NEW EXAMPLES OF PROBABILISTIC WELL-POSEDNSS FOR NONLINEAR WAVE EQUATIONS

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NEW EXAMPLES OF PROBABILISTIC WELL-POSEDNSS FOR NONLINEAR WAVE

EQUATIONS

Chenmin Sun, Nikolay Tzvetkov

To cite this version:

Chenmin Sun, Nikolay Tzvetkov. NEW EXAMPLES OF PROBABILISTIC WELL-POSEDNSS FOR NONLINEAR WAVE EQUATIONS. 2019. �hal-02064322�

NONLINEAR WAVE EQUATIONS

CHENMIN SUN, NIKOLAY TZVETKOV

Abstract. We consider fractional wave equations with exponential or arbitrary poly- nomial nonlinearities. We prove the global well-posedness on the support of the cor- responding Gibbs measures. We provide ill-posedness constructions showing that the results are truly super-critical in the considered functional setting. We also present a result in the case of a general randomisation in the spirit of the work by N. Burq and the second author.

1. Introduction

Our goal in this work is to give new examples of probabilistic well-posedness for non- linear wave equations with data of super-critical regularity. More precisely, we consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We will prove the global well-posedness on the support of the corresponding Gibbs measures (and also a result for more general random initial data). We will also provide ill-posedness constructions showing that the considered problem is super-critical in the sense that the obtained solutions crucially depend on the particular regularisations of the initial data.

Let us recall that in the case of a deterministic low regularity well-posedness for dispersive PDE’s, the obtained solutions can be seen as limits of approximated smooth solutions, independently of the choice of the approximation of the low regularity initial data (see e.g.

[13, 16, 17]).

1.1. The case of exponential nonlinearity. Let (M, g) be a compact smooth riemann- ian manifold of dimensiondwithout boundary. Let ∆gbe the associated Laplace-Beltrami operator. For σ ∈ R, we set D^σ = (1−∆g)^σ/2.Consider the following (fractional) wave equation

(1.1) ∂_t²u+D^2αu+e^u = 0,

whereu :R×M → R. The case α = 1 corresponds to the usual wave (or Klein-Gordon) equation posed on M. Let u be a smooth solution of (1.1). If we multiply (1.1) by ∂tu and integrate over M, we get

d dt

hZ

M

1

2(∂tu)²+1

2(D^αu)²+e^ui

= 0.

Let (ϕ_n)n≥0 be an orthonormal basis ofL²(M) of eigenfunctions of −∆_g associated with increasing eigenvalues (λ²_n)n≥0. By the Weyl asymptotics λ_n ≈n^d−12 . With v =∂_tu, we rewrite (1.1) as the following first order system:

(1.2) ∂_tu=v, ∂_tv=−D^2αu−e^u.

1

The system (1.2) is a Hamiltonian system of PDEs with the Hamiltonian:

(1.3) H(u, v) = 1

2 Z

M

(D^αu)²+v² +

Z

M

e^u.

The Hamiltonian H(u, v) controls the H^α norm of (u, v), where we denote H^s≡H^s(M)×H^s−α(M)

for any s∈R, and H^s(M) is the classical Sobolev space of orders. Forα≥ ^d₂,

Z

M

e^u

≤Ce^Ckuk2Hα(M)

and therefore for these values ofα the potential energy can be seen as a perturbation. In particular we can show the global well-posedness of (1.2) for data inH^α.

Theorem 1.1. Letα≥ ^d₂. The Cauchy problem associated with (1.2)is (deterministically) globally well-posed for data in H^s, s≥α.

Let (g_n, h_n)n≥0 be a family of independent standard gaussians on the probability space (Ω,F,P). The gaussian measure µ is the image measure under the map ω 7→ (u^ω, v^ω) defined by

(1.4) u^ω(x) =

∞

X

n=0

gn(ω)

hλ_ni^αϕn(x), v^ω(x) =

∞

X

n=0

hn(ω)ϕn(x).

We can see µ as a probability measure on H^s(M), σ < α−d/2. One has the following key property ofµ.

Proposition 1.2. Let α > d/2. For θ ∈ [0, α−d/2), kD^θuk_L^∞_(M) is finite µ-almost surely. More precisely

µ{u: kD^θuk_L∞(M)> λ} ≤C e^−cλ2 for some C, c >0 independent of λ≥1.

Applying Proposition 1.2 with θ= 0 we obtain that R

Me^u is finiteµalmost surely and we can define the Gibbs measure ρ associated with (1.2) as

dρ(u, v) =e^−RMeudµ(u, v). Indeed, using that ifu=P

nc_nϕ_n then 1

2 Z

M

(D^αu)²=

∞

X

n=0

hλ_ni^2αc²_n,

we deduce that one may interpretµ as a renormalisation of the formal measure Z⁻¹e⁻¹²

R

M((D^αu)²+v²) du dv

and therefore ρ becomes Z⁻¹e^−H(u,v)dudv which reminds the Gibbs measure for finite dimensional systems. We have the following result.

Theorem 1.3. Let α > ^d₂ and 0 < σ < α− ^d₂. The problem (1.2) is µ almost surely globally well-posed. Moreover ρ is invariant under the resulting flow Φ(t) in the following sense. There exists a measurable set Σ⊂ H^σ with full µ measure, such that Φ(t)Σ = Σ and for any measurable set A⊂Σ, we have ρ(A) =ρ(Φ(t)A) for allt∈R.

The main difficulty in Theorem 1.3 comes from the fact that the random data (1.4) is not in the scope of applicability of the deterministic well-posedness result of Theorem 1.1.

We however have the following connection between Theorem 1.1 and Theorem 1.3.

Theorem 1.4. Let (u^ω_N(t, x), v^ω_N(t, x))be the solution of (1.2)given by Theorem 1.1 with smooth data

u^ω_N(x) = X

λn≤N

gn(ω)

hλ_ni^αϕn(x), v_N^ω(x) = X

λn≤N

hn(ω)ϕn(x).

Then almost surely in ω, (u^ω_N(t, x), v_N^ω(t, x)) converges to the solution (u, v) of (1.2) con- structed in Theorem 1.3, inC(R;H^σ)∩L^∞_loc(R×M).

The restrictionα > d/2 in Theorem 1.3 and Theorem 1.4 is optimal in the sense that for α=d/2 the construction of the measure ρ fails because R

e^u is ill defined on the support ofµ. However, ford= 2 one may suitably renormaliseR

e^u. Such a renormalisation would unfortunately lead to a change of the equation. In the case of the renormalisation used in [14] one would obtain the wave equation, without the mass term, but with a source term, related to the curvature ofM. One can also use a renormalisation as in [11] which would avoid the source term in the equation but the mass term should be kept. In the case of both renormalisations we have just mentioned, one can apply compactness techniques as employed in [11, 20]. The obtained solutions would however be non unique and an approximation result as the one of Theorem 1.4 is completely out of reach of the scope of the applicability of these weak solution techniques. We believe that obtaining a result as Theorem 1.4 in the case α= 1 (d= 2) for the above mentioned renormalised equations is an interesting and challenging problem.

A novelty in the proof of Theorem 1.3 and Theorem 1.4 compared to [2, 3, 9, 10, 25]

is that because of the exponential nonlinearity, we need to prove probabilistic Strichartz estimates involvingL^∞ norms with respect to the time variables.

1.2. The case of arbitrary polynomial nonlinearity. The strategy to prove Theo- rem 1.3 and Theorem 1.4 works equally well for power type nonlinearity of an arbitrary degree, as follows

(1.5) ∂_t²u+D^2αu+u^2k+1= 0.

We have the following statement in the context of (1.5).

Theorem 1.5. Let α > ^d₂ and 0 < σ < α−^d₂. Then (1.5) is µ almost surely globally well-posed.

Note that from the scaling consideration: u(t, x) 7→ λ^αku(λ^αt, λx), the critical index is s_c = ^d₂ − ^α_k. Therefore, if s < ^d₂ − ^α_k, (1.5) is super-critical with respect to H^s. As a

consequence, we have an ill-posedness result, Proposition 6.1, proved in Section 6.

We underline that Theorem 1.5 is really a super-critical result, in the sense that the way to approximate the solution inC(R;H^σ) by smooth solutions is very sensitive. More precisely, as a consequence of Theorem 1.5 and Proposition 6.1, we have the following remarkable statement.

Corollary 1.6. Assume that (2k−1)d

2k+ 1 < α < kd

2 , k∈N, and d

2− 2α

2k−1 ≤σ < α−d 2.

• For almost every (u^ω₀, u^ω₁)∈ H^σ, there exists a sequence u^ω_N(t, x)∈C(R;C^∞(M)), N = 1,2,· · · of global solutions to (1.5) such that

N→∞lim k(u^ω_N(0), ∂_tu^ω_N(0))−(u^ω₀, u^ω₁)kH^σ →0, while for every T >0,

N→∞lim ku^ω_n(t)k_L∞([0,T];H^σ(M))→ ∞.

• Let (u^ω_N(t, x), v_N^ω(t, x)) be the solution of (1.5) with smooth data u^ω_N(x) = X

λn≤N

gn(ω)

hλ_ni^αϕn(x), v_N^ω(x) = X

λn≤N

hn(ω)ϕn(x).

Then almost surely in ω, (u^ω_N(t, x), v_N^ω(t, x)) converges to the solution (u, v) of (1.5) constructed in Theorem 1.5, in C(R;H^σ)∩L^∞_loc(R×M).

Remark 1.7. For fixed k∈N, if

(2k−1)d

2k+ 1 < α < kd 2 , then

d

2 − 2α 2k−1,d

2 −α k

∩

0, α−d 2

6=∅.

Remark 1.8. The first assertion of this corollary will follow from the strong ill-posedness result of Proposition 6.1. Unlike usual ill-posedness construction near the zero initial data, we prove norm-inflation near any smooth data of arbitrary size. The restriction α > ^(2k−1)d_2k+1 here is only a technical assumption (see case 2 in the proof of Lemma 6.5 for detailed discussion). It would be interesting to decide whther the same conclusion holds for the full range ^d₂ < α < ^kd₂ .

1.3. General randomisations. We remark that for the polynomial nonlinearity, if the underlying manifold M = T^d, we could also treat the general randomization introduced in [9]. More precisely, for any (u0, v0)∈H^s(T^d)×H^s−1(T^d),

u0(x) =a0+ X

n∈Z^d\{0}

(an,1cos(n·x) +an,2sin(n·x)),

v₀(x) =b₀+ X

n∈Z^d\{0}

(b_n,1cos(n·x)x+b_n,2sin(n·x)), we consider the randomization around (u0, v0):

u^ω₀(x) =a0g0(ω) + X

n∈Z^d\{0}

(an,1gn,1(ω) cos(n·x) +an,2gn,2(ω) sin(n·x)), v₀^ω(x) =b₀h₀(ω) + X

n∈Z^d\{0}

(b_n,1h_n,1(ω) cos(n·x)x+b_n,2h_n,2(ω) sin(n·x)), (1.6)

where {g₀(ω), g_n,j(ω), h₀(ω), h_n,j(ω) : n ∈ Z^d\ {0}, j = 1,2} are independent standard Gaussian variables. Denote by

u^ω_0,N(x) =a0g0(ω) + X

|n|≤N,n6=0

(an,1gn,1(ω) cos(n·x) +an,2gn,2(ω) sin(n·x)), v_0,N^ω (x) =b0h0(ω) + X

|n|≤N,n6=0

(bn,1hn,1(ω) cos(n·x)x+bn,2hn,2(ω) sin(n·x)), We have the following almost surely global existence as well as uniqueness theorem:

Theorem 1.9. Assume that M = T^d, α > ^d₂. Let (u0, v0) ∈ H^s with ^(k−1)α_k < s <

α. Then almost surely in ω ∈ Ω, (1.5) with initial data (u^ω₀, v₀^ω) is globally well-posed.

Moreover, the sequence of smooth solutions uN(t) to (1.5) with initial datum (u^ω_0,N, v_0,N^ω ) converges in C(R;H^s(T^d)×H^s−1(T^d)) to the solutionu^ω(t) with initial data (u^ω₀, v₀^ω).

We will sketch the proof in Section 7. The only additional ingredient in the proof of Theorem 1.9 is an energy a priori estimate, following the method of Oh-Pocovnicu [21]

(see also [23]). The crucial fact we use to prove the energy estimate is the almost sure L^∞bound for the linear evolution of the Gaussian random initial data. This is the reason to restrict our consideration to M =T^d in Theorem 1.9. For general randomizations on arbitrary manifold, the almost sureL^∞bound does not always hold true (see for example [1]). However, as in [22], using the idea of Burq-Lebeau [6], such an L^∞ bound can be achieved by imposing some assumptions on the variations of the Fourier coefficients of (u₀, v₀).

It is worth mentioning that there are many situations when the energy method of Oh- Pocovnicu does not cover the results obtained by exploiting the Gibbs measure. Indeed, for general randomizations, we needs→α ask→ ∞while for data on the support of the Gibbs measure we need 0< s < α−^d₂,independentlyof k.

Acknowledgement. The authors are supported by the ANR grant ODA (ANR-18-CE40- 0020-01). The problem considered in this paper is inspired by a talk of Vincent Vargas at the University of Cergy-Pontoise and by a discussion of the second author with R´emi Rhodes and Vincent Vargas at ENS Paris.

2. Construction of the Gibbs measure

2.1. Notations. Denote by Π_N the sharp spectral projector onE_N = span{ϕ_n:λ_n≤N}.

LetπN be a smooth projector where πN

∞

X

n=0

cnϕn

:=

∞

X

n=0

ψ(λn/N)cnϕn,

where ψ ∈ C₀^∞(R) ψ(r) ≥ 0 andψ(r) ≡1 if r ≤1/2, ψ(r) ≡ 0 if r > 1. By convention π∞= Id. Clearly, we have

πNΠN =πN, πNΠ_N/2= Π_N/2. We will also use the notations:

π_N^⊥ := Id−πN, Π^⊥_N := Id−ΠN.

2.2. Definition of Gibbs measure. Denote by µe_N the distribution of the E_N ×E_N valued random variable

ω7→



 X

λn≤N

g_n(ω)

hλ_ni^αϕ_n(x), X

λn≤N

h_n(ω)ϕ_n(x)



.

Consider the Gaussian measureµeN induced by this map, which is the probability measure on R^{2 dim(EN)} defined by

deµN = Y

λn≤N

hλ_ni^α 2π e⁻

(λ2 n+1)αa2

n 2 −^b2n

2 dandbn= 1 ZN

e^−H0(an,bn) Y

λn≤N

dandbn, whereH₀ is the free Hamiltonian

H0(u, v) = 1 2

Z

M

(|D^αu|²+|v|²).

Now we define a Gaussian measure on H^σ(M)(σ < α−d/2) be the induced probability measure by the map

ω 7→

∞

X

n=0

g_n(ω) hλ_ni^αϕ_n(x),

∞

X

n=0

h_n(ω)ϕ_n(x)

! .

The measureµcan be decomposed intoµ=µ^N⊗µe_N for allN, whereµ^N is the distribution of the random variable onE_N^⊥×E_N^⊥. Now we define the Gibbs measure ρ by

dρ(u) = exp

− Z

M

e^u

dµ(u).

We denote by

FN(u) = Z

M

e^πNu, F(u) = Z

M

e^u

and

G(u) = exp

− Z

M

e^u

, G_N(u) = exp

− Z

M

e^πNu

. To be precise, we firstly define its finite dimensional approximations

(2.1) dρeN(u) =GN(u)dµeN(u), dρN(u) =GN(u)dµ(u) =dµ^N⊗dρeN. The following proposition justifies our definition ofdρ(u).

Proposition 2.1. We have the following statements :

(1) The sequences (FN(u))N≥1 and (GN(u))N≥1 converge to the limits F(u), G(u) in L^p(dµ(u)), 2≤p <∞, respectively. In particular, G(u) exists almost surely with respect to µ.

(2) G(u)⁻¹ = exp R

Me^u

is almost surely finite with repsect to µ.

(3) lim_N→∞ρ_N(A) =ρ(A), for every Borel set A⊂ H^σ. We need the following Moser-Trudinger type inequality.

Lemma 2.2. There exists C >0 such that for all f ∈H^d2(M), we have Z

M

e^f ≤exp

Ckfk²

H^d2(M)

. Proof. From

e^f(x)≤

∞

X

k=0

|f(x)|^k k! , we have that

Z

M

e^f ≤

∞

X

k=0

kfk^k_Lk(M) k! . Note that the inequality (see [5])

kfk_Lp(R^d)≤C√ pkfk

H^d2(R^d)

also holds true by replacingR^d toM, because its proof is only based on local arguments.

Therefore, we have

Z

M

e^f ≤

∞

X

k=0

k^k2kfk^k

H^d2(M)

k! .

From the simple observation

k^k/2

k! ≤ 2^k/2 b k/2c!,

we obtain that Z

M

e^f ≤exp

Ckuk²

H^d2

.

This completes the proof of Lemma 2.2.

Proof of Proposition 2.1. (1) As mentioned in Remark 3.8 of [24], in order to prove that F_N(u), G_N(u) converge toF(u), G(u) in L^p(dµ), it will be sufficient to show that

• kF_N(u)k_Lp(dµ),kG_N(u)k_Lp(dµ) are uniformly bounded.

• FN(u), GN(u) converge to F(u), G(u) in measure.

Now we verify the boundeness inL^p(dµ). Note thatG_N(u)≤1, we only need to check for F_N(u). We write

|F_N(u)| ≤vol(M)e^{kπNukL∞(M)}. Therefore using Proposition 1.2, we write forλ≥1,

µ(u:|F_N(u)|> λ)≤µ u:kπ_Nuk_L^∞_(M)> log(λ) C

≤C⁰e^{−C(log(λ))2} ≤C_lλ^−l, for every l ≥1. This proves the uniform in N boundedeness of kF_N(u)k_L^p_(dµ) for every p <∞. Next, we claim thatFN(u) converges in measure toF(u). Once this is justified, the convergence in measure for G_N(u) would follow automatically sinceG_N =e^−FN. For N1 ≥N2, we observe that

|F_N1(u)−F_N2(u)| ≤ Z

M

|π_N1u−π_N2u| ·e^{|πN1u|+|πN2u|}

≤Ckπ_N1u−π_N2uk_L2(M)exp kπ_N1uk_L∞(M)+kπ_N2uk_L∞(M)

. Therefore by Cauchy-Schwarz,kF_N1(u)−F_N2(u)k_L1(dµ) is bounded by

C

kπ_N1u−πN2uk_L2(M)

L²(dµ)kexp kπ_N1uk_L^∞_(M)+kπ_N2uk_L^∞_(M)

k_L2(dµ). The first factor is clearly going to zero as N1, N2 go to infinity. One can show that the second factor is uniformly bounded, exactly as in the proof of the uniform boundedeness of kF_N(u)k_Lp(dµ) . Therefore (FN(u)) is a Cauchy sequence in L¹(dµ) which implies its convergence in mesure. This in turn implies the convergence in measure of the sequence (G_N(u)) (G_N is a continuous function of F_N).

(2) To show thatR

Me^u is almost surely finite, it will be sufficient to verify that Eµ

Z

M

e^u

<∞.

From the proof of (1), Eµ[e^{kπNukL∞(M)}] is uniformly bounded inN. Thus we conclude by the dominated convergence.

(3) It will be sufficient to check that, for all Borel set A⊂H^σ(M), we have

N→∞lim Z

H^σ(M)

1u∈A|G_N(u)−G(u)|dµ(u) = 0.

This is a simple consequence ofL²(dµ(u)) convergence, since Z

H^σ(M)

1u∈A|G_N(u)−G(u)|dµ(u)≤ kG_N(u)−G(u)k_L2(dµ)µ(A)^1/2 →0,

asN → ∞. This completes the proof of Proposition 2.1.

3. Probabilistic local well posedness

3.1. Deterministic local well-posedness result. Consider the following truncated ver- sion of (1.2)

(3.1) ∂_tu=v, ∂_tv=−D^2αu−π_N(e^πNu), with initial data

(3.2) u|_t=t0 =u₀, v|_t=t0 =v₀. Let us next define the free evolution. The solution of

∂_tu=v, ∂_tv =−D^2αu, subject to initial data

u(0, x) =u₀(x), v(0, x) =v₀(x) is given by

S(t)(u¯ 0, v0) = (S(t)(u0, v0), ∂tS(t)(u0, v0)), where

S(t)(u0, v0) = cos(tD^α)u0+D^−αsin(tD^α)v0

and

∂_tS(t)(u₀, v₀) =−D^αsin(tD^α)u₀+ cos(tD^α)v₀. Assume that₀1, 1r₀ <∞ such that

d r0

< 0 < α−d 2,

which is possible sinceα > ^d₂. In order to establish the probabilistic local well-posednesss as well as globalizing the dynamics, we need some auxillary functional spacesX^s,β,Y^β,Z^β, definining via the norms

k(u, v)k_Yβ :=X

l∈Z

(1 +|l|)^−β sup

l≤t<l+1

kS(t)(u, v)k_W0,r0(M),

k(u, v)k_Xs,β :=k(u, v)k_H^s +k(u, v)k_Yβ

and

k(u, v)k_Zβ :=X

l∈Z

(1 +|l|)^−β sup

l≤t≤l+1

kS(t)(u, v)k_L∞(M). Note that

α−s₀

d = 1

2 − 1 r0

withs₀=α−d 2 + d

r0

> ₀,

thus H^α(M) ⊂W^0,r0(M)⊂ L^∞(M) (with continuous inclusions). Moreover, Y^β ⊂ Z^β. The definition of this weighted in time spaces Y^β,Z^β is inspired by the work of N. Burq and the second author (a similar definition is appeared in [9]). The weight β >1 will be fixed in the sequel to ensure the l¹ summation, without other importance. These norms are only designed to treat the linear evolution part of the solution, since unlike [10], the linear evolution is not periodic in time.

Denote by Φ_N(t) the flow of the truncated equation (3.1),(3.2). In components, we write

Φ_N(t)(u₀, v₀) = (Φ¹_N(t)(u₀, v₀),Φ²_N(t)(u₀, v₀))

with Φ¹_N(t)(u0, v0) =uN(t),Φ²_N(t)(u0, v0) =∂tuN(t). Similarly, we denote by Φ(t)(u0, v0) = (Φ¹(t)(u0, v0),Φ²(t)(u0, v0)),

where Φ¹(t)(u₀, v₀) =u(t),Φ²(t)(u₀, v₀) =∂_tu(t) for solutions of the untruncated equation

∂_t²u+D^2αu+e^u= 0, (u, ∂_tu)|_t=0 = (u₀, v₀).

We also denote the nonlinear evolution part by

W(t)(u₀, v₀) := Φ¹(t)(u₀, v₀)−S(t)(u₀, v₀),

W_N(t)(u_0,N, v_0,N) := Φ¹_N(t)(u_0,N, v_0,N)−S(t)(u_0,N, v_0,N).

The next proposition contains the local theory for (3.1) and the original system (1.2).

Proposition 3.1. LetN ∈N∪ {∞}andβ >1. There existc >0andκ >0such that the following holds true. The Cauchy problems (3.1)-(3.2)and (1.2) are locally well-posed for data (u₀, v₀) such that S(t)(u₀, v₀) ∈ L^∞(M). More precisely for every R ≥1 if (u₀, v₀) satisfies

(3.3) k(u₀, v₀)k_Zβ < R,

then there is a unique solution of (3.1)-(3.2)on [t0−τ, t0+τ], where

(3.4) τ =ce^−κR

which can be written as

(u, v) = ¯S(t−t₀)(u₀, v₀) + (˜u,˜v), with

k(˜u,v)k˜ _H^α ≤C .

In particular, the Cauchy problems (3.1)-(3.2) and (1.2) are locally well-posed for data (u0, v0)∈ X^σ,β, for allt∈[t0−τ, t0+τ], provided that σ < α−^d₂.

Proof. We can write (3.1)-(3.2) as

(u(t), v(t)) = ¯S(t−t₀)(u₀, v₀) + F₁(u), F₂(u) , where

F1(u) =− Z t

t0

D^−αsin((t−τ)D^α)π_N e^πNu(τ) dτ,

F2(u) =− Z t

t0

cos((t−τ)D^α)πN e^πNu(τ) dτ.

If we write (u, v) = ¯S(t−t₀)(u₀, v₀) + (˜u,v), we obtain that (˜˜ u,v) solves˜

∂tu˜= ˜v, ∂tv˜=−D^2αu˜−πN(e^{πN(˜u+S(t)(u0,v0))}) with zero initial data. Therefore ˜u solves

(3.5) u(t) =˜ F₁(˜u+S(t−t₀)(u₀, v₀)).

Once we solve (3.5), we recover ˜v by ˜v=∂tu. Define the map Φ˜ u0,v0 by Φ_u0,v0(u) :=F₁(u+S(t−t₀)(u₀, v₀)).

It follows from the definition that

kΦ_u0,v0(u)k_L^∞_([t0,t0+T];Hα(M))≤CTke^{πN(u+S(t−t0)(u0,v0))}k_L^∞_(I;L2(M))

≤CT(vol(M))¹²ke^{πN(u+S(t−t0)(u0,v0))}k_L∞(I;L^∞(M))

(3.6)

where I = [t₀, t₀ +T]. Since π_N is bounded on L^∞ (see e.g. [4]), a use of the Sobolev embedding H^α(M)⊂L^∞(M) yields the estimate

kΦ_u0,v0(u)k_L^∞_([I;Hα(M))≤CT e^{CkS(t−t0)(u0,v0))kL∞(I×M)}e^{kukL∞(I;Hα(M))}. Note that

kS(t−t₀)(u₀, v₀)k_L∞(I×M)≤Ck(u₀, v₀)k_Zβ, hence under (3.3) we deduce that

(3.7) kΦ_u0,v0(u)k_L∞(I;H^α(M))≤CT e^CRe^{kukL∞(I;Hα(M))}. Similarly we obtains that

kΦ_u0,v0(u)−Φu0,v0(v)k_L^∞_(I;Hα(M))

≤CT e^CRexp kuk_L^∞_(I;Hα(M))+kvk_L^∞_(I;Hα(M))

ku−vk_L^∞_(I;Hα(M)). (3.8)

Define the space X_T as

X_T ={u∈C(I;H^α(M)) : kuk_L∞(I;H^α(M))≤1}.

Using (3.7), we obtain that forT as in (3.4) the map Φ_u0,v0 enjoys the property Φ_u0,v0(X_T)⊂X_T.

Under the same restriction on T, thanks to (3.8), we obtain that the map Φu0,v0 is a contraction on X_T. The fixed point of this contraction is the solution of (3.5). This

completes the proof of Proposition 3.1.

In the same spirit of the proof, we establish a local convergence result, which will be needed to construct global dynamics in Section 4.

Lemma 3.2. Assume that 0 < σ < α− ^d₂ and β >1. There exist R0 >0, c > 0, κ > 0 such that the following holds true. Consider a sequence (u_0,Np, v_0,Np) ∈ E_Np ×E_Np and (u0, v0)∈ X^σ, withNp→ ∞. Assume that there existsR > R0 such that

k(u_0,Np, v0,Np)k_Xσ,β ≤R, k(u₀, v0)k_Xσ,β ≤R, and

p→∞lim kπ_Np(u0,Np, v0,Np)−(u0, v0)kH^σ = 0.

Then if we set τ = ce^−κR, the flow Φ_Np(t)(u_0,N0, v_0,Np),Φ(t)(u₀, v₀) exist for t ∈ [−τ, τ]

and satisfy

kΦ_Np(t)(u_0,Np, v_0,Np)k_L∞([−τ,τ];X^σ,β) ≤R+ 1, kΦ(t)(u₀, v0,)k_L∞([−τ,τ];X^σ,β)≤R+ 1.

Furthermore,

p→∞lim kπ_NpΦNp(t)(u0,Np, v0,Np)−Φ(t)(u0, v0)k_L^∞_{([−τ,τ];H}σ)= 0, and

p→∞lim kπ_NpWNp(t)(u0,Np, v0,Np)−W(t)(u0, v0)k_L^∞_{([−τ,τ];H}α(M))= 0.

Proof. The existence of ΦNp(t)(u0,Np, v0,Np) and Φ(t)(u0, v0) as well as the bound on [−τ, τ]

are guranteed by the local well-posedness result, Propsotion 3.1. We only need to prove the convergence.

Denote by

(u_p(t), ∂_tu_p(t)) = Φ_Np(t)(u_0,Np, v_0,Np), (u(t), ∂_tu(t)) = Φ(t)(u₀, v₀).

From local theory, we can write

up(t) =S(t)(u0,Np, v0,Np) +wp(t), u(t) =S(t)(u0, v0) +w(t), such that

k(w_p, ∂_tw_p)k_L^∞_([0,τ];Hα) ≤R+ 1, k(w, ∂_tw)k_L^∞_([0,τ];Hα) ≤R+ 1.

The convergence of the linear part

kπ_NpS(t)(u_0,Np, v_0,Np)−S(t)(u₀, v₀)k_L∞([0,τ];H^α)= 0 follows from the assumption and the boundeness of π_Np on W^0,r0(M).

Next we estimate the nonlinear part

kπ_Npwp(t)−w(t)k_Hα(M).

Writting w_p, w by the Duhamel formula and using triangle inequality, we can bound the quantity above by the three contributions:

I(t) :=

Z t 0

e^{πNp(wp(t0)+S(t0)(u0,Np,v0,Np))}−e^{πNp(w(t0)+S(t0)(u0,v0))}

L²(M)dt⁰, II(t) :=

Z _t

0

e^{w(t0)+S(t0)(u0,v0)}−e^{πNp(w(t0)+S(t0)(u0,v0))}

_L2(M)dt⁰, III(t) :=

Z t 0

π_N^⊥e^{w(t0)+S(t0)(u0,v0)}

L²(M)dt⁰. Since

ew(t)+S(t)(u0,v0)∈L^∞([−τ, τ]×M), we have that

kIIIk_L^∞_([−τ,τ])=op→∞(1).

For I(t) and II(t), we can bound them by C(R, τ)

op→∞(1) + Z t

0

kπ_Npw_p(t⁰)−w(t⁰)k_L2(M)dt⁰

.

By applying Grwonwall inequality, the proof of Lemma 3.2 is complete.

3.2. Large deviation estimate for linear evolutions. Proposition 3.1 is deterministic.

The probabilistic part of the analysis comes from the following statement.

Proposition 3.3. Assume that β >1. There are positive constantsC andcsuch that for every R≥1,

µ (u0, v0) : k(u₀, v0)k_Yβ ≥R

≤C e^−cR2. As a consequence, we also have a similar bound for the Gibbs measure

ρ (u0, v0) : k(u₀, v0)k_Yβ ≥R

≤C e^−cR2.

Observe that in the case of an exponential nonlinearity, we need a large deviation estimate for L^∞ norms in time (in [9] onlyL^p in time for a finite p were established).

Proof of Proposition 3.3. Letη∈C₀^∞be a bump function localising in the interval [−2,2].

Denote by η_l(t) :=η(t−l) for l∈Z. We need to show

X

l∈Z

hli^−βkη_l(t)S(t)(u₀, v₀)k_L∞(R;W^0,r0(M)

L^p(dµ)

≤C√ p .

Coming back to the definition of S(t), we observe that it suffices to prove the bound

X

l∈Z

hli^−β η_l(t)

∞

X

n=0

c_ng_n(ω)

hλ_ni^α e^itλαnϕ_n(x)

L^∞(R;W^0,r0(M)

L^p(Ω)

≤C√ p ,

where |c_n|is bounded. Thanks to β >1, we have P

lhli^−β <∞, thus it suffices to prove the bound

η_l(t)

∞

X

n=0

cngn(ω)

hλ_ni^α e^itλαnϕn(x)

L^∞(R;W^0,r0(M))

L^p(Ω)

≤C√ p,

uniformly in l∈Z.

To simplify the notation, we only write down the case where l = 0. For other l, the arguments are exactly the same. Using the Sobolev embedding with a largeq and a small δ (depending on q) we obtain that its is sufficient to obtain

D^δ_tD⁰ η(t)

∞

X

n=0

cngn(ω)

hλ_ni^α e^itλαnϕn(x) _Lq(

R;L^r0(M)

_Lp(Ω)≤C√ p ,

whereD^δ_t = (1−∂_t²)^δ/2. Since D^δϕn=hλ_ni^δϕn, we are reduced to prove the bound

∞

X

n=0

c_ng_n(ω)

hλ_ni^α−δα_n(t)ϕ_n(x)

L^q(R;L^r0(M)

L^p(Ω)

≤C√ p ,

where αn(t) =D^δ_t(η(t) exp(itλ^α_n)).Applying the Minkowski inequality, we deduce that it suffices to prove that forp≥q ≥r0,

∞

X

n=0

c_ng_n(ω)

hλ_ni^α−δα_n(t)ϕ_n(x) L^p(Ω)

L^r0(M;L^q(R))

≤C√ p .

Observe that in this discussionqis large but fixed andpgoes to∞. Now for a fixed (t, x), we can apply the Khinchin inequality and write

∞

X

n=0

c_ng_n(ω)

hλ_ni^α−δαn(t)ϕn(x)

L^p(Ω) ≤C√ p

X^∞

n=0

|α_n(t)|²|ϕ_n(x)|² hλ_ni^2(α−δ)

¹₂ . Therefore, we reduce the matters to the deterministic bound

∞

X

n=0

|α_n(t)|²|ϕ_n(x)|² hλ_ni^2(α−δ)

L^r0/2(M;L^q/2(R)

≤C .

For a fixed x, we apply the triangle inequality to obtain that (3.9)

∞

X

n=0

|α_n(t)|²|ϕ_n(x)|² hλ_ni^2(α−δ)

L^q/2(R)≤

∞

X

n=0

|ϕ_n(x)|²

hλ_ni^2(α−δ)kα_nk²_Lq(R). Lemma 3.4. There is C such that for every n≥1, kα_nk_Lq(R) ≤Cλ^αδ_n . Proof. Letβ(t) =η(t)e^itλαn, we have that

kα_nk_L^q_(R) ≤Ckβk_L^q_(R)+Ckβk_B˙δ q,2(R). From the characterization of Besov spaces (see [5]), we have that

kβk²_˙

B_q,2^δ ∼ Z

R

Z

R

|β(t+τ)−β(t)|^qdt ²_q

dτ

|τ|^1+2δ. We write

η(t) exp(itλ^α_n)−η(τ) exp(iτ λ^α_n) = exp(itλ^α_n)(η(t)−η(τ)) +η(τ)(exp(itλ^α_n)−exp(iτ λ^α_n)) which yields two contributions. The first one is again uniformly bounded. Therefore the issue is to check that

Z

R

Z

R

|η(t)(e^i(t+τ)λαn −e^itλαn)|^qdt ²_q

dτ

|τ|^1+2δ ≤(Cλ^αδ_n )². For|τ| ≤cλ^−α_n , we use

|e^{iτ λαn} −1| ≤ |τ|λ^α_n, and thus

Z

|τ|<cλ^−αn

1

|τ|^1+2δ Z

R

|η(t)(e^i(t+τ)λαn−e^itλαn)|^qdt ²_q

dτ

≤ Z

|τ|<cλ^−αn

1

|τ|^1+2δ Z

R

|η(t)|^q|τ|^qλ^αq_n ²_q

dτ ≤Cλ^2αδ_n . The other contribution for |τ|> cλ^−α_n can be bounded by

Z

|τ|>cλ^−α_n

Z

R

|2η(t)|^qdt ²_q

dτ ≤Cλ^2αδ_n . This completes the proof of Lemma 3.4.

Coming back to (3.9) and using Lemma 3.4, we deduce that it suffices to majorize

∞

X

n=0

|ϕ_n(x)|² hλ_ni^{2(α−δ−αδ)}

L^r0/2(M).

Using the compactness of M, it suffices to get the following estimate sup

x∈M

∞

X

n=0

|ϕ_n(x)|²

hλ_ni^{2(α−δ−αδ)} <∞.

Fix δ sufficiently small such that β := 2(α−δ−αδ) > d (here we fix the value of q as well). Therefore we need to show that

(3.10) sup

x∈M

∞

X

n=0

|ϕ_n(x)|²

hλ_ni^β <∞, β > d .

Estimate (3.10) is direct if|ϕ_n(x)|are uniformly bounded (this is the case of the torus). In the case of a general manifold, it is not true that|ϕ_n(x)|are uniformly bounded. However (3.10) is true thanks to [15]. More precisely for a dyadic N, we can write

X

N≤hλni≤2N

|ϕ_n(x)|²

hλ_ni^β ≤CN^−β X

N≤hλni≤2N

|ϕ_n(x)|².

Thanks to [15] there is C such that for every dyadic N ≥1 and every x∈M,

(3.11) X

N≤hλni≤2N

|ϕ_n(x)|² ≤CN^d.

This readily implies (3.10). The proof of Proposition 3.3 is completed.

We complete this section by proving following probabilistic bound for the tails of sharp spectral truncation, which will be used in the proof of Theorem 1.4.

Proposition 3.5. Assume that 2≤r₀ <∞, ₀ < α−^d₂, 0< s < α−^d₂ −₀ and β >1.

Then there exist C >0, c >0,such that for any R >0, we have µ

(u₀, v₀) :N^skΠ^⊥_N(u₀, v₀)k_Yβ > R

≤Ce^−cR2.

Proof. Following the same notations as in the proof of the Proposition 3.3. It suffices to prove that

(3.12)

η(t) X

λn≥N

cngn(ω)

hλ_ni^α e^itλαnϕn(x)

L^∞(R;W^0,r0(M))

L^p(Ω)

≤CN^−s√ p,

forp large enough. From Sobolev embedding, we are reduced to prove the bound (3.13)

X

λn≥N

c_ng_n(ω)

hλ_ni^α−0α_n(t)ϕ_n(x)

L^q(R;L^r0(M))

_Lp(Ω)

≤CN^−s√ p,

whereα_n(t) =D_t^δ(η(t)e^itλαn). From Minkowski inequality, it will be sufficient to prove that forp≥q, p≥r₀,

(3.14)

X

λn≥N

c_ng_n(ω)

hλ_ni^α−0α_n(t)ϕ_n(x) L^p(Ω)

_Lq(

R;L^r0(M))

≤CN^−s√ p