Elements of a Qualitative Theory

of Hamiltonian PDEs

Sergei B. Kuksin

We discuss nonlinear Hamiltonian partial differential equations (PDEs) and consider the finite-volume case only. That is, we are concerned with equations for functions (or vector-functions) u(t, x), where the space-variable x belongs to a bounded domain and the equations are supplemented by appropriate boundary conditions. We treat them as ordinary differential equations in infinite-dimensional function spaces formed by functions of x and assume that they can be written in the Hamiltonian form:

(1)  u̇(t) = J∇H(u(t)).

Here J is an anti self-adjoint operator in the space of square-integrable functions, H is a hamiltonian of the equation and ∇H is its L_{2}-gradient (if H is a functional of the calculus of variations, then ∇H equals to its variational derivative). The equation (1) is Hamiltonian with respect to a symplectic structure, defined in the function space by the form α_{2},

α_{2}(ξ(x), η(x)) = ⟨(−J)^{−1}ξ(x), η(x)⟩_{L_{2}}.

Hamiltonian PDEs are of extreme physical importance since they describe processes without dissipation of energy: (usually) the system’s energy equals the hamiltonian H and preserves due to the same trivial arguments as in the finitedimensional case.

Below we discuss three groups of results concerning qualitative behaviour of Hamiltonian PDEs. We have selected them according to our own taste, the references are by no means complete.

1. Nearly integrable PDEs

Some of nonlinear Hamiltonian PDEs with one-dimensional space variable x in a finite segment, supplemented by appropriate boundary conditions, are known to be integrable. For example, the Korteweg - de Vries equation (KdV) under zero-meanvalue periodic boundary conditions:

(KdV)  u̇ = \frac{∂}{∂x}(−u_{xx} + 3u^{2}),  x ∈ S^{1} = ℝ/2πℤ, ∫_{0}^{2π} u(t, x) dx = 0,

and the Sine-Gordon equation (SG) under Dirichlet boundary conditions:

(SG)  ü = u_{xx} − A sin Bu,  u(t, 0) = u(t, π) = 0,

where A, B > 0. We view the equations as dynamical systems in appropriate Sobolev spaces Z^{s}, formed by functions u(x) which respect the boundary conditions. (For the KdV, Z^{s} is the Sobolev space H_{0}^{s}(S^{1}), formed by zero-meanvalue functions on the circle S^{1}. For the SG equation Z^{s} is the Sobolev space formed by odd 2π-periodic functions – these functions vanish for x = 0 and x = π).

Integrability of the KdV equation manifests itself in the following properties of a dynamical system which the equation defines in the spaces Z^{s}, discovered twenty years ago by P.Lax and S.P.Novikov (see [DMN]): For n = 1, 2, … the space ∩Z^{s} contains a smooth 2n-dimensional manifold T^{2n}, invariant for the KdV-flow, such that:

a) restriction of the equation to T^{2n} defines a Liouville-Arnold integrable Hamiltonian system,

b) T^{2n} ⊂ T^{2m} if m > n,

c) union of all manifolds T^{2n} is dense in each space Z^{s}.

For the SG equation everything is much the same but the manifolds T^{2n} have algebraic singularities and their union is only proven to be dense in the vicinity of the origin.

The invariant manifolds T^{2n} are filled with time-quasiperiodic solutions u_{n}(t, x) (so-called n-gap solutions). An n-gap solution u_{n} depends on an n-dimensional action p ∈ ℝ_{+}^{n} and on n-dimensional angle q ∈ 𝕋^{n}: u_{n}(t, x; p, q) = Φ_{n}(W_{p}t+q, x, p) . The function Φ_{n}(q, x, p) is analytic and can be explicitly written in terms of thetafunctions (the Its-Matveev formula, see [DMN]); this is another manifestation of integrability of KdV and SG equations. The n-vector W_{p} is called the frequency vector. The union in q and t of the curves u_{n}(t, ·; p, q) is a smooth invariant n-torus in the space ∩Z^{s}, called the n-gap torus.

1.1. The problem of persistence. Since both KdV and SG equations do not arise in mathematical physics in their exact form (as, for example, Navier - Stokes equations do), but only present simplified forms of some real physical equations, then it is important to understand if the finite-gap solutions u_{n} have something to do with “real” equations. Assuming that a “real” equation is Hamiltonian, that (say) the KdV equation comprises its highest derivatives and that the equation is local^{1} (i.e. it does not contain integral terms), we write it as

(2)  u̇ = \frac{∂}{∂x}(−u_{xx} + 3u^{2} + ε\frac{∂}{∂u}h(u, x)),

where the function h is assumed to be analytic in u.

Elements of a Qualitative Theory of Hamiltonian PDEs 821

1.2. KAM for PDEs. The question we posed in the previous section can be understood in the following way: Does a finite-gap solution u_{n}(t, x) of the KdV equation persist as a time-quasiperiodic solution for equation (2) (i.e., does (2) have a time-quasiperiodic solution u_{n}^{ε}, close to u_{n})? The affirmative answer is given by the following KAM for PDEs theorem: For most (in the sense of measure) values of the action p, the n-gap solution u_{n}(t, x; p, q) for the KdV-equation persists as a time-quasiperiodic solution u_{n}^{ε}(t, x; p, q) for equation (2). The solution u_{n}^{ε} is linearly stable. Its closure in any space Z^{s} forms an invariant smooth n-torus.

1this assumption is imposed only for simplicity

Documenta Mathematica · Extra Volume ICM 1998 · II · 819–829

The persisted solutions u_{n}^{ε} have the form u_{n}^{ε}(t, x; p, q) = Φ_{n}^{ε}(W_{p}^{ε}t+q, x, p). The new frequency vector W_{p}^{ε} is O(ε)-close to W_{p} and the function Φ_{n}^{ε} is O(ε^{ρ})-close to Φ_{n} for any ρ < 1. In particular, for most p the theta-formula for an n-gap solution with the corrected frequency vector gives the function Φ_{n}(W_{p}^{ε}t + q, x, p), which is forever O(ε^{ρ})-close to an exact solution of (2). The corrected frequency vector is W_{p}^{ε} = W_{p} + εW_{p}^{1} + o(ε), where W_{p}^{1} equals to averaging along the corresponding n-gap torus of a vector-function, constructed in terms of a hamiltonian of the perturbation.

A union (in n, p and q) of all persisted solutions, treated as curves in a space Z^{s}, becomes dense in Z^{s} as ε → 0.

Similar results hold for the perturbed SG equation:

(3)  u_{tt} = u_{xx} − A sin Bu + εg(u, x) = 0.

The differences are that, first, large-amplitude solutions both for SG equation and for (3) are not linearly stable and, second, we do not know if the persisted solutions jointly are asymptotically dense as ε → 0.

The KAM-theorem for PDEs is an infinite-dimensional version of the classical finite-dimensional theorem due to Kolmogorov-Arnold-Moser. Essential difference is that in the finite-dimensional case persisted time-quasiperiodic solutions fill the phase-space up to a set of small measure, while in the PDE-case the solutions which persist due to the theorem jointly have zero measure (for any reasonable measure in the corresponding function space).

The theorem we discussed in this section applies to quasilinear perturbations of all “classical” integrable PDEs with one-dimensional space variable, including all equations from the KdV hierarchy, etc. It is based on an abstract infinitedimensional KAM-theorem. For exact statements and proofs see [K1, K3, K4, P1].

It is unknown what happens to infinite-gap solutions (and the corresponding infinite-gap tori) under Hamiltonian perturbations of the integrable equations.

1.3. Small oscillations. The persistence problem posed in section 1.1 admits another understanding: does a small-amplitude finite-gap solution for the KdV or for SG equation persist after we have perturbed the equation by a higher-order at zero term? The affirmative answer follows from the same abstract KAM-theorem which implies the results of the previous section, see [BoK]. In particular, since sin Bu = Bu − B^{3}u^{3}/6 + O(u^{5}), then most of small amplitude finite-gap solutions of the SG equation (with appropriate A and B) persist in the φ^{4}-equation:

(φ^{4})  u_{tt} = u_{xx} − mu + γu^{3}, m, γ > 0.

Small solutions for this (and similar) equations can be also constructed treating (φ^{4}) as a perturbation of another integrable infinite-dimensional system, namely its Birkhoff normal form at zero, see [KP, P2] (also see [W] and the Introduction to [K2] for related results).

Nothing is known about small time-quasiperiodic solutions for the φ^{4}-equation with m = 0.

1.4. Closely related persistence problem arises when we examine an equation (1) with a small nonlinearity and with a linear part with pure imaginary spectrum in order to prove that time-quasiperiodic solutions of the linear equation persist in the nonlinear equation (1). If the linear equation depends on an additional finitedimensional parameter in a non-degenerate way, then any its time-quasiperiodic solution persists in the nonlinear equation for most values of the parameter, provided that: 1) the space-variable x belongs to a finite segment and 2) the perturbed equation is quasi-linear (i.e., the nonlinear term of (1) contains less derivatives than its linear part). – This follows from the same abstract KAM-theorem as above, see [K2, P1].

Some years ago J.Bourgain [B1] developed another KAM-approach, originally proposed by Craig - Wayne in [CW], and successfully used it to study the persistence problem which we discuss in this subsection. The main advantage of this approach is that it applies to two-dimensional (in space) Schrödinger equation. A disadvantage is that it applies only to semilinear equations (i.e. to equations where the nonlinear term contains no derivatives).

We do not know what happens to invariant tori of an n-dimensional (in space) linear Schrödinger equation with n ≥ 3 and of a linear wave equation with n ≥ 2 under Hamiltonian perturbations.

1.5. Averaging theorems. Due to the KAM-results presented in section 1.2, the perturbed KdV equation (2) contains invariant finite-dimensional tori, filled with linearly stable time-quasiperiodic solutions, and union of these tori is asymptotically dense in any space Z^{s} as ε → 0. Hence, any solution of equation (2) with sufficiently small ε for long time stays close to some n-gap torus. This result does not specify the persistence time. For a finite-dimensional nearly-integrable system this time is known to grow at least as exp ε^{−a}, a > 0 (Nekhoroshev’s theorem). To obtain an analogy of this result for equation (2) is an intriguing open problem. What is known is a local theorem which applies to a class of nearly integrable PDEs and states that for solutions of these equations with small analytical initial data the persistence time is bigger than C_{M}ε^{−M} for each M, see [Bam] (also see there references for related results concerning some parameter-depending equations with small nonlinearities).

2. Symplectic invariants and Gromov’s non-squeezing property.

2.1. Gibbs measure. Flow-maps {S_{t}} of any Hamiltonian PDE (1) preserve the symplectic form α_{2} (see the introduction), provided that they are C^{1}-smooth. For a finite-dimensional Hamiltonian system in the space (ℝ_{p,q}^{2N}, dp ∧ dq) symplecticity of the flow-maps of a Hamiltonian vector-field yields that they preserve the Lebesgue measure dpdq as well as the Gibbs measure exp(−H(p, q))dpdq, where H is the hamiltonian. In an infinite-dimensional function space {u(x)} a Lebesgue measure du(·) does not exist, but the Gibbs measure μ_{H} = exp(−H(u(·))) du(·), where H is a hamiltonian of the PDE, often is well defined if dim x = 1 or 2. Its construction is well known from the quantum field theory (see [GJ]). A difficulty is that the measure μ_{H} is supported by a space of functions of low smoothness. To prove invariance of μ_{H}, a flow of the equation (1) has to be proven to exist in the corresponding low-smoothness space and to possess some regular properties. This can be done for many one-dimensional and for some two-dimensional equations, see [B2, B5, MV] and references therein.

It is an open problem whether a non-integrable Hamiltonian PDE has an invariant measure, supported by smooth functions (this measure should not be supported by a trivial invariant set like a periodic trajectory of the equation). This problem is closely related to the following question: is it true that high Sobolev norms of typical solutions for a non-integrable Hamiltonian PDE grow with time unboundedly, see [B3, B4].

2.2. Symplectic capacity. The Gibbs measure μ_{H} corresponds to a subset of the function phase-space of a Hamiltonian PDE a flow-invariant quantity, namely its measure. This is not a unique invariant characteristic of subsets. Existence of another symplectic invariant for finite-dimensional Hamiltonian systems, called symplectic capacity, follows from Gromov’s non-squeezing theorem (or can be constructed independently to prove the theorem), see in [HZ]. To discuss a version of this invariant applicable to (1), we need a notion of a Darboux phase-space Z_{D} for this equation^{2} : Z_{D} is a Hilbert space which admits an orthonormal Hilbert basis {φ_{j} | j ∈ ℤ_{0}} (ℤ_{0} is the set of non-zero integers), which is a Darboux basis for the equation’s symplectic structure, i.e., α_{2}[φ_{j}, φ_{−k}] = δ_{j,k} for any j ∈ ℕ and for all k.

Examples. 1) For the KdV equation (and its perturbation (2)), Z_{D} is the Sobolev space H_{0}^{−1/2}(S^{1}). 2) A nonlinear wave equation

ü − δ△u + mu + f(u, x) = 0, u = u(t, x), x ∈ 𝕋^{n},

where m > 0 and f is a smooth function, can be written in the following Hamiltonian form:

(4)  u̇ = −Lw, ẇ = Lu + L^{−1}f(u, x),

where L = (−δ△ + m)^{1/2}. For this equation Z_{D} = Z^{1/2} = H^{1/2}(𝕋^{n}) × H^{1/2}(𝕋^{n}) (see [K5, K6]). 3) If f = 0 (so the equation (4) is linear), then any space Z^{s} is a Darboux space. (On the contrary, for a typical nonlinear equation (4) a space Z^{s} with s ≥ 5 is not Darboux. It is plausible that Z^{1/2} is the unique Darboux space).

Let Z_{D} be a Darboux space for a symplectic form α_{2} and {φ_{j} | j ∈ ℤ_{0}} be its basis as above. A map c which corresponds to an open subset O ⊂ Z_{D} a number c(O) ∈ [0, ∞] is called a (symplectic) capacity if

2in fact, for its symplectic structure.

Documenta Mathematica · Extra Volume ICM 1998 · II · 819–829

α) c is translationary invariant, i.e., c(O) = c(O + ξ) for ξ ∈ Z_{D}; β) c is monotonic, i.e. a bigger set has a bigger capacity; γ) c is 2-homogeneous, i.e. c(τO) = τ^{2}c(O); δ) c(B_{r}) = c(Π_{r}^{(k)}) = πr^{2}, where B_{r} is the r-ball in Z_{D}, centered at the origin, and Π_{r}^{(k)} is the cylinder formed by all vectors ∑ z_{l}φ_{l} such that z_{k}^{2} + z_{−k}^{2} ≤ r^{2}.

A finite-dimensional symplectic space (ℝ_{p,q}^{2n}, dp ∧ dq) admits a symplectic capacity, invariant for symplectomorphisms [HZ]. A Darboux space Z_{D} also admits one. This capacity is invariant for flow-maps {S_{t}} of a Hamiltonian equation (1), provided that

(5)  S_{t} = linear operator + compact smooth operator,

where the linear operator is a direct sum of rotations in the planes, spanned by the vectors φ_{j} and φ_{−j}, j = 1, 2, … (see [K5]).

The assumption (5) is met by the nonlinear wave equation (4) if n = 1 and f(u, x) has a polynomial growth in u, or n = 2, 3 and f as a function of u is a polynomial of a sufficiently low degree, see [K5,K6] and [B4].

The symplectic capacity is an invariant of the flow of a Hamiltonian PDE in a function space of low smoothness, as well as the Gibbs measure. An essential difference between these two invariants is that the former is constructed in terms of the equation’s symplectic structure, while the latter – in terms of its hamiltonian (the same is true for the corresponding function spaces, so usually they are different).

An immediate consequence of existence of a symplectic capacity is that the flow-maps {S_{t}}, satisfying (5), can not squeeze a ball in a Darboux space Z_{D} to a cylinder of a smaller radius^{3}; cf. the properties α), β) and δ). This is Gromov’s non-squeezing property.

On the contrary, the squeezing (and a closely related pulling-through phenomenon, see below) both are possible (and are typical under some circumstances) if we consider the equation in a function space of high smoothness, i.e. study its classical solutions rather than generalised ones. In particular, the flow {S_{t}} of equation (4) in a Sobolev space Z^{s}, s ≥ 5, squeezes a typical ball of a radius of order one to a cylinder Π_{ρ}^{(k)} with ρ ∼ (ln δ^{−1})^{−1}, provided that the nonlinearity f is also typical,^{4} see [K6].

3. Small-dispersion/dissipation equations

Let us consider the following class of PDEs:

⟨non-linear homogeneous Hamiltonian equation⟩ + ⟨δ_{1}-small linear damping⟩

(6) + ⟨δ_{2}-small linear dispertion⟩ = ζ(t, x),

where δ_{1} ≥ 0, δ_{2} ≥ 0 and δ := \sqrt{δ_{1}^{2} + δ_{2}^{2}} > 0. If δ_{1} = 0, then this equation is Hamiltonian. Still, the most important are equations with δ_{1} > 0 since they describe turbulence in different physical media.

3It is unknown if the assumption (5) is superfluous and can be dropped.

4Clearly, ρ ≪ 1 if δ ≪ 1. So Gromov’s property fails in this space.

Documenta Mathematica · Extra Volume ICM 1998 · II · 819–829

The Navier-Stokes (NS) equations have the form (6) with the Euler equations for the homogeneous Hamiltonian equation and with δ_{1} > 0, δ_{2} = 0. Another good example of equation (6) is given by the damped/driven nonlinear Schrödinger equation:

(7)  u̇ − δ_{1}△u + iδ_{2}△u − i|u|^{2p}u = ζ(t, x), p ∈ ℕ, δ_{1}, δ_{2} ≥ 0,

which we shall consider for x ∈ ℝ^{n}, n ≤ 3, under the odd periodic boundary conditions:

u(t, x) = u(t, x_{1}, …, x_{j} + 2, …) = −u(t, x_{1}, …, −x_{j}, …) ∀j

(they imply that u(t, x) vanishes at the boundary of the cube of half-periods {0 ≤ x_{j} ≤ 1}). It is known that (7) has a unique smooth in x solution for any smooth odd periodic initial data u(0, x) = u_{0}(x) (and for any continuous in t, smooth odd periodic in x function ζ). We shall discuss qualitative behaviour of solutions for equation (7) in the turbulent limit, i.e. when δ ≪ 1. We shall state results for equation (7), using some terminology which comes from the hydrodynamical turbulence, i.e. from the NS equations.

3.1. Essential part of a phase-space. Let us first consider equation (7) with ζ = 0, supplemented by an order-one initial condition

(8)  u|_{t=0} = u_{0}(x) ∈ C^{∞}, |u_{0}|_{L_{∞}} = U,

U ∼ 1. Due to a trivial a priori estimate, L_{2}-norm in x of a solution u decays with t at least as exp − δ_{1}t. Hence, the solution practically vanishes by a time ≫ δ_{1}^{−1}. We are interested in its behaviour for 0 ≤ t ≤ δ^{−a} with 0 < a ≤ 1.

Denoting by |u|_{m} the C^{m}-norm of a function u(x), we define the essential part of the smooth phase-space of equation (7)|_{ζ=0} (with respect to the C^{m}-norm, m ≥ 2) as

𝔄_{m} = {u(x) ∈ C^{∞} | u is odd periodic and |u|_{0}^{2mκ+1} < K_{m}δ^{mκ}|u|_{m}}.

Here κ is any fixed number < 1/3 and K_{m} = K_{m}(κ) is some specific constant. This set is formed by fast oscillating functions since |u|_{m} ≫ |u|_{0} for any u ∈ 𝔄_{m} if ∥u∥_{0} ≳ 1 (when δ ≪ 1). The set looks like a narrow tube with respect to the C^{m}-norm since its intersection with a ball {|u|_{m} ≤ R} is contained in the narrow cylinder Π_{ρ}^{(k)}, formed by complex functions u = ∑ u_{s}e^{πis·x} such that |u_{k}| < ρ, where ρ = C_{m}δ^{1/2+O(m^{−1})}R^{O(m^{−1})}.

The set 𝔄_{m} is important to understand dynamics of the equation (7) since: by the time C_{m}δ^{−1} the flow of equation (7)|_{ζ=0} will pull the whole space of smooth odd periodic functions through 𝔄_{m}. This pull-through phenomenon can be specified: a solution u for (7)_{ζ=0}, (8) will visit the set 𝔄_{m} by the time δ^{−1/3}U^{−4/3}. By the moment of a first entry to 𝔄_{m} the solution will change its supremum-norm no more than twice.

Hence, by the time δ^{−1/3} any solution u(t, x) for (7)_{ζ=0}, (8)_{U=1} will make its C^{m}-norm as big as C_{m}δ^{−mκ}.

Equation (7) with ζ = 0, δ_{1} = 0 takes the Hamiltonian form

(7′)  u̇ + iδ△u − i|u|^{2p}u = 0.

It has two integrals of motion: the hamiltonian and the L_{2}-norm |u(t, ·)|_{L_{2}}. Since |u(t, ·)|_{L_{∞}} ≥ |u(t, ·)|_{L_{2}} = const, then any non-zero solution for (7′) will visit 𝔄_{m} during any time-interval longer than δ^{−1/3}|u|_{L_{2}}^{−4/3}. I.e., 𝔄_{m} is a recursion subset for this Hamiltonian PDE.

3.2. Bounds for averaged high norms. It turns out that since C^{m}-norms of a solution for (7)_{ζ=0}, (8)_{U=1} become big at least once, then they are big at the average; hence, its Sobolev norms are big at the average as well:

(9)  δ^{a} ∫_{0}^{δ^{−a}} ∥u(t, ·)∥_{m}^{2}dt ≥ C_{m}δ^{−2mκ_{m}}.

Here a ≥ 1/3, κ_{m} = κ_{m}(a) ↗ 1/3 and ∥ · ∥_{m} stands for the norm in the Sobolev space of odd periodic functions. This estimate is essentially nonlinear since it obviously fails if p = 0.

The norms of the solution u satisfy usual upper estimates: if δ_{2} = 0 and ζ = 0, then

(10)  δ_{a} ∫_{0}^{δ^{−a}} ∥u(t, ·)∥_{m}^{2}dt ≤ C′_{m}δ^{−m},

where the constants C′_{m} depend on C^{m}-norms of the initial condition u_{0}. We stress that the exponents for δ in the r.h.s.’s of (9) and (10) are universal: they do not depend on the nonlinearity |u|^{2p}u, the dimension n and the initial condition u_{0}.

Estimates similar to (9), (10) remain true for solutions of equation (7) with non-zero forcing ζ if we assume that ζ = ζ^{ω}(t, x) is a random field, smooth odd periodic in x and stationary in t (such equations are believed to present right mathematical description of physical turbulence, see in [EKMS, K8]): If u^{ω}(t, x) is a solution for (7) with, say, zero initial condition at t = 0 and ⟨∥u∥_{m}^{2}⟩ is its averaged squared Sobolev norm, ⟨∥u∥_{m}^{2}⟩ = δ^{a} ∫_{0}^{δ^{−a}} E∥u(t, ·)∥_{m}^{2}dt, then

(11)  C_{m}^{−1}δ^{−2mν_{m}} ≤ ⟨∥u∥_{m}^{2}⟩ ≤ C_{m}δ^{−2mμ_{m}} if a ≥ 1,

where μ_{m} ↗ B < ∞ and ν_{m} ↗ A > 0. Moreover, we know that \frac{3}{17} < A, B ≤ \frac{3}{2} and that (11) remains true if in the definition of ⟨∥u∥_{m}^{2}⟩ we replace the time-segment [0, δ^{−a}] by any segment in [0, ∞), longer than δ^{−a}.

A popular mathematical idealisation of the physically correct forcings ζ as above is given by a random field ζ which is white noise in time [EKMS]. For forcings like that the estimates (11) hold with A = \frac{1}{2}, B = 1.

An important feature of turbulent behaviour of a solution u_{δ}^{ω}(t, x) is a short size of its space-scale l_{x} (see e.g. [LL], § 33 and [CDT]). Defining the space-scale as l_{x} = δ^{γ}, where

γ = γ(u^{ω}) = lim inf_{m→∞} lim inf_{δ→0} \frac{ln⟨∥u_{δ}∥_{m}^{2}⟩^{1/2m}}{ln δ^{−1}}

(see [K8]), we get from (11) that A ≤ γ ≤ B.

Documenta Mathematica · Extra Volume ICM 1998 · II · 819–829

3.3. Asymptotical spectral properties of solutions and the Kolmogorov - Obukhov law. The estimates for the space-scale l_{x} of a solution u^{ω}(t, x), discussed above, characterise its infinitesimal in x behaviour. Arguments of Tauberian kind transform these estimates to information on asymptotical as s → ∞ behaviour of Fourier coefficients û_{s}^{ω}(t) of the solution.^{5} To present it we denote by E_{s} the averaged squared Fourier coefficient û_{s}, E_{s} = δ^{a} ∫_{0}^{δ^{−a}} E|û_{s}^{ω}(t)|^{2} dt. (We remark that if u^{ω} was a space-periodic solution for the NS equations, then E_{s} would be the energy of the fluid, corresponding to the wave vector s).

The numbers E_{s} obey the following asymptotic, which hold for any ε > 0 with A, B and γ as in the previous section:

1. E_{s} = o(|s|^{−M}) for |s| ≥ δ^{−B−ε} with every M, if δ is sufficiently small. If |s| ≥ δ^{−γ−ε}, then the same holds true for all δ from an appropriate sequence {δ_{j} ↘ 0}.

2. There exist c(ε) and C(ε) such that

δ^{c} ≤ |A_{ε}|^{−1} ∑_{s∈A_{ε}} E_{s} ≤ δ^{C},

where A_{ε} = {δ^{−A+ε} ≤ |s| ≤ δ^{−B−ε}}, for all small δ. The same holds true for the smaller set A_{ε} = {δ^{−γ+ε} ≤ |s| ≤ δ^{−γ−ε}} with appropriate exponents c(ε) and C(ε), for all δ from a sequence {δ_{j} ↘ 0}.

The heuristic Kolmogorov - Obukhov (K-O) law (see [LL], § 33) states that the energy E_{s} of a wave-vector s is o(|s|^{−M}) for every M if |s| > δ^{−γ^{K}}, and

\frac{1}{C} ∑_{r≤|s|≤r+C} E_{s} ∼ const · r^{θ} for δ^{−γ^{0}} < r < δ^{−γ^{K}} .

The inverse threshold δ^{γ^{K}} is called Kolmogorov’s inner scale of the turbulent flow. For 3-dimensional NS equations the exponent θ = 5/3 and γ^{K} = 3/4, see [LL].

The properties 1 and 2 of a solution u^{ω}(t, x) for (7) present a weak form of the K-O law. In particular, if any solution u^{ω} for (7) satisfies the K-O law, then γ^{K} must equal the exponent γ(u^{ω}). Consequently, γ^{K} must meet the estimate A ≤ γ^{K} ≤ B. It is curious to note that for the forcing ζ^{ω}(t, x) which is white noise in time, the results of section 3.2 imply the bounds \frac{1}{2} ≤ γ^{K} ≤ 1 which remarkably agree with the value γ^{K} = 3/4, prescribed by K-O for the 3-dimensional hydrodynamic turbulence.

The property 1 shows that the Fourier modes û_{s}^{ω}e^{πis·x} with |s| > δ^{−γ} can be ignored when a solution u is calculated numerically, while the modes with |s| < δ^{−γ} are essential. Hence, a numerical scheme to calculate u has to have dimension of order δ^{−2γn}. This is a very big number since δ corresponding to a turbulent regime is very small (for turbulence in water and in air it is as small as 10^{−7} − 10^{−4}). – This is why it is so difficult to study turbulence numerically.

Proofs of the results presented in sections 3.1-3.3 see in [K7, K8]. See [EKMS] for the turbulence-limit δ → 0 in a randomly forced Burgers equation.

5We write u^{ω}(t, x) as ∑_{s∈ℤ^{n}} û_{s}^{ω}(t)e^{πis·x}.

Documenta Mathematica · Extra Volume ICM 1998 · II · 819–829

References

[B1] Bourgain J., Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Annals of Mathematics 145 (1998).

[B2] Bourgain J., Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), 1-26.

[B3] Bourgain J., Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations, Geometric and Functional Analysis 5 (1995), 105-140.

[B4] Bourgain J., On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Mathematics Research Notes (1996), 277-304.

[B5] Bourgain J., Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), 421-445.

[Bam] Bambusi D., Nekhoroshev theorem for small amplitude solutions in NLS equations, Math. Z. (to appear).

[BoK] Bobenko A.I, Kuksin S.B., The nonlinear Klein-Gordon equation on an interval as a perturbed Sine-Gordon equation, Comment. Math. Helv. 70 (1995), 63-112.

[CDT] Constantin P., Doering Ch., Titi E., Rigorous estimates of small scales in turbulent flow, J. of Math. Physics 37 (1996), 6142-6156.

[CW] Craig W., Wayne C.E., Newton’s method and periodic solutions of nonlinear wave equation, Comm. Pure. Appl. Math. 46 (1993), 1409–1501.

[DMN] Dubrovin B.A., Matveev V.F., Novikov S.P., Nonlinear equations of Korteweg-de Vries type, finite zone linear operators, and Abelian varieties, Uspekhi Mat. Nauk 31:1 (1976), 55-136; English transl. in Russ. Math. Surv. 31:1 (1976).

[EKMS] E W., Khanin K., Mazel A., Sinai Ya., Invariant measures for 1-D Burgers equation with stochastic forcing (to appear).

[GJ] Glimm J., Jaffe A., Quantum physics: a functional integral point of view, 2nd ed., Springer, 1987.

[HZ] Hofer H., Zehnder E., Symplectic invariants and Hamiltonian dynamics, Birkhäuser, 1994.

[K1] Kuksin S.B., Perturbation theory for quasiperiodic solutions of infinitedimensional Hamiltonian systems, and its applications to the Kortewegde Vries equation, Matem. Sbornik 136(178):3 (1988); English transl in Math. USSR Sbornik 64 (1989), 397-413.

[K2] Kuksin S.B., Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math. 1556 (1993), Springer.

[K3] Kuksin S.B., A KAM-theorem for equations of the Korteweg - de Vries type, Rev. Math. & Math. Phys. 10:3 (1998), 1-64.

[K4] Kuksin S.B., Analysis of Hamiltonian PDEs, Preprint of a book to be accepted by Oxford University Press.

[K5] Kuksin S.B., Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs, Comm. Math. Phys. 167 (1995), 531-552.

[K6] Kuksin S. B., On squeezing and flow of energy for nonlinear wave equations, Geometric and Functional Analysis 5 (1995), 668-701.

[K7] Kuksin S. B., On turbulence in nonlinear Schrödinger equations, Geometric and Functional Analysis 7 (1997), 783-822.

[K8] Kuksin S. B., Spectral properties of solutions for nonlinear PDEs in the turbulent regime, Geometric and Functional Analysis (to appear).

[KP] Kuksin S.B., Pöschel J., Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Annals of Mathematics 143 (1996), 149-179.

[MV] McKean H., Vaninsky K., Statistical mechanics of nonlinear wave equations, in the “Trends and perspectives in applied mathematics”, collection of articles dedicated to Fritz John, 1994.

[P1] Pöschel J., A KAM-theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15 23 (1996), 119-148.

[P2] Pöschel J., Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici 71 (1996), 269-296.

[W] Wayne, C.E., Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys. 127 (1990), 479-528.

Sergei B. Kuksin

Department of Mathematics Heriot-Watt University Riccarton EH14 4AS Edinburgh. UK and Steklov Institute Moscow S.B.Kuksin@ma.hw.ac.uk 830

Documenta Mathematica · Extra Volume ICM 1998 · II ·