A posteriori KAM theorems withapplications to ill-posed PDEs

Yannick Sire

Johns Hopkins University, USA

Joint work with Rafael de la Llave

Goals and overview

1 KAM for tori with hyperbolic directions to be applied toHamiltonian PDEs

2 KAM in an a posteriori format: no need to be close tointegrable (no use of action-angle variables) and validationof approximations of quasi-periodic solutions

3 KAM for PDEs in particular "ill-posed" PDES4 Example: Boussinesq equation (theory of water waves,

long wave approximation)

utt = µuxxxx + uxx + (u2)xx on T

where µ > 0 and t ∈ R.

Construct infinite dimensional analogues of Whiskered tori.

Several proofs of whiskered tori in finite dimensionsassuming close to integrable and using transformationtheory. (Moser, Graff, Zehnder).

For PDE’s the notion of integrability is very subtle.Equations with strong instability are usually not integrable.(We are interested in equations that do not define anevolution).Transformation theory is hard – but not impossible – togeneralize to PDE, specialy ill posed ones. (see forexample the work of Kuksin, Eliasson, Pöschel, X.P. Yuan,Si, H.Y. Cheng)

Proofs without transformation theory and working far fromintegrable have been done in finite dimensional systems:E. Fontich, R. de la Llave, Y. S. JDE (2013)or in infinite dimensional onesE. Fontich, R. de la Llave, Y. S. JDE (2015)

The Boussinesq equation: Hamiltonian structure and linear theory

utt = µuxxxx + uxx + (u2)xx on T, µ > 0, t ∈ R

Other interesting examples are:Boussinesq system of water waves

∂t

(uv

)=

(0 −∂x − µ∂xxx−∂x 0

)(uv

)+

(∂x (uv)

0

)(1)

Elliptic equations in cylindrical domains (considered beforeby Kirgchassner, Mielke)Complex Ginzburg Landau (and derivative CGL) for somevalues of the parameters (considered before by X. Yuan,G. Si, H.Y Chen).

∂tu = u + (1 + iµ)∆u − (1 + iα)|u|2u

Hamiltonian structure:

z = J∇Hµ(z), z = (u, v = ut )

J =

(0 ∂x∂x 0

)

Hµ(u, v) =

∫ 1

0

12

{u2 + v2 − µ(∂xu)2

}+

13

u3.

Linear theoryu(x , t) = e2πi(kx+ω(k)t)

ω2(k) = −µ|k |4(2π)2 + |k |2

k large: ω(k) ≈ ±iµ1/2|k |2 (hence ill-posedness of the Cauchyproblem)

Formulation of the abstract problem

We consider an evolutionary PDE, which we write symbolically,

dudt

= X ◦ u (2)

X (u) = Au +N (u) (3)

where A is linear and N is a nonlinear.X ,A,N unbounded differential operators.(N will be of lower order than A)

We will consider two spaces

X a space of smooth functionsY a space of rough functions.

We will assumeN is analytic from X to YDN (0) = 0.

The Theorem: notations and crucial quantities

Diophantine numbers Given κ > 0 and ν ≥ `− 1, we defineD(κ, ν) as the set of frequency vectors ω ∈ R` satisfying theDiophantine condition:

|ω · k |−1 ≤ κ|k |ν , for all k ∈ N` − {0}

where |k | = |k1|+ ...+ |k`|.Parametrization method: the embedding K

X ◦ K = DK · ω (4)

z(t) = K (ω t + θ), K : Dρ → X

Dρ ={

z ∈ C`/Z` : |Im zi | < ρ i = 1, ..., `}.

The linearized equation

Given a smooth curve ξ(t) in X – not necessarily a solution –we can define a linearized evolution for ξ.

dudt

= DX ◦ ξ(t) u(t) ≡ Au + DN (ξ(t))u(t) (5)

In particular, for a curve parameterized by a hull function:

dudt

= DX ◦ K0(θ + ωt)u (6)

A particular case, which will be important is the case K0 ≡ 0. Inthis case, we just need to study properties of A.

Hyperbolicity assumptions

Our hyperbolicity assumption is a generalization of the usualtrichotomy assumption in finite dimensions.There are some directions that contract in the future, somedirections that contract in the past, and some neutral ones(which are finite dimensional).We assume that projecting in the contracting directions in thefuture (resp. in the past) and evolving in the future (resp. in thepast) smooths the functions. This is formulated rigorously assaying that the evolution operators send the spaces of roughfunctions into the space of smooth functions, with good bounds(depending on time).

This structure is persistent under perturbations of lower order –even if the perturbations are unbounded

We will be considering analytic functions of the T` into spaces.We will denote

||f ||ρ = sup|=(θ)|<ρ

||f (θ)||

(and consider the Banach space of functions which extendcontinuously to the boundary.)

Non-degeneracy conditions: Spectral conditions

An embedding K is spectrally non degenerate if for every θ inR`, there is a splitting (with bounded analytic (in θ)) projectionsΠs,c,u)

X = X sθ ⊕ X c

θ ⊕ X uθ

in such a way that:dim X c

θ = 2` and J|X cθ

induces a symplectic form on X cθ .

There are families of operators

Usθ (t) : Y s

θ → X sθ+ωt t ≥ 0

Uuθ (t) : Y u

θ → X uθ+ωt t ≥ 0

Usθ (t) : Y c

θ → X cθ+ωt t ∈ R

such that

Us,c,uθ+ωt ′(t)U

s,c,uθ (t ′) = Us,c,u

θ (t + t ′)

There exist α1, α2 ∈ [0,1), β1, β2, β3 > 0 and Ch > 0independent of θ such that

‖Usθ (t)||ρ,Y ,X ≤ Che−β1t t−α1 , t > 0, (7)

‖Uuθ (t)‖ρ,Y ,X ≤ Cheβ2t |t |−α2 , t < 0, (8)

‖Ucθ (t)‖ρ,X ,X ≤ Cheβ3|t |, t ∈ R. (9)

and the operators Us,u,cθ are fundamental solutions of the

variational equations

ddt

Usθ (t) = DX (θ + ωt)Us

θ (t) t > 0

ddt

Uuθ (t) = DX (θ + ωt)Uu

θ (t) t < 0

ddt

Ucθ (t) = DX (θ + ωt)Uc

θ (t) t ∈ R

(10)

Non-degeneracy: twist condition

Denote N(θ) the `× ` matrix such that N(θ)−1 = DK (θ)⊥DK (θ)Denote P(θ) = DK (θ)N(θ)Let Jc stand for restriction of symplectic operator J to X c

θ .We now define the twist `× `-matrix S(θ)

S(θ) = N(θ)DK (θ)⊥[J−1c ∂ω(DK N)− D(J∇ ◦ K )J−1

c (DK N)](θ)

Twist condition: the average on T` of S(θ) is non singular.

Main Hyperbolic Lemma

The assumption of hyperbolic smoothing splitting is stable. Thatis: if it is true for some function K , it is true for all the functionsK close to it.Moreover, if the perturbation N is small (in the sense of analyticfunctions from X to Y , we can bound the changes of thesplittings and of the constants by the size of the perturbations.

Notice that hyperbolicity assumption is easy to verify for K0 ≡ 0in many cases, since it is just the corresponding splittling for thelinear operator A.

Previous results by S.N. Chow, H. Leiva.

An abstract Theorem

Theorem (R. de la Llave, S.)Let ω ∈ D(κ, ν) for some κ > 0 and ν ≥ `− 1. Assume that

K0 satisfies the non-degeneracy conditions describedbeforeRegularity properties on the vector field

Define the initial error E0 = ∂ωK0 −X (K0) in YThen there exists a constant C > 0 such that, if

Cκ4δ−4ν‖E0‖ρ0,Y < 1, Cκ2δ−2ν‖E0‖ρ0,Y < r ,

where 0 < δ ≤ min(1, ρ0/12) is fixed, then there exists anembedding K∞ analytic in Dρ∞:=ρ0−6δ such that

∂ωK∞ = X (K∞), ‖K∞ − K0‖ρ∞,X ≤ Cκ2δ−2ν‖E0‖ρ0,Y .

Strategy: Newton-Nash/Moser iteration (Projecting the equation)

Fω(K ) = ∂ωK − J∇H ◦ K .

Lemma

Consider the linearized equation DFω(K )∆ = −E .Then there exists a constant C such that assuming thatδ ∈ (0, ρ/2) satisfies Cκδ−(ν+1)‖E‖ρ,Y < 1 we have that hereexists an approximate solution ∆: ∃E

DKFω(K )∆ = −E + E

‖E‖ρ−δ,Y ≤ Cκ2δ−(2ν+1)‖E‖ρ‖Fω(K )‖ρ,YFor all δ ∈ (0, ρ/2)

‖∆‖ρ−2δ,X ≤ Cκ2δ−2ν‖E‖ρ,Y , ‖D∆‖ρ−2δ,X ≤ Cκ2δ−2ν−1‖E‖ρ,Y .

On the (un)stable spaces we can use just the Duhamelformula going either in the future or in the past.In the center directions, we use that the restriction issymplectic (it is finite dimensional).The geometry reduces the equations with constantcoefficients that can be solved using only small divisorestimates (automatic reducibility).The solution of the small divisors entails loss of domainand the usual Nash-Moser tame estimates.One can iterate following the estimates already inKolmogorov’s paper.The error comes only from the lower order terms becausethe higher order term is linear.

Note that the above a-posteriori theorem requires only someapproximate solutions and that we can verify on them thenon-degeneracy assumptions.

One can take as approximate solutions the output of anon-rigorous procedure. For example some numericalcomputation or some asymptotic expansion.

The approximate solution for the Boussinesq equation

utt = µuxxxx + uxx + ε(u2)xx on T, µ > 0, t ∈ R

uε(t , x) = Uε(ωεt , x)

where Uε : T` × T 7→ R with ` =dimX c

02 . Then

(ωε · ∂)2Uε = ∂2xxUε + µ∂4

xxxxUε + ε(U2ε )xx . (11)

We consider the formal Lindstedt series

Uε(θ, x) ∼∞∑

k=1

εkUk (θ, x)

and

ωε ∼∞∑

k=0

εkωk .

The Theorem

Theorem (de la Llave, S. )Fix the Diophantine exponent ν and X a suitable Hilbert(Sobolev type ) space. There exist three explicit functionsa,bd ,ba : R+ → R+ such that

a(s)→ 0,bd (s),ba(s)→∞, s → 0

and for ε sufficiently small, denote by Ba(ε)(ω0) ⊂ R` the ball of

radius a(ε) around ω0 and let ω ∈ D(b(ε), ν) ∩ Ba(ε)(ω0).

Then, there exists K analytic from Db(ε) into X solving theBoussinesq equation with frequency ω. Furthermore,

|D(b(ε), ν) ∩ Ba(ε)(ω0)|

|Ba(ε)(ω0)|→ 1, ε→ 0.