Discretely Self-Similar Solutions for 3D MHD Equations and Global Weak Solutions in Weighted \(L^2\) Spaces

Abstract

This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces \(L^2_{w_\gamma }\), with \(w_\gamma (x)=(1+\vert x\vert )^{-\gamma }\) and \(0 \le \gamma \le 2\). Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work (Fernández-Dalgo et al. in Arch Rational Mech Anal 237:347–382, 2020) for the Navier–Stokes equations.

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Correspondence to Pedro Gabriel Fernández-Dalgo.

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Pedro Gabriel Fernández-Dalgo and Oscar Jarrín declare that they have no conflict of interest.

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Communicated by D. Chae

Approximated System

Approximated System

Let \(\theta \in {\mathcal {D}}({\mathbb {R}}^{3})\) be a non-negative, radial and radially decreasing function such that \(\int _{{\mathbb {R}}^{3}}\theta (x)dx=1\), and \(Supp (\theta ) \subset B(0,1)\). For \(\varepsilon >0\) we let \(\theta _\varepsilon (x) = \frac{1}{\varepsilon ^3} \theta (\frac{x}{\varepsilon })\). o

Proposition 1

Let \(\mathbf{u }_0 \in L^2({\mathbb {R}}^{3})\), \(\mathbf{b }_0 \in L^{2}({\mathbb {R}}^{3})\) be divergence free vector fields. Let \({\mathbb {F}} = (F_{i,j})_{1 \le i,j \le 2 } \) be a tensor such that \({\mathbb {F}} \in L^{2}((0, T), L^2)\), for all \(T< T_\infty \).

Then there exists a unique solution \((\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon , p_\varepsilon , q_\varepsilon )\) of the following approximated system

$$\begin{aligned} (MHD_\varepsilon ) \left\{ \begin{array}{ll} \partial _t \mathbf{u }= \Delta \mathbf{u }- [(\mathbf{u }*\theta _{\varepsilon }) \cdot \nabla ] \mathbf{u }+ [(\mathbf{b }*\theta _{\varepsilon }) \cdot \nabla ] \mathbf{b }- \nabla p + {{\varvec{\nabla }}}\cdot {\mathbb {F}}, \\ \partial _{t} \mathbf{b }= \Delta \mathbf{b }- [(\mathbf{u }_\varepsilon *\theta _{\varepsilon }) \cdot \nabla ] \mathbf{b }+ [(\mathbf{b }*\theta _\varepsilon ) \cdot \nabla ] \mathbf{u }- {{\varvec{\nabla }}}q ,\\ {{\varvec{\nabla }}}\cdot \mathbf{u }=0, \, {{\varvec{\nabla }}}\cdot \mathbf{b }=0, \\ \mathbf{u }(0,\cdot )=\mathbf{u }_0, \, \mathbf{b }(0,\cdot )=\mathbf{b }_0, \end{array} \right. \end{aligned}$$

on \([0,T_\infty )\) such that:

  1. (1)

    \(\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon \in L^{\infty } ([0, T),L^{2}({\mathbb {R}}^{3}))\cap L^{2}([0, T), {\dot{H}}^{1}({\mathbb {R}}^{3}))\), \( p_\varepsilon ,q_\varepsilon \in L^2((0,T), \dot{H} ^{-1}) +L^2((0,T), L^2) \), for all \(0<T < T_\infty \)

  2. (2)

    the pressure \(p_\varepsilon \) and the term \(q_\varepsilon \) are related to \(\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon \) and \({\mathbb {F}}\) by

    $$\begin{aligned} p_\varepsilon = \sum _{1 \le i,j\le 3} {\mathcal {R}}_i {\mathcal {R}}_j ((u_{\varepsilon , i}*\theta _\varepsilon ) u_{\varepsilon ,j} - (b_{\varepsilon ,i}*\theta _\varepsilon ) b _{\varepsilon ,j} - F_{i,j} ),\end{aligned}$$

    and

    $$\begin{aligned} q_\varepsilon = \sum _{1 \le i,j\le 3} {\mathcal {R}}_i {\mathcal {R}}_j ( (u_{\varepsilon ,i}*\theta _\varepsilon ) b_{\varepsilon ,j} - (b_{\varepsilon ,j}*\theta _\varepsilon ) u _{\varepsilon ,i}), \end{aligned}$$

    where \({\mathcal {R}}_i =\frac{\partial _i}{\sqrt{-\Delta }}\) denote always the Riesz transforms. In particular, \( p_\varepsilon ,q_\varepsilon \in L^{4}((0,T),L^{6/5}) + L^{2}((0,T),L^{2})\).

  3. (3)

    The functions \((\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon , {\mathbb {F}}, {\mathbb {G}})\) verify the following global energy equality:

    $$\begin{aligned} \begin{aligned} \partial _t(\frac{\vert \mathbf{u }_\varepsilon \vert ^2 + |\mathbf{b }_\varepsilon |^2 }{2}) =&\, \Delta (\frac{\vert \mathbf{u }_\varepsilon \vert ^2 + |\mathbf{b }_\varepsilon |^2 }{2})-\vert {{\varvec{\nabla }}}\mathbf{u }_\varepsilon \vert ^2 - |{{\varvec{\nabla }}}\mathbf{b }_\varepsilon |^2 \\&- {{\varvec{\nabla }}}\cdot \left( (\frac{\vert \mathbf{u }_\varepsilon \vert ^2}{2} + \frac{\vert \mathbf{b }_\varepsilon \vert ^2}{2})(\mathbf{u }_\varepsilon * \theta _\varepsilon ) + p_\varepsilon \mathbf{u }_\varepsilon \right) \\&+ {{\varvec{\nabla }}}\cdot ((\mathbf{u }_\varepsilon \cdot \mathbf{b }_\varepsilon ) (\mathbf{b }_\varepsilon * \theta _\varepsilon ) + q_\varepsilon \mathbf{b }_\varepsilon ) \\&+ \mathbf{u }_\varepsilon \cdot ({{\varvec{\nabla }}}\cdot {\mathbb {F}}). \end{aligned} \end{aligned}$$
    (19)

    and

    $$\begin{aligned} \begin{aligned}&\Vert \mathbf{u }_\varepsilon (t) \Vert ^{2}_{L^2} + \Vert \mathbf{b }_\varepsilon (t) \Vert ^{2}_{L^2} + 2 \int _{a}^{t} (\Vert {{\varvec{\nabla }}}\mathbf{u }_\varepsilon (s) \Vert ^{2}_{L^2} + \Vert {{\varvec{\nabla }}}\mathbf{b }_\varepsilon (s) \Vert ^{2}_{L^2} ) ds \\&\quad = \Vert \mathbf{u }_\varepsilon (a) \Vert ^{2}_{L^2} + \Vert \mathbf{b }_\varepsilon (a) \Vert ^{2}_{L^2} - 2 \sum _{1 \le i,j \le 3} \int _{a}^{t} \int F_{i,j}\partial _iu_{_\varepsilon ,j}\ \, dx\, ds , \end{aligned} \end{aligned}$$
    (20)

    which implies in particular

    $$\begin{aligned} \begin{aligned}&\Vert \mathbf{u }_\varepsilon (t) \Vert ^{2}_{L^2} + \Vert \mathbf{b }_\varepsilon (t) \Vert ^{2}_{L^2} + \int _{0}^{t} (\Vert {{\varvec{\nabla }}}\mathbf{u }_\varepsilon (s) \Vert ^{2}_{L^2} + \Vert {{\varvec{\nabla }}}\mathbf{b }_\varepsilon (s) \Vert ^{2}_{L^2} ) ds \\&\quad \le \Vert \mathbf{u }_0 \Vert ^{2}_{L^2} + \Vert \mathbf{b }_0 \Vert ^{2}_{L^2} + C \Vert {\mathbb {F}}\Vert ^{2}_{L^{2}_{t} L^{2}_{x}}. \end{aligned} \end{aligned}$$
    (21)

Proof

We consider \(0< T< T_1 < T_\infty \) and the space \(E_T= {\mathcal {C}}([0,T], L^2({\mathbb {R}}^{3})) \cap L^2((0, T) {\dot{H}}^{1}({\mathbb {R}}^{3}))\) doted with the norm \(\Vert \cdot \Vert _{T}= \Vert \cdot \Vert _{L^{\infty }_{t}L^{2}_{x}}+ \Vert \cdot \Vert _{L^{2}_{t}{\dot{H}}^{1}_{x}}\). We will construct simultaneously \(\mathbf{u }_\varepsilon \) and \(\mathbf{b }_\varepsilon \). For this we will consider the space \(E_T \times E_T\) with the norm \(\Vert (\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon ) \Vert _{T}= \Vert \mathbf{u }_\varepsilon \Vert _{T}+ \Vert \mathbf{b }_\varepsilon \Vert _{T}\).

We use the Leray projection operator in order to express the problem \((MHD_\varepsilon )\) in terms of a fixed point problem. We let

$$\begin{aligned} a= e^{ t \Delta } (\mathbf{v }_0, \mathbf{c }_0)+ \int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}( {{\varvec{\nabla }}}\cdot {\mathbb {F}}, (0,0,0) )(s,\cdot ) ds \end{aligned}$$

and

$$\begin{aligned} B((\mathbf{u },\mathbf{b }), (\mathbf{v },\mathbf{c })) = ( \, B_1((\mathbf{u },\mathbf{b })\, , \, (\mathbf{v },\mathbf{c })), B_2((\mathbf{u },\mathbf{b }), (\mathbf{v },\mathbf{c })) \, ) , \end{aligned}$$

where

$$\begin{aligned} B_1((\mathbf{u },\mathbf{b }), (\mathbf{v },\mathbf{c })) =&\int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}([(\mathbf{u }*\theta _{\varepsilon }) \cdot \nabla ) \mathbf{v }- [(\mathbf{v }*\theta _{\varepsilon }] \cdot \nabla ] \mathbf{c })(s,\cdot ) ds ,\\ B_2((\mathbf{u },\mathbf{b }), (\mathbf{v },\mathbf{c })) =&\int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}([(\mathbf{u }*\theta _{\varepsilon }) \cdot \nabla ] \mathbf{c }- [(\mathbf{b }*\theta _{\varepsilon }) \cdot \nabla ] \mathbf{v })(s,\cdot ) ds . \end{aligned}$$

Then

$$\begin{aligned} (\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon , p_\varepsilon , q_\varepsilon ) \in E_T^2 \times \left( L^2((0,T), \dot{H} ^{-1}) +L^2((0,T), L^2) \right) ^2 \end{aligned}$$

is a solution of \((MHD_\varepsilon )\) if and only if \((\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon )\) is a fixed point for the application \((\mathbf{u },\mathbf{b }) \mapsto a + B((\mathbf{u },\mathbf{b }),(\mathbf{u },\mathbf{b }))\) and

$$\begin{aligned} p_\varepsilon = \sum _{1 \le i,j\le 3} {\mathcal {R}}_i {\mathcal {R}}_j ((u_{\varepsilon , i}*\theta _\varepsilon ) u_{\varepsilon ,j} - (b_{\varepsilon ,i}*\theta _\varepsilon ) b _{\varepsilon ,j} - F_{i,j} ) \end{aligned}$$

and

$$\begin{aligned} q_\varepsilon = \sum _{1 \le i,j\le 3} {\mathcal {R}}_i {\mathcal {R}}_j ( (u_{\varepsilon ,i}*\theta _\varepsilon ) b_{\varepsilon ,j} - (b_{\varepsilon ,j}*\theta _\varepsilon ) u _{\varepsilon ,i}) .\end{aligned}$$

We will use the Picard’s point fixed theorem. In order to study the linear terms, recall the following estimates, for a proof see [12], Theorem 12.2, page 352.

Lemma A.1

Let \(f \in L^{2}({\mathbb {R}}^{3})\) and \(g \in L^{2}_{t}{\dot{H}}^{-1}_{x}\). We have:

  1. (1)

    \(\Vert e^{ t \Delta } f \Vert _{T} \le c \Vert f \Vert _{L^2}\).

  2. (2)

    \(\left\| \int _{0}^{t} e^{(t-s)\Delta } g(s,\cdot )ds \right\| _{T} \le c(1+\sqrt{T}) \Vert g \Vert _{L^{2}_{t} {\dot{H}}^{-1}_{x}}\).

By this lemma we have

$$\begin{aligned} \Vert e^{ t \Delta } (\mathbf{u }_0, \mathbf{b }_0) \Vert _{T} \le c (\Vert \mathbf{u }_0 \Vert _{L^2} + \Vert \mathbf{b }_0 \Vert _{L^2}) \end{aligned}$$
(22)

and

$$\begin{aligned}&\left\| \int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}( {{\varvec{\nabla }}}\cdot {\mathbb {F}}, (0,0,0))(s,\cdot ) ds \right\| _{T} \nonumber \\&\le c (1+\sqrt{T}) \Vert {\mathbb {P}}( {{\varvec{\nabla }}}\cdot {\mathbb {F}}) \Vert _{L^{2}_{t} {\dot{H}}^{-1}_{x}} \nonumber \\&\le c (1+\sqrt{T}) \Vert {\mathbb {F}}\Vert _{L^{2}_{t} L^{2}_{x}}. \end{aligned}$$
(23)

Now, to study the bilinear terms recall the following estimate given in [12] (Theorem 12.2, page 352):

Lemma A.2

Let \( \mathbf{u }, \mathbf{b }\in E_T\). We have

$$\begin{aligned} \begin{aligned} \displaystyle {\left\| \int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}((( \mathbf{u }*\theta _{\varepsilon }) \cdot \nabla ) \mathbf{b })(s,\cdot )ds \right\| _{T} \le c \sqrt{T} \varepsilon ^{-3/2} \Vert \mathbf{u }\Vert _{T} \Vert \mathbf{b }\Vert _{T}}. \end{aligned} \end{aligned}$$

Applying this lemma to each bilinear term in the equation (22) we get

$$\begin{aligned} B((\mathbf{u },\mathbf{b }), (\mathbf{v },\mathbf{c })) \le c \sqrt{T}\varepsilon ^{-3/2} \Vert (\mathbf{u }, \mathbf{b }) \Vert _{T}\, \Vert (\mathbf{v }, \mathbf{c }) \Vert _{T}. \end{aligned}$$
(24)

Once we have inequalities (22), (23) and (24), for a time \(0< T_0 <T_1 \) such that

$$\begin{aligned} T_0 = \min \left( T_1, \frac{c \varepsilon ^3 }{(\Vert (\mathbf{u }_0, \mathbf{b }_0)\Vert _{L^2} + \Vert {\mathbb {F}}\Vert _{L^2((0,T_1), L^2)})^2} \right) , \end{aligned}$$

by the Picard’s contraction principle, we obtain \((\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon , p_\varepsilon , q_\varepsilon )\) a local solution of \((MHD_\varepsilon )\), where \( \mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon \in E_T \) and \(p_\varepsilon , q_\varepsilon \in L^2((0,T), \dot{H} ^{-1}) +L^2((0,T), L^2)\). We can verify that this solution is unique.

Now, we prove that \( (\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon , p_\varepsilon , q_\varepsilon )\) is a global solution. We define the maximal existence time of the solution \(\mathbf{u }_\epsilon , \mathbf{b }_\epsilon \) by

$$\begin{aligned} T_{MAX} = \sup \{ 0< T \le T_\infty \, : \, \mathbf{u }_\epsilon , \mathbf{b }_\epsilon \in E_T \} \end{aligned}$$

If \( T_{MAX} < T_{\infty }\) we take \( 0< T< T_{MAX}< T_1 < T_{\infty }\), then \((\mathbf{u }_\epsilon , \mathbf{b }_\epsilon )\) is a solution of (\(MHD_\varepsilon \)) on [0, T] and \((\mathbf{u }_\epsilon , \mathbf{b }_\epsilon )\) is a solution on \([T, T+ \delta ]\), where

$$\begin{aligned} \delta = \min \left( T_1 - T , \frac{c \varepsilon ^3 }{(\Vert (\mathbf{u }_\epsilon (T), \mathbf{b }_\epsilon (T))\Vert _{L^2} + \Vert {\mathbb {F}}\Vert _{L^2((T,T_1), L^2)} )^2} \right) , \end{aligned}$$

which implies that \(\lim _{T \rightarrow T_{MAX}^{-} } \Vert (\mathbf{u }_\varepsilon (T), \mathbf{b }_\varepsilon (T) ) \Vert _{L^2} = + \infty \), however, we will see that it is not possible.

As \( ((\mathbf{b }_{\varepsilon } * \theta _\varepsilon ) \cdot {{\varvec{\nabla }}}) \mathbf{b }_{\varepsilon } ) \mathbf{u }_{_\varepsilon } = {{\varvec{\nabla }}}\cdot ( \mathbf{b }_{\varepsilon } \otimes ( \mathbf{b }_{\varepsilon } * \theta _\varepsilon )) \mathbf{u }_{_\varepsilon }\) belongs to \(L^2((0,T), \dot{H} ^{-1})\), and the same for the other non linear terms, we can write

$$\begin{aligned} \frac{d}{dt} \Vert \mathbf{u }_\varepsilon (t) \Vert ^{2}_{L^2}= & {} 2 \langle \partial _t \mathbf{u }_\varepsilon (t), \mathbf{u }_\varepsilon (t) \rangle _{{\dot{H}}^{-1}\times {\dot{H}}^{1}} \\= & {} -2 \Vert {{\varvec{\nabla }}}\mathbf{u }_\varepsilon (t) \Vert ^{2}_{L^2} + 2 \sum _{1\le i,j \le 3} \int b_{\varepsilon ,i} ( b_{\varepsilon ,j} * \theta _\varepsilon ) \partial _i u_{_\varepsilon ,j} dx \\&- 2 \sum _{1\le i,j \le 3} \int F_{i,j}\partial _iu_{_\varepsilon ,j}\ \, dx , \end{aligned}$$

and

$$\begin{aligned} \frac{d}{dt} \Vert \mathbf{b }_\varepsilon (t) \Vert ^{2}_{L^2}= & {} 2 \langle \partial _t \mathbf{b }_\varepsilon (t), \mathbf{b }_\varepsilon (t) \rangle _{{\dot{H}}^{-1}\times {\dot{H}}^{1}} \\= & {} -2 \Vert {{\varvec{\nabla }}}\mathbf{b }_\varepsilon (t) \Vert ^{2}_{L^2} + 2 \sum _{1\le i,j \le 3} \int u_{\varepsilon ,i} ( b_{\varepsilon ,j} * \theta _\varepsilon ) \partial _i b_{_\varepsilon ,j} dx . \end{aligned}$$

where we have used the fact that

$$\begin{aligned} \int ( (\mathbf{u }_\varepsilon * \theta ) \cdot \nabla ) \mathbf{b }_\varepsilon \cdot \mathbf{b }_\varepsilon \, dx&= \int \sum _{1\le i,j \le 3} ( (u_{j, _\varepsilon } *\theta ) \partial _j b_{i,_\varepsilon }) b_{ i, _\varepsilon }\, dx \\&= -\frac{1}{2} \int (\mathbf{u }_{ \varepsilon } *\theta ) \cdot {{\varvec{\nabla }}}( |\mathbf{b }_\varepsilon |^2 ) \, dx \\&=-\frac{1}{2} \int \nabla \cdot (\mathbf{u }_\varepsilon *\theta _\varepsilon ) |\mathbf{b }_\varepsilon |^2 dx = 0. \end{aligned}$$

Then, an integration by parts gives

$$\begin{aligned} \sum _{1\le i,j \le 3} \int u_{\varepsilon ,i} ( b_{\varepsilon ,j} * \theta _\varepsilon ) \partial _i b_{\varepsilon ,j} dx = - \sum _{1\le i,j \le 3} \int b_{\varepsilon ,i} ( b_{\varepsilon ,j} * \theta _\varepsilon ) \partial _i u_{\varepsilon ,j} dx, \end{aligned}$$

so we have

$$\begin{aligned} \frac{d}{dt} ( \Vert \mathbf{u }_\varepsilon (t) \Vert ^{2}_{L^2}+ \Vert \mathbf{b }_\varepsilon (t) \Vert ^{2}_{L^2})= & {} -2 (\Vert {{\varvec{\nabla }}}\mathbf{u }_\varepsilon (t) \Vert ^{2}_{L^2} + \Vert {{\varvec{\nabla }}}\mathbf{b }_\varepsilon (t) \Vert ^{2}_{L^2} ) \\&- 2 \sum _{1 \le i,j \le 3} \int F_{i,j}\partial _iu_{j}\ \, dx\, ds . \end{aligned}$$

By integrating on the time interval [0, T] we obtain the control (20) which implies by Grönwall inequality that \(\Vert (\mathbf{u }_\varepsilon , \mathbf{b }_\varepsilon )(T) ) \Vert _{L^2}\) does not converges to \(+\infty \) when T goes to \(T_{MAX}\) if \(T_{MAX}< T_\infty \), hence the solution is defined on \([0, T_\infty )\). Finally, remark that we can write

$$\begin{aligned} {{\varvec{\nabla }}}\cdot ((\mathbf{b }_\varepsilon \cdot \mathbf{u }_\varepsilon ) (\mathbf{b }_\varepsilon *\theta _\varepsilon ))&= {{\varvec{\nabla }}}(\mathbf{b }_\varepsilon \cdot \mathbf{u }_\varepsilon ) \cdot (\mathbf{b }_\varepsilon *\theta _\varepsilon ) \\&=((\mathbf{b }_\varepsilon *\theta _\varepsilon ) \cdot {{\varvec{\nabla }}}) \mathbf{b }_\varepsilon \cdot \mathbf{u }_\varepsilon + ((\mathbf{b }_\varepsilon *\theta _\varepsilon ) \cdot {{\varvec{\nabla }}}) \mathbf{u }_\varepsilon ) \cdot \mathbf{b }_\varepsilon \end{aligned}$$

so that

$$\begin{aligned} \partial _t(\frac{\vert \mathbf{u }_\varepsilon \vert ^2}{2})=&\, \Delta (\frac{\vert \mathbf{u }_\varepsilon \vert ^2}{2})-\vert {{\varvec{\nabla }}}\mathbf{u }_\varepsilon \vert ^2- {{\varvec{\nabla }}}\cdot \left( \frac{\vert \mathbf{u }_\varepsilon \vert ^2}{2} (\mathbf{u }_\varepsilon * \theta _\varepsilon ) + p_\varepsilon \mathbf{u }_\varepsilon \right) \\&+ {{\varvec{\nabla }}}\cdot ((\mathbf{u }_\varepsilon \cdot \mathbf{b }_\varepsilon ) (\mathbf{b }_\varepsilon * \theta _\varepsilon )) - ((\mathbf{b }_\varepsilon *\theta _\varepsilon ) \cdot {{\varvec{\nabla }}}) \cdot \mathbf{u }_\varepsilon ) \mathbf{b }_\varepsilon + \mathbf{u }_\varepsilon \cdot ({{\varvec{\nabla }}}\cdot {\mathbb {F}}), \end{aligned}$$

similarly we find

$$\begin{aligned} \partial _t(\frac{\vert \mathbf{b }_\varepsilon \vert ^2}{2})=&\, \Delta (\frac{\vert \mathbf{b }_\varepsilon \vert ^2 }{2})-\vert {{\varvec{\nabla }}}\mathbf{b }_\varepsilon \vert ^2 -{{\varvec{\nabla }}}\cdot \left( \frac{\vert \mathbf{b }_\varepsilon \vert ^2}{2}(\mathbf{u }_\varepsilon * \theta _\varepsilon ) + q_\varepsilon \mathbf{b }_\varepsilon \right) \\&+ ((\mathbf{b }_\varepsilon *\theta _\varepsilon ) \cdot {{\varvec{\nabla }}}) \cdot \mathbf{u }_\varepsilon ) \mathbf{b }_\varepsilon . \end{aligned}$$

By adding these equations we obtain the energy equality (19). \(\square \)

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Fernández-Dalgo, P.G., Jarrín, O. Discretely Self-Similar Solutions for 3D MHD Equations and Global Weak Solutions in Weighted \(L^2\) Spaces. J. Math. Fluid Mech. 23, 22 (2021). https://doi.org/10.1007/s00021-020-00551-1

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Keywords

  • MHD equations
  • Weighted \(L^2\) spaces
  • Discretely self-similar solutions
  • Energy controls

Mathematics Subject Classification

  • 35Q30
  • 76D05