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Original citation:
Debussche, Arnaud and Weber, Hendrik (2018) The Schrödinger equation with spatial white noise
potential. Electronic Journal of Probability, 23. pp. 1-16. 28. doi:
10.1214/18-EJP143
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El e c t ro n ic
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Electron. J. Probab.23(2018), no. 28, 1–16. ISSN:1083-6489 https://doi.org/10.1214/18-EJP143
The Schrödinger equation with spatial white noise
potential
Arnaud Debussche
*Hendrik Weber
†Abstract
We consider the linear and nonlinear Schrödinger equation with a spatial white noise as a potential over the two dimensional torus. We prove existence and uniqueness of solutions to an initial value problem for suitable initial data. Our construction is based on a change of unknown originally used in [13] and conserved quantities.
Keywords:nonlinear Schrödinger equation; spatial white noise; renormalization.
AMS MSC 2010:60H15; 35Q55.
Submitted to EJP on December 12, 2016, final version accepted on January 16, 2018.
1
Introduction
In this work we study a linear or nonlinear Schrödinger equation on a periodic domain with a random potential given by a spatial white noise in dimension2. This equation is important for various purposes. In the linear case, it is used to study Anderson localisation. It is a complex version of the famous PAM model. In the nonlinear case, it describes the evolution of nonlinear dispersive waves in a totally disorder medium (see for instance [9], [12] and the references therein).
Ifudenotes the unknown, the equation is given by:
idu
dt = ∆u+λ|u|
2u+uξ, x∈
T2, t≥0,
whereT2denotes the two dimensional torus, identified with[0,2π]2, andξis a real-valued
spatial white noise. Of course,λ= 0for the linear equation. A positiveλcorresponds to the focusing case andλ <0to the defocusing case. For simplicity, we takeλ=±1in the nonlinear case. The qualitative properties of the solutions are completely different in these two cases.
We are here interested in the question of existence and uniqueness of solutions. This is a preliminary but important step before studying other phenomena, such as solitary
*Univ Rennes, CNRS, IRMAR - UMR 6625, F- 35000 Rennes, France. E-mail:arnaud.debussche@
ens-rennes.fr
waves, blow up phenomena or Anderson localisation. The main difficulty is due to the presence of the rough potential. Recall that in dimension2, a white noise has a negative regularity which is strictly less than−1. In the parabolic case (i.e. replaceidudt by dudt) renormalized solutions were recently constructed in [13]. However, their argument relies strongly on the strong regularising properties of the heat semigroup which are not available for the Schrödinger equation. Some smoothing properties for the Schrödinger equation are known when it is set on the plane. For instance in [6], section 2.5, it is shown that localized initial data yield smooth solutions. Further results are mentioned in Remark 2.7.8 of the same book. See also [8, 14, 16, 18] for further results as well as [4, 17] for deterministic nonlinear smoothing in the euclidean setting. In the periodic case, considered here however, similar linear smoothing results do not hold. Some deterministic nonlinear smoothing hold for the cubic NLS on the one dimensional torus (see [10, 11, 15]), but there is no such deterministic smoothing for the quintic or higher, and it is believed not to hold in dimensions bigger than 2 regardless of the nonlinearity. Instead of using regularising estimates for the Schrödinger semigroup we rely on the Hamiltonian structure of the equation and its conserved quantities. We use the same transformation introduced in [13]. On the level of this transformed equation, the conservation of mass and energy gives enough control to construct solutions taking values in almostH2(
T2), the standard Sobolev space of functions with derivatives up to
order2inL2(
T2). Inverting the transformation this yields solutions in almostH1(
T2).
This is rather surprising since the regularity is comparable to what is obtained in the parabolic case, when strong smoothing properties are available. However, we have to assume more structure on the initial datum than is needed in the parabolic case: On the level of the transformed equation the initial datum has to haveH2(
T2)regularity,
which translates to the assumption that the initial datum for the original equation is the product of anH2(
T2)function and a explicit function of regularity almostH1(
T2)which
depends on the specific realisation of the noiseξ.This may be surprising at a first glance but, remembering that the domain of the linear operator appearing in the equation is random ([1]), this is in fact natural.
As in the parabolic case, a renormalization is necessary and at the level of the original equation, the renormalized equation rewrites formally:
idu
dt = ∆u+λ|u|
2u+u(ξ
− ∞), x∈T2, t≥0.
The transformationu→ei∞tutransforms the original equation into the renormalized one. Therefore, the renormalization amounts to renormalize only the phase. A similar remark was made in [3] in a related but different context.
We first consider the linear equation,λ= 0. In this way the ideas can be explained in a simpler setting. We use a transformation introduced in [13] in the parabolic case, it transforms the equation into a less singular one. Conservation of theL2(T2)andH1(T2)
norm imply some bounds in these spaces, but these are not yet sufficient to construct a solution. We then use conservation of theL2(
T2)norm of the time derivative to get
bounds inH2on the transformed equation with a smoothed noise. The bound explodes
when the smoothing disappears but thanks to an idea reminiscent of interpolation theory we show that this implies uniformHγ,γ <2, bounds on the smoothed solutions. This
bound is sufficient to prove that they converge to a solution of our transformed equation. Going back to the original equation yields a solution inH1.
Forλ <0, we obtain global solutions for any initial data satisfying some smoothness assumptions. Forλ >0, as in the deterministic case, we need a smallness assumption on the initial data. The ideas are similar but the estimates are more delicate.
We could of course consider the equation with a more general nonlinearity: |u|2σu
Another easy generalization is to consider a general bounded domain and Dirichlet boundary conditions, as long as they are sufficiently smooth and the properties of the Green function of the Laplace operator are sufficiently good so that Lemma 2.1 below holds.
The study of the linear equation is closely related to the understanding of the Schrödinger operator with white noise potential. This is the subject of a recent very interesting article by Allez and Chouk ([1]) where the paracontrolled calculus is used to study the domain and spectrum of this operator. It is not clear how this can be used for the nonlinear equation.
Notation
We use the classical Lp = Lp(
T2) spaces forp ∈ [1,∞], as well as the L2 based
Sobolev spaces Hs = Hs(T2) for s ∈ R and the Besov spaces Bsp,q = Bsp,q(T2), for
s∈R, p, q∈[1,∞]. These are defined in terms of Fourier series and Littlewood-Paley theory (see [2]). Recall that fors >2
p the spaceB s
p,q(T2)embeds intoL∞(T2).
Throughout the article, cdenotes a constant which may change from one line to the next. Also, we use a small parameter0 < ε < e−1 andK
ε is a random constant
which can also change but such thatEKp
ε is uniformly bounded inεfor allp. Similarly,
for0 < ε1, ε2 < e−1, the random constant K
ε1,ε2 may depend on ε1, ε2 butEK p ε1,ε2 is
uniformly bounded inε1, ε2for allp.
2
Preliminaries
We consider the following nonlinear Schrödinger equation in dimension 2 on the torus, that is periodic boundary conditions are assumed, for the complex-valued unknown
u=u(x, t):
idu
dt = ∆u+λ|u|
2u+uξ, x∈
T2, t≥0. (2.1)
It is supplemented with initial data
u(x,0) =u0(x), x∈T2.
We need to choose the initial data of special form which depends on the realisation of the noiseξ. This will be made precise below. In the focusing caseλ >0we need an extra assumption on the size ofku0kL2 which has be small enough (see (4.3) below).
The potentialξis random and is a real valued spatial white noise onT2. For simplicity, we assume that it has a zero spatial average. The general case could be recovered by adding an additional Gaussian random potential which is constant in space. This would not change the analysis below.
Formally equation (2.1) has two invariant quantities. Given a solutionuof (2.1), the mass:
N(u(t)) =
Z
T2
|u(x, t)|2dx
is constant in time as well as the energy:
H(u(t)) =
Z
T2
1
2|∇u(x, t)| 2
−λ 4|u(x, t)|
4 −1
2ξ(x)|u(x, t)| 2dx.
This is formal because the noiseξis very rough. In dimension1, the noise has regularity
−1/2−and belongs toBα
∞,∞for anyα <−1/2, therefore the productξ|u|2can be defined
rigorously foru∈H1and this provides a bound inH1. Existence and uniqueness through
In dimension2, the noise lives in any space with regularity−1−, that is any regularity strictly less than −1, and the solution is not expected to be sufficiently smooth to compensate this. In fact, the product is almost well defined foru∈H1and we are in
a situation similar to the two dimensional nonlinear heat equation with spatial white noise (i.e. the two dimensional continuous parabolic Anderson model). We expect that a renormalization is necessary.
Inspired by [13], we introduce:
Y = ∆−1ξ
(note that this is well defined since we consider a zero average noise, we chooseY also with a zero average) andv=ueY. Then the equation forvreads
idv
dt = ∆v−2∇v· ∇Y +v|∇Y| 2+
λ|v|2ve−2Y. (2.2)
The random fieldY has regularity1−and∇Y is of regularity0−. Thus this transformation has lowered the roughness of the most irregular term on the right hand side. At this point it is easier to see why we need a renormalization: the term|∇Y|2is not well defined
since∇Y is not a function. However, the roughness is mild here and it has been known for long that up to renormalization by a log divergent constant this square term can be defined in the second order Wiener chaos based onξ.
Let us be more precise. Letρε=ε−2ρ(ε·)be a compactly supported smooth mollifier
and consider the smooth noiseξε=ρε∗ξ. We denote byYε= ∆−1ξε. Then it is proved in
[13] that for everyκ >0,ξbelongs almost surely toB−1−κ
∞,∞ and, asε→0,ξεconverges
in probability toξinB−1−κ
∞,∞ .
Also, denoting by
Cε=E |∇Yε|2
the quantity:|∇Yε|2:=|∇Yε|2−Cεconverges inLp(Ω;B∞−κ,∞)for anyp≥1,κ >0to a
random variable:|∇Y|2:in the second Wiener chaos associated toξ. It is easy to see
thatCεgoes to∞as|lnε|asε→ ∞:
E |∇Yε|2∼K0|lnε|
for someK0>0. By stationarity this quantity does not depend onx∈T2.
This discussion is summarised in the following Lemma whose proof can be found in [13, Lemma 1.1 and Proposition 1.3] in the more difficult case of the space variable in
R2.
Lemma 2.1.For1≥κ0> κ00>0and anyp≥1, there exist a constantcindependent of εsuch that:
E
kYε−Yk p
B1−∞,κ∞0
1p
≤cεκ00−2p
and
E
k:|∇Yε|2:−:|∇Y|2:k p
B−∞κ,∞0
1p
≤cεκ00−2p.
Remark 2.2.Using the monotonicity of stochastic Lp norms inp, one can drop the
exponent−2
p in the right hand side. We state the result in this way because this is the
bound that one actually proves. Below, we use this bound withκ00−2
p = κ0
2.
Note that fors <˜s,p, r≥1, we haveBs˜
∞,∞⊂Bp,rs . Thus, bounds in the latter Besov
Instead of solving equation (2.2) forvε, we consider:
idvε
dt = ∆vε−2∇vε· ∇Yε+vε:|∇Yε|
2: +λ|v
ε|2vεe−2Yε (2.3)
and settinguε=vεe−Yε:
iduε
dt = ∆uε+λ|uε| 2u
ε+uε(ξε−Cε), x∈T2, t≥0. (2.4)
SinceYεis smooth, it is classical to prove that these equations have a unique global
solution inC([0, T];Hk)for anyT >0for an initial data inHk, k= 1,2, provided the
L2norm is small forλ >0(see for instance [6], Section 3.6). More details are given in
Section 4.
The mass and energy are transformed into the two following quantities which are invariant under the dynamics forvε:
˜
Nε(vε(t)) =
Z
T2
|vε(x, t)|2e−2Yε(x)dx
and
˜
Hε(vε(t)) =
Z
T2
1
2|∇vε(x, t)| 2+1
2|vε| 2:|∇Y
ε|2:−
λ
4|vε(x, t)|
4e−2Yε(x)
e−2Yε(x)dx.
Since the most irregular term:|∇Yε|2:here is not as rough asξ, this transformed energy
is a much better quantity than the original one. It is possible to give a meaning to it for
ε= 0and use it to get bounds inH1.
Below, we use the following simple results.
Lemma 2.3.For anyκ∈(0,1)and anyp≥1, there exists a constant independent onε such that:
h
Eke−2Yε−e−2Ykp B1−κ
∞,∞
i1p
≤cεκ2.
Proof. SinceB1−κ
∞,∞is equal to the Hölder spaceC1−κ(T2)we have:
ke−2Yε−e−2Yk
B∞1−,κ∞=ke
−2Y(e−2(Yε−Y)−1)k
B∞1−,κ∞ ≤ ke −2Yk
B∞1−,κ∞ke
−2(Yε−Y)−1k B∞1−,κ∞.
Then we write:
ke−2Yk
B1−∞,κ∞ ≤2ke −2Yk
L∞kYk
B1−∞,κ∞+ke −2Yk
L∞,
ke−2(Yε−Y)−1k
B1−∞,κ∞ ≤2ke −2Yk
L∞ke−2YεkL∞kYε−Yk B1−∞,κ∞
The result follows by Hölder inequality, Lemma 2.1 (in the form of Remark 2.2 with
κ00=κ) and Gaussianity to bound exponential moments ofY andYε.
Lemma 2.4.There exists a constantcindependent ofεsuch that:
E k∇Yεk4L4
≤c|lnε|2
and
E k:|∇Yε|2:k4L4
≤c(|lnε|)4.
Proof. It suffices to write:
E
Z
T2
|∇Yε(x)|4dx
=
Z
T2E
|∇Yε(x)|4
dx= 12π2Cε2.
Similarly:
E
Z
T2
|:|∇Yε(x)|2:|4dx
=E
Z
T2
(|∇Yε(x)|2−Cε)4dx
3
The linear case
In this section, we start with the linear case:λ= 0. Then the equation forvεreads
idvε
dt = ∆vε−2∇vε· ∇Yε+vε:|∇Yε|
2:. (3.1)
There exists a unique solution in C([0, T];H2)if vε(0) ∈ H2. Indeed, forε > 0, Yε is
smooth so thatuε,0=v0e−Yε is also inH2. Existence and uniqueness of a global solution
inC([0, T];H2)∩C1([0, T];L2)of
iduε
dt = ∆uε+uε(ξε−Cε), x∈T 2, t
≥0, uε(0) =uε,0, (3.2)
is easy to prove, for instance by a fixed point argument on a mild form of the equation - recall that(eit∆)
tis a strongly continuous group of isometries on any Hs. This mild
solution satisfies (3.1) as an inequality inL2, see for instance [7], chapter 4. Settingv
ε=
uεeYε gives a solution inC([0, T];H2)∩C1([0, T];L2)to (3.1). Uniqueness ofuεimplies
uniqueness ofvε. In fact, the same argument gives a unique solution inC([0, T];Hk+2)∩
C1([0, T];Hk)ifvε(0)∈Hk,k∈N.
We take the initial data
vε(0) =v0=u0eY
and assume below that it belongs toH2. Note that this gives an initial data depending
onεforuεfor (3.2), but we recover the good initial data at the limitε→0.
The mass and energy of a solution are now:
˜
Nε(vε(t)) =
Z
T2
|vε(x, t)|2e−2Yε(x)dx
and
˜
Hε(vε(t)) =
Z
T2
1
2|∇vε(x, t)| 2+1
2|vε| 2:|∇Y
ε|2:
e−2Yε(x)dx.
They are constant in time under the evolution (3.1). This follows from standard computa-tions. Since the solution is inC([0, T];H2)∩C1([0, T];L2), these can be done rigorously.
For instance, thanks to this property we have:
d dt
Z
T2
1
2|∇vε(x, t)|
2e−2Yε(x)dx=Re
Z
T2
∆vε(x, t)
d¯vε
dt e
−2Yε(x)dx.
SinceYεconverges inB∞1−,κ∞for anyκ >0asεtends to zero, we see that the mass
gives a uniform bound inL2onv
ε. More precisely:
kvε(t)k2L2 ≤ ke
2Yεk
L∞ke−2YεkL∞kv0k2L2 =Kεkv0k2L2 (3.3)
with
Kε=ke2YεkL∞ke−2YεkL∞. (3.4)
The energy enables us to get a bound on the gradient.
Proposition 3.1.Letκ∈(0,1/2), there exists a random constantKεbounded inLp(Ω)
with respect toεfor anyp≥1such that ifv0∈H1:
Z
T2
|∇vε(x, t)|2dx≤Kε
˜
Hε(v0) +kv0k2L2
Proof. SinceB−∞κ,2is in duality withBκ
1,2we deduce by the standard multiplication rule
in Besov spaces (see e.g. [2], Section 2.8.1)
Z
T2
|vε|2:|∇Yε|2:dx
≤ kv2
εkBκ
1,2k:|∇Yε|
2:k
B−∞κ,2 ≤Kεkv
2
εkBκ
1,2 ≤Kεkvεk
2
Bκ
2,2.
Then we note thatkvεk2Bκ
2,2 =kvεk
2
Hκ so that by interpolation
Z
T2
|vε|2:|∇Yε|2:dx
≤Kεkvεk
2(1−κ)
L2 kvεk2Hκ1.
It follows
Z
T2
|∇vε(x, t)|2dx ≤KεH˜(vε(t)) +Kεkvε(t)k
2(1−κ)
L2 kvε(t)k2Hκ1
≤KεH˜ε(v0) +Kεkvε(t)k2L2+Kεkvε(t)k
2(1−κ)
L2 k∇vεk2Lκ2
≤KεH˜ε(v0) +Kεkv0k2L2+
1 2k∇vεk
2
L2
and hence, by absorbing the last term in the left hand side,
Z
T2
|∇vε(x, t)|2dx≤Kε
˜
Hε(v0) +kv0k2L2
.
SinceH˜ε(v0)is bounded forv0∈H1, we obtain a (random) bound onvεinH1using
similar arguments as above.
Corollary 3.2.There exists a random constantKεbounded inLp(Ω)with respect toε
for anyp≥1such for anyv0∈H1:
kvε(t)kH1 ≤Kεkv0kH1, t≥0.
Unfortunately, this regularity is not sufficient to control the product∇vε· ∇Yεon the
right hand side of (3.1).
The next observation is that,wε= dvdtε is formally a solution of:
idwε
dt = ∆wε−2∇wε· ∇Yε+wε:|∇Yε|
2:, (3.5)
and sincewεsatisfies the same equation asvε, it has the same invariant quantities. We
use in particular the mass:
˜
N(wε(t)) = ˜N(wε(0)).
Hence:
kwε(t)k2L2 ≤KεN(w˜ ε(0))≤Kεkwε(0)k2L2. (3.6)
This formal argument can be justified as follows. We take a sequence (vε,η(0))η>0 in H4 which converges tovε(0)inH2. Then, as noted above, the corresponding solution
vε,η of (3.2) lives inC([0, T];H4)∩C1([0, T], H2)and converges tovε inC([0, T];H2)∩
C1([0, T], L2). The argument above holds forw
ε,η= dvε,η
dt and
kwε,η(t)k2L2 ≤KεN(w˜ ε,η(0))≤Kεkwε,η(0)k2L2.
Lettingη→0, we obtain that (3.6) is true under the assumption thatv0∈H2.
Proposition 3.3.There exists a random constantKεbounded inLp(Ω)with respect to
εfor anyp≥1such that ifv0∈H2:
Proof. From (3.1), we have:
wε(0) =−i(∆v0−2∇v0· ∇Yε+v0:|∇Yε|2:),
so that, thanks to the embeddingH1/2⊂L4,
kwε(0)kL2 ≤c kv0kH2+kv0kH3/2k∇YεkL4+kv0kH1/2k:|∇Yε|2:kL4.
By interpolation we deduce:
kwε(0)kL2 ≤c
kv0kH2+kv0k3/4 H2kv0k
1/4
L2 k∇YεkL4+kv0k1/4 H2kv0k
3/4
L2 k:|∇Yε|2:kL4
≤ckv0kH2+kv0kL2k∇Yεk4L4+kv0kL2k:|∇Yε|2:k4/3 L4
≤c kv0kH2+Kεkv0kL2|lnε|2,
(3.7) where:
Kε=k∇Yεk4L4|lnε|−
2+
k:|∇Yε|2:k
4/3
L4|lnε| −2.
By Lemma 2.4 and gaussianity, we know that the moments of this random variable are bounded with respect toε.
It follows
kwε(t)kL2 ≤Kε kv0kH2+kv0kL2|lnε|2
.
This in turn allows us to controlkvεkH2. Indeed, from (3.1),
k∆vεkL2 ≤ kwε(t)kL2+ 2k∇vε· ∇YεkL2+kvε:|∇Yε|2:kL2
and by similar arguments as above
k∆vεkL2≤ kwε(t)kL2+
1
2k∆vεkL2+cKεkvε(t)kL2|lnε| 2
and finally
k∆vεkL2 ≤Kε kv0kH2+kv0kL2|lnε|2
.
The result follows thanks to (3.3).
This bound does not seem to be very useful since it explodes asε→0. To use it, we consider the difference of two solutions.
Proposition 3.4.Letε2> ε1>0then forκ∈(0,1],p≥1there exists a random constant Kε1,ε2 bounded inL
p(Ω)with respect toε1, ε2for anyp≥1such that ifv0∈H2
sup
t∈[0,T]
kvε1(t)−vε2(t)k
2
L2 ≤Kε1,ε2ε κ/2 2 |lnε2|
1+2κ
kv0k2H2
Proof. We setr=vε1−vε2 and write:
idr
dt = ∆r−2∇r· ∇Yε1+r:|∇Yε1|
2:−2∇v
ε2· ∇(Yε1−Yε2) +vε2(:|∇Yε1|
2:−:|∇Y
ε2|
2:).
By standard computations, we deduce:
1 2
d dt
Z
T2
|r(x, t)|2e−2Yε1(x)dx
=Im
Z
T2
−2∇vε2· ∇(Yε1−Yε2) +vε2(:|∇Yε1|
2:−:|∇Y
ε1|
2:)
¯
re−2Yε1(x)dx
≤2k∇vε2re¯
−2Yε1kBκ
1,2k∇(Yε1−Yε2)kB∞−κ,2+kvε2¯re
−2Yε1kBκ
1,2k:|∇Yε1|
2:−:|∇Y
ε2|
2:k
the first term of the right hand side is bounded thanks to interpolation and paraproduct inequalities (see [2]) and we have
k∇vε2¯re
−2Yε1kBκ
1,2 ≤ckvε2kH1+κ(kvε1kHκ+kvε2kHκ)ke
−2Yε1kBκ
∞,2.
Then, by Proposition 3.3 and interpolation:
kvε2kH1+κ ≤ckvε2k
(1+κ)/2
H2 kvε2k
(1−κ)/2
L2 ≤Kε2(kv0kH2+kv0kL2|lnε2|
2)(1+κ)/2kv0k(1−κ)/2
L2
and, by Corollary 3.2, fori= 1,2
kvεikHκ ≤ kvεik κ H1kvεik
1−κ
L2 ≤Kεikv0k κ H1kv0k
1−κ
L2 ≤Kεikv0k κ/2
H2kv0k
1−κ/2
L2 .
It follows
k∇vε2re¯
−2Yε1kBκ
1,2 ≤Kε1,ε2ke
−2Yε1kBκ
∞,2kv0k 3 2−κ
L2 (kv0kH2+kv0kL2|lnε2|2) 1 2+κ
The second term is bounded by the same quantity and we deduce:
1 2 d dt Z T2
|r(x, t)|2e−2Yε1(x)dx
≤Kε2ke
−2Yε1kBκ
∞,2kv0k 3 2−κ
L2 (kv0kH2+kv0kL2|lnε2|2) 1 2+κ
×k∇(Yε1−Yε2)kB∞−κ,2+k:|∇Yε1|
2:
−:|∇Yε2|
2: kB−κ
∞,2
.
The result follows thanks to integration in time and Lemma 2.1. For instance, we have:
k∇(Yε1−Yε2)kB∞−κ,2 ≤ck∇(Yε1−Yε2)k B−
7κ
8 ∞,∞
≤ckYε1−Yε2k B1−
7κ
8 ∞,∞
≤Kε1,ε2ε
κ
2
2,
thanks to Lemma 2.1 withκ0= 78κ,κ00= 34κ andp= 8κ.
By interpolation, we deduce from Proposition 3.3 and 3.4 the following result.
Corollary 3.5.Letε2> ε1>0then forκ∈(0,1],γ∈[0,2),p≥1there exists a random constantKε1,ε2bounded inL
p(Ω)with respect toε1, ε2for anyp≥1such that ifv0∈H2:
sup
t∈[0,T]
kvε1(t)−vε2(t)k
2
Hγ ≤Kε1,ε2ε
κ
2(1−
γ
2)
2 |lnε1| 4
kv0k2H2
We are now ready to state and prove the main result of this section.
Theorem 3.6.Assume that v0 = u0eY ∈ Lp(Ω;H2) for some p > 1. For any T ≥ 0, q < p,γ∈(1,2), whenε→0, the solutionvεof (3.1)satisfyingvε(0) =v0converges in
Lq(Ω;C([0, T];Hγ))tovwhich is the unique solution to
idv
dt = ∆v−2∇v· ∇Y +v:|∇Y|
2: (3.8)
inC([0, T];Hγ)such thatv(0) =v
0. Here, by solution, we mean that (3.8)holds as an equality inC([0, T];Hγ−2)
Proof. Let us first prove pathwise uniqueness. Since the equation is linear, this amounts to prove that a solution withv(0) = 0is0. Let us consider such a solution and write:
Z
T2
|v(x, t)|2e−2Y(x)dx= lim
ε→0
Z
T2
as can be seen from Lemma 2.1. Then,v∈C([0, T];Hγ)implies that dv
dt ∈C([0, T];H γ−2)
and we may differentiate forε >0:
d dt
Z
T2
|v(x, t)|2e−2Yε(x)dx= 2(dv
dt, ve
−2Yε) =−2(i∆v−2i∇v· ∇Y +iv:|∇Y|2:, ve−2Yε),
where (·,·) has to be understood as the duality betweenHγ−2 andH2−γ - note that
2−γ≤γ. Recall that:
(i∆v−2i∇v· ∇Yε+iv:|∇Yε|2:, ve−2Yε) = 0,
and deduce:
d dt
Z
T2
|v(x, t)|2e−2Yε(x)dx= (−2i∇v· ∇(Y −Y
ε) +iv(:|∇Yε|2:−:|∇Yε|2) :, ve−2Yε).
We then repeat the estimate of the proof of Proposition 3.4 but estimate theH1+κnorm
by theHγ norm and obtain
d dt
Z
T2
|v(x, t)|2e−2Yε(x)dx
≤Kεε κ
2.
The random constantKεdepends on the norm ofv1, v2inC([0, T];Hγ)and onYε. We
take ε = 2−k. By Lemma 2.1 and Borel-Cantelli, we know that sup
kK2−k < ∞ a.s.. Integrating in time and lettingk→ ∞, we getR
T2|v(x, t)|
2e−2Y(x)dx= 0.
Now letεk = 2−k, Corollary 3.5 implies that(vεk)is Cauchy inL
q(Ω;C([0, T];Hγ)). It
is not difficult to prove that the limitvis a solution of (3.8) and
E sup
t∈[0,T]
kvεk(t)k q Hγ
!
≤cE kv0kpH2
q/p
,
E sup
t∈[0,T]
kv(t)kqHγ
!
≤cE kv0kpH2
q/p
.
Letε >0andksuch thatεk< ε. By interpolation, withγ <˜γ <2,
sup
t∈[0,T]
kvε(t)−vεk(t)k
2
Hγ
≤c sup
t∈[0,T]
kvε(t)−vεk(t)k
2(1−γ/˜γ)
L2 sup
t∈[0,T]
kvε(t)−vεk(t)k
2γ/˜γ Hγ˜
≤c sup
t∈[0,T]
kvε(t)−vεk(t)k
2(1−γ/˜γ)
L2 sup
t∈[0,T]
(kvε(t)kH2+kvεk(t)kHγ˜)
2γ/˜γ
.
By Proposition 3.4, Proposition 3.3 and the above inequality, we deduce:
E sup
t∈[0,T]
kvε(t)−vεk(t)k q Hγ
!
≤cεκ2(2−γ/γ˜))|lnε|4E kv0kp H2
q/p
.
Letting k → ∞, we deduce that the whole family(vε)ε>0 converges tov in Lq(Ω; C([0, T];Hγ)). It remains to letε→0in each term of (3.1) to prove thatvis a solution.
4
The nonlinear equation
We now study the nonlinear equation (2.2) and consider its approximation (2.3) with initial condition
and assume below that it belongs toH2. Using the same argument as in [5], we can
prove existence of a solution to (2.4) inC([0, T∗);H2)∩C1([0, T∗);L2)whereT∗ is the
maximal time of existence. It is either infinite or finite and in the latter case theH2norm
blows up. This implies local existence and uniqueness for (2.3). Proposition 4.2 below shows that, under condition 4.2, theH2norm is bounded on finite time intervals and thus implies thatT∗=∞.
Again the solution is sufficiently regular to justify the computations yielding conser-vation of the mass and energy.
The mass gives a uniform bound inL2onv
ε:
kvε(t)k2L2≤Kεkv0k2L2. (4.1)
The estimate on theH1norm using the energy is similar to the linear case. Recall
that the energy is given by:
˜
Hε(vε(t)) =
Z
T2
1
2|∇vε(x, t)| 2+1
2|vε| 2:|∇Y
ε|2:−
λ
4|vε(x, t)|
4e−2Yε(x)
e−2Yε(x)dx
and it can be checked that for allt≥0we haveH˜ε(vε(t)) = ˜Hε(v0).
Proposition 4.1.There exists a constantKεbounded inLp(Ω)for anyp≥1such that if
v0∈H1and
ke−2Yεk3
L∞ke2YεkL∞kv0kL2 ≤1ifλ= 1 (4.2)
then
kvε(t)k2H1 ≤Kε kv0k2H1+kv0k2L2kv0k2H1
.
Moreover, the sequence(Kk) = (K2−k)is bounded almost surely, i.e.supkK2−kis almost
surely finite.
Proof. We proceed as in the proof of Proposition 3.1. We first have
Z
T2
v2ε:|∇Yε|2:dx
≤Kεkvεk
2(1−κ)
L2 kvεk2Hκ1
and
Z
T2
|∇vε(x, t)|2dx
≤KεH˜(vε(t)) +Kεkvε(t)k
2(1−κ)
L2 kvε(t)k2Hκ1+
λ 4
Z
T2
|vε(x, t)|4e−2Yε(x)e−2Yε(x)dx
≤KεH˜ε(v0) +Kεkv0k2L2+
1 2k∇vεk
2
L2+
λ 4
Z
T2
|vε(x, t)|4e−2Yε(x)e−2Yε(x)dx.
Forλ=−1, the result follows after dropping the last term and using
Z
T2
|v0(x)|4e−4Yε(x)dx≤K
εkv0k4L4≤Kεkv0k4H1/2 ≤Kεkv0k2L2kv0k2H1
thanks to the Sobolev embeddingH1/2⊂L4and interpolation.
Forλ= 1, Gagliardo-Nirenberg inequality (see for instance [5] for a simple proof with the constant1/2used below):
Z
T2
|vε(x, t)|4e−4Yε(x)dx ≤ ke−2Yεk2L∞
Z
T2
|vε(x, t)|4dx
≤ 1 2ke
−2Yεk2 L∞
Z
T2
|vε(x, t)|2dx
Z
T2
|∇vε(x, t)|2dx
≤ 1 2ke
−2Yεk3
L∞ke2YεkL∞kv0k2L2
Z
T2
where we have made use ofkvε(t)k2L2 ≤ ke2YεkL∞ke−2Yεk
L∞kv0k2
L2 according to (3.3).
The result follows easily under assumption (4.2).
The constant Kε is a polynomial in k : |∇Yε|2 : kB−κ
∞,2, ke −2Yεk
L∞ and ke2YεkL∞.
By Lemma 2.1 and Lemma 2.3 and Borel-Cantelli, we know that : |∇Y2−k|2 : and
Y2−k converge almost surely inB∞−κ,∞ andB1∞−,∞κ so thatKk is indeed bounded almost
surely.
We now proceed with theH2bound.
Proposition 4.2.There exists a random constantsKεbounded inLp(Ω)with respect to
εfor anyp≥1such that ifv0=u0e−Y ∈H2and (4.2)holds:
kvε(t)kH2
≤cKε 1 +kv0kH2+kv0kL2|lnε|4+kv0k3H1+kv0k3L2kv0k3H1
exp(Kε(kv0k2H1+kv0k2L2kv0k2H1)t).
MoreoverKk =K2−k is bounded almost surely.
Proof. As in Section 3, we setwε= dvdtε which now satisfies:
idwε
dt = ∆wε−2∇wε· ∇Yε+wε:|∇Y| 2: +λ
|vε|2wε+ 2Re(vεw¯ε)vε
e−2Yε.
From (2.3), we have:
wε(0) =−i(∆v0−2∇v0· ∇Yε+v0:|∇Yε|2:) +λ|v0|2v0e−2Yε,
and as in Proposition 3.3 and using the embeddingH1⊂L6:
kwε(0)kL2≤ckv0kH2+kv0kL2
k∇Yεk4L4+k:|∇Yε|2:k
4/3
L4
+ke−2Yεk
L∞kv0k3H1.
By Lemma 2.4:
P(k∇Yεk4L4+k:|∇Yε|2:k
4/3
L4 ≥ |lnε|
4)≤c|lnε|−2
it follows that
kwε(0)kL2≤ckv0kH2+Kεkv0kL2|lnε|4+Kεkv0k3H1
withKεhaving all moments finite and such thatK2−kis almost finite by Borel-Cantelli.We have taken |lnε|4 instead of |lnε|2 in the estimate above in order to have this latter
property. Recall thatY2−kconverges a.s. inL∞. We do not have preservation of theL2norm but:
1 2
d dt
Z
T2
|wε(x, t)|2e−2Yε(x)dx
= 2λ
Z
T2
Re(vε(x, t) ¯wε(x, t))Im(vε(x, t) ¯wε(x, t))e−4Yε(x)dx
≤Kεkvε(t)k2L∞
Z
T2
|wε(x, t)|2e−2Yε(x)dx
≤Kεkvε(t)k2H1(1 + ln(1 +kvε(t)kH2))
Z
T2
|wε(x, t)|2e−2Yε(x)dx
≤Kε kv0k2H1+kv0k2L2kv0k2H1
(1 + ln(1 +kvε(t)kH2))
Z
T2
|wε(x, t)|2e−2Yε(x)dx
thanks to the Brezis-Gallouet inequality ([5]) and to Proposition 4.1.
This computation is not rigorous. We proceed as in the linear case to justify it. We take a sequence of smooth initial data, say inH4. In this case, using Theorem 5.4.1 in
theorem is proved in theR2framework but the proof in fact works on any domain. The
estimate above is obtained at the limit for an initial data inH2.
Then as above we have:
k∆vε(t)kL2 ≤2kwε(t)kL2+cKεkvε(t)kL2|lnε|4+λk|vε(t)|2vεe−2YεkL2
≤2kwε(t)kL2+Kεkvε(t)kL2|lnε|4+Kεkvε(t)k3L6.
By the embeddingH1⊂L6and Proposition 4.1, we deduce
kvε(t)kH2 ≤ckwε(t)kL2+Kε kv0kL2|lnε|4+kv0k3H1+kv0k3L2kv0k3H1
.
Again, the almost sure boundedness of the different constantKk is obtained thanks to
Lemma 2.3, Lemma 2.4 and Borel-Cantelli.
To lighten the following computation, we use the temporary notations:
˜
wε=kwεe−Yεk2L2, αε=Kε kv0k2H1+kv0k2L2kv0k2H1
,
βε=Kε kv0kL2|lnε|4+kv0k3H1+kv0k3L2kv0k3H1
.
Then we have:
d
dtw˜ε ≤αε(1 + ln(1 +kvεkH2)) ˜wε ≤αε(1 + ln(1 +ckwεkL2+βε)) ˜wε
≤αε(1 + ln(1 +Kεw˜ε+βε)) ˜wε.
Hence
d
dt(1 + ln(1 +Kεw˜ε(t) +βε))≤αε(1 + ln(1 +Kεw˜ε(t) +βε)).
By Gronwall’s Lemma we deduce:
1 + ln(1 +Kεw˜ε(t) +βε)≤(1 + ln(1 +Kεw˜ε(0) +βε)) exp(αεt)
and taking the exponential
kwε(t)kL2 ≤Kεw˜ε(t)≤cKε(1 +Kεw˜ε(0) +βε)exp(αεt)
≤Kε(1 +Kεkwε(0)kL2+βε)exp(αεt)
≤Kε(1 +Kεkv0kH2+βε)exp(αεt).
The result follows.
We see that thisH2 bound is not as good as in the linear case. Due to the double
exponential, we do not have moments here. However, this is sufficient to prove existence and uniqueness. We now state the main result of this section.
Theorem 4.3.Assume thatv0=u0eY ∈H2and
ke−2Yk3
L∞ke2YkL∞kv0kL2<1ifλ= 1. (4.3)
For anyT≥0,p≥1,γ∈(1,2), whenε→0, the solutionvεof (2.3)satisfyingvε(0) =v0
converges in probability inC([0, T];Hγ))tovwhich is the unique solution to
idv
dt = ∆v−2∇v· ∇Y +v:|∇Y|
2: +λ|v|2ve−2Y (4.4)
Proof. Pathwise uniqueness is proved as in the linear case. Letv1, v2be two solutions of
(4.4) with paths inC([0, T];Hγ)and starting with the same initial data. We setr=v
1−v2
which satisfies
idr
dt = ∆r−2∇r· ∇Y +r:|∇Y|
2: +λ(|v1|2v1− |v2|2v2)e−2Y.
The following computation can be done thanks to the regularity ofv1andv2:
d dt
Z
T2
|r(x, t)|2e−2Yε(x)dx
= 2(dr dt, re
−2Yε)
=−2(i∆r−2i∇r· ∇Y +ir:|∇Y|2: +λ(|v
1|2v1− |v2|2v2)e−2Y, re−2Yε)
≤(−2i∇r· ∇(Y −Yε) +ir(:|∇Y|2:−:|∇Yε|2) : +λ(|v1|2v1− |v2|2v2)e−2Y, re−2Yε).
After similar manipulations as above:
d dt
Z
T2
|r(x, t)|2e−2Yε(x)dx≤K
ε(εκ/2+
Z
T2
|r(x, t)|2e−2Yε(x)dx)
withKεsuch thatsupkK2−k <∞a.s.. It remains to use Gronwall Lemma, takeε= 2−k and letk→ ∞to conclude thatr= 0a.s.
Under assumption (4.3), we know that (4.2) holds forεsmall enough. Thus we may use Propositions 4.1 and 4.2.
We takeε2> ε1>0, setr=vε1−vε2 and write:
idr
dt = ∆r−2∇r· ∇Yε1+r:|∇Yε1|
2:
−2∇vε2· ∇(Yε1−Yε2) +vε2(:|∇Yε1|
2:
−:|∇Yε2|
2:)
+λ|vε1|
2re−2Yε1 −λ(|v ε2|
2− |v
ε1|
2)v
ε2e
−2Yε1 +λ|v ε2|
2v
ε2(e
−2Yε1 −e−2Yε2).
By the same arguments as in Section 3 and standard estimates we obtain;
1 2
d dt
Z
T2
|r(x, t)|2e−2Yε1(x)dx
≤cke−2Yε1kBκ
∞,2kvε2(t)kH1+κ(kvε1(t)kHκ+kvε2(t)kHκ)
×k∇(Yε1−Yε2)kB−κ
∞,2+k:|∇Yε1|
2
:−:|∇Yε2|
2 :kB−κ
∞,2
+Kε1(kvε1kL∞+kvε2kL∞)kvε2kL∞
Z
T2
|r(x, t)|2e−2Yε1(x)dx
+Kε1kvε2k
3
L∞(kvε1kL2+kvε2kL2)ke
−2Yε1 −e−2Yε2kL2
≤Kε1,ε2kvε2(t)k κ
H2(kvε1(t)kH1+kvε2(t)kH1)
2−κεκ/2 2 +Kε1(kvε1kH1+kvε2kH1)
2(1 + ln(1 +
kvε1kH2) + ln(1 +kvε2kH2))
×
Z
T2
|r(x, t)|2e−2Yε1(x)dx
+Kε1,ε2kvε2k
3
H1(1 + ln(1 +kvε1kH2))
3/2(kv
ε1kL2+kvε2kL2)ε κ
2,
by the Brezis-Gallouet inequality. We have used forκ∈(0,1):
ke−2Yε1 −e−2Yε2kL2 ≤2kYε1−Yε2kL2(ke
−2Yε1kL∞+ke−2Yε2kL∞)
≤c|ε2|κkYk Bκ
∞,∞(ke
−2Yε1kL∞+ke−2Yε2kL∞)
Again, the constantsKε1,Kε1,ε2 above have all moments bounded independently ofε1, ε2
are almost surely bounded inkforε1= 2−(k+1),ε2= 2−k.
We deduce from Gronwall’s Lemma:
lnkrkL2 ≤Kε1,ε2P(v0)(1 + lnP(v0) + ln|lnε1|)
exp(Kε1,ε2P(v0)t)
+κexp(Kε1,ε2P(v0)t)(lnKε1,ε2+ lnP(v0) + ln|lnε1|)−
κ 2|lnε2|,
whereP(v0)denotes a polynomial inkv0kH2. By interpolation, we have
lnkrkHγ≤c+ (1−
γ
2) lnkrkL2+ γ
2 lnkrkH2
so that a similar estimate holds forlnkrkHγ,γ <2.
The constantsKε1,ε2 are bounded almost surely whenε1= 2
−(k+1), ε2= 2−kso that
kv2−(k+1)−v2−kkHγ ≤C(v0)|ln 2−(k+1)|C(v0)2− κ
2(k+1),
where now C(v0) is a random constant depending on kv0kH2. It follows thatv2−k is Cauchy inC([0, T], Hγ).
Finally, we reproduce the estimate above forkvε(t)−v2−kkL2 but boundkv2−kkL∞by
kv2−kkHγ instead of using Brezis-Gallouet inequality. We obtain:
lnkvε(t)−v2−kkL2 ≤K˜ε,2−kP(v0)(1 + lnP(v0) + ln|lnε|)exp( ˜Kε,2−kP(v0)t)
+κexp( ˜Kε,2−kP(v0)t)(ln ˜Kε,2−k+ lnP(v0) + ln|lnε|)− κ 2|lnε|,
where K˜ε,2−k are constants with moments bounded independently onεandk. Again, a similar bound holds for theHγ norm thanks to an interpolation argument. Letting
k→ ∞yields:
lnkvε(t)−vkL2 ≤K˜εP(v0)(1 + lnP(v0) + ln|lnε|)exp( ˜KεP(v0)t)
+κexp( ˜KεP(v0)t)(ln ˜Kε+ lnP(v0) + ln|lnε1|)−
κ 2|lnε|,
where againK˜εare constants with bounded moments.
The conclusion follows.
Remark 4.4.Condition (4.3) is probably not optimal. It can easily be weakened to
λK3
2,εK1,εkv0kL2 <2, but this is probably not optimal either.
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