Modified dissipativity for a non-linear evolution equation arising in

Modified Dissipativity for a Non-Linear Evolution Equation Arising in Turbulence C. BARDOS, P. PENEL, U. FRISCH & P . L . SULEM

Communicated by C. DAFERMOS

Abstract

We are concerned with the regularity properties for all times of the equation

~u(t,x)=

~2

[ r [u(t,o)-u(t,x)]2-v \-~x21 u(t,x)

which arises, with ~= 1, in the theory of turbulence. Here U(t,') is of positive type and the dissipativity ~ is a non-negative real number. It is shown that for arbitrary v > 0 and ~ >0, there exists a global solution in L~ [0, ~ ; H~-~(IR)]. If v > 0 and ~ > c% = 1, smoothness of initial data persists indefinitely. If 0 < ~ < ~cr, there exist positive constants va(~) and v2(~), depending on the data, such that global regularity persists for v>v~(~), whereas, for 0 5/4. FRISCH [6] has conjectured the global regularity of solutions of the modified Navier-Stokes equation for arbitrary v > 0 and ~>~cr with 0 < ~cr < 1. It will be shown in this paper that the modified dissipativity problem can be completely solved for the Burgers M R C M equation (1.1). It is easily seen that for this equation the critical value ~c~ does indeed lie between zero and one. For ~ = 1, we already know that global regularity holds. For c~= 0, the change of 1 -e

variables U ( x , t ) = e - ~ t V ( x , t ) , t ' -

-~t

reduces the problem to the inviscid v case, for which we know that the second derivative at the origin becomes infinite --,

at t ' = t , . This corresponds to a time t = - i - l n ( 1 - v t , ) , which will actually be 1 v attained if and only if v < - - . t, The technical aspects of this study can be outlined as follows. The existence theorem is established by the Galerkin method using an L 2 a priori estimate and a compactness theorem to pass to the limit. A global regularity result, uniform in the dissipativity and the viscosity, is proved from an L1 a priori estimate for the second derivative, using a method of KRUZKOV [14]. This "optimal" regularity is not strong enough to ensure uniqueness (Chapter 2). The main theorem (Chapter 3), which gives the critical dissipativity ~cr-~, 1 has two aspects. First, to show global regularity for ~ >89 we use a priori estimates in the form of differential inequalities which are the analogs of KATO'S inequalities [9] for the Sobolev norms of the solutions of the Euler or Navier-Stokes equation. Second, to prove loss of regularity for 0 < ~ < 89and sufficiently small

Non-linear Evolution Equations

239

viscosity, we use a lemma of FOIAS and PENEL [5] to show that persistence of c6t regularity for all times would lead to a contradiction. In the last chapter we discuss the physical meaning of our results and show how some of the techniques can be carried over (in part) to a modified NavierStokes equation and, finally, to the true Navier-Stokes equation.

2. Existence and Optimal Global Regularity 2.1. Statement of the problem and notations We shall deal with the Burgers M R C M equations (OU t , x ) + v \(- 02 ~U x+,)t(~x2U [O ,t(U -)x2],)=t(O~.xZ]

d2 (2.1)

The function U is defined for x e R and t e R + and satisfies the boundary and initial conditions (2.2) lira U(t,x)=O, Ixl~oo U (O, x) = Uo(x ). (2.3) The viscosity v and the dissipativity ~ are non-negative parameters. Because of the probabilistic origin of the problem, we shall be interested only in real solutions of positive type in x, i.e., +oo

c(.,x)= ~ e-'kxO(',k)dk, where the Fourier transform in space, 0(.,k), called the energy spectrum, is positive and even. Notice that real functions of positive type are even and attain their absolute maximum at the origin. The Fourier transform of the Burgers M R C M equation (2.1) is +09

8t

t~(t,k)+vlkl 2~t?(t,k)-k 2 ~ [O(t,p) O ( t , k - p ) - 2 0 ( t , p ) U(t,k)] dp=O. (2.4) _ oo

Many of the a priori estimates in this paper make use of the energy moments defined by +oo

IU(t)ls= f Ikl2SO(t,k)dk,

s>O.

(2.5)

-oo

~2 Here [U(t)lo=U(t,O) is called the energy and I U ( t ) l l = - ~ x 2 U ( t , O ) the enstrophy. Morever, ~L2(IR) will denote the intersection of all Sobolev spaces H~(R) with s > 0 , while B.V.(R) will mean the space of functions of bounded variation. Remark 2.1. It is easily checked that if UeNL2(IR ) then all the energy moments are finite.

240

C. BARDOS et al. 2.2. A regularized problem

The construction of a solution of the Burgers M R C M equation (2.1) will be done by the method of elliptic-regularization, namely by first studying the problem _

_

(t,x)+v ~ ax2! U(t,x)-#-ff~x2(t,x)+~ix2[U(t,O)-l](t,x)]2=O

,

(2.6)

lim U(t, x) = 0,

(2.7)

U(x, O) = Uo(x )

(2.8)

where # > 0. 2.3. An L 2 a priori estimate

Theorem 2.3. Let U be a sufficiently smooth solution of (2.1)-(2.3). Then, for arbitrary v >=O, ~ >=0 and every t >__O, we have d +OO - =

=-4

[U(t,O)-U(t,x)]

(2.9)

(t,x)

dx0) and - #

82

are, according to [25, p. 259], the gene-

rators of contraction semigroups in L~, it follows from Lemma 2.5 that the first integral on the right-hand side of (2.15) is negative. To show that the second integral is also negative, we introduce the following smooth regularization of the sign function: sign.({)=

!l " ({/q)

if {>qn/2 if - q n / 2 < { < q n / 2 if ~ < - rln/2.

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C. BARDOSet al.

Denoting the derivative of sign, by sign',, we observe that + o~ / O 17\2 I sign', i ~ x ) (U(t,O)-U(t,x))dx2. Indeed, it will be shown in Chapter3 U(t, k ) ~ C(t)k -2 as k ~ ~ for e0, there exists a positive constant 2 s such that d d t IU(t)ls + v lU(t)ls+=_- 1 we have d dt IU(t)l"+ vlU(t)l"+'>(Zn+ l)(2n+ 2)lU(t)lt IU(t)l..

(3.9)

Proof of (3.5). Starting from (2.4), we integrate with respect to k to obtain d

+o~ +o~

dt IU(t)lo+vlU(t)l== I -

oo

I k2 U(t,p) U ( t , k - p ) d p d k -

-2

ct3

~ --

0~3

I k2 [~(t, p) U(t, k) dp dk.

(3.10)

--of?

* (3.5) requires only cgl regularity in space. When this regularity is not present, energy is removed faster than can be accounted for solely by the dissipative term (energy catastrophe); the energy equation then reads [21]

dt IU(Olo+vlU(t)l=+2

Llim~x (t, x)

=0.

(3.5')


247

To show that the right-hand side of (3.10) is identically zero, we replace k - p by q and k 2 by (p + q)2, and then use the eveness of U with respect to the Fourier variable. (These operations would be false if U(t, k) did not decay fast enough as k ~ oo (e.g., if U(t, k)~k-2).)

Proof of (3.6). Starting from (2.4), we multiply by k 2, integrate with respect to k, and use the same method as before to obtain d

dt Ig(t)ll+vlS(t)ll+~=

+~

+m

-oo

-oo

~

~ (p4+q,) O(t,p) U(t,q)dpdq

+ oo

-2 ~

+oo

~ k40(t,p) U(t,q)dpdq

-oo

-oo

+oo

+oo

~ p2q2U(t,p) U(t,q)dpdq.

+6 ~

(3.11)

--oo - o o

Equation (3.6) follows since the two first terms on the right-hand side cancel.

Proof of (3.7). Write

l~(t,k)dk+

IU(t)ls < Ikl_-l

< ~ O(t,k)dk+ -oe

~ Ikl2*O(t,k)dk.

(3.12)

[k[>l

The result then follows easily from (2.14') and (3.5).

Proof of (3.8). We multiply equation (2.4) by Ikl 2~ and integrate with respect to k. After suitable changes and relabellings of variables, this yields d d-t IU(t)ls + v IU(t)ls+~

= ~ [Ip + ql2 +2~- IPl2+2S- lql 2 +2s3 t2 (t, p) ~2(t, q) dp dq ~2

= 4 ~ dp i dq[(P + q)2 +2~-(P-q)2 +2~- 2p2 +2~- 2q z +2~] U (t, p) U (t,q). o o (3.13) Let us introduce the function (p + q)2 + 2s - -

Fs(p, q) -

(p

--

q)2 + 2,_ 2p2 + 2s - - 2q2 + 2,

p2 q2S + q2 p2S (1 +X)2 +2s-+-(1 --X)2+2s--2--2X 2+2s (3.14)

X 2 --[- X 2 s

where x=q/p is in [0, 1]. Then, F,(x)-Fs(p, q) is bounded in [0, 1] by a positive constant 2s. Indeed, for x--,0, we have

F~(x)

(2 + 2s)(1 +2s) x 2

x2 +x2S

(2-2n)(1 +2n)(p2q2"+p2"q2),

(3.17)

which is a consequence of the binomial formula for any integer n > 1, we then proceed as in (3.16). Remark 3.2.1. In (3.6) we let v ~ 0 and integrate, thus obtaining (cf. [3], [1],

I Uol~

[2], [21])

(3.18)

IU(t)ll =l_61Uol~ t' which implies that the enstrophy becomes infinite at t.=l/(6[Uoll), As a consequence, since the dissipative term can only decrease, the enstrophy IU(t)[ 1 remains uniformly bounded in ~ and v for t~[0, T], T < t . . Remark 3.2.2. It is seen from (3.10) that the regularity of IU(t)l~ is governed by the behaviour of IU(t)ll, namely as long as IU(t)l~ remains finite, regularity holds for any IU(t)[~, s > 0, provided [Uols is finite. Corollary 3.2 (loss of analyticity). For v = 0 and for initial data of positive type which are analytic but not identically zero, the analyticity of the solution is lost for any t > 0 . Proof. Assume that analyticity holds for some time interval, and expand

U(t, x) in powers of x. Since the 2n th derivative at the origin is equal to the energy moment t U(t)l,, we have of)

U(t, x)= ~ x 2" IU(t)l, n= 0

(3.19)

(2 n) ! "

Furthermore, integration of (3.9) for v = 0 gives, for n > 1.

{

'

}

Ig(t)l,>lgol, exp (2n+ 1)(2n+2) I Ig(~)ll d~ .

(3.20)

0

Since Uo is analytic, not identically zero, and of positive type, there exists a constant c > 0 such that S Uo(k) dk=a>O. Hence c +co

IUo[,,=2 I k2"Uo(k)dk>=2c2"a. 0


249

Using this and (3.20), we see easily that the series (3.19) has a zero radius of convergence for any t > 0.

3.3. Global regularity for c~> 1/2 and v > 0 In view of Remark 3.2.2, it is sufficient to study the regularity of IU(t)l~. For this, we start from (3.6) and use the lower bound for the dissipative term v lUR~+~ given by Lemma 3.1.2 (with s~ = s < 8 9 s 2 = 1, s3= 1 +e), namely l+~-s

a

v l Ula +,, >_-v IUl~~---W-/l UIA ,.

(3.21)

Then, using (3.7), we have V

vlUIl+>c~

l+~--s

IUla 1-,

(3.22)

with

1 cs= [l/51Volo~ rc(1-2s)

Vo[

11L

dx 2 L'~]

By (3.22), (3.6) becomes d~-lgll 0). For global regularity to hold, the dissipative term must be able to counterbalance the non-linear transfer of energy. Why this requires ~ > 89may be understood by the following heuristic argument. The rate of energy dissipation is d

+oo

~lu(t)10=v ~ k2"t?(t,k)dk. -o0

Since U(t,k)< Ck -2, the integral will be uniformly bounded in v for ~< 89 and the dissipation will go to zero with v. It is not possible therefore to remove energy by this mechanism. For true turbulence, according to present ideas, the energy spectrum should behave like k-", where n is slightly in excess of 5/3; see [ 11], [ 13], [23], [26]. The above heuristic argument thus suggests possibly a critical value of the dissipativity for the Navier-Stokes equation should be close to 89

C~c,=--~

which


253

4.2. The M R C M Navier-Stokes equation The procedure which leads from the ordinary Burgers equation to the Burgers MRCM equation (1.1) may be applied to the Navier-Stokes equation, giving a corresponding Navier-Stokes MRCM equation for the energy spectrum of homogeneous isotropic turbulence. The energy spectrum E(t, k) is defined for t and k positive, and satisfies (cf. [7], [17]) the equation

~E & (t,k)+vk2~E(t,k) 1 k 2 =~!!~-~bkpq[k E(t,q)E(t,p)-p2E(t,q)E(t,k)] dpdq.

(4.1)

In (4.1), A k is defined by the triangle inequalities Ak= {(P,q)lp + q >=k, IP- ql < k} and P 3 bkpq=~(xY+z ),

x,y, and z being the cosines of the interior angles of the (k, p, q)-triangle (modified dissipativity has been assumed). Contrary to the equation (1.1) (with ~--1), equation (4.1) continues to be of integrodifferential type when it is written in configuration-space due to the presence of a non-local pressure term in the Navier-Stokes equation. Nevertheless several of the a priori estimates which we have established for equation (1.1) are still valid for (4.1). Indeed, defining oo

IE(t)ls= j" k2"E(t,k)dk, 0

the energy equation d

~ IE(t)lo + v rE(t)l~ = 0

(4.2)

and the enstrophy equation [16] d ~ l E ( t ) l l +vlE(t)l I +~-glE(t)ll _2 2

(4.3)

remain valid for sufficiently smooth solutions. From these a priori estimates, and other analogous to (3.8), one can deduce global smoothness for the Navier-Stokes MRCM equation with the usual dissipativity ~ = 1 and for arbitrary positive viscosity. The essential step of the proof goes as follows. Using the Schwarz inequality Ig(t)12 ~ IE(t)l~/lE(t)lo,

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C. BARDOS et al.

we obtain d

IE(t)l~ ~ IE(t)la {~ IE(t)ll - vlE(t)l~ lE(t)lo

1}.

Upon integration and use of (4.2), we then get

IE(t)l~