma 2.5 below). Compared with the linear equation (1.10), the system (1.1)–(1.4) contains complicated nonlinearities, but they do not make the decay of solutions worse (see Sect. 6). Thus, if \(b_0\in L^2\cap R_\beta \), the component b of the weak solutions (u, b) of the system in general cannot decay faster than \((1+t)^{-3/4}\).
Remark 1.6
The space \(R_\beta \) is strictly contained in \(L^1\). This means that there are functions that contained in \(L^1\) but they do not belong to \(R_\beta \). In fact, let \(\chi :\mathbb {R}^3\rightarrow [0,1]\) be a smooth function that satisfies \(\chi (x)=1\) for \(|x|\le 1\) and \(\chi (x)=0\) for \(|x|\ge 2\). Let \(\phi (x)=\chi (x)-\chi (2x)\). Then both \(\chi \) and \(\phi \) are Schwartz functions on \(\mathbb {R}^3\). It is clear that \(\mathcal {F}^{-1}\chi \in R_\beta \), and \(\mathcal {F}^{-1}\phi \in L^1\) whereas \(\mathcal {F}^{-1}\phi \notin R_\beta \), here \(\mathcal {F}^{-1}\) denotes the Fourier inverse transform.
The proof of Theorems 1.3 and 1.4 are based on the Fourier splitting method [19]. The main task is to estimate \(\hat{b}\) for its lower frequency part. We point out that there is a difficulty similar as that has been stated above. That is, the absence of \(\partial _tu\) in (1.1) leads to the absence of a priori bound of u in \(L^\infty (\mathbb {R}_+,L^2(\mathbb {R}^3))\), and this is much unlike the Navier–Stokes equations [17, 19, 20] or MHD equations [1, 21]. However, we overcome the difficulty by energy estimate of solutions and applications of interpolation inequalities.
2 Preliminaries
Throughout the paper, c represents a positive constant (depending only on \(\nu \), \(\eta \)) whose value may change at each occurrence. \(A\lesssim B\) denotes the inequality \(A\le cB\). \(c(\alpha _1,\alpha _2,\ldots )\) stands for a positive constant that depends on \(\alpha _1\), \(\alpha _2\), ...etc. We denote by \(\hat{f}\) the Fourier transform of f, while the inverse Fourier transform of f is denoted by \(\mathcal {F}^{-1}f\). We consider function spaces on \(\mathbb {R}^3\), for instance, \(C_c^\infty :=C_c^\infty (\mathbb {R}^3)\), \(L^p:=L^p(\mathbb {R}^3)\), \(H^s:=H^s(\mathbb {R}^3)\). \(L^{p,\infty }\) denotes the weak \(L^p\) space. We will use \(\int :=\int _{\mathbb {R}^3}\), \(\Vert \cdot \Vert _p:=\Vert \cdot \Vert _{L^p}\) and \(\Vert \cdot \Vert :=\Vert \cdot \Vert _{2}\) for convenient. We define \(\mathcal {D}_\sigma =\{\mathbf{f}\in C_c^\infty :\nabla \cdot \mathbf{f}=0\}\). Let \(L_\sigma ^2\) and \(H_\sigma ^1\) be the closure of \(\mathcal {D}_\sigma \) in the \(L^2\) and \(H^1\) norm, respectively.
Definition 2.1
[14] Let \(T>0\) and let \(b_0\in L_\sigma ^2\). A function (u, b) is called a weak solution of the equation (1.1)–(1.4) on (0, T), if
-
(i)
\(u\in L^\infty (0,T;L^{3/2,\infty })\cap L^2(0,T;H^1_\sigma )\) and \(b\in L^\infty (0,T;L^2)\cap L^2(0,T;H^1_\sigma )\),
-
(ii)
(u, b) verifies:
$$\begin{aligned}&\int \nu \nabla u:\nabla \phi _1+(b\cdot \nabla )\phi _1\cdot b\mathrm{d}x=0, \\&\int b_0\cdot \phi _2(0)\mathrm{d}x{-}\int _0^T\int b\cdot \partial _t\phi _2{-}\eta \nabla b:\nabla \phi _2{+}(u\cdot \nabla ) \phi _2\cdot b{-}(b\cdot \nabla )\phi _2\cdot u\mathrm{d}x\mathrm{d}t{=}0, \end{aligned}$$for all test functions \(\phi _1\), \(\phi _2\in C_c^\infty ([0,T);\mathcal {D}_\sigma )\).
Motivated by Ogawa, Rajopadhye and Schonbek [17] about the decay of weak solutions of forced Navier–Stokes equations or Agapito and Schonbek [1] about the analysis of decay of MHD equations, we formulate the following technical lemma:
Lemma 2.2
Let \(b_0\in L^2_\sigma \). Assume that (u, b) is a weak solution of (1.1)–(1.4). Then for \(E(t)\in C^1(\mathbb {R};\mathbb {R}_+)\) with \(E(t)\ge 0\) and \(\psi \in C^1(\mathbb {R};C^1\cap L^2)\) such that \(\psi (t)\) is radial on \(\mathbb {R}^3\), the solution (u, b) satisfies the following equations:
and
for all \(0\le s\le t\le \infty \).
Proof
We first give the proof of (2.2). Taking the inner product of (1.2) with \(2E(t)\psi *\psi *b(t)\), then integrating in \([s,t]\times \mathbb {R}^3\), we obtain that
The main task is to deal with the terms on the left-hand side of (2.3). For the first term, by integration by parts, we have
here \(\psi \) is radial has been used to derive the third equation. Thus, we apply Parseval’s relation to obtain that
Since \(\psi \) is radial, it follows that \(\bar{\hat{\psi }}=\hat{\psi }\) and \((\bar{\hat{\psi }})^2=|\hat{\psi }|^2\). Hence the previous equation turns to the following:
here Plancherel’s theorem has been used to deduce the second equation.
For the second term, a similar computation gives that
Substituting (2.6) and (2.7) into (2.3), we conclude that (2.2) holds.
The proof of (2.1) is slightly simpler and can be shown in a way similar to that of (2.2). Taking the inner product of (1.1) with \(2E(t)\psi *\psi *u(t)\), then integrating in \([s,t]\times \mathbb {R}^3\), we obtain that
By repeating the manipulation of the derivation of (2.7), we know that
Substituting (2.9) into (2.8), it follows that (2.1) holds. The proof of Lemma 2.2 is completed. \(\square \)
The following result is a straightforward of Lemma 2.2.
Corollary 2.3
Let \(b_0\in L^2_\sigma \). Assume that (u, b) is a weak solution of (1.1)–(1.4). Then for a radial function \(\varphi \in L^2\), (u, b) satisfies
for all \(0\le s\le t\le \infty \).
Proof
For any \(\epsilon >0\), set \(\psi (\tau )=\mathcal {F}^{-1}(e^{-\eta |\xi |^2(t+\epsilon -\tau )}\varphi (\xi ))\) and \(E(t)=1\) in (2.2), we deduce that
By integration by parts, it is seen that
Substituting (2.12) into (2.11), we have
By passing to the limit as \(\epsilon \rightarrow 0\) in (2.13), we finally conclude that (2.10) holds true. This completes the proof of Corollary 2.3. \(\square \)
The following \(L^p\)-\(L^q\) estimate for heat operator will be frequently used in the rest of the paper.
Lemma 2.4
[11]. Let \(\mu >0\), \(1\le p\le q\le \infty \), \(f\in L^p\) and let \(m\ge 0\). Then the following \(L^p\)-\(L^q\) estimate holds
Lemma 2.5
[20]. Let \(\mu >0\), \(f\in L^2\cap R_\beta \) for some \(\beta >0\). Let \(e^{\mu t\Delta }f=\int K_t(x-y)f(y)dy\) with \(K_t(x)=\frac{1}{(4\pi \mu t)^{3/2}}e^{-\frac{|x|^2}{4\mu t}}\). Then there exists \(c(\mu ,\beta )>0\) such that \(\Vert e^{\mu t\Delta }f\Vert \ge c(\mu ,\beta )(1+t)^{-\frac{3}{4}}\).
3 Proof of Theorem 1.1
We begin with the \(L^{3/2,\infty }\times L^2\) estimate of (u, b). Taking the inner product of (1.1) and (1.2) with u and b, respectively, then integrating in \(\mathbb {R}^3\) and summing the resultant equations, we use integration by parts and (1.3) to obtain that
Integrating with respect to t, we deduce that for any \(t\ge 0\),
Based on this \(L^2\) estimate of b, we can deduce that u(t) is bounded in \(L^{3/2,\infty }\) (see [10, 14, 22]). In fact, consider the following nonhomogeneous Stokes equation
we know that \((u,p_*)\) is solved by
here \((\mathbf{U}(\cdot ),\mathbf{q}(\cdot ))\) is the fundamental solution of Stokes equations and \(\mathbf{U}(x)=O(|x|^{-1})\) as either \(|x|\rightarrow 0\) or \(|x|\rightarrow \infty \), see Section IV.2 in [9] for details. Thus, \(\nabla \mathbf{U}\in L^{3/2,\infty }\). Moreover, by \(\nabla \cdot b=0\) and Young inequality in weak \(L^p\) spaces, we deduce that for any \(t\ge 0\),
Let \(\varphi :\mathbb {R}^3\rightarrow [0,1]\) be a smooth, radial cutoff function such that \(\varphi (\xi )=1\) for \(|\xi |\le 1\), \(\varphi (\xi )=0\) for \(|\xi |\ge 2\). By Plancherel’s theorem, \(\Vert b(t)\Vert =\Vert \hat{b}(t)\Vert \le \Vert \varphi \hat{b}(t)\Vert +\Vert (1-\varphi )\hat{b}(t)\Vert \). It suffices to show that \(\lim _{t\rightarrow \infty }\Vert \varphi \hat{b}(t)\Vert =0\) and \(\lim _{t\rightarrow \infty }\Vert (1-\varphi )\hat{b}(t)\Vert =0\), respectively.
We apply (2.10) and Plancherel’s theorem to obtain
By Young inequality, the second term on the right-hand side of (3.5) is bounded by (in the sense of ‘\(\lesssim \)’)
Combining (3.5) and (3.6), and passing to the limit as \(t\rightarrow \infty \), we use Lebesgue’s dominated convergence theorem and (3.2) to deduce that
This implies that \(\lim _{t\rightarrow \infty }\Vert \varphi \hat{b}(t)\Vert =0\).
Next, we prove that \(\lim _{t\rightarrow \infty }\Vert (1-\varphi )\hat{b}(t)\Vert =0\). Let \(\gamma >0\). Setting \(E(t)=(1+t)^\gamma \) and \(\psi =\mathcal {F}^{-1}(1-\varphi )\) in (2.1) and (2.2), then summing the resultant equations, we apply Plancherel’s theorem to deduce that
The bound of \(I_2+I_3\) is obtained by Fourier splitting method (see [19, 20]). In fact, for each \(t>0\), we define \(r(t)=\sqrt{\frac{\gamma }{2\eta (1+t)}}\), here \(\gamma >0\) is given in the previous paragraph. Thus, there holds
Since \(E'(\tau )-2\eta E(\tau )r^2(\tau )=0\) for all \(\tau \in [s,t]\), we deduce that
Now we estimate \(I_4+I_5+I_6\). Let \(\zeta =-2\varphi +\varphi ^2\). Then \(\zeta \in C^\infty _c\) and \(\mathcal {F}^{-1}\zeta \) belongs to Schwartz space. By the divergence free condition (1.3), there hold \(\langle \widehat{u\cdot \nabla b},\hat{b}\rangle =0\), \(\langle \widehat{b\cdot \nabla b},\hat{u}\rangle +\langle \widehat{b\cdot \nabla u},\hat{b}\rangle =0\). Thus, we use Plancherel’s theorem to deduce that
Substituting (3.9) and (3.10) into (3.8), then multiplying the resultant equation by \(E(t)^{-1}\), we obtain
Passing to the limit as t, \(s\rightarrow \infty \), we have
This completes the proof of Theorem 1.1.
4 Proof of Theorem 1.2
We assume that \(b_0\) belongs to some Sobolev spaces, that is, \(b_0\in H^1\) with \(\Vert b_0\Vert =\alpha \). For \(\lambda >0\), consider the scaling transformation \(\delta _\lambda :L^2\rightarrow L^2\) via \(f(\cdot )\mapsto \lambda ^{3/2}(\lambda \cdot )\). It’s clear that it preserves the \(L^2\) norm, that is, \(\Vert \delta _\lambda f\Vert =\Vert f\Vert \). Let \((u^\lambda ,b^\lambda )\) be a solution of (1.1)–(1.4) with the initial data \(\delta _\lambda b_0\), then \((u^\lambda ,b^\lambda )\) satisfies following equations:
here \(p_*^\lambda =p^\lambda +\frac{1}{2}|b^\lambda |^2\).
Lemma 4.1
(i) For any \(\lambda >0\), there exists a \(T_{\lambda }>0\), such that (4.1)–(4.4) admits a unique strong solution \((u^\lambda ,b^\lambda )\in C([0,T_\lambda );H^{1})\cap L^2(0,T_\lambda ;H^2)\). (ii) There exist \(\lambda _0=\lambda _0(\nu ,\eta , \Vert b_0\Vert _{H^1})>0\) and \(c(\nu ,\eta ,\Vert b_0\Vert )>0\), such that for any \(0<\lambda <\lambda _0\), (4.1)–(4.4) admits a unique global solution \((u^\lambda ,b^\lambda )\in C([0,\infty );H^{1})\cap L^2(0,\infty ;H^2)\), and there holds
for all \(t\ge 0\).
Proof of Lemma 4.1.
-
(i)
The local-in-time existence of strong solutions are ensured by the global existence of weak solutions \((u^\lambda ,b^\lambda )\) (see [14, page 521]) together with a priori \(H^1\) estimate near the initial time (see (4.7)–(4.9) below). This a priori estimate also leads to the uniqueness of strong solutions, as well as the continuity with respect to t or initial data. The proof is similar to that of Navier–Stokes equations and thus it is omitted here.
-
(ii)
By repeating the manipulation of (3.1)–(3.4), we have the following bounds:
$$\begin{aligned} \Vert u^\lambda (t)\Vert _{L^{3/2,\infty }}\lesssim \Vert b^\lambda (t)\Vert ^2,\ \ \ \ \Vert b^\lambda (t)\Vert ^2 +2\int _0^t\big (\nu \Vert \nabla u^\lambda (\tau )\Vert ^2+\eta \Vert \nabla b^\lambda (\tau )\Vert ^2\big )\mathrm{d}\tau \le \Vert b_0\Vert ^2, \end{aligned}$$(4.6)for all \(t\ge 0\).
Taking the inner product of (4.1) with \(u^\lambda \) and integrating in \(\mathbb {R}^3\), we use interpolation inequality and (4.6) to obtain \(\Vert \nabla u^\lambda (t)\Vert \le c(\nu ,\eta ,\Vert b_0\Vert )\Vert \nabla b^\lambda (t)\Vert ^{3/2}\). Thus, it sufficient to show \(\Vert \nabla b^\lambda (t)\Vert \le c(\nu ,\eta ,\Vert b_0\Vert )\Vert \nabla \delta _\lambda b_0\Vert \).
Taking the inner product of (4.1) and (4.2) with \(-\Delta u^\lambda \) and \(-\Delta b^\lambda \) respectively, integrating in \(\mathbb {R}^3\) and then summing the resultant equations, we use \(\nabla \cdot u=0\) to obtain that
By (4.3) and integration by parts, the absolute value of the right-hand side of (4.7) is bounded (in the sense of ‘\(\lesssim \)’) by
where we have used interpolation inequality to derive the second and the fourth inequalities. Substituting (4.8) into (4.7), we apply (4.6) to deduce that
Since \(\Vert \nabla \delta _\lambda b_0\Vert =\lambda \Vert \nabla b_0\Vert \), we choose \(\lambda _0\) small enough such that \(c(\nu ,\eta ,\Vert b_0\Vert )\Vert \nabla \delta _{\lambda _0}b_0\Vert ^2<\eta \). For any \(\lambda \in (0,\lambda _0)\), by the fact that \(b^\lambda \in C([0,T_\lambda );H^{1})\), we deduce that there exists a \(\tilde{T}_\lambda \in (0,T_\lambda )\) such that \(c(\nu ,\eta ,\Vert b_0\Vert )\Vert \nabla \delta _\lambda b_0(t)\Vert ^2<\eta \) for \(t\in [0,\tilde{T}_\lambda ]\). Thus, it follows from (4.9) that \(\Vert \nabla b^\lambda (t)\Vert ^2\le \Vert \nabla \delta _\lambda b_0\Vert ^2\) for \(t\in [0,\tilde{T}_\lambda ]\). By induction, we deduce that (4.5) holds for all \(t\ge 0\). \(\square \)
Assume that \(\lambda <\lambda _0\). By Lemma 4.1 (ii), (4.1)–(4.4) admits a unique global strong solution, which is denoted by \((u^\lambda ,b^\lambda )\). By Fourier transformation, \(b^\lambda \) is solved as follows:
Hence
The main task is to calculate the limit (as \(\lambda \rightarrow 0^+\)) of terms on the right-hand side of (4.11). For the first term, by Plancherel’s theorem and Lebesgue’s dominated convergence theorem, we know that
While for the second term, we claim that
In fact, for small enough \(\lambda >0\), one applies Lemma 2.4 and (4.5) to deduce that
Passing to the limit as \(\lambda \rightarrow 0^+\), it follows that (4.13) holds true.
Multiplying (4.11) by \(\Vert b^\lambda (0)\Vert ^{-1}\) and passing to the limit as \(\lambda \rightarrow 0^+\), we use \(\Vert b^\lambda (0)\Vert =\Vert b_0\Vert \), (4.12) and (4.13) to deduce that
This completes the proof of Theorem 1.2.
5 Proof of Theorem 1.3
Step 1. A formal proof of \(L^{3/2,\infty }\times L^2\) decay. Assume that (u, b) is a smooth solution of (1.1)–(1.4). Removing \(2\nu \Vert \nabla u\Vert ^2\) from (3.1) and using Plancherel’s theorem, it follows that
Let \(r(t)=\sqrt{\frac{3}{2\eta (1+t)}}\), we deduce that
Substituting (5.2) into (5.1), and multiplying the resultant equation by \((1+t)^3\), we obtain
Now we prove that for all \(|\xi |<r(t)\), there holds
In fact, by taking Fourier transformation to (1.2), \(\hat{b}\) is solved as
Since \(b_0\in L^1\), we deduce from (5.5) that
By interpolation inequality and (3.2), the second term on the right-hand side of (5.6) is bounded by
Substituting (5.7) into (5.6), this proves (5.4).
Integrating (5.3) with respect to t and using (5.4), we deduce that
for all \(t\ge 0\). Hence, we have
Now we prove that, by a iteration process, the decay rate for b(t) in (5.9) can be improved to a much faster decay rate, which is proportional to that of solutions of heat equation. More precisely, we show that \(\Vert b(t)\Vert \le c(\nu ,\eta ,\Vert b_0\Vert _{L^1\cap L^2})(1+t)^{-3/4}\) for all \(t\ge 0\).
By a calculation similar to that of (5.7), we apply (5.9) to deduce that
The bound of \(I_1\) is obtained by dividing the integration interval [0, t] into [0, t/2] and [t/2, t].
Similarly, \(I_2\) is bounded by
Substituting (5.11) and (5.12) into (5.10), we deduce from (5.5) that
This implies that
Integrating (5.3) with respect to t and using (5.14), we obtain the desired decay estimate of b(t):
Furthermore, the equation (5.15), together with (3.4), implies that \(\Vert u(t)\Vert \le c(\nu ,\eta ,\Vert b_0\Vert _{L^1\cap L^2})(1+t)^{-3/2}\).
Step 2. The decay of weak solutions. Let \(\epsilon >0\) be arbitrary. For \(1\le p\le \infty \), we define a mollification \(\mathcal {J_\epsilon }:L^p\rightarrow L^p\) via \(f\mapsto \rho _\epsilon *f\), where \(\rho _\epsilon (\cdot )=\frac{1}{\epsilon ^3}\rho (\frac{\cdot }{\epsilon })\) and \(\rho \in C_0^\infty \) is a nonnegative, radial function that satisfies \(\int \rho (x)\mathrm{d}x=1\).
Consider the following equations:
This system is obtained by adding an artificial diffusion term \(\epsilon (-\Delta )^{3/2}\) to the b-equation of (1.1)–(1.4), then replacing the initial datum \(b_0\) by a smooth function \(\mathcal {J}_\epsilon b_0\). The fractional Laplacian \((-\Delta )^{3/2}\) is defined by the Fourier transform, namely, \((-\Delta )^{3/2}=\mathcal {F}^{-1}(|\xi |^3\mathcal {F})\).
We say that \((u^\epsilon ,b^\epsilon )\) is a global-in-time strong solution of the equations (5.16)–(5.19) if it is a weak solution of the system (this means that \((u^\epsilon ,b^\epsilon )\) belongs to a proper integrable space and it solves the system in the sense of distribution) and it satisfies
It can be proved that (5.16)–(5.19) admits a unique global-in-time strong solution \((u^\epsilon ,b^\epsilon )\). In fact, it was recently shown by the authors in [10] that (5.16)–(5.19) admits a unique global-in-time strong solution when \(\eta =0\). This global well-posedness result is absolutely true for the same equations when \(\eta >0\).
Thus, for (5.16)–(5.19), we repeat the manipulation of Step 1 to deduce that for all \(\epsilon >0\) and \(t\ge 0\),
Note that the presence of \(\epsilon (-\Delta )^{3/2}b^\epsilon \) does not affect the decay of \((u^\epsilon ,b^\epsilon )\) in \(L^{3/2,\infty }\times L^2\).
It remains to show that \(\{(u^\epsilon ,b^\epsilon )\}_{\epsilon >0}\) (or a subsequence) converges in some sense to (u, b) as \(\epsilon \) tends to zero, and (u, b) is a weak solution of (1.1)–(1.4) as well as (u, b) satisfies (1.8).
By the energy method, it is clear that
here \(H^{-1}\) denotes the dual space of \(H^1_\sigma \). Thus, it follows from Banach-Alaoglu theorem that there exists a subsequence which is still denoted by \(\{(u^\epsilon ,b^\epsilon ,\partial _tb^\epsilon )\}_{\epsilon >0}\) and an element \((u,b,\partial _tb)\), such that
Furthermore, for any bounded Lipschitz domain \(\Omega \subset \mathbb {R}^3\), since \(H^1(\Omega )\hookrightarrow L^2(\Omega )\) is compact, it follows from Aubin-Lions theorem, (5.26) and (5.27) that there exists a subsequence which is still denoted by \(\{b^\epsilon \}\) such that
This convergence, together with the \(L^{3/2,\infty }\)-estimate of \(u^\epsilon \) (see (3.4)), implies that
Hence, by (5.28), (5.29) and the uniform boundedness of \(\{(u^\epsilon ,b^\epsilon )\}_{\epsilon >0}\), we deduce that
By passing to the limit as \(\epsilon \rightarrow 0\), we conclude that (u, b) is a weak solution of (1.1)–(1.4).
Moreover, by (5.28) and Fatou’s lemma, we deduce from (5.22) that
This equation, together with (3.4), implies that
This completes the proof of Theorem 1.3.
6 Proof of Theorem 1.4
Let \(b_1(t)=e^{\eta t\Delta }b_0\), and let \(w=b-b_1\). A direct computation yields \(\Vert b(t)\Vert \ge \Vert b_1(t)\Vert -\Vert w(t)\Vert \). The lower bound of decay ratio for b(t) is obtained by establishing the lower bound and upper bound for \(b_1(t)\) and w(t), respectively. By Lemma 2.5, it is seen that \(\Vert b_1(t)\Vert \ge c(\eta ,\beta )(1+t)^{-3/4}\). Thus, (1.9) holds if one shows that there exists a \(\rho >3/4\) such that \(\Vert w(t)\Vert \le c(1+t)^{-\rho }\).
Consider the equations that satisfied by (u, w):
Taking the inner product of (6.1) and (6.2) with u and w, respectively, then integrating in \(\mathbb {R}^3\) and summing the resultant equations, we use integration by parts and \(\nabla \cdot u=\nabla \cdot b=0\) to obtain that
Let \(r(t)=\sqrt{\frac{3}{2\eta (1+t)}}\). By repeating the manipulation of the derivation of (5.3), we find that
Applying interpolation inequality, Lemma 2.4 and (1.8), we bound the integration of the second term on the right-hand side of (6.6) as follows:
Substituting (6.7) into (6.6), we know that
Now we estimate \(|\hat{w}|\) on the ball \(\{\xi \in \mathbb {R}^3:|\xi |<r(t)\}\). Taking Fourier transformation to (6.2), we use (6.4) to deduce that
Thus, by repeating the manipulation of (5.7) or (5.10), we apply (1.8) to obtain
Substituting (6.10) into (6.8) and integrating the resultant equation in [0, t], we see that
Thus, we finally obtain the desired upper decay rate for w(t):
This completes the proof of Theorem 1.4.
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Acknowledgements
The authors would like to thank the referee for the carefully reading of the earlier version of the manuscript and insightful comments which greatly improved the scope of this paper. The first author was partially supported by NSFC Grant (#11701099), the second author was partially supported by Guangdong Basic and Applied Basic Research Foundation (#2020A1515110299).
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Ji, Y., Tan, W. Large time behavior of solutions to a Stokes-Magneto equations in three dimensions. J. Evol. Equ. 21, 2449–2470 (2021). https://doi.org/10.1007/s00028-021-00689-z
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DOI: https://doi.org/10.1007/s00028-021-00689-z