1 Introduction: the spatial setting

In a series of remarkable works [2, 3] Ball and Murat, starting from previous ideas of Meyers [5], introduce solutions of a self-similar type, unorthodox in one sense, which show how from a given variational solution of a static problem in elasticity, one can generate a plenty of new solutions, more precisely, attributing the same numerical value to the energy functional of the given initial solution. The key tool turns out to be Vitali’s covering theorem. A substantially exhaustive discussion (arising around delicate quasi-convexity questions) on the correct regularity to choose for this type of problems, involving \(W^{1,p}\) spaces, has been elaborated in [1,2,3].

Below we propose a version of that order of ideas for static problems, together with a statement of Vitali’s theorem. In the next Section, we introduce a space time version of this setting, obtaining a plenty of turbulence self similarity solutions in a full conservative environment.

Consider an elastic body \({{\mathcal {B}}}\), which we identify with the reference configuration \(\Omega \subset {\mathbb {R}}^3\), homogeneous with mass density \(\mu \equiv 1\) in \(\Omega \) and stored energy density

$$\begin{aligned} w:\text {Lin}^+ =\{F\in \text {GL}({\mathbb {R}},3):\det F>0\}\longrightarrow {\mathbb {R}},\ \ \ F\longmapsto w(F). \end{aligned}$$
(1)

Let

$$\begin{aligned} \begin{array}{rcl} x:\ \ \Omega &{}\longrightarrow &{} {\mathbb {R}}^3 \\ \ &{} \ &{} \ \\ X&{}\longmapsto &{} x(X),\\ \end{array} \end{aligned}$$
(2)

be a minimizing of the functional

$$\begin{aligned} J(x)=\int _{X\in \Omega } w\left( \frac{\partial x}{ \partial X}(X)\right) dX \end{aligned}$$
(3)

with some natural boundary conditions.

Ball and Murat [2, 3] highlight the following (see also [7]).

Vitali Covering Theorem in the spatial domain  \(\Omega \subset {\mathbb {R}}^3\). Given any \(\varepsilon \in (0,1)\), there exists a finite or countable family of real numbers \(0< \varepsilon _i\le \varepsilon \), \(i\in I\subseteq {\mathbb {N}}\), and vectors  \(a_i \in {\mathbb {R}}^3\), such that

(i):

\(a_i + \varepsilon _i \Omega \) are disjoint subsets of \(\Omega \),

and

(ii):

\(meas\left( \bigcup _{i\in I} \big [a_i + \varepsilon _i \Omega \big ]\setminus \overline{\Omega }\right) =0.\)

Hence, they show that if x(X) is a minimum for J then

$$\begin{aligned}{} & {} {\bar{x}}:\Omega \longrightarrow \Omega \end{aligned}$$
(4)
$$\begin{aligned}{} & {} {\bar{x}}(X):=a_i + \varepsilon _i \ x\left( \frac{X-a_i}{\varepsilon _i}\right) , \quad \text {if}\quad X\in a_i + \varepsilon _i \Omega , \end{aligned}$$
(5)

is also a minimum for J: \(J(x)=J({\bar{x}})\); we do not verify the above statement since this may be easily done inside the next space-time setting. Again, in this order of ideas, for us it is sufficient to take into account the above \({\bar{x}}\) in (5), which is a slight simplified version of the proposed in [2, 3].

Fig. 1
figure 1

Crystals of carbonate and fluorite

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It is surprising to realize that these amazing new self-similarity solutions so generated have a remarkable physical crystallographic relevance. The pictures in Fig. 1 showFootnote 1 very clearly natural realizations (crystals of carbonate and fluorite) of such predicted mathematical behaviors.

2 The space-time setting

Once again we consider an elastic body \({{\mathcal {B}}}\) identified with the reference configuration \(\Omega \subset {\mathbb {R}}^3\), e.g. a non-viscous homogeneous fluid (mass reference density \( \mu \equiv 1\)) and stored energy density as in (1).

Let

$$\begin{aligned} \begin{array}{rcl} x:[0,T] \times \Omega &{}\longrightarrow &{} \Omega \\ \ &{} \ &{} \ \\ (t,X)&{}\longmapsto &{} x(t,X),\\ \end{array} \end{aligned}$$
(6)

be a T-periodic motion of \(\Omega \) into itself, minimizing the functional

$$\begin{aligned} J(x)=\int _{(t,X)\in [0,T] \times \Omega }\left[ \frac{1}{2} \left| \frac{\partial x}{ \partial t}(t,X)\right| ^2 - w\left( \frac{\partial x}{ \partial X}(t,X)\right) \right] \ dt\, dX \end{aligned}$$
(7)

in some appropriate (see [1,2,3]) functional space, with velocities tangent to the boundary \(\partial \Omega \) :

$$\begin{aligned} \Big (\frac{\partial x}{ \partial t} (t,X)\cdot n(z) \Big )=0 \quad \text {for any} \ t, X, z\quad \text {such that}\quad z=x(t,X)\in \partial \Omega \end{aligned}$$
(8)

Inspired by the above framework of Ball and Murat, we extend into the space-time their self-similarity construction, and rephrase the Vitali covering theorem—see [7]—for the space-time cylinder \([0,T] \times \Omega \) of \( {\mathbb {R}}^4\).

Vitali Covering Theorem in the space-time cylinder \([0,T] \times \Omega \subset {\mathbb {R}}^4\). Given any  \(\varepsilon \in (0,1)\), there exists a finite or countable family of real numbers \(0< \varepsilon _i\le \varepsilon \), \(i\in I\subset {\mathbb {N}}\), and vectors \((\tau _i,a_i) \in {\mathbb {R}}\times {\mathbb {R}}^3\), such that

(a):

\((\tau _i, a_i) + \varepsilon _i ( [0,T] \times \Omega )\) are disjoint subsets of \([0,T] \times \Omega \),

and

(b):

\(meas\left( \bigcup _{i\in I} \big [(\tau _i,a_i ) + \varepsilon _i ([0,T] \times \Omega )\big ]\setminus \overline{( [0,T] \times \Omega )}\right) =0.\)

By reparametrizing any cylinder \((\tau _i, a_i) + \varepsilon _i ( [0,T] \times \Omega )\) into \([0,T]\times \Omega \),

$$\begin{aligned}{} & {} \tau _i + \varepsilon _i [0,T] \ni t\mapsto s(t)=\frac{t-\tau _i}{\varepsilon _i}\in [0,T],\nonumber \\{} & {} a_i + \varepsilon _i \Omega \ni X \mapsto Y(X)=\frac{X-a_i}{\varepsilon _i}\in \Omega , \end{aligned}$$
(9)

we see that

$$\begin{aligned} meas \ [0,T]\times \Omega =\sum _i\int _{t\in \tau _i +\varepsilon _i[0,T]}\ \int _{ X\in a_i + \varepsilon _i \Omega }dt\ dX= \sum _i\int _{s\in [0,T]}\int _{Y\in \Omega }\varepsilon _i^4\ \ ds\ dY \end{aligned}$$

and necessarily the sequence \(\{\varepsilon _i\}\) satisfies

$$\begin{aligned} \sum _i \varepsilon _i^4 =1. \end{aligned}$$
(10)

Starting from a motion (6), x(t, X), we define a new candidate (non smooth) motion \({\bar{x}}(t,X)\):

$$\begin{aligned}{} & {} {\bar{x}}:[0,T]\times \Omega \longrightarrow \Omega \end{aligned}$$
(11)
$$\begin{aligned}{} & {} {\bar{x}}(t,X):=a_i + \varepsilon _i \ x\left( \frac{t-\tau _i}{\varepsilon _i},\frac{X-a_i}{\varepsilon _i}\right) , \quad \text {if} \quad t\in \tau _i + \varepsilon _i [0,T] \quad \text {and} \quad X\in a_i + \varepsilon _i \Omega ,\nonumber \\ \end{aligned}$$
(12)

It follows that, \(\text {for}\ t\in \tau _i + \varepsilon _i [0,T] \ \text {and }\ X\in a_i + \varepsilon _i \Omega \),

$$\begin{aligned} \left\{ \begin{array}{c} \frac{\partial {\bar{x}}}{ \partial t}(t,X) ={\varepsilon _i}\frac{\partial x}{ \partial t}\left( \frac{t-\tau _i}{\varepsilon _i},\frac{X-a_i}{\varepsilon _i}\right) = \frac{\partial x}{ \partial s}\left( s,Y\right) _{s=\frac{t-\tau _i}{\varepsilon _i},Y=\frac{X-a_i}{\varepsilon _i}}\\ \frac{\partial {\bar{x}}}{ \partial X}(t,X) ={\varepsilon _i}\frac{\partial x}{ \partial X}\left( \frac{t-\tau _i}{\varepsilon _i},\frac{X-a_i}{\varepsilon _i}\right) = \frac{\partial x}{ \partial Y}\left( s,Y\right) _{s=\frac{t-\tau _i}{\varepsilon _i},Y=\frac{X-a_i}{\varepsilon _i}} \end{array}\right. \end{aligned}$$
(13)

In the interior of any space-time i-cylinder, we see that \({\bar{x}}(t,X)\) satisfies the expected Euler-Lagrange system if x(t, X) does:

$$\begin{aligned} {{\mathcal {L}}}(v, F)= & {} \frac{1}{2}\left| v\right| ^2-w(F),\left( \frac{d}{dt}\frac{\partial {{\mathcal {L}}}}{\partial v^\ell } (v, F)+\frac{\partial }{\partial X^L}\frac{\partial {{\mathcal {L}}}}{\partial F^\ell _L} (v, F)\right) {\Big \vert }_{v=\frac{\partial {\bar{x}}}{\partial t},F=\frac{\partial {\bar{x}}}{\partial X} }\\= & {} \frac{d}{dt}\frac{d {\bar{x}}^\ell }{dt} +\frac{\partial }{\partial X^L} \left[ \frac{\partial w}{\partial F^\ell _L} \left( \frac{\partial {\bar{x}}}{\partial X}\right) \right] \\= & {} \frac{d}{dt}\frac{d {\bar{x}}^\ell }{dt} +\frac{\partial ^2 w}{\partial F^m_N \partial F^\ell _L} \frac{\partial ^2{ {\bar{x}}}^m}{\partial X^L\partial X^N}= \frac{1}{\varepsilon _i}\left( \frac{d^2 x^\ell }{ds^2} +\frac{\partial ^2 w}{\partial F^m_N \partial F^\ell _L} \frac{\partial ^2{ x}^m}{\partial Y^L\partial Y^N}\right) \\= & {} \frac{1}{\varepsilon _i}\left( \frac{d}{ds}\frac{\partial {{\mathcal {L}}}}{\partial v^\ell }(v, F)+\frac{\partial }{\partial Y^L}\frac{\partial {{\mathcal {L}}}}{\partial F^\ell _L}(v, F) \right) {\Big \vert }_{v=\frac{\partial x}{\partial s},F=\frac{\partial x}{\partial Y} }. \qquad \qquad \qquad \square \end{aligned}$$

Next to that, we check that \(J({\bar{x}})=J( x)\).

Indeed, if in some suitable functional space x(t, X) realizes a minimum for J, then \({\bar{x}}(t,X)\) does too:

$$\begin{aligned} J({\bar{x}})= & {} \sum _{i\in I} \int _{t\in \tau _i +\varepsilon _i[0,T]}\ \int _{ X\in a_i + \varepsilon _i \Omega } \left[ \frac{1}{2}\left| \frac{\partial {\bar{x}}}{ \partial t}(t,X) \right| ^2 - w\left( \frac{\partial {\bar{x}}}{ \partial X}(t,X) \right) \right] \ dt\ dX, \\= & {} \sum _{i\in I} \int _{t\in \tau _i +\varepsilon _i[0,T]}\ \int _{ X\in a_i + \varepsilon _i \Omega }\\{} & {} \quad \left[ \frac{1}{2}\left| \frac{\partial x}{ \partial t} \left( \frac{t-\tau _i}{\varepsilon _i},\frac{X-a_i}{\varepsilon _i}\right) \right| ^2 - w\left( \frac{\partial x}{\partial X}\left( \frac{t-\tau _i}{\varepsilon _i},\frac{X-a_i}{\varepsilon _i}\right) \right) \right] \ dt\ dX,\\= & {} \sum _{i\in I} \int _{s\in [0,T]} \int _{Y\in \Omega } \left[ \frac{1}{2}\left| \frac{\partial x}{ \partial s}(s,Y) \right| ^2 - w\left( \frac{\partial x}{ \partial Y}\left( s,Y\right) \right) \right] \ \varepsilon _i^4 \,ds\ dY =J(x), \end{aligned}$$

because (10) does hold. \(\square \)

Fig. 2
figure 2

Turbulence generation

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This expected ‘phenomenology’ seems to evoke something like the cascade theoryFootnote 2 of Richardson [6] subsequently taken up by Kolmogorov (see K41 in [4]), although definitely different, since our environment is not dissipative (Fig. 2).