Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

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1 Commun. Math. Phys. 184, (1997) Communications in Mathematical Physics c Springer-Verlag 1997 Remarks on Singularities, imension an Energy issipation for Ieal Hyroynamics an MH Russel E. Caflisch, Isaac Klapper, Gregory Steele Mathematics epartment, UCLA, Los Angeles, CA , USA. Receive: 21 March 1995 / Accepte: 6 August 1996 Abstract: For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B 3 s with s greater than 1/3.B p s consists of functions that are Lip(s) (i.e., Höler continuous with exponent s) measure in the L p norm. Here this result is applie to a velocity fiel that is Lip(α ) except on a set of co-imension κ 1 on which it is Lip(α 1 ), with uniformity that will be mae precise below. We show that the Frisch-Parisi multifractal formalism is vali (at least in one irection) for such a function, an that there is energy conservation if min α (3α + κ(α)) > 1. Analogous conservation results are erive for the equations of incompressible ieal MH (i.e., zero viscosity an resistivity) for both energy an helicity. In aition, a necessary conition is erive for singularity evelopment in ieal MH generalizing the Beale-Kato-Maja conition for ieal hyroynamics. 1. Introuction In turbulent flow at high Reynols number, the energy issipation rate is observe to be approximately inepenent of the coefficient of viscosity. If the Euler equations for ieal hyroynamics are to correctly escribe the infinite Reynols number limit for turbulent flow, which is a major open question of flui mechanics, then energy issipation an singularities must occur in their solutions. The situation is similar for magneto-hyroynamics (MH) at high Reynols an magnetic Reynols number [2]. Although the available evience is not as clear-cut, caflisch@math.ucla.eu. Research supporte in part by the ARPA uner URI grant number # N1492-J klapper@math.montana.eu. Current aress: Mathematics epartment, Montana State University, Bozeman, MT 59717, USA. Research supporte in part by an NSF postoctoral fellowship. steeleg@wl.lmco.com. Current aress: Lockhee Martin Western evelopment Laboratories, 32 Zanker R. MS X-2, San Jose CA 95134, USA. Research supporte in part by the NSF uner grant # MS

2 444 R.E. Caflisch, I. Klapper, G. Steele energy issipation is apparently constant in the ieal limit. In contrast, accoring to the Taylor conjecture, magnetic helicity oes not issipate in the ieal limit. If the ieal MH equations are to allow reasonable limits of incompressible MH, these two observations must be reflecte in properties of the solutions. In 1949 Onsager [12] state that energy is conserve for weak solutions u Lip(α) with α>1/3. This result is containe in a famous paper that initiate the statistical theory of point vortices, an receive little attention until the work of Eyink [7], which gave it a rigorous mathematical proof in a certain function class. The proof was consierably simplifie an extene to the Besov function space Bs p (= Bs p, ) in subsequent work of Constantin, E an Titi [5]. In this note, we shall specialize the result of [5] to explicitly show the epenence on both the egree of singularity of the velocity an the imension of the singular set. In particular, we consier a velocity which is Lip(α ) everywhere (i.e. with co-imension κ = ) except on a set of co-imension κ 1 on which it is Lip(α 1 ). Our main result for ieal hyroynamics, which is state formally in Corollary 3.1 below, is that there is energy conservation for weak solutions of the Euler equations if inf(3α + κ(α)) > 1 (1.1) α in which the inf is taken over α = α,α 1. As shown below, this criterion is vali for negative, as well as positive, values of α. In fact, we show that for this class of functions, the multifractal formalism of Frisch- Parisi [9] is vali (at least in one-irection), an that the functions are in the Besov space Bs p for any s>s p = inf α pα + κ(α) an for all 1 p<. So the energy conservation criterion (1.1), using p = 3, then follows from [5]. In fact, the criterion (1.1) is correct whenever the Frisch-Parisi formalism is vali. The energy conservation criterion (1.1) is implicit in the work of Eyink [8] on multifractals an Besov spaces. Nevertheless, we believe that an explicit statement of this criterion an its valiation for a particular class of velocities is noteworthy. In particular, it shoul be helpful in preicting the type of singularities for Euler flows, an in assessing their flui ynamic significance if they o occur. Note, however, that there is no proof that the Euler velocity fiel will have the smoothness escribe above. We then present two results on singularities an energy issipation for ieal incompressible MH. First, we erive criteria for energy conservation an helicity conservation for weak solutions of ieal MH. Secon, we show that if smooth initial ata for the ieal MH equations leas to a singularity at a finite time t, then t ω + J t = (1.2) in which ω = u is the flui vorticity, J = B is the electrical current, an is the L norm in space. This result is analogous to the theorem of Beale-Kato-Maja [1] for singularity formation in ieal hyroynamics. 2. Singularity an imension Consier a function f, efine on a set R m, an assume that f is smooth except on a manifol S of co-imension κ (an integer) on which it is Lip(α); e.g., f(x) = ist(x, S ) α. efine sets S(r) consisting of points in within istance r of S. Then

3 Singularities, imension an Energy issipation for Hyroynamics 445 S(r) vol(s(r)) ar κ (2.1) for some constant a, which will be ajuste for use in subsequent bouns. Next consier the ifference of f(x) an f(x + y) for two points x an x + y that are at least istance r from S, i.e. with x, x + y S(r). Since the erivative of f blows up like r (1 α) then f(x) f(x + y) ar (1 α) y. (2.2) Alternatively, f is everywhere Lip(α)ifα, while f is of size r α if α<; i.e. f(x) f(x + y) { a y α if α ar α if α< (2.3) for x, x + y S(r). This can be generalize to a function that is Lip(α )in S,, with α >α 1,in which case the bouns can be combine as f(x) f(x + y) (r, α,α 1 ) (2.4) if x, x + y S(r), in which a y α r (α α1) if y r (r, α,α 1 )= a y α1 if r< y an α 1 ar α1 if r< y an α 1 < (2.5) efinition. A function f satisfying the bouns (2.4) with α >α 1 an <κwill be sai to be in class Lip(α,α 1,,κ). Next we erive L p estimates for any function in Lip(α,α 1,,κ). These estimates show that such functions are in Besov space. Lemma 2.1. Let f Lip(α,α 1,,κ 1 ), let 1 p, an enote κ =. efine s p = min i=,1 (α i + κ i /p) (2.6) an assume that s p >. Then for any s p >s>there is a constant b (epening on s p s) such that f( + y) f( ) L p <b y s. (2.7) Proof of Lemma 2.1. First assume that α 1 an rewrite the efining inequality (2.4) in a smooth way as f(x + y) f(x) (r) a(r+ y ) α+α1 y α (2.8) for x, x + y S(r). Also enote V (r)=vol(s(r)) a(r + y ) κ1 Ṽ (r)=vol(s(r) (S(r) y)) 2V (r). (2.9) Write the integral of the Höler ifference as a Stieljes integral over r, then integrate by parts to estimate (omitting constant factors)

4 446 R.E. Caflisch, I. Klapper, G. Steele f(x + y) f(x) p x = 1 1 (r) p Ṽ (r) (r) p x r ( (r)p )Ṽ (r)r + (1) p Ṽ (1) = { } 1 y αp (r+ y ) 1 p(α α1)+κ1 r +1 y sp { log y if α1 + κ 1 /p = α 1 otherwise (2.1) in which s = min(α,α 1 +κ 1 /p). This proves (2.7) for α 1. On the other han, if α 1 < then (r) = min(r (α α1) y α,r α1 ) (2.11) Then, repeating the first few steps of the previous estimation, the boun becomes 1 f(x + y) f(x) p x = 2 r ( (r)p )V (r)r +2 (1) p V (1) y 1 r 1+pα1+κ1 r + y αp r 1 p(α α1)+κ1 r + a y αp y y sp { log y if α1 + κ 1 /p = α 1 otherwise (2.12) in which s = min(α,α 1 +κ 1 /p) >. This proves (2.7) for f Lip(α,α 1,,κ 1 ). The Besov spaces are characterize by the L p bouns prove in Lemma 2.1, which leas to the following result: Corollary 2.1. Assume that function f Lip(α,α 1,,κ 1 ) an that 1 p<. efine s p = min(α + κ(α)/p). (2.13) If s p >, then f Bs p for any s p s>. This is exactly the formula for s p in the Frisch-Parisi formalism, which shows onesie valiity of the Frisch-Parisi formalism for this function class. 3. Energy Conservation for Ieal Hyroynamics For simplicity assume that = [,1] 3 with perioic bounary conitions. A weak solution of the incompressible Euler equation is a function u =(u 1,u 2,u 3 ) satisfying T u j t ψ j +( i ψ j )u i u j +( j ψ j )px t = u j ψ j (t =)x u j ( j ϕ)x = (3.1)

5 Singularities, imension an Energy issipation for Hyroynamics 447 for all test functions ψ =(ψ 1,ψ 2,ψ 3 ) C ( R + ) an ϕ C () with compact support. Energy is conserve for an Euler solution if u (x, t) 2 x = u (x, ) 2 x (3.2) for t [,T]. The following energy conservation theorem for ieal hyroynamics is a consequence of Corollary 2.1 an the theorem of [5]. Corollary 3.1. (Energy Conservation for Euler). Let u be a weak solution of the Euler equations on =[,1] 3. Suppose that u C([,T],B()) in which B() = Lip(α,α 1,,κ 1 )). Then energy is conserve if min(3α i + κ i ) > 1. (3.3) i Note that here an in the next section, the function space C([,T],B()) coul be replace by L 3 ([,T],B()) C([,T],L 2 ()) or something similar, as in [5]. 4. Energy Conservation for Ieal MH The energy conservation results of [5] can be extene to ieal MH in a straightforwar manner. The equations for ieal MH are ( t + u )u = p 1 2 b 2 + b b ( t +u )b =b u (4.1) u = b =. Actually, incompressibility of b ( b = ) nee only be require at t =, an it then hols for all t. Let u =(u 1,u 2,u 3 ) an b =(b 1,b 2,b 3 ) be functions satisfying the weak form of the ieal MH equations, namely, T [ ] u j t ψ (1) j (b i b j u i u j ) i ψ (1) j +(p+b 2 /2) i ψ (1) i x t = T [ ] b j t ψ (2) j +(ɛ jkl u k b l )(ɛ jmn m ψ n (2) ) x t = u j j ξ (1) x = b j j ξ (2) x =,t=,t= u j ψ (1) j x b j ψ (2) j x for all test functions ψ (β) =(ψ (β) 1,ψ(β) 2,ψ(β) 3 ) C ( R + ) an ξ (β) C (), with β =1,2. Again, the incompressibility conition on b nee only be impose at t = an it then follows for all t. In analogy to the conservation of energy for the Euler equations, energy conservation for ieal MH hols if ( u (x,t) 2 + b(x,t) 2) ( x = u(x,) 2 + b (x, ) 2) x (4.2) for t [,T]. For simplicity we assume that =[,1] n. Whereas singularity formation an energy issipation is only possible for three-imensional hyroynamics, for MH it is a possibility for imension n =2orn=3.

6 448 R.E. Caflisch, I. Klapper, G. Steele Theorem 4.1. (Energy Conservation for Ieal MH). Let u an b be a weak solution of the ieal MH equations in =[,1] n. Suppose that u C([,T],B α1 3 ) an b C([,T],B α2 3 ).If α 1 >1/3, α 1 +2α 2 >1, (4.3) then (4.2) hols. Proof. The proof follows that of [5] but will be briefly repeate here. efine ϕ ɛ (x )= (1/ɛ n )ϕ(x /ɛ) to be a positive, smooth mollifier with support in B(, 1) an total mass 1. We make use of the efinitions r ɛ (f,g)(x )= ϕ ɛ (y)(δ y f(x ) δ y g(x ))y, q ɛ (f,g)(x )= ϕ ɛ (y)(δ y f(x ) δ y g(x ))y, where δ y h(x )=h(x y) h(x). The proof relies critically on the following ientities (first observe in [5]): (f g) ɛ = f ɛ g ɛ + r ɛ (f,g) (f f ɛ ) (g g ɛ ) (4.4) (f g) ɛ = f ɛ g ɛ + q ɛ (f,g) (f f ɛ ) (g g ɛ ). (4.5) In aition the following estimates hol for functions in B α 3 : f( + y) f( ) L 3 c y α, (4.6) f ɛ L 3 Cɛ α 1 f L 3, (4.7) f f ɛ L 3 Cɛ α f L 3. (4.8) Using ψ (1)ɛ (x )= ϕ ɛ (y x)u ɛ (y,t)y an ψ (2)ɛ (x )= ϕ ɛ (y x)b ɛ (y,t)y as test functions results in the equations u ɛ (x,t) 2 x u ɛ (x,) 2 x t = Tr [ (u u) ɛ u ɛ (b b) ɛ u ɛ] (x,t)xt b ɛ (x,t) 2 x b ɛ (x,) 2 x t [ = (u b )ɛ b ɛ] (x,t)xt. The ientities (4.6), (4.7) an (4.8) then yiel the estimates u ɛ (x,t) 2 + b ɛ (x,t) 2 x u ɛ (x,) 2 + b ɛ (x, ) 2 x t Tr [ (r ɛ (u,u) r ɛ (b,b) (u u ɛ ) (u u ɛ ) +(b b ɛ ) (b b ɛ ) ) u ɛ] x τ

7 Singularities, imension an Energy issipation for Hyroynamics 449 t + ( q ɛ (u, b ) (u u ɛ ) (b b ɛ ) ) b ɛ xτ t [( r ɛ (u, u ) 2/3 3/2 + r ɛ(b, b ) 2/3 3/2 ) u ɛ 1/3 + u u ɛ 2/3 3/2 + b b ɛ 2/3 3/2 ) ( + q ɛ (u, b ) 2/3 3/2 + u u ɛ 1/3 3/2 b b ɛ 1/3 3/2 u ɛ 1/3 3 3 ] τ C 1 ɛ 3α1 1 + C 2 ɛ α1+2α2 1. The result (4.2) follows in the limit ɛ, which finishes the proof of Theorem 4.1. A similar theorem for magnetic helicity can be proven. The time evolution of the magnetic helicity for smooth ieal MH is given by [a t b + a b t ] x = [b (u b )+b α+a (u b)]x = [b α+a (u b)]x, = where α is some smooth function an b = a. Then for ψ C ( R + ), T ( ψ(x,t)) (u (x,t) b(x,t))x t = (4.9) implies weak conservation of helicity. Using arguments ientical to those of the previous proof we obtain Theorem 4.2. (Magnetic Helicity Conservation for Ieal MH). Let u an b be a weak solution of the ieal MH equations in =[,1] n. Suppose that u C([,T],B α1 3 ) an b C([,T],B α2 3 ).Ifα 1+2α 2 >, then (4.9) hols. In 2 imensions the magnetic helicity vanishes ientically. In its place the quantity a 2 x serves as a non-trivial invariant. In 2 imensions, a satisfies (up to a graient) an we have a t + u a = Theorem 4.3. Let u an b be a weak solution of the ieal MH equations in = [, 1] 2. Suppose that u C([,T],B α1 3 )an a C([,T],Bα2+1 3 ).Ifα 1 +2α 2 > 1, then a 2 x is conserve. We remark that Theorems 4.1, 4.2, an 4.3 specialize easily to functions u an b in Lip(α,α 1,,κ 1 ), as in Corollary 3.1. In these cases the bouns of Theorem 4.1 become s 1 > 1/3, s 1 +2s 2 > 1, the boun for Theorem 4.2 becomes s 1 +2s 2 >, an the boun for Theorem 4.3 becomes s 1 +2s 2 > 1. Here

8 45 R.E. Caflisch, I. Klapper, G. Steele s 1 = min(α 1 + κ 1 (α 1 )/3), α 1 s 2 = min(α 2 + κ 2 (α 2 )/3), α 2 where κ 1, κ 2 are efine as in the introuction. For the commonly observe phenomenon of coimension 1 current sheets, κ 2 = 1 so that s 1 +2α 2 >1/3 implies energy conservation an s 1 +2α 2 > 2/3 implies helicity conservation ( 5/3 in 2). We also remark that while the flui result (1.1) picks out the Kolmogorov exponent 1/3 naturally, the classical MH exponent (namely 1/4 [1, 11]), while consistent with the bouns of Theorems 4.1 an 4.2, oes not rop out as naturally. This shoul not be a surprise since important non-local MH effects are not inclue in the argument. Aitionally, Theorems 4.1 an 4.2 are consistent with recent intermittency moels (see, e.g., [3]). Analogous results can be obtaine in terms of the Elsasser (characteristic) variables z ± = u ± b for the MH equations. The system (4.1) can be rewritten as ( t + z + )z = Π, ( t + z )z + = Π, (4.1) z ± = in which Π = p b 2. The following theorem gives two variants of the previous energy conservation result for MH. Theorem 4.4. (Energy Conservation for Ieal MH in Characteristic Variables). For a weak solution of the MH equations in [, 1] n, there is energy conservation if either of the following conitions are satisfie: (i) For some p, q with values in (1, ) an with 1/p +2/q =1 in which u C([,T],B α 3 B α1 p ), b C([,T],B α2 q ), (4.11) 3α > 1, α 1 +2α 2 >1. (4.12) (ii) For some p i,q i (i =1,2) with values in (1, ) an with 1/p 1 +2/q 1 =2/p 2 + 1/q 2 =1, in which z + C([,T],B α1 p 1 z C([,T],B β1 q 1 B α2 p 2 ), B β2 q 2 ) (4.13) α 1 +2β 1 >1, 2α 2 +β 2 >1. (4.14) Similar statements can be mae with regars to magnetic helicity.

9 Singularities, imension an Energy issipation for Hyroynamics Singularity Formation for Ieal MH We will show the analogue of the Beale-Kato-Maja theorem for ieal MH. Theorem 5.1. For the system (4.1) with initial ata u, b H s, with s 3, the solution u (t), b (t) is in the class as long as an T C([,T],H s ) C 1 ([,T],H s 1 ) T (The 2 inequalities are in fact equivalent.) ω (t) + j (t) t <, z + + z t <. Here j = b an H s is the L 2 Sobolev space. The approach closely follows that of [1]. Assume that T z + + z t = M<. (5.1) The proof consists of three parts: First, we erive energy estimates on z ± s in terms of z ±. Secon, we estimate z ± L2. Finally, we utilize an inequality erive in [1] an Gronwall s lemma to boun z ± s Energy estimates. We begin by eriving energy estimates for the system (4.1) with t [,T]. Let α be a multi-inex with α s. Let η = α x z +. Apply α x to the secon equation in (4.1) to obtain in which Π = α x Π an ( t + z )η = Π F F = α [(z z + )] z α z +. A boun on F in the L 2 norm can be base on the general inequality α (fg) f α g L 2 c( f s g + f g s 1 ), which was erive in [1] base on the Gagliaro-Nirenberg inequalities. Application of this to F yiels F L 2 c( z s z + + z z + s 1 ). (5.2) This leas to the following boun on η t η 2 L c( z 2 s z + + z z + s 1 ) η L 2. Summing over α leas to

10 452 R.E. Caflisch, I. Klapper, G. Steele t z + 2 s c( z s z + + z z + s ) z + s. (5.3) There is a similar result for z ; i.e., t z 2 s c( z + s z + z + z s ) z s. (5.4) A these two inequalities to obtain an thus t ( z 2 s + z + 2 s) c( z + + z )( z + 2 s + z 2 s), (5.5) z + 2 s + z 2 s ( z + 2 s + z 2 s)exp ( C t ) ( z + + z )τ. (5.6) 5.2. L 2 bouns on z ±.Take the curl of (4.1) to obtain ( t + z + )ζ = z + A z, (5.7) ( t + z )ζ + = z A z +. where ζ ± = z ± an A is a constant matrix. Multiplying the first equation in (5.7) by ζ an integrating gives t ζ 2 L C z + z ζ x 2 C ζ ( z + L 2 z L 2) C ζ ( z + 2 L + z 2 2 L2). (5.8) Since z ± =, z ± an ζ ± are relate by z ± = ( 1 ζ ± ) an their Fourier transforms are relate by ( z ± )(k) = S(k)ζ ± (k) where S(k) is boune inepenent of k. Thus z ± L 2 C ζ ± L 2, so that (5.8) leas to t ζ 2 L C ζ 2 ( ζ + 2 L + ζ 2 2 L 2). We obtain a similar result for ζ + ; that is t ζ + 2 L C ζ + 2 ( ζ + 2 L + ζ 2 2 L 2). A these two equations to obtain so that t ( ζ + 2 L 2 + ζ 2 L 2) c( ζ + + ζ )( ζ + 2 L 2 + ζ 2 L 2)

11 Singularities, imension an Energy issipation for Hyroynamics 453 ( t ) ζ + 2 L + ζ 2 2 L ( ζ L + ζ 2 2 L exp C ( ζ + (τ) 2 + ζ (τ) )τ. By Assumption (5.1) we have ζ + 2 L + ζ 2 2 L M( ζ L + ζ 2 2 L2), (5.9) where M = exp(cm) Final estimates. In [1] it was prove, via the Biot-Savart law, that f C{1+(1+log + f 3 ) f + f L 2} (5.1) where Thus log + a = { log a if a 1 otherwise. (5.11) z + + z C{1+(1+log + z + 3 ) ζ + + ζ + L 2 + (1 + log + z 3 ) ζ + ζ L 2}. Using the result from Section 5.2, we have z + + z C{1+( ζ + + ζ )(log + z log + z 3 +2). Combining this with the result from Sect. 5.2 gives { t z + s + z s c( z + s + z [ s)exp C (1+( ζ + + ζ )) (log( z e) + log( z 3 + e)) ] τ }. Let y ± (t) = log( z ± s + e) then y + (t)+y (t) log c( z + s + z s) +C t (1+( ζ + + ζ )(y + (τ)+y (τ))τ. Application of Gronwall s lemma then shows that y + (t)+y (t) is boune by a constant which epens only on M,T an z ± (, ) s. This conclues the proof of Theorem 5.1.

12 454 R.E. Caflisch, I. Klapper, G. Steele 6. Conclusions At present, there are only a few analytical results on singularities in ieal hyroynamics: The Beale-Kato-Maja theorem is a necessary conition for the formation of singularities from smooth initial ata. Constantin [4] an Constantin & Fefferman [6] have obtaine aitional necessary conitions in terms of the geometry of the vorticity fiel. Finally, Onsager s energy conservation criterion provies a necessary conition for energy issipation ue to singularities in an ieal flui. The first part of this paper has refine Onsager s criterion by explicitly showing the effect of singularity type an imension on the necessary conition for energy issipation. The result is an example of the Frisch-Parisi multi-fractal formalism, which has been prove to be vali for functions in the class Lip(α,α 1,,κ 1 ). In the remaining parts of the paper two analytical results the Beale-Kato-Maja theorem an Onsager s energy conservation theorem have been extene to ieal MH. Since energy issipation but helicity conservation are expecte, this suggests a limite range of values for the uniform singularity spectrum in MH. The appearance of the Elsasser variables z + an z in the extension of the Beale-Kato-Maja inequality shoul also be note. We expect these results to be useful in two ways: First, as a sufficient conition for regularity of ieal hyroynamic an MH solutions. They shoul also serve as a guie in investigation of possible singularities an their physical significance. For example in 3 hyroynamics with singularities of type α on a smooth set S, nonzero energy issipation requires α for a 2 singularity surface (κ = 1), α 1/3 for a curve of singularities (κ = 2), an α 2/3 for a point singularity (κ = 3). In particular, in the point an curve cases, infinite velocities are require. These results also help to inicate the relation between the smoothness of b an that of u. Theorem 5.1 suggests that b an u shoul have the same egree of smoothness, while Theorems 4.1, 4.2, an 4.3 suggest a traeoff between smoothness of u an that of b. References 1. Beale, J.T., Kato, T., Maja, A.: Remarks on the breakown of smooth solutions for the 3- Euler equations. Commun. Math. Phys. 94, 61 66, (1984) 2. Biskamp,.: Nonlinear Magnetohyroynamics. Cambrige: Cambrige Univ. Press, Carbone, V.: Cascae moel for intermittency in fully evelope magnetohyroynamic turbulence. Phys. Review Letters 71, (1993) 4. Constantin, P.: Geometric statistics in turbulence. SIAM Rev.,36, (1994) 5. Constantin, P., Weinan E., Titi, E.S.: Onsager s conjecture energy conservation for solutions of Euler s equation. Commun. Math. Phys. 165, (1994) 6. Constantin. P., Fefferman, C.: irection of vorticity an the problem of global regularity for the Navier-Stokes equations. Iniana U. Math. J. 42, (1994) 7. Eyink, G.: Energy issipation without viscosity in ieal hyroynamics. 1. Fourier analysis an local energy transfer. Physica 78, (1994) 8. Eyink, G.: Besov spaces an the multifractal hypothesis. J. Stat. Phys (1995) 9. Frisch, U., Parisi, G.: On the singularity structure of fully evelope turbulence. In M.Ghil, R.Benzi, an G.Parisi, eitors, Turbulence an Preictability in Geophysical Flui ynamics an Climate ynamics. Proc. Internatial Summer School of Physics Enrico Fermi, Amsteram: North-Hollan, 1985, pp Iroshnikov, P.S.: Turbulence of a cucting flui in a strong magnetic fiel. Soviet Astromy 7, (1964)

13 Singularities, imension an Energy issipation for Hyroynamics Kraichnan, R.H.: Inertial range spectrum in hyromagnetic turbulence. Physics of Fluis A 8, (1965.) 12. Onsager, L.: Statistical hyroynamics. Nuovo Cimento (Supplemento) 6, 279 (1949) Communicate by J.L. Lebowitz