ICES REPORT August Thomas J. R. Hughes

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1 ICES REPORT August 213 Amplitude-Phase Decompositions and the Growth and Decay of Solutions of the Incompressible Navier-Stokes and Euler Equations by Thomas J. R. Hughes The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas Reference: Thomas J. R. Hughes, Amplitude-Phase Decompositions and the Growth and Decay of Solutions of the Incompressible Navier-Stokes and Euler Equations, ICES REPORT 13-21, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, August 213.

2 Amplitude-Phase Decompositions and the Growth and Decay of Solutions of the Incompressible Navier Stokes and Euler Equations by Thomas J. R. Hughes Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin 21 East 24th Street, C2 Austin, Texas Abstract We introduce the concept of amplitude phase decompositions and apply it to the study of growth and decay of solutions of the incompressible Navier Stokes and Euler equations. Amplitudes coincide with functionals of physical and mathematical interest. We are able to explicitly solve for the amplitudes in terms of the phases. The results obtained provide insights into the growth and decay of enstrophy and viscous dissipation, and identify new criteria for solutions to remain smooth for all time or blow up at finite time. Computational and Applied Mathematics Chair III, Professor of Aerospace Engineering and Engineering Mechanics i

3 Contents Abstract i 1 Introduction 1 2 Preliminaries 2 3 Initial-value Problem 3 4 Amplitude phase Decomposition of the Velocity 4 5 Amplitude phase Decomposition of the Velocity Gradient, Rate-of-deformation and Vorticity 6 6 Discussion 19 7 Conclusions 23 Acknowledgements 23 References 24 ii

4 1 Introduction There is broad interest throughout science, engineering and mathematics in the growth and decay of solutions of the incompressible Navier Stokes and Euler equations. The classical approach to the study of growth and decay is to derive equations governing various quantities of interest, such as linear momentum, angular momentum, vorticity, kinetic energy, viscous dissipation, enstrophy, helicity, etc. See the books of Pope [1] and Majda and Bertozzi [2]. A shortcoming of this approach is that the derived equations are not closed in the sense that one cannot solve for the quantities of interest in terms of other independent quantities. We address this issue herein by introducing amplitude phase decompositions. Our focus is on periodic flows on the n-dimensional torus T n. The amplitude is the quantity of interest. It is a function of time, but not of spatial coordinates. The phase is a function of both space and time. It records space time structure, but it is insensitive to amplitude. In fact, its amplitude is unity, by definition. The benefit of the amplitude phase decomposition is that typical growth and decay equations become scalar ordinary differential equations in time with the amplitude as unknown and with coefficients that are functionals of only the phase. The quantities studied herein are Frobenius L 2 norms of velocity, velocity gradient, rate-of-deformation (i.e., the symmetric velocity gradient) and vorticity (i.e., the skew-symmetric velocity gradient). Physically speaking, these are equivalent to kinetic energy, viscous dissipation and enstrophy. In the case of the velocity, the ordinary differential equation that arises is linear and of first order, hence simply solved. In each of the other cases, the ordinary differential equation that arises is nonlinear, but it is a Bernoulli equation, and is also simply solved. The exact solution of the Bernoulli equation explicitly exhibits the potential for the solution to become unbounded in finite time (i.e., blow up ). Based on relationships among the coefficients of the Bernoulli equations, we are able to collapse the solutions to a single and canonical form. The growth and decay of all the amplitudes can be expressed entirely in terms of invariants of the rate-of-deformation phase. The amplitudes are all independent of the vorticity. In particular the so-called vortex stretching term and the enstrophy are determined by the rate-of-deformation phase, and do not depend in any way on the vorticity. The only quantity that has the potential to create growth is the trace of the cube of the rate-of-deformation phase. This term may be interpreted as the skewness of the symmetric velocity gradient tensor. In three dimensions it becomes simply three times the 1

5 determinant of the rate-of-deformation phase and the potential for growth resides exclusively with it. It is important to emphasize that the growth/decay results obtained herein are equations rather than inequalities. The canonical form of solutions of the Bernoulli equations gives a precise condition for a solution to become unbounded in finite time. Specifically, a functional of the skewness and gradient of the phase must attain the value 1.. This functional may prove useful in studies aimed at determining whether or not blow-up occurs. An outline of the rest of the paper follows. Some notational conventions and preliminary results are presented in Section 2. The initial-value problem for the incompressible Navier Stokes and Euler equations is stated in Section 3. As mentioned previously, we focus on the spatially periodic case. However, most results hold for the initial-value problem on all of Euclidean space, assuming sufficiently strong assumptions are made on decay at spatial infinity. In Section 4 we present an initial application of the amplitude phase decomposition that leads to an explicit formula for the growth and decay of kinetic energy. In Section 5 we perform the amplitude phase decompositions of the velocity gradient, rate-of-deformation and vorticity. These lead to the growth and decay results described above. In Section 6 we discuss the implications of the results obtained and in Section 7 we draw conclusions. 2 Preliminaries Vector and tensor quantities are written in bold-face italic font. The components of vector and tensor quantities are written in light-face italic index notation. For example, the velocity vector is u and its components are denoted u i, where 1 i n; n is the number of spatial dimensions. Likewise, def its gradient is written as u and its components are u i,j = u i / x j, where i, j = 1,..., n and x j is the j th spatial coordinate. Throughout we employ the summation convention on all repeated indices i, j, k, l, m. For example, u i,j u i,j = n ( ) 2. ui,j By virtue of our periodicity and smoothness i,j=1 assumptions, all boundary terms that arise when using integration-by-parts are zero. For indexed quantities we use Frobenius-type norms. For example, the Frobenius norm of u is given by u = (u i,j u i,j ) 1/2. We define L 2 (T n ) norms of indexed quantities in terms of their Frobenius norms as, for 2

6 ( ) 1/2. example, u = u 2 dx We use both index-free direct notation T and indexed notations n interchangeably throughout. In the midst of detailed computations it is generally clearer to work with indices. To express basic ideas, direct notation is usually preferable. 3 Initial-value Problem We consider the l-periodic initial-value problem for the incompressible Navier Stokes equations in R n. Let T n denote the n-torus of characteristic dimension l R +, that is, T n = { x = (x 1,..., x n ) xi < l, i = 1,..., n } (1) The initial-value problem consists of solving the following equations for u : T n [, T ) R n, the velocity vector, and p : T n (, T ) R, the pressure (divided by the constant density), u t + (u u) + p = ν u + f on Tn (, T ) (2) u = on T n (, T ) (3) u(x, ) = u (x) x T n (4) where f : T n (, T ) R n is the prescribed body force (per unit volume), ν is the kinematic viscosity, assumed positive and constant, u : T n R n is the given initial velocity, is the gradient operator, = T is the divergence operator, = is the Laplace operator, and denotes the tensor product (in components, (u u) ij = u i u j, i, j = 1,..., n). It is assumed that u ( C (T n ) ) n, u =, and f ( C (T n (, T )) ) n. Equations (2) and (3) are the balance of linear momentum and incompressibility constraint, respectively. The initial-value problem for the incompressible Euler equations consists of (2) (4) with ν = in (2). We will assume that the solution is initially smooth (i.e., u ( C (T n (, T )) ) n ) in our calculations. In three dimensions it is known that the solution of the initial-value problem remains smooth for at least a finite time interval (, T ), < T <, for both the incompressible Navier Stokes and Euler equations. The question is whether or not T =. Our results hold with less stringent regularity assumptions, but we need greater regularity 3

7 than that possessed by classical Leray Hopf weak solutions [3, 4] and socalled suitable weak solutions [5]. Our aim is to provide insight into the growth and decay behavior of solutions of the Navier Stokes and Euler equations, and provide new criteria for solutions to either remain smooth for all time (i.e., T = ) or blow up in finite time. 4 Amplitude phase Decomposition of the Velocity Let u 2 = u T u = u i u i (5) ( ) 1/2 u = u 2 dx (6) T n u 2 = tr ( ( u) T u ) = u i,j u i,j (7) ( ) 1/2 u = u 2 dx (8) T n (u, f) = u T f dx (9) T n The amplitude phase decomposition of u consists of U = u, the amplitude, and û = u/u, the phase. Note that the amplitude of the phase is 1, that is, û = 1. As an initial illustration of the use of an amplitude phase decomposition, we state the following theorem. Theorem 4.1. Assume u is a smooth solution of the Navier Stokes initialvalue problem (2) (4) for all t [, T ). Let U(t) = u(t) denote the velocity amplitude at time t, let U() = u() = u denote its value at t =, and let û(x, t) = u(x, t)/u(t) be the velocity phase. Then ( U(t) = U() exp ν + t ( exp ν t t τ ) û(σ) 2 dσ û(σ) 2 dσ) (û(τ), f(τ) ) dτ (1) 4

8 Proof Multiply (2) by u T and use (3) to obtain ( 1 ( 1 t 2 u 2) + u) 2 u 2 + (pu) = ν ( 1 2 u 2) ν u 2 + u T f (11) Integrate (11) over T n and use the amplitude phase decomposition: 1 d u 2 dx + + = ν u 2 dx + u T f dx (12) 2 dt T n T n T n 1 d 2 dt U 2 = ν û 2 U 2 + (û, f) U (13) d dt U = ν û 2 U + (û, f) (14) This is an inhomogeneous linear first-order ordinary differential equation for U. Its solution is given by (1). Corollary 4.2. When f =, (1) becomes ( t ) U(t) = U() exp ν û(σ) 2 dσ In the case of the Euler equations, (i.e., ν = ), (1) becomes U(t) = U() + and when f =, U(t) = U(). Remarks t (15) (û(τ), f(τ) ) dτ (16) 1. Results obtained in the next section will verify that û 2 in (1) and (13) (15) can be replaced by either 2 symm( û) 2 or 2 skew( û) 2, where and symm( û) = 1 2( û + ( û) T ) (17) skew( û) = 1 2( û ( û) T ) (18) 2. The kinetic energy is given by U 2 /2. Consequently, the growth and/or decay of kinetic energy can be determined from (1) and any of its special cases. 5

9 5 Amplitude phase Decomposition of the Velocity Gradient, Rate-of-deformation and Vorticity The Euclidean decomposition of a second-rank tensor uniquely sum-decomposes it into its symmetric and skew-symmetric parts. Let a be a second-rank tensor. Then symm(a) = 1 2 (a + at ) = symm(a T ) (19) skew(a) = 1 2 (a at ) = skew(a T ) (2) If a = u, then d = symm(a) is the rate-of-deformation tensor and w = skew(a) is the vorticity tensor. (We work exclusively throughout with the vorticity tensor rather than the vorticity vector.) The amplitudes and phases of these quantities are ( ) 1/2 ( ) 1/2 A = a = a 2 dx = a ij a ij dx (21) T n T ( ) n 1/2 ( ) 1/2 D = d = d 2 dx = d ij d ij dx (22) T n T ( ) n 1/2 ( ) 1/2 W = w = w 2 dx = w ij w ij dx (23) T n T n â = 1 A a (24) ˆd = 1 D d (25) ŵ = 1 W w (26) â = ˆd = ŵ = 1 (27) 6

10 Let α a = ν a 2 = ν a 2 dx = ν a ij,k a ij,k dx (28) T n T n α d = ν d 2 = ν d 2 dx = ν d ij,k d ij,k dx (29) T n T n α w = ν w 2 = ν w 2 dx = ν w ij,k w ij,k dx (3) T n T n ˆα a = ˆα d = ˆα w = 1 A α 2 a = ν â 2 (31) 1 D α 2 d = ν ˆd 2 (32) 1 W α 2 w = ν ŵ 2 (33) β a = tr(a T a 2 ) dx = a ji a jk a ki dx (34) T n T n β d = tr(d a 2 ) dx = d ij a jk a ki dx (35) T n T n β w = tr(w T a 2 ) dx = w ji a jk a ki dx (36) T n T n Remarks ˆβ a = ˆβ d = ˆβ w = 1 A β 3 a (37) 1 D β 3 d (38) 1 W β 3 w (39) 1. All phase quantities, such as the ˆαs and ˆβs, are insensitive to changes in their corresponding amplitudes. 2. The enstrophy is W 2 and the viscous dissipation is 2νD The ˆβs are measures of velocity gradient skewness. It is known from the statistical theory of turbulence [6] that the velocity derivative skewness is related to the production of viscous dissipation and enstrophy. 7

11 Definition 5.1. A force f is said to be conservative if it is derivable from a scalar potential, that is, there exists ϕ : T n (, T ) R such that f = ϕ. With these definitions, we have the following theorem. Theorem 5.2. Assume u is a smooth solution of the Navier Stokes initialvalue problem (2) (4) for all t [, T ) with f conservative. Let A(t) = a(t), D(t) = d(t) and W (t) = w(t), and â(x, t) = a(x, t)/a(t), ˆd(x, t) = d(x, t)/d(t) and ŵ(x, t) = w(x, t)/w (t) denote amplitudes and phases at time t of a = u, d = symm( u) and w = skew( u), respectively. Let A() = a() = u, D() = d() = symm( u ) and W () = w() = skew( u ) be the corresponding initial values. Then A(t) = D(t) = W (t) = ( t A() exp ) ˆα a (σ) dσ t ( τ 1 A() ˆβ a (τ) exp ( t D() exp t ( τ 1 D() ˆβ d (τ) exp ( t W () exp t ( τ 1 W () ˆβ w (τ) exp ) ˆα a (σ) dσ dτ ) ˆα d (σ) dσ ) ˆα d (σ) dσ dτ ) ˆα w (σ) dσ ) ˆα w (σ) dσ dτ (4) (41) (42) Proof To prove (4), take the gradient of (2), pre-multiply by ( u) T, take the trace of the result, and use (3) repeatedly to obtain ( 1 t 2 u 2) + tr ( ( u) T ( u) 2) ( 1 + u) 2 u 2 + tr ( ( u) T Hp ) = ν ( u) 2 + ν ( 1 2 u 2) + tr ( ( u) T Hϕ ), (43) where H denotes the Hessian operator. Simplify the notation using a = u and then integrate over T n : 8

12 ( 1 t 2 a 2) + tr ( a T a 2) ( 1 + u) 2 a 2 + tr ( a T Hp ) = ν a 2 + ν ( 1 2 a 2) + tr ( a T Hϕ ) (44) 1 d 2 dt A2 β a + + = α a + + (45) Use the amplitude phase decomposition to arrive at 1 d 2 dt A2 = ˆβ a A 3 + ˆα a A 2 (46) d dt A = ˆβ a A 2 + ˆα a A (47) (47) is a Bernoulli equation and its solution is (4). To prove (41), take the symmetric part of the gradient of (2), pre-multiply by symm( u), take the trace, use (3) repeatedly, and simplify the notation using a = u and d = symm(a). This results in ( 1 ( 1 t 2 d 2) + tr(d a 2 ) + u) 2 d 2 + tr(d Hp) Integrate this result over T n, = ν d 2 + ν ( 1 2 d 2) + tr(d Hϕ) (48) 1 d 2 dt D2 β d + + = α d + + (49) Use the amplitude phase decomposition to obtain 1 d 2 dt D2 = ˆβ d D 3 + ˆα d D 2 (5) d dt D = ˆβ d D 2 + ˆα d D (51) This is another Bernoulli equation and its solution is (41). The proof of (42) follows in a similar manner. Take the skew-symmetric part of the gradient of (2), pre-multiply by skew( u) T, take the trace of the result, and simplify the notation by using a = u and w = skew(a): 9

13 ( 1 ( 1 1 t 2 w 2) + tr(w T a 2 ) + u) 2 w 2 = ν w 2 + ν ( 2 w 2) (52) Integrating this result over T n yields 1 d 2 dt W 2 β w + = α w + (53) Employing the amplitude phase decomposition results in 1 d 2 dt W 2 = ˆβ w W 3 + ˆα w W 2 (54) d dt W = ˆβ w W 2 + ˆα w W (55) Again we arrive at a Bernoulli equation. Its solution is (42). Corollary 5.3. In the case of the Euler equations, (4) (42) become A(t) = D(t) = W (t) = 1 A() 1 D() 1 W () A() t D() t W () t ˆβ a (τ) dτ ˆβ d (τ) dτ ˆβ w (τ) dτ (56) (57) (58) Remark If forcing is non-conservative, the situation is more complicated because the ordinary differential equations become inhomogeneous, that is, Riccati equations, for which there is no general exact solution. Nevertheless, many solutions to special cases exist and various approximation procedures may be fruitful. It would seem that judicious design of forcing might increase the possibility of blow-up. However, the statement of the Navier Stokes Millenium Problem (Fefferman [7]) is quite restrictive with respect to the possibilities. Nevertheless, this would represent an interesting direction for future research. 1

14 In a series of propositions, we will show that A = 2 D = 2 W (59) ˆα a = ˆα d = ˆα w (6) and Defining ρ = ˆβ a = 1 2 ˆβd = 1 2 ˆβw (61) A A() = D D() = W W (), (62) γ = A() ˆβ a = D() ˆβ d = W () ˆβ w (63) and ˆα = ˆα a = ˆα d = ˆα w, (64) we can collapse (4) (42) to the single result ρ(t) = 1 t ( t ) exp ˆα(σ) dσ ( τ γ(τ) exp ) ˆα(σ) dσ dτ (65) We will discuss (65) further after establishing (59) (64), and after obtaining versions of γ specific to n = 2 and n = 3. Proposition 5.4. a 2 = d 2 + w 2 (66) Proof The result follows from the fact that the trace of the product of a symmetric and skew-symmetric tensor is zero: a 2 = tr(a T a) = tr ( (d + w) T (d + w) ) = tr(d 2 + d w + w T d + w T w) = tr(d 2 ) + tr(d w) + tr(w T d ) + tr(w T w) = d w 2. 11

15 From (34) (36) and (66), it follows that A 2 = D 2 + W 2 (67) Proposition 5.5. A 2 = 2 D 2 (68) Proof First, note that d 2 = tr(d a), viz., d 2 = tr(d 2 ) = tr ( d(a w) ) = tr(d a) tr(d w) = tr(d a) (69) Then 1 ) 2 tr(d a) = 2 tr( 2 (a + at ) a = tr(a 2 ) + tr(a T a) = tr(a 2 ) + a 2 (7) Integrating over T n, 2 D 2 = 2 d 2 dx T n = 2 tr(d a) dx T n = tr(a 2 ) dx + a 2 dx (71) T n T n = tr(a 2 ) dx + A 2 (72) T n 12

16 The result follows from integration-by-parts and u =, tr(a 2 ) dx = tr( u u) dx T n T n = u i,j u j,i dx T n = u i u j,ij dx T n = u i (u j,j ),i dx T n = u ( u) dx T n = (73) Corollary 5.6. From (67) and (68) it follows that D = W (74) and A 2 = 2 W 2 (75) In summary, A = 2 D = 2 W, and we have shown (59). Remark Similar procedures can be used to show where s = } {{ }, s =, 1, 2,..., and s times s a 2 = s d 2 + s w 2 (76) s a 2 = a ij, kl... m }{{} s indices s d 2 = d ij, kl... m }{{} s indices s w 2 = w ij, kl... m }{{} s indices a ij, kl... m }{{} s indices d ij, kl... m }{{} s indices w ij, kl... m }{{} s indices (77) (78) (79) 13

17 and s a 2 = s d 2 + s w 2 (8) s d 2 = s w 2 (81) s a 2 = 2 s d 2 = 2 s w 2 (82) From definitions (28) (3) and (8) (82) with s = 1, we get α a = α d + α w (83) α d = α w (84) α a = 2α d = 2α w (85) From definitions (31) (33) and (59) we see that ˆα a = ˆα d = ˆα w, and we have proved (6). Proposition 5.7. (i) tr(a T a 2 ) = tr(d 3 ) + tr(w T d w) (86) (ii) tr(d a 2 ) = tr(d 3 ) tr(w T d w) (87) (iii) tr(w T a 2 ) = 2 tr(w T d w) (88) (iv) 3 tr(w T d w) = tr(d 3 ) tr(a 3 ) (89) Proof First note that d 2 and w 2 are symmetric and dw + wd is skewsymmetric. (i) tr(a T a 2 ) = tr ( (d + w) T (d + w) 2) = tr ( (d + w T ) (d 2 + d w + w d + w 2 ) ) = tr ( d 3 + d 2 w + d w d + d w 2 + w T d 2 + w T d w + w T w d + w T w 2) = tr(d 3 ) + tr(d 2 w) + tr(d w d) + tr(dw 2 ) + tr(w T d 2 ) + tr(w T d w) + tr(w T w d) + tr(w T w 2 ) = tr(d 3 ) + + tr(w T d w) + + tr(w T d w) + tr(w T d w) + = tr(d 3 ) + tr(w T d w) (9) 14

18 (ii) tr(d a 2 ) = tr ( (d(d + w) 2) = tr ( d (d 2 + d w + w d + w 2 ) ) = tr(d 3 + d 2 w + d w d + d w 2 ) = tr(d 3 ) + tr(d 2 w) + tr(d w d) + tr(d w 2 ) = tr(d 3 ) tr(w d w) = tr(d 3 ) tr(w T d w) (91) (iii) tr(w T a 2 ) = tr ( (a d) T a 2) = tr(a T a d a 2 ) = tr(a T a 2 ) tr(d a 2 ) = 2 tr(w T d w) (92) (iv) tr(a 3 ) = tr ( (d + w) 3) = tr(d 3 + d w d + d 2 w + d w 2 + w d 2 + w 2 d + w d w + w 3 ) = tr(d 3 ) + tr(d w d) + tr(d 2 w) + tr(d w 2 ) + tr(w d 2 ) + tr(w 2 d) + tr(w d w) + tr(w 3 ) (93) = tr(d 3 ) tr(w d w) + + tr(w d w) + tr(w d w) + = tr(d 3 ) 3 tr(w T d w) (94) Proposition 5.8. T n tr(a 3 ) dx = (95) 15

19 Proof For this proof it is more convenient to use coordinate representations. Integration-by-parts and u = u i,i = are used repeatedly. tr(a 3 ) dx = a ij a jk a ki dx T n T n = u i,j u j,k u k,i dx T n = u i,j u j,ki u k dx T n = + u i,jk u j,i u k dx T n = u i,k u j,i u k,j dx T n = u i,k u k,j u j,i dx T n = a ik a kj a ji dx T n = tr(a 3 ) dx (96) T n Corollary 5.9. tr(w T d w) dx = 1 tr(d 3 ) dx (97) T 3 n T n β a = tr(a T a 2 ) dx = 4 tr(d 3 ) dx (98) T 3 n T n β d = tr(d a 2 ) dx = 2 tr(d 3 ) dx (99) T 3 n T n β w = tr(w T a 2 ) dx = 2 tr(d 3 ) dx (1) T 3 n T n 16

20 β a = β d + β w (11) β d = β w (12) β a = 2β d = 2β w (13) ˆβ a = 1 A β 3 a = 1 ( 2 tr ( ˆd ) ) 3 dx (14) 2 3 T n ˆβ d = 1 D β 3 d = 2 tr ( ˆd ) 3 dx (15) 3 T n ˆβ w = 1 W β 3 w = 2 tr ( ˆd ) 3 dx (16) 3 T n Therefore, ˆβ a = 1 2 ˆβd = 1 2 ˆβw, and we have shown (61). Remarks 1. (97) generalizes the three-dimensional result of Chae [8] to n dimensions. Chae [8] obtained (97) using a spectral decomposition of the 3 3 matrix d. (97) states that the so-called vortex stretching term is completely determined by the rate-of-deformation. 2. If the sign of the velocity gradient is reversed, the ˆβs and γ change sign, whereas α does not. 3. Note that, due to (6), (61), (14), and (15), ρ(t) can be written as but Proposition 5.1. ρ(t) = F ( A(), â ) = F ( D(), ˆd ) (17) ρ(t) F ( W (), ŵ ) (18) (i) For n = 2, tr(d 3 ) =, and so β a = β d = β w =. 17

21 (ii) For n = 3, tr(d 3 ) = 3 det(d), and so β d = β w = 2 det(d) dx (19) T 3 β a = 4 det(d) dx (11) T 3 Proof (i) The Cayley-Hamilton theorem for the case n = 2 states that d 2 tr(d) d + det(d) I 2 =, (111) where I 2 is the 2 2 identity tensor. Multiplying (111) by d yields d 3 tr(d) d 2 + det(d) d =. (112) Taking its trace results in tr(d 3 ) tr(d) tr(d 2 ) + det(d) tr(d) =. (113) The trace of d is zero. Consequently, tr(d 3 ) =. (ii) The Cayley-Hamilton theorem for the case n = 3 states that d 3 tr(d) d 2 + 2( 1 (tr(d)) 2 tr(d 2 ) ) d det(d) I 3 =, (114) where I 3 is the 3 3 identity tensor. Taking its trace yields tr(d 3 ) tr(d) tr(d 2 ) + 2( 1 (tr(d)) 2 tr(d 2 ) ) tr(d) det(d) tr(i 3 ) =, (115) and again using tr(d) =, we get tr(d 3 ) = 3 det(d). (116) 18

22 Remarks 1. In three dimensions there are two independent tensor invariants of d, namely, d and det(d). d is sign insensitive, that is, reversing the sign of d has no effect on d, whereas det(d) is sign sensitive, that is, reversing the sign of d reverses the sign of det(d). In two dimensions there is only one independent invariant of d, namely d. 2. Equation (115) also holds for the velocity gradient tensor a. Consequently, tr(a 3 ) = 3 det(a) in three dimensions. By (95), T 3 det(a) dx =. 6 Discussion 1. By virtue of (59), (6), (63), and (14) (16), the right-hand side of (65) can be expressed entirely in terms of the initial value of D, and ˆd, the phase of the rate-of-deformation. Thus the growth and/or decay of all the amplitudes, A, D and W, depend only on the symmetric part of the velocity gradient. In three dimensions, the crucial term governing growth and decay of these quantities is det( ˆd) dx, the T 3 rate-of-deformation skewness. 2. Clearly, the viscosity terms, as represented by ˆα, can only produce decay of the amplitudes. The role of γ is more complex. Only it can create growth, but it is also capable of producing decay. It is sensitive to the sign of the velocity gradient. If γ is positive, reversing the sign of the velocity gradient makes γ negative, and vice versa. 3. In the two-dimensional case, by virtue of Proposition 5.1, part (i), ( t ) ρ(t) = exp ˆα(σ) dσ (117) As ˆα, by definition, solutions of the two-dimensional Navier Stokes and Euler equations cannot grow in amplitude, a well-known result. In fact, it is also well known that solutions of the two-dimensional Navier Stokes and Euler initial-value problems remain smooth for all t [, ) (Ladyzhenskaya [9]). 19

23 4. In the three-dimensional case, the potential for growth is determined by the behavior of det d, as may be concluded from Proposition 5.1, part (ii). It is instructive to examine the local structure of det d. For this purpose it is convenient to work in the eigenspace of d. In eigencoordinates, d = diag(λ 1, λ 2, λ 3 ) (118) where the λs are the eigenvalues of d, and so λ 1 + λ 2 + λ 3 = (119) If one of the eigenvalues of d is zero, then det d =, and so this case is uninteresting. There are only two other possibilities, either one eigenvalue is positive and the other two are negative, or one eigenvalue is negative and the other two are positive. In the former case det d >, whereas in the latter case det d <. In the case of det d >, without loss of generality, assume λ 3 >, which we refer to as the dominant eigenvalue, and λ 1 < and λ 2 <. A fluid line element aligned with the principal direction corresponding to the dominant eigenvalue is undergoing extension, whereas fluid line elements in the other two orthogonal principal directions are being compressed. We refer to this as the tensile case, in analogy with the tensile test of a specimen in strength of materials testing. In the case of det d <, assume the dominant eigenvalue is λ 3 <, and λ 1 > and λ 2 >. In this case, a fluid line element aligned with the principal direction corresponding to the dominant eigenvalue is undergoing compression, whereas fluid line elements in the other two orthogonal principal directions are experiencing extension. We refer to this as the compression case, again in analogy with terminology used in strength of materials testing. Spatial locations corresponding to the compression case contribute to the growth of ρ, as may be gleaned from (65), whereas spatial locations corresponding to the tensile case contribute to its decay. It was first observed in numerical tests by Woodward et al. [1] that where there is compression, the flow tends to be unstable and turbulent, while where there is tension, the flow tends to be stable and smooth. The situation is schematically illustrated in Figure 1 for the case in which λ 1 = λ 2 = λ, λ 3 = 2λ. The preceding observations suggest that the growth and potential blow up of solutions of the incompressible Navier Stokes and Euler equations occur in regions of turbulent flow. 2

24 (a) λ3 = (λ1 + λ2) = 2λ > (b) λ3 = (λ1 + λ2) = 2λ < λ2 = λ < λ2 = λ > λ1 = λ < λ1 = λ > det d = λ1 λ2 λ3 = 2λ 3 > det d = λ1 λ2 λ3 = 2λ 3 < Figure 1: Local structure of the rate-of-deformation tensor in eigencoordinates; λ1, λ2 and λ3 are the eigenvalues. λ1 is assumed equal to λ2, and λ3 is the dominant eigenvalue. (a) is the tensile case, which contributes to decay. (b) is the compression case, which contributes to growth.

25 5. Since, by definition, ρ(t), it follows from (65) that t ( τ ) δ(t) def = γ(τ) exp ˆα(σ) dσ dτ = 2 t 3 D() tr ( ˆd(x, ) ( τ ) τ) 3 dx exp ν ˆd(, σ) 2 dσ dτ T n = 2 t 3 D() 1 tr ( d(x, τ) 3) ( τ 1 ) dx exp ν d(, σ) 2 dσ dτ D(τ) 3 T n D(σ) 2 1, (12) where we recall that D = d = symm u. In three dimensions, this becomes t δ(t) = 2D() det ˆd(x, ( τ ) τ) dx exp ν ˆd(, σ) 2 dσ dτ T 3 t 1 ( τ 1 ) = 2D() det d(x, τ) dx exp ν d(, σ) 2 dσ dτ D(τ) 3 T 3 D(σ) 2 1 (121) This provides an a priori estimate, or realizability constraint, on solutions of the incompressible Navier Stokes and Euler equations. From (65), (12) and (121), we have the following theorem. Theorem 6.1. If u is smooth for all t (, ), then δ(t) < 1 for all t (, ). However, if there exists T < such that δ(t) 1 as t T, then for all ɛ >, u is not smooth on the interval (, T + ɛ) and T is the time of blow-up. Remarks 1. This blow-up criterion is unusual in that it characterizes the solution becoming singular in terms of a global functional of the time history of the solution attaining a finite value. All other criteria that have been advanced heretofore involve local quantities becoming unbounded at a time instant. 2. (12) and (121) may also prove useful in numerical investigations of blow-up. It may be noted that if a Fourier Galerkin formulation is used and strictly interpreted, and a divergence-zero basis is employed, as is usually done, then (65), (12) and (121) also hold for the numerical solution. 22

26 7 Conclusions We have introduced the concept of amplitude-phase decompositions and applied it to the study of growth and decay of solutions of the incompressible Navier Stokes and Euler equations. The methodology is illustrated by deriving growth/decay identities for Frobenius L 2 norms of velocity, velocity gradient, rate-of-deformation and vorticity in a periodic n-torus. The results obtained provide insights into the behavior of these quantities and new criteria for solutions of the initial-value problem to remain smooth for all time or blow up in finite time. Amplitude-phase decompositions enable the extraction of scalar ordinary differential equations governing quantities of physical and/or mathematical interest from systems of nonlinear partial differential equations. As such, amplitude-phase decompositions may provide a general methodology for investigating the growth and decay characteristics of a wide variety of nonlinear evolution equations. Acknowledgements I would like to express my appreciation to Luis Caffarelli, John Evans, Bob Moser and Paul Woodward for helpful remarks, and to Fred Nugen for help in preparing the manuscript. 23

27 References [1] S.B. Pope, Turbulent Flows, Cambridge University Press, 2. [2] A. Majda and A.L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 21. [3] J. Leray, Sur le mouvement d un liquide visqueux emplissant l espace, Acta Math. 63, , [4] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4, , [5] V. Scheffer, Hausdorff measure and the Navier Stokes equations, Comm. Math. Phys. 55, , [6] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press, Cambridge, Massachusetts, [7] C.L. Fefferman, Existence and Smoothness of the Navier Stokes Equation, J. Carlson, A. Jaffe, and A. Wiles, editors, The Millenium Prize Problems, American Mathematical Society, 26. [8] D. Chae, On the spectral dynamics of the deformation tensor and new a priori estimates for the 3D Euler equations, Comm. Math. Phys. 263, , 25. [9] O.A. Ladyzhenskaya, The Mathematical theory of viscous incompressible flow, Gordon and Breach, New York-London, [1] P.R. Woodward, D.H. Porter, I. Sytine, S.E. Anderson, A.A. Mirin, B.C. Curtis, R.H. Cohen, W.P. Dannevik, A.M. Dimits, D.E. Eliason, K.-H. Winkler, and S.W. Hodson, Very High Resolution Simulations of Compressible Turblent Flows, in Computational Fluid Dynamics, Proc. of the Fourth UNAM Supercomputing Conference, Mexico City, June 27 3, 2, Eds. E. Ramos, G. Cisneros, R. Fernando-Flores, and A. Santillan-Gonzalez, pp. 3 15, World Scientific,