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On analyticity rate estimates to the magneto-hydrodynamic equations in Besov-Morrey spaces
Boundary Value Problems volume 2015, Article number: 155 (2015)
Abstract
In this article, we establish higher-order regularizing rate estimates of solutions to generalized magneto-hydrodynamic equations in Morrey spaces with initial data \((u_{0}, d_{0})\) in Besov-Morrey spaces \(\dot {\mathbf{N}}_{r, \lambda,\infty}^{-s}\times\dot{\mathbf{N}}_{r, \lambda , \infty}^{-s}\), where \(n\geq2\), \(1\leq r<\infty\), \(0\leq\lambda< n\), \(r>n-\lambda\), \(\frac {1}{2}+\frac{n-\lambda}{4r}<\sigma< 1+\frac{n-\lambda}{4r}\), and \(s=2\sigma-1-\frac{n-\lambda}{r}\), for which under some smallness condition, the solution of the Cauchy problem is analytic in the spatial variable. Our class of initial data contains strongly singular functions and measures and extends the ones in early work.
1 Introduction and main results
In this article, we investigate the generalized magneto-hydrodynamic equations in the whole space \(\mathbb{R}^{n}\),
Here u is the velocity field of the flow, \(d(\cdot,t)\) is the magnetic field. \(p(\cdot,t): \mathbb{R}^{n}\rightarrow\mathbb{R}\) represents the pressure function. \(\nabla\cdot u = 0\) and \(\nabla\cdot d = 0 \) represent the incompressible conditions. \((u_{0}, d_{0})\) is for given initial data with \(\nabla\cdot u_{0} = 0 \) and \(\nabla \cdot d_{0} = 0 \) in the distribution sense.
When \(\sigma= 1\), the equations of system (1.1) become the usual MHD equations, which govern the dynamics of the velocity and magnetic fields in electrically conducting fluids. The system plays a fundamental role in applied sciences such as astrophysics, geophysics, and plasma physics. The first equation of system (1.1) reflects the conservation of momentum, the third equation of system (1.1) is the magnetic induction equation and the second equation of system (1.1) specifies the conservation of mass.
For general σ, system (1.1) is a generalization of the usual incompressible MHD system. As observed in [1], a fractional power of Laplacian can, in principle, be used as a mild dissipation in MHD equations. Besides their physical applications, system (1.1) is also mathematically significant.
According to Duhamel’s principle, the mild solution \((u, d)\) for system (1.1) can be represented as
Here \(\mathbb{P}\) is the Leray projection operator, which can be expressed as an \(n\times n\) matrix: \(\mathbb{P}=\{\mathbb{P}_{j,k}\}_{1\leq j,k\leq n}=\{\delta_{j,k}+\mathbb {R}_{j}\mathbb{R}_{k}\}_{1\leq j,k\leq n} \) with \(\delta_{j,k}\) being the Kronecker symbol, \(\mathbb{R}_{j}=\partial _{j}(-\Delta)^{-\frac{1}{2}}\) being the Riesz transform. \(\mathfrak{L}:=(-\Delta)^{\sigma}\) denotes the fractional Laplacian, which is defined as \(\widehat{[(-\Delta)^{\sigma}f]}(\xi)=|\xi| ^{2\sigma}\hat{f}(\xi)\).
To give a clearer introduction to our results in this article, we first note that system (1.1) enjoys scaling properties. Clearly, if \((u(x, t), d(x, t))\) is a solution to system (1.1), then \((u^{\lambda}(x,t), d^{\lambda}(x,t))\) is also a solution of (1.1) corresponding to the initial data \((u_{0}^{\lambda}, d_{0}^{\lambda})\), where
We say that the solution \((u, d)\) is self-similar for system (1.1), if \((u^{\lambda}, d^{\lambda})=(u, d )\) for each \(\lambda>0\).
A function space \(\mathbb{Y}\) is called a critical space for (1.1) if it satisfies invariance under the scaling \(\| u( \cdot, t)\|_{\mathbb{Y}}=\| u^{\lambda }(\cdot, t)\|_{\mathbb{Y}}\) for all \(u\in\mathbb{Y}\).
Before going further, we recall the functional spaces we are going to use. Let \(\mathscr{S}\) be the Schwartz class of rapidly decreasing functions and \(\mathscr{S}'\) be the space of tempered distributions. Here \(\mathcal{F}\) and \(\mathcal {F}^{-1}\) denote the Fourier and inverse Fourier transforms of \(L^{1}\) functions, respectively, defined by \(\mathcal{F}f=\hat{f}(\xi)=(2\pi)^{-\frac{n}{2}}\int_{\mathbb {R}^{n}}e^{-ix\cdot\xi}f(x)\, dx\) and \(\mathcal{F}^{-1}f=\check{f}(x)=(2\pi)^{-\frac{n}{2}} \int_{\mathbb{R}^{n}}e^{ix\cdot\xi}f(\xi)\, d\xi\). More generally, the Fourier transform of any \(f\in\mathscr{S}'\) is given by \((\mathcal{F}f,g)=(f,\mathcal{F}g)\), for any \(g\in\mathscr{S}\). First, we recall the definition of Morrey space introduced in [2]: for \(1\leq p <\infty\) and \(0\leq\lambda< n\), the Morrey space \(M_{p,\lambda}=M_{p,\lambda }(\mathbb{R}^{n})\) is defined as
with the norm given by
where \(B(x_{0},r)\) denotes the ball in \(\mathbb{R}^{n}\) with center \(x_{0}\) and radius r. The space \(M_{p,\lambda}\) endowed with the norm \(\|\cdot\| _{p,\lambda}\) is a Banach space and has the following nice scaling property: for \(\mu> 0\),
Set \(S_{h}=\{\phi\in\mathscr{S}, \partial^{\alpha}\mathcal {F}f(0)=0\}\) for any multi-index \(\alpha\in\mathbb{N}_{0} :=\mathbb {N}^{n}\cup\{ 0\}\), \(\mathbb{N}\) is the set of all positive integers. The dual space of \(S_{h}\) is given by \(S_{h}'=\mathscr {S}/\mathcal{P}\), where \(\mathcal{P}\) is the space of polynomials. We now introduce a dyadic partition of \(\mathbb{R}^{n}\). Let \(\varphi\in\mathscr{S}\) be a radially symmetric function with support in \(\{\xi\in\mathbb{R}^{n}:\frac{3}{4}\leq|\xi |\leq\frac{8}{3}\}\) and such that
Furthermore, we define \(\varphi_{k}=\varphi(2^{-k}\xi)\) for every \(k\in \mathbb{Z}\).
For any \(f\in S_{h}'\), setting \(\Delta_{k}f=(\varphi_{k}\hat{f})^{\check {}}\), \(k=0, \pm1, \pm2,\ldots\) , and \(S_{j}f=\sum_{k\leq j-1}\Delta _{k}f\). We have the Littlewood-Paley decomposition,
In [3], Kozono and Yamazaki introduced the homogeneous Besov-Morrey space \(\dot{\mathbf{N}}_{p,\lambda,q}^{s}\). Recall that the space \(\dot{\mathbf{N}}_{p,\lambda,q}^{s}\) is defined by
where
When \(\lambda=0\), \(\dot{\mathbf{N}}_{p, 0, q}^{s}=\dot{\mathbf{B}}_{p, q}^{s}\), where \(\dot{\mathbf{B}}_{p, q}^{s}\) is the homogeneous Besov space (see [4]).
If \(\sigma= 1\), \(d=0\), system (1.1) is the well-known Navier-Stokes equations (NS), Foias and Temam [5] proved spatial analyticity for solutions in Sobolev spaces of periodical functions in an elementary way. The analyticity of solutions in \(L^{p}\) for NS was first shown by Grujič, and Kukavica [6] and Lemarié-Rieusset [7] gave a different approach based on multilinear singular integrals. In a very interesting paper [8], Kahane established the spatial analyticity of weak solutions in Serrin’s class \(L_{t}^{p} L_{x}^{q}\) with \(n/q+2/p<1\). In cylindrical domains, Komatsu [9] showed that the solutions have global spatial analyticity up to the boundary. Using iterative derivative estimates, in [10], Giga and Sawada considered the regularizing rates of the higher-order derivatives and analyticity for the NS for the initial velocity in \(L^{n}\). Similar results for the Navier-Stokes equations have been established by Sawada [11] when initial value \(u_{0}\in\dot{H}^{\frac{n}{2}-1}(\mathbb{R}^{n})\) and by Miura and Sawada [12] when \(u_{0}\in {BMO}^{-1}\). Recently, Bae et al. [13] obtained the analyticity of the solutions of NS for the sufficiently small initial data in critical Besov spaces \(\dot{\mathbf{B}}_{p, q}^{-1+3/p}\), and Huang and Wang [14] showed the analyticity of the local solutions of NS with large initial data in critical Besov spaces \(\dot{\mathbf{B}}_{p, q}^{-1+n/p}\) and modulation spaces \(M_{p, 1}^{-1}\).
For general σ and \(d=0\), the equations of system (1.1) reduce to generalized Navier-Stokes equations (GNS). Dong and Li [15] showed that solutions are analytic in space variables for \(1/2<\sigma<1\) with initial data in \(L^{n/(2\sigma-1)}\). Huang and Wang [14] showed the analyticity of the solutions of GNS in critical Besov spaces \(\dot {\mathbf{B}}_{p, q}^{1-2\sigma+n/p}\) and modulation spaces \(M_{p, 1}^{1-2\sigma}\) for \(1/2<\sigma<1\). When \(\sigma=\frac{1}{2}\), Huang and Wang [14] showed the analyticity of the solutions of GNS in critical Besov spaces \(\dot{\mathbf{B}}_{p, 1}^{n/p}\) and modulation spaces \(M_{\infty, 1}^{0}\cap\dot{\mathbf{B}}_{\infty, 1}^{0}\).
When \(\sigma=1\), Liu and Cui in [16] show the analyticity of the usual MHD with initial data in \(L^{n}\), \(\dot{H}^{\frac{n}{2}-1}\) and \({BMO}^{-1}\). When \(1/2<\sigma<(n+2)/4\), Liu et al. in [17] show that the solution is analytic in the spatial variable of system (1.1) with the initial velocity in \({PM}^{n-2\sigma+1}\).
In [18], Yamamoto considered the regularizing rates and analyticity for the drift-diffusion equation for the initial data in \(L^{\frac{n}{\theta}}\) (\(1<\theta\leq n\)) and extended the results to Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces.
Inspired by the interesting work above, especially [10–12, 16–18] and motivated by the work of Mazzucato [19], and Kozono and Yamazaki [3] on the Navier-Stokes equations and a particular class of semi-linear heat equations with initial data in a certain Besov-Morrey space, our goal in the present article is to establish regularizing decay rate estimates and show space analyticity of mild solutions of system (1.1) with initial data in certain Besov-Morrey spaces. For more information on Besov-Morrey spaces, we also refer to [20–22]. The question of the largest Besov-type spaces on initial data for which the solutions of (1.1) have well-posedness and analyticity is still open.
We give our main results in the following theorem.
Theorem 1.1
Let \(n\geq2\), \(1\leq r<\infty\), \(0\leq\lambda< n\), \(r>n-\lambda\), \(\frac {1}{2}+\frac{n-\lambda}{4r}<\sigma< 1+\frac{n-\lambda}{4r}\), \(s=2\sigma -1-\frac{n-\lambda}{r}\), \(\alpha=\frac{2\sigma-1}{\sigma}-\frac{n-\lambda }{2r\sigma}\), \(\nabla\cdot u_{0}=0\), \(\nabla\cdot d_{0}=0\), \((u_{0},d_{0})\in \dot{\mathbf{N}}_{r,\lambda,\infty}^{-s}\times\dot{\mathbf {N}}_{r,\lambda,\infty}^{-s}\), \(q\in[r, \infty]\). Assume further that there exist positive constants \(M_{1}\) and \(M_{2}\), such that the solutions \((u, d)\) to system (1.1) exist globally in time and satisfy
for any \(t>0\) and \(M_{1}\) sufficiently small. Then there exist positive constants \(K_{1}\), \(K_{2}\) such that
where \(\tilde{\beta}\in\mathbb{N}_{0}^{n}\) is a multi-index and \(| \tilde{\beta}|=m\).
Remarks
(I) The assumptions in Theorem 1.1 are natural. Indeed, let \(1\leq r<\infty\), \(0\leq\lambda< n\), \(r>n-\lambda\), \(\frac{1}{2}+\frac{n-\lambda }{4r}<\sigma< 1+\frac{n-\lambda}{4r}\), \(\alpha=\frac{2\sigma-1}{\sigma }-\frac{n-\lambda}{2r\sigma}\), \(s=2\sigma-1-\frac{n-\lambda}{r}\). The Banach spaces E are defined by \(E=\{ u: \nabla\cdot u=0, u \in {BC}((0, \infty), \dot{\mathbf{N}}_{r, \lambda, \infty}^{-s}), t^{\frac{\alpha}{2}}u\in {BC}((0, \infty), M_{2r, \lambda})\}\), which are Banach spaces with norms given by \(\| u\|_{E}=\sup_{t>0}\| u(t)\|_{\dot {\mathbf{N}}_{r,\lambda,\infty}^{-s}}+\sup_{t>0}t^{\frac{\alpha }{2}}\| u(t)\|_{2r, \lambda}\). Let \(u_{0}\) and \(d_{0}\) be divergence free vector fields and \((u_{0},d_{0})\in\dot{\mathbf{N}}_{r,\lambda,\infty}^{-s}\times\dot {\mathbf{N}}_{r,\lambda,\infty}^{-s}\) with \(\|(u_{0}, d_{0})\|_{\dot{\mathbf{N}}_{r,\lambda,\infty}^{-s}} \) sufficiently small. Following a similar method to Theorems 3 and 4 on p.967 in [3] for Navier-Stokes equations, then there exists a globally in time solution \((u(x), d(x))\in E\times E\) to (1.1) that satisfies (1.5). The proof of this is standard by making minor modifications with Theorems 3 and 4 on p.967 in [3], and we will outline its proof in the Appendix for completeness.
(II) When \(\frac{1}{2}<\delta\leq1\), \(K_{2} \geq2\), the estimate (1.6) is equivalent to (see [10])
(III) From Remark 1.4 of [23], we get \({PM}^{n-2\sigma+1}\subset \dot{\mathbf{B}}_{p, \infty}^{1-2\sigma+\frac{n}{p}}\) for \(\frac {n}{2\sigma-1}< p<\infty\). It follows (see [3], p.964) that the space \(\dot{\mathbf{N}}_{r,\lambda,\infty}^{-2\sigma+1+\frac {n-\lambda}{r}}\) is strictly larger than \(\dot{\mathbf{B}}_{p,\infty}^{1-2\sigma+\frac{n}{p}}\), when \(p=\frac{nr}{n-\lambda}\), \(\lambda>0\). The pseudomeasure space \({PM}^{a}\) (\(a \geq0\)) introduced in [24] is defined as \({PM}^{a}:=\{f\in\mathscr{S}': \hat{f}\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n}), \| f\|_{{PM}^{a}}=\operatorname{ess}\sup_{{\xi}\in\mathbb {R}^{n}}|\xi|^{a}|\hat{f}|<\infty\}\). In view of the continuous inclusions above, we see that the initial spaces \(\dot{\mathbf{N}}_{r,\lambda,\infty}^{-s}\) (\(r>\max\{\frac{n-\lambda }{2\sigma-1}, n-\lambda\}\), \(\lambda>0\)) defined in Theorem 1.1 is larger than pseudomeasure space \({PM}^{n-2\sigma+1}\) in [17]. In [17], the authors considered the regularizing rates of the higher-order derivatives for system (1.1) for the initial velocity in \({PM}^{n-2\sigma+1}\).
(IV) In particular, when \(\sigma=1\), \(d(x, t)=0\), system (1.1) becomes the usual Navier-Stokes equations. We also notice that \({BMO}^{-1}\) may be regarded as the largest critical space for initial data, where well-posedness and spatial analyticity of the Navier-Stokes equations can be constructed (see [25]). In [12], Miura and Sawada considered the regularizing rates of the higher-order derivatives for the Navier-Stokes equations for the initial velocity \(u_{0}\in {BMO}^{-1}\). The space \({BMO}^{-1}\) is the space of tempered distributions that can be written as divergence of a vector with components in \({BMO}\), where \({BMO}\) is the space of functions of bounded mean oscillations. The norm on \({BMO}^{-1}\) is given by
But the initial data \(\dot{\mathbf{N}}_{r, \lambda,\infty}^{-s}\) (\(\sigma =1\)) given in Theorem 1.1 is not included completely with the space \({BMO}^{-1}\). Using the characterization from Lemma 2.3 below, we obtain (see [19], p.1314)
and, for \(0\leq\lambda< n\), \(n\geq2\),
Thus we note that even for the Navier-Stokes equations, our result in Theorem 1.1 is also new.
Notation
Throughout this article, we denote vector fields \(u=(u_{1}, u_{2},\ldots ,u_{n})\), \(d=(d_{1}, d_{2}, \ldots,d_{n})\). For a functional space X, we denote by \(\|(u, d)\|_{X}\),
We use \(c>0\) to denote a constant independent of the main variables, which may be different from line to line. We will employ the notation \(a\lesssim b\) to mean that \(a\leq cb\) for a universal constant \(c>0\) that only depends on the parameters coming from the problems.
2 Preliminaries
In this section, we prepare several tools from harmonic analysis to be used in the proof of Theorem 1.1.
Lemma 2.1
Assume that \(1\leq p_{j}\), \(q\leq\infty\) for all \(j=1, 2, 3\), and \(1\leq p\leq\infty\), \(1\leq r< \infty\), \(0\leq\lambda\), \(\lambda_{i} < n\) for all \(i=1, 2, 3\).
-
(1)
If \(p_{1}>p_{2}\), \(s_{1}-\frac{n-\lambda}{p_{1}}=s_{2}-\frac {n-\lambda}{p_{2}}\), \(s_{1}, s_{2}\in\mathbb{R}\), then
$$ \dot{\mathbf{N}}_{p_{2},\lambda,q}^{s_{2}}\hookrightarrow\dot{ \mathbf {N}}_{p_{1},\lambda,q}^{s_{1}} \quad \textit{and} \quad \dot{\mathbf {N}}_{r,\lambda,1}^{0}\hookrightarrow M_{r,\lambda} \hookrightarrow\dot {\mathbf{N}}_{r,\lambda,\infty}^{0}. $$ -
(2)
If \(1\leq r\leq\tilde{r}\leq\infty\), \(s\in\mathbb{R}\), then \(\dot{\mathbf{N}}_{p,\lambda,r}^{s}\hookrightarrow\dot{\mathbf {N}}_{p,\lambda,\tilde{r}}^{s}\).
-
(3)
If \(1\leq p_{1}\leq p_{2}\leq\infty\), \(0\leq\lambda_{1}, \lambda_{2}< n\), \(\frac{n-\lambda_{1}}{p_{1}}=\frac{n-\lambda _{2}}{p_{2}}\), then \(M_{p_{2}, \lambda_{2}}\subset M_{p_{1}, \lambda_{1}}\).
-
(4)
If \(\frac{1}{p_{3}}=\frac{1}{p_{2}}+\frac{1}{p_{1}}\), \(\frac {\lambda_{3}}{p_{3}}=\frac{\lambda_{2}}{p_{2}}+\frac{\lambda _{1}}{p_{1}}\), \(h_{i}\in M_{p_{i},\lambda_{i}}\) for \(i=1, 2\), then \(\| h_{1}h_{2} \|_{p_{3},\lambda_{3}}\leq\| h_{1}\|_{p_{1},\lambda_{1}}\| h_{2}\| _{p_{2},\lambda_{2}}\).
Proof
For the proof of Lemma 2.1, we refer to [2, 3, 19, 26, 27]. □
From the Calderón-Zygmund operator theory, the Riesz transform \(\mathbb{R}_{j}\) is continuous on \(M_{r,\upsilon}\) for \(1< r<\infty\) and \(0\leq\upsilon< n\), thus \(\mathbb{P}\) is bounded on \(M_{r,\upsilon}\). By the estimates for the multiplier operator, we can also see that \(\mathbb{P}\) is bounded on \(\dot{\mathbf{N}}_{p,\lambda,q}^{s}\) for \(1\leqslant p\), \(q\leqslant \infty\), \(0\leq\lambda< n\), and \(s\in\mathbb{R}\).
Lemma 2.2
Let \(\mu>0\) and \(\mathbb{N}_{0}^{n}\ni\alpha=(\alpha_{1}, \alpha _{1},\ldots,\alpha_{n}) \) be a multi-index with \(|\alpha|=\mu\), \(s_{1}\leq s_{2}\), \(1\leq q\leq\infty\), \(1\leq p_{1}\leq p_{2}\leq\infty\), \(0\leq\lambda< n\), for all \(f\in\mathscr{S}'\), then there exist \(c_{0}\), \(c_{1}\), c̃, \(\tilde{c}_{0}\), \(\tilde{c}_{1}\), c, and c̄ depending only on n such that
Further, if \(s<\rho\), the estimate
holds for every \(t>0\).
Lemma 2.2 still holds true with \((-\Delta)^{\mu}\) in place of \(\partial^{\alpha}\).
Proof
We first prove (2.1) by proceeding in the following way. For all \(1\leqslant p\leqslant\infty\), \(0\leq\lambda< n\), \(g\in M_{p, \lambda}\), \(\phi\in L^{1}\), in Morrey spaces we have
Note that (2.6) implies
According to Lemma 2.1 of [28], we have \(K_{t}\in L^{p}\) for \(1\leq p\leq\infty\), where \(K(x):=(\frac{1}{2\pi })^{\frac{n}{2}} \int_{\mathbb{R}^{n}}e^{i x\cdot\xi}e^{-|\xi|^{2\sigma}}\, d\xi\) and \(K_{t}:= t^{-\frac{n}{2\sigma}}K(\frac{x}{t^{\frac{1}{2\sigma }}})\). Thus we get \(\| e^{-t\mathfrak{L}}f\|_{p_{2},\lambda}\leq\bar {c}\| f\|_{p_{2},\lambda}\).
From Lemma 2.1 of [28], we have the point-wise estimate \(| K(x)|\leq\bar{c}(1+| x|)^{-n-2\sigma}\). Hölder’s inequality yields \(| e^{-t\mathfrak{L}}f|^{p_{1}}\leq\bar{c}K_{t}\ast| f| ^{p_{1}}\). Therefore, one has
Using the definition of Morrey spaces in (1.4), we get
Then from the interpolation inequality it follows that
Combining (2.7) and (2.8) gives
Thus, we complete the estimate (2.1).
To estimate (2.2), application of the commutativity of the semigroup and derivatives gives the following estimate:
Then, by (2.6),
With the aid of the Hörmander-Mikhlin type estimate in [29], we obtain
Applying (2.10) and (2.11), we get
Thus, one obtains the estimate of (2.2).
To estimate (2.3), we apply the frequency projection operator \(\Delta_{j}\) to \(e^{-t\mathfrak{L}}\) and take the \(M_{p, \lambda}\) norm, then by (2.1)
For every \(j\in\mathbb{Z}\), it follows from (2.13) that
By the definition of Besov-Morrey spaces, from (2.14) we get (2.3) immediately.
For (2.4), using the estimate of (2.2), we can prove (2.4) exactly in the same way as deriving (2.3). Here we omit the proof of (2.4).
Assume that \(s<\rho\), applying (2.3) with \(q=\infty\), we obtain
and
Using (2.15), (2.16), and the interpolation relation \((\dot{\mathbf{N}}_{r, \lambda,\infty}^{2\rho-s}, \dot{\mathbf{N}}_{r, \lambda,\infty}^{s})_{\frac{1}{2}, 1}=\dot{\mathbf{N}}_{r, \lambda ,1}^{\rho}\) (see Proposition 2.12 of [3]), we get the desired estimate (2.5). Thus, we complete the proof of Lemma 2.2. □
Following the method used by [4], we give the proof of Lemma 2.3.
Lemma 2.3
Suppose \(1\leq p,q\leq\infty\), \(s>0\), and \(0<\sigma <\infty\), then one has \(f\in\dot{\mathbf{N}}_{p,\lambda,q}^{-2s}\) if and only if
Proof
Let \(\mathbb{C}=\{\xi: 0< r_{1}\leq|\xi|\leq r_{2}, r_{1}>0, r_{2}>0 \}\) be an annulus, there exists a positive constant \(c>0\), such that for any \(1\leq p\leq\infty\) and any couple \((t,\lambda)\) of positive real numbers, from the same ideas from Lemma 2.4 of [4], we have
here, we omit the proof (2.17).
In the following, we only show the case \(1\leq q<\infty\). For \(q=\infty \) we have the same process. Note that, by (2.17),
Then, in virtue of \(f=\sum_{j\in\mathbb{Z}}\Delta_{j}f\), we deduce that
where \(\{c_{r, j}=\frac{2^{-2js}\|\Delta_{j}f \| _{p,\lambda}}{\| f\|_{\dot{\mathbf{N}}_{p,\lambda ,q}^{-2s}}}\}_{j \in\mathbb{Z}}\in l^{q}\). Note that \(\| c_{r,j}\|_{l^{q}}=1\), the change of variable \(\tau=ct2^{2l\sigma }\) yields
which is based on a technique developed in [30] (see (2.59) on p.27 in [30]), where \(\Gamma(s)=\int_{0}^{+\infty }x^{s-1}e^{-x}\, dx\) is the Γ function for \(s>0\).
Therefore, Hölder’s inequality with weight \(t^{\frac{s}{\sigma }}2^{2js}e^{ct2^{2j\sigma}}\), Fubini’s theorem, (2.19), and (2.20) imply that
Since \(\Gamma(\frac{s}{\sigma}+1)=\int_{0}^{+\infty}t^{\frac{s}{\sigma }} e^{-t}\, dt\), by the definition of the Fourier transform, we thus get
Taking the \(M_{p, \lambda}\) norm on (2.21), in view of (2.17), one easily sees that
The change of the variable \(x=\frac{ct}{2}2^{2m\sigma}\) implies that
Thus, we complete the proof of Lemma 2.3. □
Lemma 2.4
For all \(\delta\in(\frac{1}{2}, 1)\), then there exists a constant \(c>0\) such that
holds for every \(\gamma\in\mathbb{Z}_{+}^{n}\), \((\tilde{\alpha}_{1}, \tilde{\alpha}_{2} ,\ldots, \tilde{\alpha}_{n})=\tilde{\alpha}\leq\gamma =(\gamma_{1}, \gamma_{2} ,\ldots, \gamma_{n})\). Note that \(\tilde{\alpha }\leq\gamma\) means \(\tilde{\alpha}_{i}\leq\gamma_{i}\) for all \(i=1, 2,\ldots, n\) and \(\binom{\gamma}{\tilde{\alpha}}=\prod_{j=1}^{n}\frac{\gamma_{j}!}{\tilde{\alpha }_{j}!(\gamma_{j}-\tilde{\alpha}_{j})!}\).
Proof
For the proof of Lemma 2.4, see [8]. □
Lemma 2.5
Let \(\psi_{0}\) be a measurable and locally bounded function in \((0,T)\). Let \(\{\psi_{j}\}_{j=1}^{\infty}\) be a sequence of measurable functions in \((0,T)\). Assume that \(\alpha\in\mathbb{R}\) and \(\mu, \nu> 0\) satisfying \(\mu+\nu= 1\). Let \(B_{\eta} > 0\) be a number depending on \(\eta\in(0, 1)\), and assume that \(B_{\eta} > 0\) is nonincreasing with respect to η. Assume that there is a positive constant θ such that
and
for all \(j\geq0\), \(t > 0\) and \(\eta\in(0, 1)\). Let \(\eta_{0}\) be a unique positive number such that \(I(\eta_{0})= \min\{\frac{1}{2\theta}, I(1)\}\) with \(I(\eta) =\int_{(1-\eta)t}^{t}(t-s)^{-\mu}(s)^{-\nu-\alpha}\,ds\). Then
for all \(j\geq0\), \(0<\tilde{\eta}\leq\eta_{0}\), and \(0< t < T\).
Proof
3 Proof of Theorem 1.1
Before proving Theorem 1.1, we first follow the ideas from [10, 12] and prove a variant of Theorem 1.1 under some additional regularity assumptions.
Proposition 3.1
Suppose that the assumptions of Theorem 1.1 are satisfied. Assume furthermore that
for all \(r\leqslant q\leqslant\infty\). Then given \(\frac{1}{2}<\delta \leq1\), there exist constants \(K_{1}>0\), \(K_{2}>0\) (depending only on n, \(M_{1}\), \(M_{2}\), δ, and σ), such that
for all \(r\leqslant q\leqslant\infty\), where \(|\tilde{\beta}|=m\).
Proof
For \(1\leq p\leq\infty\) and \(0\leq\lambda< n\), by Lemma 2.2, note that (2.6), there exists a constant \(c>0\) such that
In fact, the proof of (3.3) is essentially the same as the proof of \(\| e^{-t\Delta}\mathbb{P}\nabla f\|_{p}\leq c t^{-\frac{1}{2}}\| f\|_{p}\). The process of proving \(\| e^{-t\Delta}\mathbb{P}\nabla f\|_{p}\leq c t^{-\frac {1}{2\sigma}}\| f\|_{p}\) can be found in [2, 29].
Using Lemma 2.2 and (3.3), for \(1\leq p_{1}\leq p_{2}\leq\infty\) and \(0\leq\lambda< n\), a straightforward calculation yields the following elementary estimates:
We use an induction argument with respect to m.
Step 1. We first shall prove (3.2) for \(m=0\). Taking the \(M_{q, \lambda}\) norm to the first term of (1.2), for some \(\epsilon\in(0, 1)\),
We shall estimate each term. To estimate the first term \(B_{1}\) on the right side of (3.5), we note that, by (3.4),
It follows from (4) of Lemma 2.1, (1.5), and (3.4) that
By Lemma 2.1, (1.5), and (3.4), similarly we can derive
Note that \(\alpha=\frac{2\sigma-1}{\sigma}-\frac{n-\lambda}{2r\sigma}\), denoting \(B_{\epsilon} =CM_{1}+ CM_{2}^{2}\epsilon^{-\frac{1}{2\sigma }-\frac{n-\lambda}{2\sigma}(\frac{1}{r}-\frac{1}{q})}\), and combining (3.6), (3.7), and (3.8), one obtains
Similarly, we can get the desired estimate of \(\| d\|_{q, \lambda}\),
Therefore, according to Lemma 2.5, we get (3.2) for \(|\tilde{\beta}|=m=0\).
Step 2. We next consider the case \(m=1\). The proof of (3.2) is essentially contained in Step 3. Thus here we omit the details.
Step 3. Assume that \(m\geq2\). We suppose that (3.2) holds for \(q\in[r, \infty]\) and all \(|\tilde{\beta}|\leq m-1\). We need to prove that (3.2) holds for \(|\tilde{\beta}|= m\). Then, for \(|\tilde{\beta}|= m\) and some \(\epsilon\in(0, 1)\), we see that
We shall estimate each of the above terms \(A_{1}\), \(A_{2}\), \(A_{3}\) separately. Note that \(\frac{m}{2\sigma}\leq2m-\delta\), since \(m\geq 2\) and \(0<\delta\leq1\). Observe that by (1.5), and Lemmas 2.2 and 2.3
To estimate the term \(A_{2}\), we note that, by Lemma 2.1, (1.5), and (3.4),
We now calculate \(\nabla^{m}(u\otimes u-d\otimes d)\) by Leibniz’s rule. Lemma 2.1 and (3.4) yield
Here, \(\gamma<\tilde{\beta}\) means \(\gamma_{i}\leq\tilde{\beta}_{i}\) and \(|\gamma|<|\tilde{\beta}|\) for the multi-indices \(\tilde {\beta}=(\tilde{\beta}_{1}, \tilde{\beta}_{2}, \ldots, \tilde{\beta }_{n})\) and \(\gamma=(\gamma_{1}, \gamma_{2},\ldots,\gamma_{n})\), where \(i=1,2,\ldots,n\).
In order to estimate the first term on the right hand of (3.13), according to Step 1, we note that there exists \(c_{5}>0\) such that \(\|(u, d)\|_{\infty, \lambda}=\|(u, d)\| _{L^{\infty}} \leq C_{5}K_{1}t^{\frac{1-2\sigma}{2\sigma}}\), then
By the assumption of the induction, we obtain
Applying Lemma 2.4, it follows that
where \(I(\epsilon):=\int_{1-\epsilon}^{1}(1-\tau)^{-\frac{1}{2\sigma}} \tau^{-\frac{m}{2\sigma}-\frac{1}{2\sigma}(4\sigma-2-\frac{n-\lambda }{q})}\,d\tau\).
Note that we set
Combining the above estimates for (3.11), (3.12), (3.13), (3.14), (3.15), and (3.16), we obtain
Similarly, from a computation it follows that
Thus, we have
Applying Lemma 2.5, we see that there exists \(\epsilon _{m_{0}}\in(0, 1)\), such that for any \(0<\epsilon_{m}\leq\epsilon_{m_{0}}\), we have
where \(I(\epsilon_{m_{0}})=\min\{\frac{1}{2C_{5}K_{1}}, I(1)\}\). Let \(\epsilon_{m}=\frac{1}{m^{2\sigma}}\), since \(I(\epsilon)\) is nonincreasing with respect to ϵ. We can choose \(m_{0}>2\) sufficiently large such that \(I(\frac{1}{m^{2\sigma}})\leq\frac {1}{2C_{5}K_{1}}\) for all \(m \geq m_{0}\). Hence, we obtain
By (3.17), we can choose \(K_{1}\) and \(K_{2}\) sufficiently large such that (3.2) holds for all \(|\tilde{\beta}|\leq m_{0}\). Finally, it is enough to show that \(2b_{\frac{1}{m^{2\sigma}}}\leq K_{1}(K_{2}| \tilde{\beta}|)^{2|\tilde{\beta}|-\delta}\) for any \(m>m_{0}\geq2\) with constants \(K_{1}\) and \(K_{2}\) sufficiently large.
Next, we compute \(I(\frac{1}{m^{2\sigma}})\),
Since \(\delta\leq2m-\frac{m}{2\sigma}\) (\(\frac{1}{2}<\delta\leq1\), \(m\geq 2\), \(\frac{1}{2}<\sigma<2\)), \(r>n-\lambda\), \(\frac{1}{2}+\frac{n-\lambda }{4r}<\sigma< 1+\frac{n-\lambda}{4r}<2 \), and \(m^{\delta+2}\leq8\cdot 2^{m-\delta}\sqrt{m}\leq8\cdot2^{2m-\delta}\), we thus have
We choose the constants \(K_{1}:=4C_{1}M_{1}+4C_{3}M_{2}^{2}\). We take \(K_{2}\) large enough, such that \(K_{2}\geq2C_{4}+C_{2}\) and \(C_{6}C(\sigma)K_{1}K_{2}^{-\delta}< \frac{1}{2}\). Then we obtain (3.2) immediately. □
Proposition 3.2
Suppose that the assumptions of Theorem 1.1 are satisfied. Then the mild solution \((u, d)\) of (1.2) satisfies (3.1), and there exist constants \(\tilde{K}_{1}, \tilde{K}_{2}>0\) such that
for all \(r\leqslant q\leqslant\infty\), where \(|\tilde{\beta}|=m\).
Proof
The proof is now standard, we refer the reader to [10–12]. □
Now Theorem 1.1 follows immediately from Proposition 3.1 and Proposition 3.2. We thus complete the proof of Theorem 1.1.
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The author is highly grateful for the referees’ careful reading of and comments on this paper. The author would like to thank Professor Lixin Yan for helpful discussions.
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Appendix
Appendix
In this Appendix, we will show the global existence of solution for system (1.1) mentioned in Theorem 1.1 or (I) of Remarks below Theorem 1.1. We note that, similarly to the Navier-Stokes equations ([3], Theorems 3 and 4, p.967), the proof of global existence can be obtained by making minor modifications to Theorems 3 and 4 on p.967 in [3]. Here, we give a brief argument of this proof for completeness and for convenience of the reader.
We say \((u, d)\in E^{\epsilon_{0}}\times E^{\epsilon_{0}}\) if \((u, d)\in E\times E\) and \(\|(u, d)\|_{E^{\epsilon_{0}}}=\|(u, d)\|_{E}=\sup_{t>0}\|(u(t), d(t))\|_{\dot{\mathbf{N}}_{r,\lambda ,\infty}^{-s}}+ \sup_{t>0}t^{\frac{\alpha}{2}}\|(u(t), d(t))\| _{2r, \lambda}\leq C\epsilon_{0}\). The definition of E can be found in (I) of Remarks below Theorem 1.1.
Lemma A.1
Let \(n\geq2\), \(1\leq r<\infty\), \(0\leq\lambda< n\), \(r>n-\lambda\), \(\frac {1}{2}+\frac{n-\lambda}{4r}<\sigma< 1+\frac{n-\lambda}{4r}\), \(s=2\sigma -1-\frac{n-\lambda}{r}\), \(\alpha=\frac{2\sigma-1}{\sigma}-\frac{n-\lambda }{2r\sigma}\), \(\nabla\cdot u_{0}=0\), \(\nabla\cdot d_{0}=0\), \((u_{0},d_{0})\in \dot{\mathbf{N}}_{r,\lambda,\infty}^{-s}\times\dot{\mathbf {N}}_{r,\lambda,\infty}^{-s}\), \(q\in[r, \infty]\). There exists a constant \(M_{1}>0\), such that \((u_{0},d_{0})\) satisfies (1.5), then we have \((\overline{u}_{0},\overline{d}_{0})\in E_{M_{1}}\times E_{M_{1}}\), where \(\bar{u}_{0}=e^{-t\mathfrak{L}}u_{0}\) and \(\bar {d}_{0}=e^{-t\mathfrak{L}}d_{0}\).
Proof
From (2.3) of Lemma 2.2, we thus obtain
Note that \(\alpha\sigma+\frac{n-\lambda}{2r}=s+\frac{n-\lambda}{r}\), it follows from Lemma 2.3 and a Sobolev-type embedding of Lemma 2.1 that
Hence, the proof of Lemma A.1 is now completed. □
Define
Lemma A.2
Let \(n\geq2\), \(1\leq r<\infty\), \(0\leq\lambda< n\), \(r>n-\lambda\), \(\frac {1}{2}+\frac{n-\lambda}{4r}<\sigma< 1+\frac{n-\lambda}{4r}\), \(s=2\sigma -1-\frac{n-\lambda}{r}\), \(\alpha=\frac{2\sigma-1}{\sigma}-\frac{n-\lambda }{2r\sigma}\). \(\Phi_{1}\) and \(\Phi_{2}\) were defined by (A.3), respectively. It holds true that
for all \((u, d)\in E\times E\).
Proof
From Lemmas 2.1 and 2.2, it follows that
By Lemmas 2.1, 2.2 and 2.3, we obtain the estimate \(t^{\frac{\alpha}{2}}\|\Phi_{1}(u,d)\|_{2r, \lambda} \lesssim t^{\frac{\alpha}{2}}\|\Phi_{1}(u, d)\|_{\dot {\mathbf{N}}_{2r, \lambda, 1}^{0}} \lesssim t^{\frac{\alpha}{2}}\|\Phi_{1}(u, d)\|_{\dot {\mathbf{N}}_{r, \lambda, 1}^{2\sigma-1-\alpha\sigma}} \). We thus obtain
In the following, in a similar way to the derivation of (A.4) and (A.5), we have
Thus, we complete the proof of Lemma A.2. □
Lemma A.3
Let \(n\geq2\), \(1\leq r<\infty\), \(0\leq\lambda< n\), \(r>n-\lambda\), \(\frac {1}{2}+\frac{n-\lambda}{4r}<\sigma< 1+\frac{n-\lambda}{4r}\), \(s=2\sigma -1-\frac{n-\lambda}{r}\), \(\alpha=\frac{2\sigma-1}{\sigma}-\frac{n-\lambda }{2r\sigma}\), \(\nabla\cdot u_{0}=0\), \(\nabla\cdot d_{0}=0\), \((u_{0},d_{0})\in \dot{\mathbf{N}}_{r,\lambda,\infty}^{-s}\times\dot{\mathbf {N}}_{r,\lambda,\infty}^{-s}\), \(q\in[r, \infty]\). Given a constant \(M_{2}>0\) small enough, let \((\hat{u}, \hat{d})\in E_{M_{2}}\times E_{M_{2}}\), and \((u_{0},d_{0})\) satisfy (1.5), then \((u ,d)\in E_{M_{2}}\times E_{M_{2}}\), where
Proof
We will prove \((u, d)\in E_{M_{2}}\times E_{M_{2}}\). Due to Lemmas A.1 and A.2, one thus has
provided \(0< M_{1}\leq M_{2} \) is chosen to be sufficiently small, where we have used the estimate
in the last step. Therefore, we obtain \((u, d)\in E_{M_{2}}\times E_{M_{2}}\).
Hence, the proof of Lemma A.3 is finished. □
Lemma A.4
For all \(M_{2}>0\) small enough, let \((u, d)\in E_{M_{2}}\times E_{M_{2}}\) and \((\tilde{u}, \tilde{d})\in E_{M_{2}}\times E_{M_{2}}\) with the same initial data \((u_{0}, d_{0})\), then \(\Phi=[\Phi_{1}, \Phi _{2}]\) defined in (A.3) is a contractive map.
Proof
Let \(\mathfrak{u}=u-\tilde{u}\) and \(\mathfrak{d}=d-\tilde{d}\), repeating the proof as Lemma A.2, it holds true that
Meanwhile, similar to the proof of Lemma A.2, we have
Taking \(M_{2}>0\) small enough, there exists \(0<\theta<\frac{1}{2}\), such that
Therefore, \(\Phi=[\Phi_{1}, \Phi_{2}]\) is a contractive map and we complete the proof of Lemma A.4. □
Applying Banach’s fixed pointed theorem, we finish the proof of global existence, it following directly from Lemmas A.1, A.2, A.3, and A.4.
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Yang, M. On analyticity rate estimates to the magneto-hydrodynamic equations in Besov-Morrey spaces. Bound Value Probl 2015, 155 (2015). https://doi.org/10.1186/s13661-015-0417-2
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DOI: https://doi.org/10.1186/s13661-015-0417-2