a(t)|ξ|2|u|b2 + 1
a(t)|∂tu|b2 + ∂ta
2a2Re (∂tbu · u)b
,
which slightly differs from the one used by Tarama in [24]. From now on, the computations can be performed as in the general case, explained in the previous section, and they allow us to find the same results of [24].
We stress again the fact that the previous construction can be performed for anyp∈[1,+∞].
A Appendix – Proof of Theorem 3.13
We show here the proof of Theorem 3.13. We will follow the main steps of Proposition 4 in [13], based on Meyer’s paralinearization method, performing suitable modifications, in order to adapt the arguments to the logarithmic instance.
First of all, we introduce the telescopic series
+∞
X
j=2
Fj, with Fj := F(Sj+1u) − F(Sju), where we adopted the same notations of Subsection 3.1.
Lemma A.1. Under the hypotheses of Theorem 3.13, the series P
jFj converges to F(u) − F(S2u) in S0. Moreover one has
(49) Fj = mj∆ju , where mj :=
Z 1 0
F0(Sju + σ∆ju)dσ .
Proof. Equality (49) is a straightforward consequence of the mean value theorem. So, it’s enough to prove the convergence of the series.
Hence, for any fixed N ≥2, let us estimate
F(u) − F(S2u) −
N
X
j=2
Fj Lp
= kF(u) − F(SN+1u)kLp ≤ ku − SN+1ukLp
F0 L∞ .
Let us suppose s >0 andr <+∞ for a while. Notice that, as∇u∈Bp,r(s−1)+αlog,
(50) lim
N→+∞
X
j≥N
2j(s−1)r(1 +j)αrk∆j∇ukrLp = 0.
Thanks to spectral localization, we have u−SN+1u = P
j≥N+1∆ju; then, by Bernstein inequalities we infer
ku − SN+1ukLp ≤ X
j≥N+1
k∆jukLp
(51)
≤ C X
j≥N+1
2j(s−1)(1 +j)α k∆j∇ukLp 2−js(1 +j)−α.
We now apply Hölder inequality for series, and relation (50) allows us to conclude.
Ifr= +∞, instead, we apply Proposition 3.6 to reconduct ourselves to the previous case with a different s0 >0. Ifs= 0, instead, we use the fact thatα >1in (51).
Remark A.2. Starting from inequality (51), it’s easy to see that the previous statement is still true even ifs= 0 and α >0, under the additional assumption that αr0 > 1, with r0 defined by the relation 1/r0 + 1/r = 1.
Let us now quote Lemma 2.63 of [2].
Lemma A.3. Let g: R2 −→ R smooth, and set mj(g) := g(Sju ,∆ju) for all j∈N.
Then, for any u ∈L∞ and any ν ∈ Nd, there exists a positive constant Cν = Cν(g,kukL∞) such that, for any j∈N,
kmj(g)kL∞ ≤ Cν2j|ν|.
Finally, we need the following lemma. Recall that, for s≥0,[s]denotes the biggest integer smaller than or equal to s.
Lemma A.4. Let s >0, α∈Rand (p, r)∈[1,+∞]2.
There exists a positive constant Cs,α such that, for any sequence (uj)j∈
N of smooth functions
On the other hand, for k < j (and thenj≥1), from Bernstein inequalities we infer k∆jukkLp ≤ C2−j([s]+1) sup
Putting together estimates (52) and (53), we end up with the inequality 2js(1 +j)αk∆jukLp ≤ C (a ∗ δ)j ,
where we have definedδ = (δk)k and a= (ak)k, with ak = (1 +k)|α|
2−ks + 2−k([s]+1−s) .
This concludes the proof of the lemma whens >0.
If s= 0, it’s enough to notice that, in (52), as k≥j, 1 +j
1 +k α
≤ C ,
while the term2−x(1 +x)α, arising in (53), can be obviously bounded by a constant.
Let us come back to the proof of Theorem 3.13. By Lemma A.1, we have the decomposition F(u) −F(S2u) =
+∞
X
j=2
Fj,
whereFj is given by (49).
Remark A.5. Due to spectral localization, in every term ∆ju (with j ≥2) of the relations in (49), we can replace u byu−S0u, with
ku − S0uk
Bs+αp,r log ≤ C k∇uk
Bp,r(s−1)+αlog .
As a first step, we want to prove that F(u)−F(S2u) ∈ Bp,rs+αlog: thanks to Lemma A.4, it’s enough to prove that
sup
|ν|≤[s]+1
2j(s−|ν|)(1 +j)α k∂νFjkLp
!
j
∈ `r(N).
If we set g(ζ, ω) = R1
0 F0(ζ+σω)dσ, by Leibniz formula and Lemma A.3 we infer k∂νFjkLp ≤ X
µ≤ν
Cν,µ2j|µ|Cµ(F0, J) 2j(|ν|−|µ|)k∆jukLp
≤ Cν(F0, J) 2j|ν|k∆jukLp. Hence, from this inequality and Remark A.5, we get
2j(s−|ν|)(1 +j)α k∂νFjkLp ≤ Cν(F0, J)cjk∇uk
B(s−1)+αp,r log, wherekcjk`r = 1.
So, we have proved that F(u)−F(S2u) ∈ Bp,rs+αlog, with kF(u) − F(S2u)kBs+αlog
p,r ≤ C k∇uk
B(s−1)+αp,r log . This implies, in particular, that its gradient is inBp,r(s−1)+αlog.
Now we notice that
∇ F(S2u)
= F0(S2u)∇S2u
belongs to Lp with all its derivatives. In fact, this easily follows from the chain rule and Leibniz formula, keeping in mind that u∈L∞ and ∇S2u∈Lp with all its derivatives.
From this fact, by embeddings we gather that∇ F(S2u)
∈Bp,r(s−1)+αlog, and then also∇(F◦ u) belongs to the same space.
This completes the proof of Theorem 3.13.
Acknowledgements
The first three authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Prob-abilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The third author was partially supported by the project “Instabilities in Hydrodynamics”, funded by the Paris city hall (program “Émergences”) and the Fondation Sciences Mathématiques de Paris. He is deeply grateful to J. Rauch for useful discussions about the problem.
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