On the Inertial Range Bounds of K-41-like Magnetohydrodynamics Turbulence

The spectral slope of magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; −3/2 is the spectral slope in Kraichnan–Iroshnikov–Dobrowolny (KID) theory, −5/3 in Marsch–Matthaeus–Zhou and Goldreich–Sridhar theories, also called Kolmogorov-like (K-41-like) MHD theory, the combination of the −5/3 and −3/2 scales in Biskamp, and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig (Physica D 241(2012) 426–438), we establish inertial range bounds for K-41-like phenomenon in MHD turbulent flow through a mathematical rigor; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to −5/3 is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.


Introduction
At a high Reynolds number fluid and plasma flows exhibit a complex random behavior called turbulence. Turbulence is observed in a great majority of fluids both in nature such as the atmosphere, river currents, oceans, solar wind, and interstitial bodies and in technical devices, such as laboratory installations, nuclear power plants, etc. Its importance in industry and physical sciences, such as making predictions about heat transfer in nuclear power plants, drag in oil pipelines, and the weather is tremendous. Besides these real-life relevant issues, the study of turbulence can assist mathematical researchers in understanding some aspects, such as the regularity of Euler's equation, the Navier-Stokes equation, magnetohydrodynamics equations, and so on, see for instance [1].
The literature shows that the phenomenon of turbulence has captured the attention of humankind for centuries, see for instance [2]. The discovery of the Euler equations in the mid-18th century and Navier-Stokes equations in the first half of the 19th century are the major scientific and mathematical breakthroughs. Towards the end of 19th century Osborne Reynolds laid a foundation for the theory of turbulence, see [3,4], ( [5], p. 488) and [6]. Reynolds number, a widely used criteria to classify whether a given flow is turbulent or not, and Reynolds averaged Navier-Stokes equations (RANS) are due to O. Reynolds. RANS is formulated by decomposing the velocity field u(x, t) in to average velocityū(x, t) over a time interval and fluctuation velocity u (x, t) = u(x, t) −ū(x, t), and finally rewriting the Navier-Stokes equations in terms of the average velocityū. In fact, RANS is still one of the most widely used models to study turbulence in fluids, see [7,8] and the references there.
where u = u(x, t) is the flow velocity, b = b(x, t) is the magnetic field, π = P + 1 2 |b| 2 is the total pressure on the system with P representing the pressure function from the equation of motion, ν > 0 is the kinetic viscosity of the fluid, η > 0 is the resistivity of the fluid, and the spatial domain D is the Euclidean space R 3 . The non-homogeneous external forces f 1 = f 1 (x, t), f 2 = f 2 (x, t) are assumed to be divergence-free and satisfy f 1 , f 2 ∈ L ∞ loc ([0, ∞); H −1 (D) ∩ L 2 (D)), where L ∞ loc is the space of locally bounded functions, H −1 and L 2 are the usual Sobolev and Lebesgue spaces, respectively. The derivation of Equation (2) is done by combining the Navier-Stokes equations and the Maxwell equations in some way, see [37][38][39].
We now introduce the spectral energy function, denoted by E(k, t); the spectral energy of the MHD flow model (2) is given by the surface integral where u and b represent the Fourier transforms of u and b, respectively. Of great scientific interest is the question of rigorous mathematical proof of the spectral theory, K-41 or otherwise, under physically admissible conditions. Therefore, our main goal will be to set the conditions on the data and to show that the spectral energy (3) satisfies −5/3 law when such conditions are met.
Before we give a formal definition to the weak solution of (2), we introduce some function spaces and their notations as they appear in [40]. We denote by C ∞ 0,σ the set of all divergence-free smooth functions with compact support in D. L p σ is the closure of C ∞ 0,σ with respect to the L p norm in the usual sense. For 1 ≤ p ≤ ∞ the space L p stands for the usual (vector-valued) Lebesgue space over R 3 . For s ∈ R, we denote by H s σ the closure of C ∞ 0,σ with respect to the H s norm.
is said to be a weak solution to (2) on D × [0, ∞) if it satisfies the following conditions: 1.
for any T > 0 the vector function (u, b) lies in the following function space, 2. the pair (u, b) is a distributional solution of (2); i.e., for every (Φ, Ψ) in 3. the following energy inequality is satisfied, The rest of the paper is divided into three main sections; Sections 2-4. In Section 2 we briefly discuss Fourier transform and its properties, rewrite Equation (2) in Fourier variables, and derive prior estimates. In Section 3 we present and prove our main results whereby we drive the bounds of the spectral energy function (3) and spectral energy bounds. Finally, Section 4 is conclusion.

The Fourier Transform
The Fourier transform of an integrable function u, denoted by u, is defined by The Fourier transform has several interesting properties, among them the following three are of great importance to this work; and In (6), ∂ α x and ∂ α ξ indicate the α th order derivative with respect to space variables in the Euclidean and Fourier spaces respectively, * in (7) is the convolution operator and Equation (5) is the Parseval-Plancherel identity. For the detail of these and other properties of the Fourier transform we refer to [41][42][43].
In fact, (5) implies that the energy of the system (2) in Fourier space is equal to the energy of the system in Cartesian space. To take advantage of (5) we give an equivalent formulation for (2) in Fourier space. This is done in two steps; first we eliminate the pressure term by applying the Leray projector given by (8).
The application of P together with the fact that the fields u and v and the non-homogeneous terms f 1 and f 2 are divergence free reduces the system (2) to Next, we take the Fourier transform of (9) to get Thus (10) is an equivalent formulation of (2) in Fourier space.

A Prior Estimates
This section is devoted to finding estimates in Fourier space for solutions of (2). For ease of calculations, we define an operator where C 3 the usual three dimensional complex space and Observe that for ξ ∈ C 3 and u divergence free, we have Now plugging (12) in (10) we get, ). If an appropriate frame is chosen and the total pressure Π is suitably normalized so that is bounded, then for any T > 0 there is a non negative function R(T) such that Furthermore, when f 1 ≡ f 2 ≡ 0, the bound R(T) = R is a constant fully determined by the initial data (u 0 , b 0 ). In this case one could actually take R to be the right hand side (RHS) of (4) and B R (0), a ball of radius R and center 0, becomes an invariant (set A is said to be an invariant (future invariant) set with respect to a function ϕ or family of functions {ϕ(t ) set for the weak solution.
Assuming that the non-homogeneous terms f 1 and f 2 are appropriately chosen so that (15) holds. With no lose of generality, one may assume from (5) that However, the problem is, since u, b are only distributional (weak) solutions, their Fourier transforms are not well defined at particular points, say (ξ, t), in Fourier space-time. We address the problem by taking a smooth cutoff of u and b over a cube of finite length and making use of the Paley-Wiener theorem ( [42], p. 193).
. Define χ k (·) to be a smooth cutoff function of a cube Q k about k of side length 2δ such that on a cube of the same center with side δ and Consider the following three smooth cutoff functions defined to suit our purpose; We now have enough preparation to start working on estimating our solution in Fourier space. To establish necessary estimates, we first need to establish estimates on e p , for p = 2 followed by estimate for e p (k, t) for all 2 ≤ p ≤ ∞. Lemma 1. Suppose that (15) holds and there exists a non-decreasing function R 1 (t) such that for all t ∈ [0, ∞) and δ < |k| Proof of Lemma 1. By definition Differentiating (22) with respect to time and using Equation (13), we get Applying elementary properties of complex numbers, it follows that For ease of calculations, we now deal with the terms on RHS of (23) separately.
The estimate in (25) is due to the fact that u and b are divergence free and elementary properties of complex numbers. Hölder's and Young's inequalities are also used. We know from construction of χ k and Hölder's inequality that Thus, combining (25) and (26) we get, Proceeding similarly with I 4 , I 5 and I 6 we get Thanks to Hölder's inequality, the integral I 7 is estimated as follows; Similarly, we have Now combining the estimates (24)-(32) we obtain Here we used Serrine's inequality ( [44], Lemma 1) to estimate upper bounds for χ k u + χ k b and χ k f 1 /|ξ| + χ k f 2 /|ξ| respectively as; , Now define the set B R 1 by, When e(k, t) = e 2 (k, t) = R 1 (t) |k| in (34), we get Then by chain rule and from the fact that e 2 (k, t) ≥ 0, we conclude that Indeed, (35) implies that B R 1 is an attracting set for e 2 (k, t). Therefore, if e 2 (k, 0) < R 1 (0) |k| , then e 2 (k, t) < R 1 (t) |k| for all t ∈ (0, ∞).

Lemma 2.
Suppose that for a given k ∈ R 3 and 2 ≤ p < ∞ there is a non-decreasing function R 1 (t) that satisfies the condition for 0 < δ < |k|/2 √ 3. If a solution to (2) initially satisfies e p (k, 0) < R 1 (0)/|k|, then for all 0 < t < ∞, Proof of Lemma 2. The proof follows same procedure as the proof of Lemma 1. We begin by taking the time derivative of e p p (k, t).
In the derivation of (36) we have used the following fact; We now estimate the integrals at the RHS of (36).
Here we used the fact that for ξ ∈ supp χ, |k| 2 ≤ |ξ| ≤ 3 2 |k|. Finally, thanks to Hölder's and Young's inequalities, we have Following a similar approach yields, We now remain to estimate I 7 and I 8 .
A similar approach yields, Now plugging the estimates (37)- (43) in (36) and rearranging the terms we get, We know from the property of χ k that ( |ξ| p | χ k (ξ)| p dξ) 1/p is bounded from above as Furthermore, we have We next put (44)-(46) together to get, Once again we consider the set Setting e(k, t) = e p (k, t) = R 1 (t) |k| , on the boundary such that |k|e p (k, t) = R 1 (t), Here we used the condition that 2 1 implies B R 1 is an attracting set for e p (k, t). Therefore, if e p (k, 0) < R 1 (0) |k| , then e p (k, t) < R 1 (t) |k| for all t ∈ R + .
The following two theorems are the main results of this section, which are direct consequences of Lemmas 1 and 2. Theorem 1. Let the assumptions of Lemma 2 hold. If the weak solution (u, b) of (2) satisfies the initial condition then for all t > 0, holds.
Theorem 2. Suppose the weak solution (u, b) of (2) satisfies (15) and sup 2≤p<∞ e p (k, 0) < Then for all T ∈ R + , we have and Proof of Theorem 1. The proof is very direct. Lemma 2 implies that e p (k, t) is bounded uniformly in p. Then taking the supremum over all 2 ≤ p < ∞ concludes the proof.

Proof of Theorem 2.
Recalling the definition of e p (k, t) from (18), we have . Now taking the derivative in time, We now plug (36) in (51) to get, For the sake of calculation simplicity, we split the RHS of (52) in to the following integrals.
We now proceed to estimating each of these integrals (I 1 )-(I 7 ).
the spectral energy; it is shown that the average is always bounded and decays over time.
Finally, the third theorem gives the inertial range bounds and formulates the conditions expected from the parameters, such as the dissipation rate, the universal constant, and viscosity coefficients so that the spectral energy decays accordingly with K-41. This is done by comparing E(k, t) with Kolmogorov's spectral function E K (k) given by (1), i.e., defined over a range of wave numbers called the inertial range; where C 0 is a universal constant called Kolmogorov constant and is the energy dissipation rate. (75) is similar to Equation (106) of ( [19], p. 267) where C 0 and were referred to as Kolmogorov constants for MHD turbulence and energy flux, respectively, instead of Kolmogorov's constant and energy dissipation rate.

Remark 3. Equation
Recall that the spectral energy function for the MHD system (76) is given by the spherical integral where 0 ≤ k < ∞ is a radial coordinate in Fourier space.
Theorem 3. Let the assumptions of Theorem 1 hold, f i ≡ 0 for all i = 1, 2 and the initial data (u 0 , b 0 ) ∈ B R (0), where R satisfies (16). Then, the estimate holds for all k and all t, where R 1 is as in Theorem 1. Moreover, when f i ≡ 0 for some i = 1, 2, (78) still holds with R 1 replaced by R 1 (t) which is still finite and possibly grows in time.
Proof of Theorem 3. When f i ≡ 0, we have from (77) and Theorem 1 that Here we used the fact that the surface area of a sphere with radius k is equal to 4πk 2 . When the external forces on the system, f i ≡ 0, for some i = 1, 2 the proof above remains same with R 1 replaced with R 1 (t). With this we complete the proof.
be part of the graph of E K (k) that lies in region S. Figure 1 shows how sets S and A are related. Due to Theorem 3 we know that the spectral energy of our system is bounded from above by 4πR 2 1 when f i ≡ 0 for all i = 1, 2 or 4πR 2 1 (T) f i ≡ 0 for some i = 1, 2. Furthermore, from Theorem 4 the time average is bounded by 4πR 2 2 (T) min (ν, η)Tk 2 .
Thus, set S represents the behavior of the function E(k, t), and set A is a set where E(k, t) behaves accordingly with K-41. Therefore, if A = ∅ then E(k, t) does not exhibit K-41like phenomenon.
Note that for A to be non-empty the point where graphs of E K (k) and 4πR 2 2 min(ν,η)Tk 2 must intersect below the line E = 4πR 2 1 , as in Figure 1, and the intersection occurs when Moreover, the graph of E K (k) intersects the line E = 4πR 2 1 below the graph of 4πR 2 2 min(ν,η)Tk 2 , as in Figure 1, which occurs when where k 1 is the intersection of the graphs of E K (k) and the constant function 4πR 2 1 and k 2 is the intersection of E K (k) and 4πR 2 2 (T) min(ν,η)Tk 2 . Thus the portion of the graph of E K (k) remains in region S as long as k is between k 1 and k 2 and k 1 ≤ k 2 , see Figure 1.
Observe from Figure 2 that if we push the graph of 4πR 2 2 (T) min(ν,η)Tk 2 to the left so that it intersects E K (k) above the graph of 4πR 2 1 , then we get k 1 > k 2 and the graph of E K (k) will not pass through region S which in turn gives A = ∅. Hence, A remains non-empty only when k ∈ [k 1 , k 2 ].
This completes the proof Theorem 5.

Conclusions
In this work, we have investigated the Leray weak solution of the deterministic MHD model (2) for the K-41-like MHD phenomenon in the presence and absence of external forces. In the process it is shown in Section 2.2 that when the external the solution field (u, b) is bounded in the Fourier space (Theorems 1 and 2) and the bound depends on the data. When the external forces f 1 and f 2 are identically 0, the bound is uniform. It is also shown that the spectral energy of the system E(k, t) is bounded, and when the external forces f i ≡ 0 for i = 1, 2 the bound is uniform (Theorem 3) and the average in time decreases in time and decays proportional to k −2 . When f i ≡ 0 for some i = 1, 2 the bonds of E(k, t) possibly depend on time. The other important result of this work is the explicit formulation of the inertial range bounds and setting the necessary condition on the parameters for the model to behave accordingly with K-41 (Theorem 5). The lower bound , is a constant in time when f i ≡ 0 for i = 1, 2 and possibly decreases in time when f i ≡ 0 for some i = 1, 2. The upper bound of the inertial range k 2 = 4πR 2 2 (T) min(ν, η)TC 0 2/3