Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 183–191

Blow-Up in a Modified Constantin-Lax-Majda Model for the Vorticity Equation

E. Wegert and A. S. Vasudeva Murthy

Dedicated to Prof. L. von Wolfersdorf on the occasion of his 65th birthday

Abstract. We propose a one-dimensional model for the vorticity equation involving viscosity. Complex methods are utilzed in order to study finite time blow-up of the solutions. In particular, it is shown that the blow-up time depends monotoneously on the viscosity.

Keywords: Vorticity equation, Hilbert transform, blow-up AMS subject classification: 76 C 05, 35 Q 35

1. Introduction

Physical arguments (e.g. Frisch [7: p. 115]) and numerical computations (e.g. Grauer and Sideris [8]) strongly suggest that finite time singularities develop in three-dimension-al inviscid incompressible flow. The equations governing such a flow are the Euler equations

u_{t} + (u · ∇)u + ∇p = 0  (1)
∇ · u = 0  (2)

for the velocity u = u(x, t) and the pressure p = p(x, t) on ℝ^{3} × ℝ_{+}. A basic question is if smooth solutions of initial value problems for (1) - (2) do exist for all time. Beale, Kato and Majda [1] have proved the following. Suppose the initial velocity field

u(x, 0) = u_{0}(x)  (3)

is smooth. Then there exists a global smooth solution if and only if the vorticity ω = ∇ × u satisfies ∫_{0}^{T} ||ω(·, t)||_{∞} dt < ∞ for every T > 0. Further, they showed that if a solution which is initially smooth loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches. Thus the formation of singularities in Euler equations depends on vorticity production or vortex stretching.

A. Wegert: TU Bergakademie, Fak. Math. & Inform., D-09596 Freiberg A. S. Vasudeva Murthy: TIFR Centre, Indian Inst. Sci., Bangalore 560 012, India ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

The interest in these possible singularities, as pointed out by Caflisch [2] in 1993, is of physical, numerical and mathematical nature: physical because singularity formation may signify the onset of turbulence and may be a primary mechanism of energy transfer from large to small scales, numerical because special methods to solve Euler equations would be required for tackling this singularity formation, mathematical because singularities in Euler equations would prevent an establishment of global existence theorems for equations (1) and (2).

The need to understand the precise mechanism of formation of singularities in finite time has led to some model problems that mimic the Euler equations. These models should not only be simpler than (1) and (2) but also possess some of their important features.

In this direction Constantin et al. [4] proposed a very simple model for the vorticity equation. We shall briefly explain the motivation for their proposal. With ω := ∇ × u, the vorticity, the Euler equations (1) and (2) can be written in the form

ω_{t} + (u · ∇) ω = (ω · ∇) u.  (4)

The initial condition u(x, 0) = u_{0}(x) is transformed into

ω(x, 0) = ω_{0}(x)  (5)

where ω_{0} = ∇ × u_{0}. Now u can be written in terms of ω by the Biot-Savart formula

u(x, t) = −\frac{1}{4π} ∫_{ℝ^{3}} \frac{x − y}{|x − y|^{3}} × ω(y, t) dy.  (6)

By substituting (6) into (4) the latter equation is reduced to

ω_{t} + (u · ∇) ω = (Dω)ω  (7)

where Dω is the symmetric part of the matrix ∇u expressed in terms of ω. The operator ω ↦ Dω is a strongly singular integral operator. The explicit formula for D is not of interest here, but some properties are worth noting.

In two space dimensions, (Dω) ω = 0 which implies conservation of vorticity. In three dimensions, D is a convolution operator with a (matrix) kernel homogeneous of order -3 and vanishing mean value on the unit sphere. Constantin et al. [4] made the remarkable observation that in one space dimension there is only one such operator, the Hilbert transform

Hω(x) = \frac{1}{π} p.v. ∫_{−∞}^{+∞} \frac{ω(y)}{x − y} dy.

By replacing the convective derivative ω_{t} +(u·∇) ω by the partial derivative ω_{t} and Dω by the Hilbert transform Hω, Constantin et al. [4] arrive at a simple one-dimensional analogue of (4) and (5):

ω_{t} = ωHω  (8)
ω(x, 0) = ω_{0}(x).  (9)

In this model the “velocity” is determined from the vorticity by

u(x, t) = ∫_{−∞}^{x} ω(y, t) dy.  (10)

Problem (8) - (9) is explicity solvable and its solution is given by

ω(x, t) = \frac{4ω_{0}(x)}{(2 − t Hω_{0}(x))^{2} + t^{2} ω_{0}^{2}(x)}.  (11)

From this formula it is clear that the solution ω blows up in a finite time T_{0} if and only if there exists an x_{0} such that ω_{0}(x_{0}) = 0 and Hω_{0}(x_{0}) > 0. Constantin et al. [4] also showed that if x_{0} is a simple zero of ω_{0}, then for 1 ≤ p < ∞

lim_{t→T_{0}} ∫_{−∞}^{+∞} |ω(x, t)|^{p}dx = ∞  and  lim_{t→T_{0}} ∫_{−∞}^{+∞} |u(x, t)|^{p}dx < M^{p} < ∞.

Thus the model vorticity equation (8) seemed to possess the most important feature of (4): finite time blow-up of vorticity with velocity remaining bounded. Now (8) - (9) with its explicit solution (11) is a challenging test problem for numerical methods designed to detect blow-up; this has been demonstrated by Stewart and Geveci [12] in 1992.

The first attempt to extend the model problem (8) - (9) to include viscous effects was made by Schochet [11], who considered the problem

ω_{t} = ωHω + ε ω_{xx} on ℝ × ℝ_{+}  (12)
ω(x, 0) = ω_{0}(x).  (13)

The solution to problem (12) - (13) was explicity constructed by Schochet, who found that it blows up at time T_{ε} with

T_{ε} < T_{0},  (14)

where T_{0} is the blow-up time for ε ≡ 0. In other words, adding diffusion makes the solution less regular! Clearly this is unsatisfactory, especially in view of the result by Constantian [3], which says that if the solution to the Euler equation is smooth, then the solutions to the slightly viscous Navier-Stokes equations with the same initial data are also smooth. Hence the simple model (12) lost most of its interest.

Improvements were suggested by De Gregorio [5, 6] who kept the convective derivative on the left-hand side and studied the equation ω_{t} + uω_{x} = ω Hω + νω_{xx} with viscosity ν ≥ 0. Note that De Gregorio defines the velocity u as a primitive of Hω and not of ω.

In the present paper we return to the Constantin et al. model (8) and introduce an alternative additive (non-local) diffusion term which results in an one-dimensional problem with an explicit solution. In contrast to Schochet’s model, the inequality in (14) is now reversed, and thus the drawback mentioned above is removed.

2. A viscous model with a non-local diffusion term

In this section we derive heuristically a proposal for including viscous effects to (8). In connection with investigations of water wave phenomenons like sharp crests and breaking of waves Whitham [14] studied the problem

u_{t} = uu_{x} on ℝ × ℝ_{+}  (15)
u(x, 0) = u_{0}(x).  (16)

It is well known that solutions to problem (15) - (16) lose regularity in finite time no matter how smooth u_{0} is. On the other hand, if we add viscosity to (15),

u_{t} = uu_{x} + νu_{xx},  (17)

then a global smooth solution exists for all time. Now Whitham asked the question if there exists a viscosity term which, when added to (15), influences the solution so that it loses regularity in finite time. He conjectured that

u_{t} = uu_{x} − K ∗ u_{x}  (18)

with the convolution kernel K having the Fourier transform K̂(ξ) = \sqrt{ξ tan h ξ} has the desired property. This conjecture has been completely settled by Naumkin and Shishmarëv [10] in 1991.

In a similar vein we ask the analogous question for the Constantin-Lax-Majda model: What is an appropriate viscosity term that when added to (8) will make the solution blow up at a finite time T_{ε} > T_{0}? Because of (14) it cannot be ε u_{xx}.

Constantin et al. [4] have shown that the blow up of (11) is different from the blow up of u_{x} where u is a solution to problem (15) - (16). Note that u_{x} satisfies along the characteristics

(u_{x})_{t} = (u_{x})^{2}  (19)
u_{x}(x, 0) = (u_{0}(x))_{x}  (20)

and hence it blows up in finite time. In other words, the equation u_{t} = uu_{x} is not a good model for the breakdown of smooth solutions to (1) and (2) but ω_{t} = ωHω is a better model. Arguing analogously one feels that −εHu_{x} would be a reasonable “viscosity” compared to εu_{xx}. So we propose

ω_{t} = ωHω − εHω_{x} on ℝ × ℝ_{+}  (21)
ω(x, 0) = ω_{0}(x)  (22)

as the viscous analogue of (8) - (9). Note that −εHω_{x} is indeed a dissipative term as can be checked by solving the linear part of (21) using Fourier transform. Such a dissipative term has also been considered by Matsuno [9] in 1992.

3. Global existence versus finite time blow-up

In the following we shall consider the periodic version of (21) - (22). More precisely, we assume that the velocity is 2π-periodic in x, which implies periodicity of the initial function ω_{0} and the solution ω (with respect to the space-variable x), as well as

∫_{0}^{2π}ω_{0}(x) dx = 0  and  ∫_{0}^{2π}ω(x, t) dx = 0.  (23)

In order to determine the exact solution we introduce the complex-valued functions

w(x, t) = Hω(x, t) + i ω(x, t)  and  w_{0}(x) = Hω_{0}(x) + i ω_{0}(x)

where H acts with respect to x. The functions w_{0} and w(·, t) extend from the real axis to (periodic) bounded holomorphic functions in the lower half-plane ℂ_{−} and tend uniformly to zero as Im z → −∞. Using the identities (recall (23))

\begin{aligned}2 H(ωHω) & = (Hω)^{2} − ω^{2} \\ H^{2}ω & = −ω \\ Hω_{x} & = (Hω)_{x} \end{aligned}

a straightforward calculation shows that problem (21) - (22) is transformed to the initial problem

w_{t} + i εw_{x} = \frac{1}{2}w^{2} on ℝ × ℝ_{+}  (24)
w(x, 0) = w_{0}(x).  (25)

Lemma 1. The unique solution of problem (24) − (25) is given by

w(x, t) = \frac{2 w_{0}(x − i εt)}{2 − t w_{0}(x − i εt)}.  (26)

Proof. Along the characteristics (24) is transformed into an ordinary differential equation. With W (t) = w(t, x + i εt) we get W′(t) = \frac{1}{2} W^{2}. This equation has the solution W (t) = \frac{2 W (0)}{2−t W (0)} = \frac{2 w(0,x)}{2−t w(0,x)}, which gives the desired result

The last lemma provides us with a simple criterion for blow-up.

Lemma 2. The solution to problem (21) − (22) blows up at (x_{0}, t_{0}) if and only if ω_{0}(x − i εt_{0}) = \frac{2}{t_{0}} .

Proof. The solution to problem (21) - (22) is given by ω(x, t) = Im w(x, t). With z := x − i εt we get from (26)

ω(x, t) = Im \frac{2w_{0}(z)}{2 − t w_{0}(z)}.

The function w_{0} is holomorphic in the lower half plane and hence the solution cannot blow up if the denominator 2 − t w_{0}(z) does not vanish. Conversely, let z_{0} = x_{0} − i εt_{0} be such that w_{0}(z_{0}) = \frac{2}{t_{0}} . Since w_{0} is holomorphic, by Taylor series w_{0}(z) = w_{0}(z_{0}) +

(z − z_{0})^{m}g(z) where g is a holomorphic function with g(z_{0}) = w_{0}^{(m)}(z_{0})/m! ≠ 0 and hence

ω(x, t) = Im \frac{4 + 2t_{0}(z − z_{0})^{m}g(z)}{2(t_{0} − t) − t t_{0} (z − z_{0})^{m}g(z)}.

If g(z_{0}) ∉ ℝ then for t = t_{0} and x → x_{0},

ω(x, t_{0}) ∼ −\frac{4}{(x − x_{0})^{m}t_{0}^{2}} · Im \frac{1}{g(z_{0})}.

If g(z_{0}) ∈ ℝ \ \{0\}, then for x → x_{0} and t = t_{0} − \frac{1}{2} t_{0}^{2} g(z_{0})(x − x_{0})^{m},

\begin{aligned}ω(x, t) & ∼ Im \frac{4}{t_{0}^{2} g(z_{0}) ((x − x_{0})^{m} − (z − z_{0})^{m})} \\ & ∼ \frac{8}{t_{0}^{4} g(z_{0})^{2} m ε (x − x_{0})^{2m−1}} . \end{aligned}

Therefore, in both cases, the solution w(x, t) is unbounded in any neighborhood of (x_{0}, t_{0})

The following technical lemma will serve to estimate the blow-up time.

Lemma 3. Let ω_{0} be a 2π-periodic Hölder-continuous function with

∫_{0}^{2π} ω_{0}(x) dx = 0.  (27)

Then w_{0} satisfies the estimate

|w_{0}(z)| ≤ M e^{−|Im z|} (z ∈ ℂ_{−})

where M = max_{x∈ℝ} |ω_{0}(x) + i Hω_{0}(x)|.

Proof. The function ζ = f(z) = exp(−i z) maps the half-strip {z ∈ ℂ_{−} : 0 ≤ Re z < 2π} onto the punctured unit disk 𝔻̇ = {ζ ∈ ℂ : 0 < |ζ| < 1}. The transplanted function w̃_{0}(ζ) = w_{0}(f^{−1}(ζ)) is holomorphic in 𝔻̇ and has a continuous extension onto the unit circle. Since the mean value along the boundary vanishes we have lim_{ζ→0} w̃_{0}(ζ) = 0. Consequently, by Schwarz’ lemma,

|w̃(ζ)| ≤ max |w_{0}| · |ζ| ≡ M |ζ|,

which together with |ζ| = e^{−|Im z|} yields the assertion We denote by T_{ε}(ω_{0}) the time of the first blow up,

T_{ε}(ω_{0}) = inf \{t > 0 : w_{0}(x − i εt) = \frac{2}{t} for some x ∈ ℝ\}.

If the set on the right-hand side is empty, T_{ε}(ω_{0}) := +∞.

In all what follows we assume that the initial function ω_{0} is not a constant. In order to study the dependence of the blow-up time T_{ε}(ω_{0}) on ε and ω_{0} we consider the images of the closed lower half-planes

ℂ_{y} := {z ∈ ℂ_{−}: Im z ≤ −y}, (y ≥ 0)

under the mapping w_{0}. More precisely, R_{y} := w_{0}(ℂ_{y}) ∪ {0}.

Lemma 4. Let ω_{0} satisfy the assumptions of Lemma 3. Then the origin lies in the interior of all sets R_{y}, and R_{y} contracts to 0 as y → +∞. The sets R_{y} form a strictly nested family,

R_{y_{2}} ⊂ int R_{y_{1}} if y_{2} > y_{1} ≥ 0.  (28)

The blow-up time T_{ε}(ω_{0}) is characterized by

T_{ε}(ω_{0}) = inf {t > 0: 2/t ∈ R_{εt}}.  (29)

Proof. First of all we note that R_{y} is the image of the closed disk D_{y} := {z ∈ ℂ: |z| ≤ exp(−y)} under the mapping w̃_{0} (see proof of Lemma 3). The first assertion follows from w̃_{0}(0) = 0 and Lemma 3.

The second assertion is a consequence of the open mapping principle for holomorphic functions.

In order to prove the third assertion, we recall that the solution blows up at time t if and only if 2/t = w_{0}(x − i εt) for some x ∈ ℝ.

Since the R_{y} are nested and R_{0} is bounded, the point 2/t lies outside R_{εt} for sufficiently small t. More precisely, there is no blow-up for all t with t < T := inf {t ∈ ℝ_{+}: 2/t ∈ R_{εt}}.

On the other hand, a continuity argument shows that 2/t ∈ R_{εt} if t = T. It follows that 2/t = w_{0}(x − i y) for some x and some y ≥ εt. Now y > εt would imply that 2/t ∈ int R_{εt} and hence 2/t ∈ R_{εt} for some t < T, which is impossible by the definition of T. Consequently y = εt

It has already been mentioned that T_{0}(ω_{0}) is finite if and only if there exists an x_{0} such that ω_{0}(x_{0}) = 0 and Hω_{0}(x_{0}) > 0. The next result shows that the solution necessarily blows up for ε = 0 if the mean value of ω_{0} vanishes, which is always the case for periodic velocity.

Theorem 1. Let ω_{0} be a non-constant Hölder-continuous 2π-periodic function. If ∫_{0}^{2π} ω_{0}(x) dx = 0 then T_{0}(ω_{0}) is finite.

Proof. Since R_{0} is a compact set, the point \frac{2}{t} lies outside R_{0} if t is sufficiently small. The origin is an interior point of R_{0} and hence 2/t belongs to R_{0} if t is large. Lemma 4 proves the assertion

The next theorem shows that the viscous term increases the blow-up time and even prevents blow-up if ε is sufficiently large.

Theorem 2. Let ω_{0} satisfy the assumptions of Theorem 1.

(i) The blow-up time T_{ε}(ω_{0}) is a monotoneously increasing function of ε. In particular, if 0 < ε ≤ δ, then T_{0}(ω_{0}) < T_{ε}(ω_{0}) ≤ T_{δ}(ω_{0}).

(ii) For each initial function ω_{0} there exists a positive ε_{∗} such that T_{ε}(ω_{0}) = +∞ if ε > ε_{∗}.

(iii) For all ε > 0 and C ∈ ℝ there exists a constant T_{∗} = T_{∗}(ε, C) such that for all ω_{0} with Hölder norm ∥ω_{0}∥_{α} ≤ C either T_{ε}(ω_{0}) ≤ T_{∗} or T_{ε}(ω_{0}) = +∞.

Moral. What survived sufficiently long will persist forever.

Proof of Theorem 2. (i) If ε < δ, then R_{δt} ⊂ int R_{εt} for all t and hence the point \frac{2}{t} (which lies outside R_{εt} for small t) meets the domain R_{εt} at an earlier time than R_{δt}.

(ii) According to Lemma 3 the intersection of R_{εt} with the real axis is contained in the interval {x ∈ ℝ : |x| < M exp(−εt)}, and hence the solution cannot blow up if \frac{2}{t} > M exp(−εt) for all t > 0. The latter condition is satisfied for all sufficiently large ε.

(iii) By what was said above, the blow-up time (if it is finite) is subject to \frac{2}{T_{ε}} ≤ M exp(−εT_{ε}) which gives an upper bound for T_{ε}

Example. For the initial function ω_{0}(x) = cos x we get w_{0}(x) = Hω_{0}(x)+i ω_{0}(x) = sin x + i cos x which has the analytical extension w_{0}(z) = i exp(−i z) onto ℂ_{−}. Thus the solution ω is the imaginary part of

w(x, t) = \frac{2 w_{0}(x − i εt)}{2 − t w_{0}(x − i εt)} = −\frac{2}{t + 2 i exp(i x) exp(εt)}.

The blow-up time T_{ε} is determined by the condition i exp(i x) exp(εT_{ε}) = −\frac{T_{ε}}{2} , which splits into

exp(εT_{ε}) = \frac{T_{ε}}{2}  and  exp(i x) = i.

Blow-Up in a Modified Constantin-Lax-Majda Model 191

The solution blows up if and only if 0 ≤ ε ≤ \frac{1}{2}ε, the blow-up time satisfies 2 ≤ T_{ε} ≤ 2ε.

The figures show the behaviour of the solution for ε = 0.21 > \frac{1}{2}ε (left, no blow-up) and ε = 0.17 < \frac{1}{2}ε (right, blow-up at T_{ε} ≈ 3.845).

References

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[4] Constatin, P., Lax, P. D. and A. Majda: A simple one-dimensional model for the three dimensional vorticity equation. Comm. Pure Appl. Math. 38 (1985), 715 – 724.

[5] De Gregorio, S.: On a one-dimensional model for the three-dimensional vorticity equation. J. Stat. Phys. 59 (1990), 1251 – 1263.

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[8] Grauer, R. and T. C. Sideris: Finite time singularities in ideal fluids with swirl. Physica D 88 (1995), 116 – 132.

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[10] Naumkin, P. I. and I. A. Shishmarëv: Nonlinear Nonlocal Equations in the Theory of Waves (Transl. Math. Mon.: Vol. 133). Providence (R.I.): Amer. Math. Soc. 1994.

[11] Schochet, S.: Explicit solutions of the viscous model vorticity equation. Comm. Pure Appl. Math. 39 (1986), 531 – 537.

[12] Stewart, K. and T. Geveci: Numerical experiments with a nonlinear evolution equation which exhibits blow up. App. Num. Math. 10 (1992), 139 – 147.

[13] Vasudeva Murthy, A. S.: The Constantin-Lax-Majda model for the vorticity equation revisited. J. Ind. Inst. Sci. 78 (1988) 109–117.

[14] Wegert, E.: On the global solvability of the diffusion equation of Satsuma and Mimura. Z. Anal. Anw. 9 (1990), 313 – 318.

[15] Whitham, G. B.: Variational methods and applications to water waves. Proc. Royal Soc. London A 299 (1967), 6 – 25.

Received 17.07.1998