The difficulty in the convergence of Assumption7is the fact that bothPtN,α∗k andμNαk depend onαk, we then need some uniformity in the convergenceuNαk(t,·)→ φtN(·) asαk→0. The following lemma gives the needed uniform convergence. To simplify the notation, we use the abuse of notation:PtN,α∗k =:PtN,k∗,μαNk =:μkNanduαk =:uk.
We postpone the proof of Lemma4.3. We remark that, using the Ito isometry and the Chebyshev inequality we have
E(1−1Sr,k(t))≤ Ct
r2 (4.14)
whereCdoes not depend on(r,k,t).
Let f : L2→Rbe a bounded Lipschitz function. Without any loss of generality we can assume that f is bounded by 1 and its Lipschitz constant is also 1. We have
PtN,k∗μkN,f − PtN∗μN, f = μNk,PtN,kf − μN,PtNf
= μNk, (PtN,k−PtN)f − μN−μkN,PtNf
=A−B.
We see thatB →0 ask→ ∞according to the weak convergenceμNk →μN. Using the boundedness of f we obtain
|A| ≤
BR(L2)|PtN,kf(w)−PtNf(w)|μkN(dw)+2μkN(L2\BR(L2))=:A1+A2.
Now from (3.16),
L2
u2L2μkN(du)≤
L2
E(u)μkN(du)≤C,
whereCdoes not depend on(N, α).Combining this with the Chebyshev inequality we obtain that
A2≤ 2C R2.
Using the boundedness and Lipschitz properties of f, we obtain A1≤ A1,1+A1,2
where
A1,1=
BR(L2)EukN(t, Nw)−φtwL21SrμkN(dw) A1,2=2
L2\BR(L2)E(1−1Sr)μkN(dw).
Using the (4.14) we obtain that
A1,2≤ 2Ct r2 . We finally obtain
|A| ≤A1,1+2Ct r2 +2C
R2.
We pass to the limit onkfirst, by applying apply Lemma4.3, we obtainA1,1→0. Then we take the limitsr→ ∞andR→ ∞, we arrive at the conclusion of Assumption7.
Now let us prove Lemma (4.3):
Proof of Lemma4.3 Letu0 ∈ BR(L2)anduk andu be the solutions of (4.5) (with viscosityαk) and (4.1) starting atu0, respectively. Recall thatukcan be decomposed asuk =vk+zkwherevkis the solution of (4.12) with initial datumu0andzkis given by (4.10) withαk. In order to prove Lemma4.3, it suffice to show that
sup
u0∈BR
sup
t∈[0,T]E
vk−uL21Sr,k(t)
→0, ask→ ∞.
Indeed, we already have by (4.11) thatEsupt∈[0,T]zk(t)L2 → 0 ask → ∞. Set wk = vk −u. We will treat the cases ≤ 1+σ which is more delicate. We then consider the equation satisfied bywk:
∂twk=iwk+N[wkf(vk,u,zk)+zkg(vk,zk)]
−αkeξ−1(vk+zkH s−)(−)s−σ(vk+zk),
where f andg is a homogeneous of degree 2 complex polynomial. We claim that limk→∞wkL2 = 0 almost surely. Indeed, by taking the dot product withwk, we obtain after the use of the Gronwall inequality,
sup In particular, we have the following two estimates:
sup
k≥1
sup
u0∈BR
wkL∞t L2x1Sr,k ≤R+3C(r,N)T, (4.16)
Hence coming back to (4.15) and using the (deterministic) conservationu(t)L2 = PNu0L2 and the estimate (4.16), we obtain
sup
u0∈BR
wk2L∞
t L2x1Sr ≤ A(R,N,r,T)(αk+ zkL1tL2x).
Therefore, using again the bound (4.11), we obtain the almost sure convergence zkL2 → 0 (ask → ∞, up to a subsequence), we obtain then the almost sure
convergence
klim→∞ sup
u0∈BR
wk2L∞
t L2x1Sr =0.
Now, we use (4.16) and the Lebesgue dominated convergence theorem to obtain E sup
u0∈BR
wkL∞t L2x1Sr →0, as k→ ∞.
The proof of Lemma4.3is finished.
4.4 Estimates
Let us recall that
M(u):=M(u,L(u))= u,L(u).
Proposition 4.4 We have that
L2
M(u)μ(du)= A0
2 . (4.17)
Proof of Proposition4.4 We present the more difficult case in which the dissipation is given by
L(u)=eξ−1(uH s−)
N|u|2u+(−)s−σu
, where 0≤s≤1+σ.
In this case we have
M(u)=eξ−1(uH s−)
u4L4+ u2Hs−σ
.
The other case can be proved using a similar procedure. We split the proof in different steps:
(1) Step 1: Identity for the(μαN).
Applying the Itô formula toM(u), whereuis the solution to (4.5), we obtain
d M(u)=#
−αM(u)dt+α 2A0N
$ dt+√
α N m=0
amu,emdβm.
Integrating intand in ensemble with repect toμαN, we obtain
L2M(u)μαN(du)= A0N
2 , (4.18)
this identity used the invarianceμNα.
Let us now establish an auxiliary bound. Apply the Itô formula toM2(u)
d M2(u)=
Integrating intand with respect respect toμNα we arrive at
L2
M(u)M(u)μNα(du)≤C, (4.19) where C does not depend in(N, α). The estimate above is obtained after the following remark
(2) Step 2: Identity for the (μN). By usual arguments, we obtain the following estimates for(μN):
We do not give details of proof of the identities. In the next step we will prove more delicate estimates whose proof is highly more difficult as the passage to the limit cannot use any finite-dimensional advantage.
(3) Step 3: Identity for theμ. In this part of the proof we perform the passage to the limitN→ ∞in (4.20) in order to obtain the identity (4.17).
The inequality
L2
M(u)μ(du)≤ A0
2
can be obtained by invoking lower semi-continuity ofM. The other way around, the analysis is more challenging. In [54] a similar identity was established with the use of an auxiliary estimate on a quantity of typeE(u)E(u). But in our con-text, such an estimate is not available. Indeed, in order to obtain it, we shall apply the Itô formula onE2, the dissipation will beE(u)E(u); but other terms includ-ing AσNE(u)have to be controlled in expectation. However, we do not have an N−independent control on
L2E(u)μN(du)because this term is smoother than
E(u)fors< σ. Therefore the latter cannot be exploited. We remark, on the other hand, that this is why the measureμis expected to be concentrated on regularities lower than the energy space Hσ. Now, without a control onE(u)E(u), we can only handle the weaker quantityM(u)M(u)from (4.19). So our proof here will be more tricky than that in [54].
Coming back to the proof of the remaining inequality, we write for some fixed frequencyF, the frequency decomposition (setting>F =1−F)
A0N
2 ≤Eeξ−1(uNH s−)
F|uN|22L2 + FuN2Hs−σ
+Eeξ−1(uNH s−)
>F|uN|22L2 + >FuN2Hs−σ
=:i+ii,
whereuNis distributed byμN. Let us follow the following sub-steps:
(a) First,iican be estimated by using the control onE (4.7) as follows ii F−σ
Eeξ−1(uNH s−)(>F|uN|22Hσ + uN2H˙s)
F−σEE(uN) F−σC.
(b) As fori, we can split it by using localization in Hs−(notice that by using the Skorokhod theorem (uN) converges almost surely on Hs− to some Hs−−valued random variable u). For any fixed R > 0 we set χR = 1{uNH s−≥R}. We have
Eeξ−1(uNH s−)(F|uN|22L2+ FuN2Hs−σ)χR
Eeξ−1(uNH s−)(F|uN|22
Hd2++ FuN2Hs−σ)χR
Eeξ−1(uNH s−)(uN4
Hd2++ FuN2Hs−σ)χR
Fd2+Eeξ−1(uNH s−)(uN4L2+ uN2Hs−σ)χR
Fmax(d2+,s−σ)R−2Eeξ−1(uNH s−)uN2Hs−
uN4L2 + uN2Hs−σ
Fmax(d2+,s−σ)+sR−2EuN2L2eξ−1(uNH s−)
#uN4L4+ uN2Hs−σ
$.
Let us use the estimate (4.20) and (4.21) to arrive at Eeξ−1(uNH s−)
F|uN|22L2+ FuN2Hs−σ
χR
Fmax(d2+,s−σ)+sR−2EM(uN)M(uN) Fmax(d2+,s−σ)+sR−2C,
whereCdoes not depend onN.
(c) On the other hand, we have the following convergence, thanks to the Lebesgue dominated convergence theorem,
Nlim→∞Eeξ−1(uNH s−)
F|uN|22L2+ FuN2Hs−σ
(1−χR)
=Eeξ−1(uH s−)
F|u|22L2 + Fu2Hs−σ
(1−χR).
Gathering all this, we obtain, after the limitN → ∞, that A0
2 ≤Eeξ−1(uH s−)
F|u|22L2 + Fu2Hs−σ
(1−χR)+F−σC1
+Fmax(d2+,s−σ)+sR−2C2. We letR→ ∞, thenF→ ∞and obtain
A0
2 ≤eξ−1(uH s−)E
u4L4 + u2Hs−σ
.
The proof of Proposition4.4is finished.
Remark 4.5 The identity (4.17) is crucial for establishing the non-degeneracy proper-ties of the measureμ.It trivially rule out the Dirac measure at 0, notice that the Dirac at 0 is a trivial invariant measure for FNLS.
Also, by considering a noiseκd W, (4.17) becomes EM(u)= κA0
2 . (4.22)
Such scaled noises provide invariant measures μκ for FNLS, all satisfying (4.22).
Define a cumulative measureμ∗by a convex combination μ∗=
∞ j=0
ρjμκj
whereκj ↑ ∞and)∞
0 ρj =1. The measureμ∗is invariant for FNLS. Moreover, for any K >0, we can find a positiveμ∗-measure set of data whoseHs-norms are larger thanK.
In order to finish the proof of Theorem1.1, we present in the section below the fulfillment of Assumptions1and2.
5 End of the proof of Theorem1.1: Local well-posedness
In this section, we present a deterministic local well-posedness result forσ ∈ [12,1] in (1.1), which heavily replies on a bilinear Strichartz estimate obtained in Sect.5.1.
See also Sect.3in [55]. We also show a convergence from Galerkin approximations of FNLS to FNLS.
Theorem 5.1 (Deterministic local well-posedness on the unit ball)The fractional NLS (1.1) withσ ∈ [12,1]is locally well-posed for radial data u0∈ Hr ads (Bd), s>sl(σ), where sl(σ)is defined as in (1.3). More precisely, let us first fix s >sl(σ)(defined as in (1.3)), and for every R > 0, we setδ = δ(R) =c R−2s for some c ∈ (0,1]. Then there exists b > 12 and C,C*> 0such that every u0 ∈ Hr ads (Bd)satisfying u0Hr ads (Bd)≤ R, there exists a unique solution of (1.1) in Xsσ,,br ad([−δ, δ]×Bd)with initial condition u(0)=u0. Moreover,
uL∞t Hxs([−δ,δ]×Bd)≤CuXs,b
σ,r ad([−δ,δ]×Bd)≤Cu* 0Hr ads (Bd).
Remark 5.2 The function f in Assumption1is found to be equal toδ(x)=cx−2s. 5.1 Bilinear Strichartz estimates
In this subsection, we prove the bilinear estimates that will be used in the rest of this section. The proof is adapted from [1] with two dimensional modification and a different counting lemma.
Lemma 5.3 (Bilinear estimates for fractional NLS)Forσ ∈ [12,1], j =1,2, Nj >0 and uj ∈L2r ad(Bd)satisfying
1√−∈[N
j,2Nj]uj =uj, we have the following bilinear estimates.
(1) The bilinear estimate without derivatives.
Without loss of generality, we assume N1≥N2, then for anyε >0 Sσ(t)u1Sσ(t)u2L2
t,x((0,1)×Bd)N
d−1 2 +ε
2 u1L2x(Bd)u2L2x(Bd). (5.1) (2) The bilinear estimate with derivatives.
Moreover, if uj ∈ H01(Bd), then for anyε >0
∇Sσ(t)u1Sσ(t)u2L2t,x((0,1)×Bd) N1N
d−1 2 +ε
2 u1L2x(Bd)u2L2x(Bd). (5.2) Remark 5.4 Notice that in (5.1) and (5.2), the upper bounds are independent on the fractional power σ. This is because a counting estimate in Claim5.7does not see difference onσ. Hence the local well-posedness index is uniform forσ ∈ [12,1). Lemma 5.5 (Bilinear estimates for classical NLS) Under the same setup as in Lemma5.3, the bilinear estimate analogue is given by
S1(t)u1S1(t)u2L2
t,x((0,1)×Bd)N
d−2 2 +ε
2 u1L2x(Bd)u2L2x(Bd)
∇S1(t)u1S1(t)u2L2t,x((0,1)×Bd)N1N
d−22 +ε
2 u1L2x(Bd)u2L2x(Bd). Notice that the proof of Lemma5.5in fact can be extended from [1], hence we will only focus on the proof of Lemma5.3and the proof of the local theory for fractional NLS in the rest of this section.
Proposition 5.6 (Lemma 2.3 in [10]: Transfer principle)For any b> 12 and for j= 1,2, Nj >0and fj ∈Xσ0,b(R×Bd)satisfying
1√−∈[N
j,2Nj]fj = fj, one has the following bilinear estimates.
(1) The bilinear estimate without derivatives.
Without loss of generality, we assume N1≥N2, then for anyε >0 f1f2L2t,x(R×Bd)N
d−12 +ε
2 f1X0,b
σ (R×Bd)f2X0,b
σ (R×Bd). (5.3) (2) The bilinear estimate with derivatives.
Moreover, if fj ∈H01(Bd), then for anyε >0
∇f1f2L2t,x(R×Bd)N1N2d−12 +εf1X0,b
σ (R×Bd)f2X0,b
σ (R×Bd). (5.4) Proof of Lemma5.3 First we write
u1=
n1∼N1
cn1en1(r), u2=
n2∼N2
dn2en2(r)
wherecn1 =(u1,en1)L2 anddn2 =(u2,en2)L2. Then Sσ(t)u1=
n1∼N1
e−it z2σn1cn1en1(r), Sσ(t)u2=
n2∼N2
e−it z2σn2dn2en2(r)
Therefore, the bilinear objects that one needs to estimate are theL2t,xnorms of E0(N1,N2)=
n1∼N1
n2∼N2
e−it(z2σn1+z2σn2)(cn1dn2)(en1en2), E1(N1,N2)=
n1∼N1
n2∼N2
e−it(z2nσ1+z2nσ2)(cn1dn2)(∇en1en2).
Let us focus on (5.1) first.
(LHS of(5.15.1))2= E0(N1,N2)2L2((0,1)×Bd)
= R×Bd Here we employ a similar argument used in the proof of Lemma 2.6 in [50]. We fix η∈C0∞((0,1)), such thatη
By expanding the square above and using Plancherel, we have (5.6)= Then by Schur’s test, we arrive at
(5.7)
(2) en1en22L2(Bd)N2d−2+. Assuming Claim5.7, we see that (5.8)
τ∈N
N2d−1+
(n1,n2)∈N1,N2,τ
cn1dn22N2d−1+εu12L2(Bd)u22L2(Bd).
Therefore, (5.1) follows.
Now we are left to prove Claim5.7.
Proof of Claim5.7 In fact, (2) is due to Hölder inequality and the logarithmic bound on theLpnorm ofenin (2.1)
en1en2L2(Bd)en1
Ld−12d −(Bd)en2L2d+(Bd)n
d−1 2 −2dd+
2 =n
d−2
2 +
2 .
For (1), we have that for fixedτ ∈Nand fixedn2∼N2
z2nσ1 +z2nσ2 −τ≤ 1
2 ⇒ zn1 ∈ [(τ−1
2 −z2nσ2)21σ, (τ +1
2 −z2nσ2)21σ] There are at most 1 integerzn1 in this interval, since by convexity
(τ+1
2 −z2nσ2)2σ1 −(τ−1
2−z2nσ2)2σ1 ≤12σ1 =1 Then
#N1,N2,τ =#{(n1,n2)∈N2:n1∼N1,n2∼N2,z2nσ1 +z2nσ2 −τ≤ 1
2} ∼O(N2)
We finish the proof of Claim5.7.
The estimation of (5.2) is similar, hence omitted.
The proof of Lemma5.3is complete now.