Magnetic Relaxation of a Voigt–MHD System

Abstract

We construct solutions of the magnetohydrostatic (MHS) equations in bounded domains and on the torus in three spatial dimensions, as infinite time limits of Voigt approximations of viscous, non-resistive incompressible magnetohydrodynamics equations. The Voigt approximations modify the time evolution without introducing artificial viscosity. We show that the obtained MHS solutions are regular, nontrivial, and are not Beltrami fields.

Appendices

Appendix A. Periodic Boundary Conditions: Setup

Let ${\mathbb {T}}^3$ be the three-dimensional torus.

1.1 A.1 Norms and spaces

We work with the spaces $H^s({\mathbb {T}}^3)$ and ${\dot{H}^s}({\mathbb {T}}^3)$, which we define as follows.

Definition A.1

Let $f \in {\mathscr {P}}'$, where ${\mathscr {P}}'$ is the space of periodic distributions on the 3-dimensional torus ${\mathbb {T}}^3$. Then, consider the Fourier coefficients ${\hat{f}}(k)$, where $k \in {\mathbb {Z}}^3$. We say that $f \in H^s({\mathbb {T}}^3)$ if the Fourier coefficients of f satisfy:

$$\begin{aligned} \Vert f \Vert _{H^s({\mathbb {T}}^3)}:= \sum _{k \in {\mathbb {Z}}^3} (1+|k|^2)^{\frac{s}{2}} |{\hat{f}}(k)|^2 < \infty . \end{aligned}$$

We also say that $f \in {\dot{H}^s}({\mathbb {T}}^3)$, if the Fourier coefficients of f satisfy:

$$\begin{aligned} {\hat{f}}(0,0,0) = 0, \qquad \Vert f \Vert _{{\dot{H}^s}({\mathbb {T}}^3)}:= \sum _{\begin{array}{c} k \in {\mathbb {Z}}^3\\ k \ne (0,0,0) \end{array}} (1+|k|^2)^{\frac{s}{2}} |{\hat{f}}(k)|^2 < \infty . \end{aligned}$$

We moreover denote

$$\begin{aligned} {{\dot{L}}}^2({\mathbb {T}}^3):= {{\dot{H}}}^0({\mathbb {T}}^3). \end{aligned}$$

Remark A.2

In the remainder of Sect. 2 (and only restricted to this section), we denote

$$\begin{aligned} {\dot{H}^s}:= {\dot{H}^s}({\mathbb {T}}^3), \qquad H^s:= H^s({\mathbb {T}}^3). \end{aligned}$$

Remark A.3

For conciseness, we denote $\Vert f \Vert _{s}:= \Vert f \Vert _{{\dot{H}^s}}$, the homogeneous Sobolev norm.

If $f, g \in {\dot{L}}^2({\mathbb {T}}^3)$, we let

$$\begin{aligned} (f,g):= \int _{{\mathbb {T}}^3} fg \text{ d }x. \end{aligned}$$

the usual $L^2$ inner product.

The dual of ${\dot{H}}^1$ is naturally identified with ${\dot{H}}^{-1}$. If $f \in {\dot{H}}^{-1}$ and $g \in {\dot{H}}^1$, we let $\langle f, g \rangle$ be the dual pairing between f and g. Note that, if $f\in {\dot{L}}^2$, then $\langle f, g \rangle = (f,g)$.

These scalar spaces extend in a straightforward manner to their vectorial counterparts. We shall use a superscript to denote vectorial spaces, i.e. if v is a 3 dimensional vector field whose components lie in a space W, then we write $v \in W^3$.

Definition A.4

We denote $H:= \{v \in L_0^2({\mathbb {T}}^3): \text {div} \ v = 0 \}$, and $V:= \{v \in {\dot{H}}^1({\mathbb {T}}^3): \text {div} \, v = 0 \}$.

Definition A.5

We define the space $D((-\Delta )^{s/2})$ for $s \in {\mathbb {R}}$ as the domain of $(-\Delta )^{s/2}$:

$$\begin{aligned} D((-\Delta )^{s/2}):= \{v \in ({\dot{H}^s})^3: \text { div } v = 0\}. \end{aligned}$$

Without ambiguity, we denote the homogeneous Sobolev s-norm on $D((-\Delta )^{s/2})$ also by $\Vert \cdot \Vert _s$.

1.2 A.2 Leray projection, the fractional Laplacian, and classical estimates

Let ${\mathbb {P}}: L_0^2 \rightarrow H$ be the Leray projector. In terms of Fourier coefficients, it can be expressed as

$$\begin{aligned} (\widehat{{\mathbb {P}} f})_k:= \Big (\text {Id}_3 - \frac{k \otimes k}{|k|^2} \Big )({\hat{f}})_k \quad \text {for } k \ne 0, \qquad (\widehat{{\mathbb {P}} f})_0:= 0. \end{aligned}$$

Here, $\textrm{Id}_3$ is the $3\times 3$ identity matrix.

If $v, w \in V$, we define

$$\begin{aligned} {\textbf{B}}(v,w):= {\mathbb {P}}(v \cdot \nabla w). \end{aligned}$$

Let us also recall the following inequality, proved in [9]. There exists a constant $C > 0$ such that, if $u,v,w \in V$ and are all divergence-free,

$$\begin{aligned} \langle {\textbf{B}}(u,v), w \rangle \le C \Vert u\Vert _0^{\frac{1}{2}} \Vert u \Vert _1^{\frac{1}{2}} \Vert v \Vert _1 \Vert w\Vert _1. \end{aligned}$$

(30)

We also recall that elements $u \in {\dot{H}}^1$ enjoy the following Poincaré inequality:

$$\begin{aligned} \Vert u \Vert _0 \le \lambda \Vert u \Vert _1, \end{aligned}$$

(31)

where $\lambda > 0$ is the Poincaré constant.

We define the fractional Laplacian as the operator $(-\Delta )^\alpha : {\dot{H}^s} \rightarrow \dot{H}^{s-2\alpha }$ whose action on the Fourier coefficients is as follows:

$$\begin{aligned} (\widehat{(-\Delta )^\alpha f})_k = |k|^{2 \alpha } {\hat{f}}_k, \quad \text {for } k \ne 0, \qquad {\hat{f}}_0 = 0. \end{aligned}$$

We define

$$\begin{aligned} {\mathbb {L}}:= (-\Delta )^\alpha . \end{aligned}$$

We have the following lemma:

Lemma A.6

Let ${\mathbb {L}}:= (-\Delta )^\alpha$, and assume that $\alpha \ge 1$. Then, ${\mathbb {L}}$ maps ${\dot{H}^s}$ onto $\dot{H}^{s-2\alpha }$, is injective and has a well defined inverse ${\mathbb {L}}^{-1}: \dot{H}^{s-2\alpha }\rightarrow {\dot{H}^s}$. Furthermore, the composition operators $\Delta {\mathbb {L}}^{-1} = {\mathbb {L}}^{-1} \Delta$ are well defined, and are both bounded operators from ${\dot{H}^s}$ to $\dot{H}^{2\alpha - 2 +s}$. In particular, there exists a constant $C > 0$ such that, for all $v \in {\dot{H}^s}$,

$$\begin{aligned} \Vert {\mathbb {L}}^{-1}\Delta v \Vert _{2\alpha - 2 + s} = \Vert \Delta {\mathbb {L}}^{-1}v \Vert _{2\alpha -2 +s } \le C \Vert v\Vert _{s}. \end{aligned}$$

Remark A.7

The lemma above holds true replacing all instances of $\dot{H}^\gamma$ with $D((-\Delta )^{\gamma /2})$.

Proof of lemma A.6

The proof follows from the spectral characterization of the operator $(-\Delta )^\alpha$. $\square$

Remark A.8

By the Fourier characterization of $(-\Delta )^\alpha$ and of ${\mathbb {P}}$, it is evident that both these operators commute with partial derivatives. Furthermore, we have that

$$\begin{aligned}{}[{\mathbb {P}}, {\mathbb {L}}] =0, \qquad [{\mathbb {P}}, (-\Delta )^\alpha ] =0. \end{aligned}$$

We also recall the definition of time-dependent spaces. Let $(X, \Vert \cdot \Vert _X)$ be a Banach space, and let $T > 0$. We say that a function $f: [0,T] \rightarrow X$ is such that $f \in L^\infty (0, T; X)$ if the following holds:

$$\begin{aligned} \sup _{t \in [0,T] } \Vert f(t) \Vert _X < \infty . \end{aligned}$$

Let $f: [0,T] \rightarrow X$ be Bochner integrable, and define $f'$ as the weak time derivative of f. We then say that $f \in {\mathcal {C}}^1(0, T; X)$ if there holds

$$\begin{aligned} \sup _{t \in [0,T] } (\Vert f(t) \Vert _X + \Vert f'(t) \Vert _X) < \infty . \end{aligned}$$

1.3 A.3 The viscous Voigt–MHD system

We focus our attention on the Voigt regularized system (VMHD-1)–(VMHD-4) (recall that ${\mathbb {L}} = (- \Delta )^\alpha$):

$$\begin{aligned}&\partial _t u + {\mathbb {L}}^{-1}(u \cdot \nabla u - B \cdot \nabla B) = \nu {\mathbb {L}}^{-1} \Delta u, \end{aligned}$$

(32)

$$\begin{aligned}&\partial _t B + {\mathbb {L}}^{-1}(u \cdot \nabla B - B \cdot \nabla u) = 0, \end{aligned}$$

(33)

$$\begin{aligned}&(u, B)|_{t = 0} = (u_0, B_0). \end{aligned}$$

(34)

Here, u(x, t) and B(x, t) are three-dimensional vector fields depending on position $x \in {\mathbb {T}}^3$ (the three-dimensional flat torus) and time t. Furthermore $\nu >0$ is a fixed parameter, and q(x, t) is the pressure term.

Let $\alpha \ge 1$. We set up initial data $u_0$ and $B_0$ such that $u_0, B_0 \in D((-\Delta )^{\alpha /2})$ (see Definition A.5). In particular, $u_0$, $B_0$ satisfy:

$$\begin{aligned} \textrm{div} \,u_0 = 0, \qquad \textrm{div} \,B_0 = 0, \end{aligned}$$

(35)

and moreover

$$\begin{aligned} \int _{{\mathbb {T}}^3} B_0 \text{ d }x = 0, \qquad \int _{{\mathbb {T}}^3} u_0 \text{ d }x = 0. \end{aligned}$$

(36)

1.4 A.4 Well-posedness, global existence and regularity

We show that the Voigt–MHD system (32)–(34) is locally well posed for strong solutions, and it moreover admits global solutions for large initial data. In the Voigt case, strong regularization easily implies well-posedness. Note that the issue is generally more involved in the case of non-regularized systems: for a discussion of local ill-posedness for a wide range of hydrodynamical systems we refer the reader to [13].

We first prove a local existence statement with a suitable continuation criterion.

Proposition A.9

(Local existence of solutions to (32)–(34)). Let $(u_0, B_0)$ in $D((-\Delta )^{\alpha /2})$ with $\alpha \ge 1$, and let $u_0$ and $B_0$ satisfy the divergence-free condition (35) and the mean zero condition (36). Then, there exists a time $T > 0$ and $u, B \in L^\infty (0,T; D((-\Delta )^{\alpha /2})$ which solve (32)–(34) in the strong sense, and such that

$$\begin{aligned} (u, B)|_{t= 0} = (u_0, B_0). \end{aligned}$$

Furthermore, if $T_*$ is the maximal time of existence, we necessarily have either

$$\begin{aligned} \limsup _{t \rightarrow T_*} \Vert u(\cdot , t) \Vert _{\alpha } = \infty , \quad or \qquad \limsup _{t \rightarrow T_*} \Vert B(\cdot , t) \Vert _{\alpha } = \infty . \end{aligned}$$

Proof of Proposition A.9

The proof follows by Picard iteration carried out in the space

$$\begin{aligned} (u, B) \in L^\infty (0,T; D((-\Delta )^{\alpha /2})). \end{aligned}$$

We follow the approach in [20] (Theorem 6.1).

We consider the following evolution equations, which are equivalent to system (32)–(34):

$$\begin{aligned} \begin{aligned}&\partial _t {\mathbb {L}}u = {\mathbb {P}}(B \cdot \nabla B - u \cdot \nabla u) + \nu \Delta u, \\&\partial _t {\mathbb {L}}B = {\mathbb {P}} (- u \cdot \nabla B + B \cdot \nabla u). \end{aligned} \end{aligned}$$

We then let $v = {\mathbb {L}}u$, $Z = {\mathbb {L}}B$, and we rewrite the system as follows:

$$\begin{aligned} \begin{aligned}&\partial _t v = {\mathbb {P}}({\mathbb {L}}^{-1}Z \cdot \nabla {\mathbb {L}}^{-1}Z - {\mathbb {L}}^{-1}v \cdot \nabla {\mathbb {L}}^{-1}v) + \nu \Delta {\mathbb {L}}^{-1}v =: N_1(v,Z), \\&\partial _t Z = {\mathbb {P}} (- ({\mathbb {L}}^{-1}v)\cdot \nabla ({\mathbb {L}}^{-1}Z) + ({\mathbb {L}}^{-1}Z) \cdot \nabla ({\mathbb {L}}^{-1}v)) =: N_2(v,Z). \end{aligned} \end{aligned}$$

The idea is to show that $N_1$ and $N_2$ are Lipschitz mappings from $D((-\Delta )^{-\alpha /2}))$ to itself, and then the Picard–Lindelöf theorem will apply. We start from $N_1$, using the conventions $v_i = {\mathbb {L}}u_i$, $Z_i = {\mathbb {L}}B_i$:

$$\begin{aligned}&\Vert N_1(v_1, Z_1) - N_1(v_2, Z_2)\Vert _{-\alpha } \\ {}&\quad \le \Vert {{\textbf {B}}}(u_1 - u_2, u_2) + {{\textbf {B}}}(u_1 , u_1 -u_2) \Vert _{-\alpha }\\ {}&\qquad + \Vert {{\textbf {B}}}(B_1 - B_2, B_2) + {{\textbf {B}}}(B_1 , B_1 -B_2) \Vert _{-\alpha } \\ {}&\qquad + \nu \Vert \Delta (u_1 - u_2) \Vert _{-\alpha } \\ {}&\quad \le C\Vert u_1 -u_2\Vert _0 \Vert u_1 - u_2\Vert _1 \Vert u_2 \Vert _1 \\ {}&\qquad + C \Vert u_1\Vert _0 \Vert u_1\Vert _1 \Vert u_1- u_2 \Vert _1 \\ {}&\qquad +C\Vert B_1 -B_2\Vert _0 \Vert B_1 - B_2\Vert _1 \Vert B_2 \Vert _1 \\ {}&\qquad + C \Vert B_1\Vert _0 \Vert B_1\Vert _1 \Vert B_1- B_2 \Vert _1 +\nu \Vert \Delta (u_1 - u_2) \Vert _{-\alpha }\\ {}&\quad \le C\lambda (\Vert v_1 \Vert _{1-2\alpha } + \Vert v_2 \Vert _{1-2\alpha } ) \Vert v_1 - v_2 \Vert _{1-2\alpha } \\ {}&\qquad + C \lambda (\Vert Z_1 \Vert _{1-2\alpha } + \Vert Z_2 \Vert _{1-2\alpha } ) \Vert Z_1 - Z_2 \Vert _{1-2\alpha } \\ {}&\qquad + C \nu \Vert v_1 - v_2\Vert _{-\alpha }\\ {}&\quad \le C\lambda (\Vert v_1 \Vert _{-\alpha } + \Vert v_2 \Vert _{-\alpha } ) \Vert v_1 - v_2 \Vert _{-\alpha } \\ {}&\qquad + C \lambda (\Vert Z_1 \Vert _{-\alpha } + \Vert Z_2 \Vert _{-\alpha } ) \Vert Z_1 - Z_2 \Vert _{-\alpha } \\ {}&\qquad + C \nu \Vert v_1 - v_2\Vert _{-\alpha }. \end{aligned}$$

Here, we used inequality (30) to go from the second to the third line, the Poincaré inequality (31), the fact that, for $\alpha \ge \gamma$, we have $\Vert f \Vert _\alpha \ge \Vert f\Vert _\gamma$, and the fact (proved in Lemma A.6) that, for $z \in D((-\Delta )^{\alpha / 2})$,

$$\begin{aligned} \Vert \Delta {\mathbb {L}}^{-1} z\Vert _{-\alpha } =\Vert {\mathbb {L}}^{-1} \Delta z\Vert _{-\alpha } \le C \Vert z \Vert _{-\alpha }, \end{aligned}$$

since $1-2\alpha \le -\alpha$ if $\alpha \ge 1$. Similarly, we have, for the terms in $N_2$,

$$\begin{aligned} \begin{aligned}&\Vert N_2(v_1, Z_1) - N_2(v_2, Z_2)\Vert _{-\alpha } \\&\quad \le \Vert {\textbf{B}}(B_1 - B_2, u_1) + {\textbf{B}}(B_2, u_1 -u_2) \Vert _{-\alpha } + \Vert {\textbf{B}}(u_2 - u_1, B_2) + {\textbf{B}}(u_1, B_2 -B_1) \Vert _{-\alpha } \\&\quad \le C\Vert B_1 -B_2\Vert _0 \Vert B_1 - B_2\Vert _1 \Vert u_1\Vert _1 + C \Vert B_2\Vert _0 \Vert B_2\Vert _1 \Vert u_1- u_2 \Vert _1 \\&\quad \quad +C\Vert u_1 -u_2\Vert _0 \Vert u_1 - u_2\Vert _1 \Vert B_2 \Vert _1 + C \Vert u_1\Vert _0 \Vert u_1\Vert _1 \Vert B_1- B_2 \Vert _1 \\&\quad \le C\lambda (\Vert v_1 \Vert _{-\alpha } + \Vert v_2 \Vert _{-\alpha } ) \Vert v_1 - v_2 \Vert _{-\alpha } + C \lambda (\Vert Z_1 \Vert _{-\alpha } + \Vert Z_2 \Vert _{-\alpha } ) \Vert Z_1 - Z_2 \Vert _{-\alpha }. \end{aligned} \end{aligned}$$

We conclude that the mapping $(N_1, N_2)$ is locally Lipschitz in $D((-\Delta )^{-\alpha /2}))$, which concludes the existence proof by an application of the Picard–Lindelöf theorem. Finally, the continuation criterion is evident from the fact that

$$\begin{aligned} \Vert {\mathbb {L}}^{-1} u \Vert _{-\alpha } \ge C \Vert u \Vert _{\alpha }. \end{aligned}$$

This concludes the proof. $\square$

We show that high norms are propagated by the system (32)–(34).

Theorem A.10

(Global existence of solutions to (32)–(34)). Let $(u_0, B_0)$ in $D((-\Delta )^{\alpha /2}))$, with $\alpha \ge 1$, satisfying (35) and (36). Then, there exist $(u, B) \in L^\infty (0,\infty ; D((-\Delta )^{\alpha /2})))$ which solve (32)–(34) in the sense of distributions, and such that

$$\begin{aligned} (u, B)|_{t= 0} = (u_0, B_0). \end{aligned}$$

Proof of Theorem A.10

Due to the continuation criterion of Proposition A.9, we only need to show that $\Vert u\Vert _{\alpha }+\Vert B\Vert _{\alpha }$ is bounded a-priori in terms of initial data. We have, taking the $L^2$ inner product of the momentum equation with u,

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\text {d}}{\text {d}t}(\Vert u \Vert ^2_{\alpha })= ( u, \partial _t {\mathbb {L}}u) = - \nu \Vert u \Vert _1^2 + ({\textbf{B}}(B,B),u)\\&\quad - ({\textbf{B}}(u,u),u) = - \nu \Vert u \Vert _0^2 - ({\textbf{B}}(B,u),B). \end{aligned} \end{aligned}$$

On the other hand, we have, taking the $L^2$ inner product of the induction equation with B,

$$\begin{aligned} \frac{1}{2} \frac{\text {d}}{\text {d}t}(\Vert B \Vert ^2_{\alpha })= ( B, \partial _t {\mathbb {L}}B) = -({\textbf{B}}(u,B),B) + ({\textbf{B}}(B,u),B). \end{aligned}$$

Recalling that $({\textbf{B}}(u,B),B)= 0$, and summing the two previous equations, we finally get, for all times $t_2 \ge t_1\ge 0$,

$$\begin{aligned} \begin{aligned}&\Vert u(\cdot , t_2) \Vert ^2_{\alpha }+ \Vert B(\cdot , t_2) \Vert ^2_{\alpha } + \nu \int _{t_1}^{t_2} \Vert \nabla u(\cdot , s) \Vert ^2 \text{ d }s \le \Vert u(\cdot , t_1) \Vert ^2_{\alpha }+ \Vert B(\cdot , t_1) \Vert ^2_{\alpha }, \end{aligned} \end{aligned}$$

which provides the required a-priori control. These a-priori estimates are only formal, but can be made rigorous by Galerkin approximation. $\square$

We show that the system (32)–(34) propagates higher regularity.

Proposition A.11

(Higher regularity of solutions to (32)–(34)). Let $(u_0, B_0)$ in $D((-\Delta )^{\beta /2}))$, with $\beta \ge \alpha \ge 1$. Then, the solution (u, B) to the system (32)–(34) constructed in Theorem (A.10) satisfies the stronger bounds:

$$\begin{aligned} \Vert (u,B) \Vert _{L^\infty (0,T; D((-\Delta )^{\beta /2})))} \le C(\Vert u_0\Vert _{\beta }, \Vert B_0\Vert _{\beta }, T). \end{aligned}$$

Sketch of proof

We only provide a sketch of the proof in the case $\beta = \alpha + k$, with $k \in {\mathbb {N}}$. Let us deal with the case $k = 1$. We formally⁴ differentiate Eqs. (32)–(34) by $\partial _i$, and obtain

$$\begin{aligned}&\partial _t (-\Delta )^\alpha \partial _i u - \Delta (\partial _i u) = {\mathbb {P}}(-\partial _i u \cdot \nabla u-u \cdot \nabla \partial _i u + \partial _i B \cdot \nabla B +B \cdot \nabla \partial _i B ),\\&\partial _t (-\Delta )^\alpha \partial _i B ={\mathbb {P}}( -\partial _i u \cdot \nabla B- u \cdot \nabla \partial _i B + \partial _i B \cdot \nabla u + B \cdot \nabla \partial _i u). \end{aligned}$$

Multiplying the first equation by $\partial _i u$ and the second equation by $\partial _i B$, summing over i, and integrating we get, using the fact that $({\textbf{B}}(a,b),c) = - ({\textbf{B}}(a,c),b)$,

$$\begin{aligned} \begin{aligned}&\partial _t ( \Vert u \Vert ^2_{\alpha +1} + \Vert B \Vert ^2_{\alpha +1} ) + \nu \Vert u \Vert ^2_2\\&= -({\textbf{B}}(\partial _i u, u), \partial _i u)+({\textbf{B}}(\partial _i B, B), \partial _i u),\\&\quad - ({\textbf{B}}(\partial _i u, B), \partial _i B) + ({\textbf{B}}(\partial _i B, u), \partial _i B). \end{aligned} \end{aligned}$$

Now, by Sobolev embedding, it is clear that

$$\begin{aligned} |({\textbf{B}}(\partial _i u, u), \partial _i u)| \le C \Vert u\Vert _2^{\frac{3}{2}} \Vert u\Vert _1^{\frac{3}{2}}, \qquad |({\textbf{B}}(\partial _i B, B), \partial _i u)| \le C \Vert u\Vert _2^{\frac{1}{2}} \Vert u\Vert _1^{\frac{1}{2}}\Vert B\Vert _2 \Vert B\Vert _1. \end{aligned}$$

Using the a-priori energy estimate, in conjunction with Grönwall’s inequality, we obtain

$$\begin{aligned} (u,B) \in L^\infty (0,T; D((-\Delta )^{(\alpha +1)/2})), \end{aligned}$$

i.e. the required bound.

The proof for larger k is similar, and we omit it here. $\square$

1.5 A.5 Magnetic potential

We state and prove a lemma on the existence of the magnetic potential in the periodic case.

Lemma A.12

(Existence of the magnetic potential). Let (u, B) such that

$$\begin{aligned} (u, B) \in L^\infty (0,T; D((-\Delta )^{\alpha /2})) \end{aligned}$$

solving the system (32)–(34), according to Theorem A.10, with divergence-free initial data $(u_0, B_0) \in D((-\Delta )^{\alpha /2})$. Let us consider the following initial value problem, for an unknown vector field $\Psi$:

$$\begin{aligned} \begin{aligned}&\partial _t {\mathbb {L}}\Psi = u \times B,\\&\Psi |_{t=0} = \Psi _0. \end{aligned} \end{aligned}$$

(37)

Here, $\Psi _0$ is the unique $\dot{H}^{\alpha +1}({\mathbb {T}}^3)$ vector field satisfying the following two properties:

$$\begin{aligned} \nabla \times \Psi _0 = B_0, \qquad \text {div } \Psi _0 = 0. \end{aligned}$$

(38)

Under these conditions, we have that the system (37) has a global solution $\Psi \in L^\infty (0,\infty ; D((-\Delta )^{(\alpha +1)/2}))$ which satisfies (37) and (38). Furthermore, the following equality holds true for all $t \ge 0$ and $x \in {\mathbb {T}}^3$.

$$\begin{aligned} \nabla \times \Psi = B. \end{aligned}$$

(39)

Proof of Lemma A.12

The existence and regularity parts are standard. To prove relation (39), we take the curl of the evolution equation in (37), in order to obtain

$$\begin{aligned} \partial _t {\mathbb {L}}(\nabla \times \Psi ) = \nabla \times (u \times B) = \partial _t {\mathbb {L}}B. \end{aligned}$$

Integrating in time and using the initial conditions, this gives ${\mathbb {L}}(\nabla \times \Psi ) = {\mathbb {L}}B$ at all times $t \ge 0$. We conclude by the fact that the kernel of ${\mathbb {L}}$ in $D((-\Delta )^{\alpha /2})$ is trivial. $\square$

Appendix B. Proof of Proposition 1.5

In this section, we provide a particular solution to the 3D MHD equation which, in the infinite time limit, creates discontinuities of B. The example is classical, and it is essentially obtained by decoupling the momentum and the induction equation, imposing that the B field is always vertical.

This example should be contrasted with the result obtained in Sect. 2, and it shows that, in general, for the full MHD system may not relax to a regular MHS equilibrium in the infinite time limit. This is different from the situation in the Voigt–MHD system.

Proof of Proposition 1.5

Let us consider the three dimensional torus obtained from the box $[-1,1] \times [-1,1] \times [-1,1]$ with opposite sides identified.

Suppose that u has the following form: $u = (u_1(x,y), u_2(x,y),0)$, and that B has the following form: $B= (0,0,B_3(t,x,y))$.

Let us consider the following stream function for u:

$$\begin{aligned} \Psi := \sin (2\pi x) \sin (2 \pi y), \end{aligned}$$

and we let $u_1:= \partial _y \Psi$, $u_2 = -\partial _x\Psi$. Under these conditions, u satisfies the steady Euler equations:

$$\begin{aligned} u \cdot \nabla u + \nabla p =0, \qquad \textrm{div} \,u = 0. \end{aligned}$$

Let us moreover evolve $B_3$ according to the following transport equation:

$$\begin{aligned} \partial _t B_3(t,x,y) + u \cdot \nabla B_3(t,x,y) = 0. \end{aligned}$$

(40)

With these choices, we have that the pair (u, B) satisfies the ideal 3D MHD equations:

$$\begin{aligned} \begin{aligned}&\partial _t u + u \cdot \nabla u+ \nabla p = B \cdot \nabla B,\\&\partial _t B + u \cdot \nabla B - B \cdot \nabla u=0,\\&\textrm{div} \,u = 0, \qquad \textrm{div} \,B = 0. \end{aligned} \end{aligned}$$

Since u is constant in time, we only need to specify initial data for B. We set $B_0 (x,y) = \chi (y)$, where $\chi$ is a smooth and periodic function, $\chi : [-1,1] \rightarrow {\mathbb {R}}$, with the property that $\chi (y) = y$ for $|y| \le 1/2$. We show that, locally around the origin, the gradient of $B_3$ grows exponentially in time.

Restricting Eq. (40) to the y-axis, we have

$$\begin{aligned} \partial _t B_3 - \sin (y) \partial _y B_3 = 0. \end{aligned}$$

Let us define a function y(t, a) by the ODE (a is the Lagrangian label): $y'(t,a) = - \sin (y(t,a))$, with $y(0) = a$. Then, it can be easily checked that, for all positive t and for all a such that $|a| \le 1/2$

$$\begin{aligned} B_3(t, 0, y(t,a), z) = B_0(0,a,z). \end{aligned}$$

(41)

Note that have the following relation satisfied by y(t, a):

$$\begin{aligned} \tan \Big ({\frac{y(t,a)}{2}}\Big ) = \tan \Big (\frac{a}{2}\Big ) e^{-t}, \end{aligned}$$

Differentiating relation (41) and calculating the result at $a = 0$, we have that

$$\begin{aligned} |\partial _y B_3 (t,0,0,z)| = C'e^t, \end{aligned}$$

for a positive constant $C'$, thereby proving the claim. $\square$