This note presents an infinite-dimensional family of exact solutions of the incompressible three-dimensional Euler equations

Transcription

1 IMRN International Mathematics Research Notices 2000, No. 9 The Euler Equations and Nonlocal Conservative Riccati Equations Peter Constantin This note presents an infinite-dimensional family of exact solutions of the incompressible three-dimensional Euler equations t u + u u + p = 0, u = 0. ) The solutions we present have infinite kinetic energy and blow-up in finite time. Blow-up of other similar infinite energy solutions of Euler equations has been proved before in [7], [2]. The particular type of solution we describe was proposed in [4]. The Eulerian- Lagrangian approach we take in [3] is not restricted to this particular case, but exact integration of the equations is. We consider a two-dimensional basic square of side L. The particular form of the solutions in [4] is ux, y, z, t) = ux, y, t),zγx, y, t) ), 2) where the scalar valued function γ is periodic in both spatial variables with period L, and the two-dimensional vector ux, y, t) =u x, y, t),u 2 x, y, t)) is also periodic with the same period. The associated two-dimensional curl is ωx, y, t) = u 2x, y, t) x u x, y, t). 3) y This represents the vertical third) component of the vorticity u of the Euler system, Received 5 November 999. Revision received 7 December 999. Communicated by Percy Deift.

2 456 Peter Constantin and, using the ansatz 2), it follows from the familiar three-dimensional vorticity equation that the equation t ω + u ω = γω 4) should be satisfied for the recipe to succeed. On the other hand, one can easily check that the vertical component of the velocity zγx, y, t) solves the vertical component of the Euler equations if γ solves the nonlocal Riccati equation γ t + u γ = γ2 + It), 5) where It) is a function that depends on time only. The divergence-free condition for u becomes u = γ. Because of the spatial periodicity of u, we must make sure that γx, t) dx = 0 7) holds throughout the evolution. This can be done provided the function It) is given by where It) = 2 γ 2 x, t) dx, 8) = dx = L 2. The velocity is determined from ω and γ using a stream function ψx, y, t) and a potential hx, y, t) by 6) u = ψ + h, 9) h = γ, ψ = ω, 0) ) with periodic boundary conditions. The ansatz ux, y, z, t) =ux, y, t),zγx, y, t)) associates to solutions of the system 4), 5), 8), 9), 0), and ) in d = 2 velocities u that obey the incompressible

3 Nonlocal Riccati 457 three-dimensional Euler equations see [4], [6]). The divergence condition 6) follows from 9) and 0). The compatibility condition ωdx = 0 is maintained throughout the evolution because of 4) and 6). We consider initial data γx, y, 0) =γ 0 x, y), ωx, y, 0) =ω 0 x, y) 2) that are smooth and have mean zero γ 0dx = ω 0dx = 0. The solutions of the system above have local existence, and the velocity is as smooth as the initial data are, as long as T 0 sup x γx, t) dt is finite the result can be proved in [5] following the idea of the proof of the well-known result in []). We consider the characteristics dx dt = ux, t), 3) and denoting Xa, t) the characteristic that starts at t = 0 from a, Xa, 0) =a, we note that, prior to blow-up, the map a Xa, t) is one-to-one and onto as a map from T 2 = R 2 /LZ 2 to itself. The injectivity follows from the uniqueness of solutions of ordinary differential equations. The surjectivity can be proved by reversing time on characteristics, which can be done as long as the velocity is smooth. Our result is an explicit formula for γ on characteristics { } γ 0 Ax, t)) γx, t) =ατt)) + τt)γ 0 Ax, t)) φτt)), 4) where Ax, t) is the inverse of Xa, t) the back-to-labels map), and the functions τt), ατ), and φτ) are computed from the initial datum γ 0. More precisely, we show the following theorem. Theorem. Consider the nonlocal conservative Riccati system see 4), 5), 8), 9), 0), and )). There exist smooth, mean zero initial data for which the solution becomes infinite in finite time. Both the maximum and the minimum values of the solution γ diverge to plus infinity and, respectively, to negative infinity at the blow-up time. There is no initial datum for which only the minimum diverges. The general solution is given on characteristics in terms of the initial data γx, 0) =γ 0 x) by see 24)) ) γ 0 a) γxa, t),t)=ατt)) + τt)γ 0 a) φτt)), where { φτ) = }{ } γ 0 a) + τγ 0 a)) 2 da + τγ 0 a) da,

4 458 Peter Constantin and ατ) = { } 2 + τγ 0 a) da, dτ = ατ), τ0) =0. dt The function τt) can be obtained from t = ) 2 γ 0 a) γ 0 b) log The Jacobian Ja, t) =Det { Xa, t)/ a} is given by Ja, t) = The moments of γ are given by ) + τγ0 a) da db. + τγ 0 b) { } da. + τt)γ 0 a) + τt)γ 0 a) γx, t)) p dx =ατ)) p The blow-up time t = T is given by where T = 2 γ 0 a) γ 0 b) log { } p γ 0 a) + τt)γ 0 a) φτt)) Ja, t) da. γ0 a) m 0 γ 0 b) m 0 ) da db, m 0 = min γ 0a) <0. We note that the curl ω plays a secondary role in this calculation and in the blow-up. Indeed, the same formula and blow-up occurs if ω 0 = 0, or if the curl ω was smooth and computed in a different fashion than via 4). Solving on characteristics We now solve the nonlocal Riccati equation on characteristics. We start with an auxiliary problem. Let φ solve τ φ + v φ = φ 2

5 Nonlocal Riccati 459 together with vx, τ) = φx, τ)+ φx, τ) dx. We consider initial data that are smooth, periodic, and have zero mean, φ 0 x) dx = 0. We also assume that the curl ζ = v 2 / x) v / y) obeys τ ζ + v ζ = Passing to characteristics φ 3 ) φx, τ) dx ζ. dy = vy, τ), 5) dτ we integrate and obtain φ Ya, τ),τ ) = which is valid as long as φ 0 a) + τφ 0 a), inf + τφ0 a) ) >0. a We need to compute The Jacobian φτ) = φx, τ) dx. Ja, τ) =Det { } Y a obeys dj = ha, τ)ja, τ), dτ where ha, τ) =φya, τ), τ) φτ).

6 460 Peter Constantin Initially, the Jacobian equals, so Ja, τ) =e τ 0 ha,s) ds. Then Ja, τ) =e τ τ 0 φs) ds d exp 0 ds log + sφ 0 a) ) ) ds and thus Ja, τ) =e τ 0 φs) ds + τφ 0 a). The map a Ya, τ) is one-to-one and onto. The change of variables formula gives and, therefore, Consequently, φx, τ) dx = φya, τ), t)ja, τ) da, φτ) =e τ φs) ds φ 0 a) 0 da. 6) + τφ 0 a)) 2 d τ dτ e 0 φs) ds = d dτ + τφ 0 a) da. Because both sides at τ = 0 equal, we have e τ 0 φs) ds = da 7) + τφ 0 a) and, using 6), { φτ) = }{ } φ 0 a) + τφ 0 a)) 2 da + τφ 0 a) da. 8) Note that the function δx, τ) =φx, τ) φτ) obeys δ τ + v δ = δ2 + 2 δ 2 dx 2φδ.

7 Nonlocal Riccati 46 We now consider the function σx, τ) =e 2 τ 0 φs) ds δx, τ) and the velocity Ux, τ) =e 2 τ 0 φs) ds vx, τ). Multiplying the equation of δ by e 4 τ 0 φs)ds, we obtain Note that e 2 τ φs)ds σ 0 τ + U σ = σ2 + 2 σ 2 dx. U = σ. Now we change the time scale. We define a new time t by the equation dt dτ = τ e 2 0 φs) ds, 9) t0) =0, and new variables and γx, t) =σx, τ) ux, t) =Ux, τ). Now γ solves the nonlocal conservative Riccati equation γ t + u γ = γ2 + 2 γ 2 dx 20) with periodic boundary conditions, u = ) [ ω + γ ] 2) and ω t + u ω = γω. 22)

8 462 Peter Constantin The initial data are given by γ 0 x) =δ 0 x) =φ 0 x). Using 7) and integrating equation 9), we see that the time change is given by the formula t = ) 2 φ 0 a) φ 0 b) log + τφ 0a) da db. 23) + τφ 0 b) Note that the characteristic system dx dt is solved by = ux, t) Xa, t) =Ya, τ), where Y solves the system 5). This implies the formula with ) φ0 a) γxa, t), t)=ατ) + τφ 0 a) φτ) 24) ατ) =e 2 τ 0 φs) ds. 25) In view of 7), 8), and 23), we have obtained a complete description of the general solution in terms of the initial data. 2 Blow-up Consider an initial smooth function γ 0 a) =φ 0 a) and assume that it has mean zero and that its minimum is m 0 <0. As it is evident from the explicit formula, the blow-up time for φya, τ),τ) is τ = m 0. It is also clear that φya, τ),τ) diverges to negative infinity for some a, and not at all for others. This of course does not necessarily mean that γ blows up in the same fashion. Let

9 Nonlocal Riccati 463 us discuss the simplest case, in which the minimum is attained at a finite number of locations a 0, and near these locations, the function φ 0 has nonvanishing second derivatives, so that locally φ 0 a) m 0 + C a a 0 2 for 0 a a 0 r. Then it follows that the integral da ɛ 2 + φ 0 a) m 0 behaves like da ɛ 2 log + + φ 0 a) m 0 for small ɛ. Taking ɛ 2 = τ τ, we deduce that, for these kinds of initial data, { } e τ 0 φs) ds log + C. τ τ ) 2 Cr ɛ, For the same kind of functions and small τ τ), the integral φ 0 a) + τφ 0 a)) 2 da C τ τ and tτ) has a finite limit t T as τ τ. The average φτ) diverges to negative infinity, [ { }] φτ) C log + C. τ τ τ τ The prefactor α becomes vanishingly small ατ) logτ τ) ) 2, and 24) becomes ) γxa, t),t) logτ τ)) 2 φ0 a) + τφ 0 a) φτ).

10 464 Peter Constantin If the label is chosen so that φ 0 a) >0,then the first term in the brackets does not blow-up and γ diverges to plus infinity. If the label is chosen at the minimum, or nearby, then the first term in the brackets dominates and the blow-up is to negative infinity, as expected from the ordinary differential equation. From equation 9) ατ)) dτ = dt, it follows that )) 2 T t τ τ) + log, τ τ and the asymptotic behavior of the blow-up in t follows from the one in τ. We end by addressing a question that was at some point raised by numerical simulations: Can there be a one-sided blow-up? From the representation in 24) of the solution, it follows that Mt) φτ)e 2 τ φs) ds 0 holds for Mt) =sup x γx, t). If we assume that, up to the putative blow-up Mt) C with some fixed constant C, then it follows that d dτ e2 τ 0 φs) ds 2C, and, integrating between τ and τ, that e 2 τ 0 φs) ds 2Cτ τ). This in turn would imply that T =, and therefore, no blow-up for γ can occur in finite t. So the answer is that for no initial datum can there exist a one-sided blow-up for γ. Acknowledgments I thank J. Gibbon and K. Ohkitani for showing me their numerical work prior to its publication. This work was partially supported by National Science Foundation grant number DMS and by the Accelerated Strategic Computing Initiative Flash Center at the University of Chicago, contract number B34495.

11 Nonlocal Riccati 465 References [] J. T. Beale,T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys ), [2] S. Childress, G. R. Ierley, E. A. Spiegel, and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech ), 22. [3] P. Constantin, An Eulerian-Lagrangian approach to fluids, preprint, 999. [4] J. D. Gibbon, A. S. Fokas, and C.R. Doering, Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations, Phys. D ), [5] S. Malham, J. Gibbon, and K. Ohkitani, preprint, [6] K. Ohkitani and J. Gibbon, personal communication, November 999. [7] J. T. Stuart, Nonlinear Euler partial differential equations: Singularities in their solution in Applied Mathematics, Fluid Mechanics, Astrophysics Cambridge, MA, 987), World Sci., Singapore, 988, Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA