Preliminaries : the functional framework

In this section, we identify the Sobolev spaces adapted to the operators H_x and H_z. More specifically, we identify the norm

kuk²_X ∼ kuk²_L2(Ω)+kH_xuk²_L2(Ω)+kH_zuk²_L2(Ω)

whenever u is smooth enough.

Moreover, we also define a regularization space that we denote Y, and, in that view, we also identify the norm

kuk²_Y:=kuk²_L2(Ω)+kH_xuk²_X+kH_z^1+α/2uk²_L2(Ω)

where α∈R, such that0< α < ¹₂.

The identification of both norms kuk_X andkuk_Y may be technically delicate, yet, it is absolutely crucial in our work. Indeed, the only uniform-in-ε bound at hand on ψ^ε, solution to (4.1.1)-(4.1.5) reads :

kψ^εk²_L2(Ω)+kH_xψ^εk²_L2(Ω)+kH_zψ^εk²_L2(Ω) =O(1)

on some non-trivial time interval [0, t]whenever the initial datum belongs to X. All other energy estimates, (obtained by simply applying the operators ∂_z,∇_x, V₁(x) or V₂(z) to equation (4.1.1)-(4.1.5) and integrating by part) give rise to commutators, hence diverging factors of order O(_ε¹2)due to the term _ε¹2H_z. Therefore, as they only give access to bounds of order O(¹_ε), they are barely useless here.

In order to identify these spaces, we refer the reader to the work done in [1]. In this paper, the authors identify the Sobolev scale adapted to their own operators that are −∆_x+V₁(x) with domain {u ∈ L²(R²), −∆_xu ∈ L²(R³), V₁u∈ L²(R²)}, and −∂_z² +V₂ with domain {u ∈ L²(R), ∂_z²u ∈ L²(R), V₂u ∈ L²(R)}. The only difference with our situation is therefore the fact that the transverse operator acts onL²(R), instead of L²(0,1)with boundary Dirichlet conditions. The key tool they use is the Weyl-Hörmander calculus, and, following the same arguments, we can state the following lemma :

Lemma 4.2.1. [Equivalence of norms]

For all u ∈ X, where (X,kk_X) is defined in (4.1.33)-(4.1.34), if u = P

pu_pχ_p with u_p(x) :=R1

0 u(x, z)χ_p(z)dz, the following equivalence holds :

kuk²_X ∼ |uk²_L2(Ω)+kH_xuk²_L2(Ω)+kH_zuk²_L2(Ω) (4.2.1)

=X

p≥0

(1 +E_p²)ku_pk²_L2(R²)+kH_xuk²_L2(Ω). (4.2.2) Moreover, fix 0< α < ¹₂ and define

Y:={u∈H₀¹(Ω), Hxu∈X, H_z^1+α/2u∈L²(Ω)}

then, for all u∈Y, if u= P

p≥0

upχp with up(x) =R1

0 u(x, z)χp(z)dz, then the following equivalence holds :

kuk²_Y:=kuk²_L2(Ω)+kHxuk²_X+kH_z^1+α/2uk²_L2(Ω) (4.2.3)

=X

p≥0

(1 +E_p^2+α)ku_pk²_L2(R²)+kH_xuk²_X (4.2.4)

∼ kuk²_H2(Ω)+k∆_xuk²_H2(Ω)+kV₁²uk²_L2(Ω)+X

p≥0

E_p^2+αku_pk²_L2(R²) (4.2.5)

∼ kuk²_H2+α(Ω)+kH_xuk²_X. (4.2.6) Proof. This lemma can be proved by combining Proposition 2.5 in [1] in order to prove the following equivalence :

H_xu∈L²(Ω) if and only if kuk²_H2(Ω)+kV₁uk²_L2(Ω)

and the following result that can be found in [17] or [18] : if 0< α < 1

2, u∈H₀¹(Ω) and H_z^1+α/2u∈L²(Ω)

⇐⇒X

p≥0

E_p^2+αkupk²_L2(R²)<∞.

Lemma 4.2.2. [Properties of the Sobolev spaces X and Y]

For any fixed 0< α < ¹₂, X and Y are continuously injected in L^∞(Ω). Moreover, X and Y are algebras, and the embedding Y⊂X is compact.

Proof. The fact that X and Y are continuously injected in L^∞(Ω) readily comes from the fact that they are continuously embedded in H²(Ω) that is continuously embedded inL^∞(Ω).

Secondly, it is clear that H²(Ω)∩H₀¹(Ω) is an algebra. Therefore, Xclearly is an algebra too according to definition (4.1.33). As far as Y is concerned, take u, v ∈Y, then, uv ∈H²(Ω)∩H₀¹(Ω), V₁²uv ∈L²(Ω) since V₁²u∈L²(Ω) and Y⊂ L^∞(Ω) with continuous embedding. The same arguments allows us to prove that, if

uv =X

p≥0

(uv)_pχ_p, with (uv)_p :=

Z 1

0

u(x, z)v(x, z)χ_p(z)dz then,

X

p≥0

E_p^2+αk(uv)_pk²_L2(R²) ≤ kuk²_L∞

X

p≥0

E_p^2+αkv_pk²_L2(R²)

≤Ckuk²_Ykvk²_Y.

In order to prove that uv ∈ Y, we need to prove that ∆x(uv)∈ Y. In that view, let us write

∆_x(uv) = (∆_xu)v+u(∆_xv) + 2 ∇_xu· ∇_xv.

However, ∆xu∈H²(Ω) and v ∈H²(Ω), therefore,

(∆_xu)v ∈H²(Ω) and k(∆_xu)vk_H²_(Ω) ≤ kuk²_Ykvk²_Y. Similarly, u(∆_xv)∈H²(Ω). Finally, asu v ∈Y, then

∇_xu,∇_xv ∈H²(Ω) and k∇_xu· ∇_xvk²_H2(Ω) ≤2kuk²_Ykvk²_Y. Consequently,

∆_x(uv)∈H²(Ω), k∆_x(uv)k_H²_(Ω) ≤Ckuk_Ykvk_Y

where C >0 does not depend on u orv. To conclude, the spaces X and Y are both algebras and we have :

∀u, v ∈X kuvk_X ≤Ckuk_Xkvk_X

∀u, v ∈Y, kuvk_Y≤Ckuk_Ykvk_Y where C >0 does not depend on u and v.

Finally, we clearly have the embedding Y⊂X. Its compactness is due to the fact that the embedding H^2+α(Ω)∩H₀¹(Ω) ⊂ H²(Ω)∩H₀¹(Ω) is locally compact since 2<2 +α < ⁵₂ together with the fact that V₁(x)goes to infinity at infinity.

We end this section by the following lemma : Lemma 4.2.3.

∀u∈Y, k∇_xuk²_X ≤Ckuk_Xkuk_Y.

Proof. In order to prove this estimate, let us consider u∈Y. Then,

k∇xuk²_X ≤ kH_x^1/2uk²_X :=kH_x^1/2uk²_L2(Ω)+kH_x^3/2uk²_L2(Ω)+kHzH_x^1/2uk²_L2(Ω). (4.2.7) However,

kH_x^1/2uk²_L2(Ω) :=hH_x^1/2u, H_x^1/2ui_L²_(Ω) =|hH_xu, ui_L²_(Ω)|

≤ kH_xuk_L²_(Ω)kuk_L²_(Ω) ≤Ckuk²_X,

kH_x^3/2uk²_L2(Ω) :=hH_x^3/2u, H_x^3/2ui_L²_(Ω) =|hH_x²u, H_xui_L²_(Ω)|

≤ kH_x²uk_L²_(Ω)kH_xuk_L²_(Ω) ≤ kuk_Ykuk_X

and, finally, using the fact that both operatorsH_x and H_z commute, we get : kH_zH_x^1/2uk²_L2(Ω) :=hH_zH_x^1/2u, H_zH_x^1/2ui_L²_(Ω)=|hH_xH_zu, H_zui_L²_(Ω)|

≤ kH_zH_xuk_L²_(Ω)kH_zuk_L²_(Ω) ≤ kuk_Ykuk_X which, combined with (4.2.7) allows us to conclude.

4.2.2 A priori estimates

In this subsection, we state a priori estimates on both nonlinearities V^ε and W defined in (4.1.4)-(4.1.5) and (4.1.13)-(4.1.14) respectively. In that view, we state the following regularity result.

Lemma 4.2.4. Consider any real number ε∈[0,1], any function f ∈X. Then, the following system

−∂_z²u^ε−ε²∆_xu^ε =f (x, z)∈Ω, (4.2.8) u^ε(x,0) =u^ε(x,1) = 0 x∈R². (4.2.9) admits a unique solution u^ε and the following holds :

ku^εk_X ≤Ckfk_X, (4.2.10)

k∂_z²u^εk_X ≤Ckfk_X. (4.2.11) To be more readable, the proof of this lemma is postponed to Appendix (4.A).

Corollary 4.2.5. Let ε∈[0,1], and define the nonlinearityF^ε(u) as eitherV^ε(|u|²) or W(|u|²). Then, the following holds :

∀u∈X, kF^ε(u)k_X ≤Ckuk²_X (4.2.12)

∀u∈X, kF^ε(u)uk_X ≤Ckuk³_X (4.2.13)

∀u, v ∈X, kF^ε(u)u−F^ε(v)vk_X ≤C kuk²_X+kvk²_X

ku−vk_X. (4.2.14) Moreover :

∀u∈Y, kW(|u|²)k_Y ≤Ckuk_Ykuk_X (4.2.15)

∀u, v ∈Y,kW(uv)k_Y ≤Ckuk_Ykvk_Y (4.2.16)

∀u∈Y, kW(|u|²)uk_Y ≤Ckuk²_Xkuk_Y (4.2.17)

∀u, v ∈Y, kW(|u|²)u−W(|v|²)vk_Y ≤C kuk²_Y+kvk²_Y

ku−vk_Y. (4.2.18) Proof of Corollary 4.2.5 In order to prove the first part of Corollary 4.2.5, we fix ε≥0,u∈X, and we apply Lemma 4.2.4 to the nonlinearityF^ε(u). Indeed,V^ε(|u|²) and W(|u|²)solve the system (4.2.8)-(4.2.9) for f =u, ε= 1 and ε = 0 respectively.

We therefore get :

kF^ε(u)k_X ≤Ck|u|²k_X ≤Ckuk²_X and

kF^ε(u)uk_X ≤Ckuk³_X

where we used the fact that X is an algebra. Estimates (4.2.12) and (4.2.13) are proved.

As far as the estimate (4.2.14) is concerned, first note that, if u, v ∈X,

F^ε(u)u−F^ε(v)v = (F^ε(u)−F^ε(v))u+F^ε(v)(v−u). (4.2.19)

Moreover, F^ε(u)−F^ε(v) readily satisfies the following system :

−∂_z²(F^ε(u)−F^ε(v))−ε²∆x(F^ε(u)−F^ε(v)) = |u|²− |v|², (F^ε(u)−F^ε(v)) (x,0) = (F^ε(u)−F^ε(v)) (x,1) = 0

Therefore, applying Lemma 4.2.4 yields :

kF^ε(u)−F^ε(v)k_X ≤Ck|u|²− |v|²k_X ≤Ck(|u|+|v|)(|u| − |v|)k_X

≤C(kuk_X+kvk_X)ku−vk_X. (4.2.20) Combining (4.2.19) and (4.2.20) with (4.2.12) finally provides us with estimate (4.2.14).

In order to prove the second part of the corollary, let us consideru∈Y. We have already proved that

kW(|u|²)uk_X ≤Ckuk³_X. (4.2.21) Applying operator H_x to equation (4.1.13)-(4.1.14) gives :

−∂_z²H_x(W(|u|²)) =H_x(|u|²), (x, z)∈Ω H_x W(|u|²)

(x,0) =H_x W(|u|²)

(x,1) = 0 Therefore, applying Lemma 4.2.4 gives :

kH_x(W(|u|²))k_X ≤CkH_x(|u|²)k_X ≤Ckuk_Ykuk_X (4.2.22) where we finally used Lemma 4.2.3. Now, note that

H_z(W(|u|²)) =−∂_z²W(|u|²) +V₂W(|u|²) = |u|²+V₂W(|u|²)∈X. Moreover, we readily have :

kH_z² W(|u|²)

k_L²_(Ω)≤ kH_z W(|u|²) k_X which, applying the estimate (4.2.11) of Lemma 4.2.4 leads to :

kH_z² W(|u|²)

k_L²_(Ω) ≤CkW(|u|²)k_X+Ck∂_z² W(|u|²)

k_X ≤Ckuk²_X. (4.2.23) Now, define

W(|u|²)p(x) = hW(|u|²)(x,·)χpi, definition (4.2.4) gives :

kH_z^1+α/2(W(|u|²))k²_L2(Ω) =X

p≥0

E_p^2+αkW(|u|²)_pk²_L2(Ω)

≤ X

p≥0

E_p⁴kW(|u|²)_pk²_L2(Ω)

!1/2

× X

p≥0

E_p^2αkW(|u|²)_pk²_L2(Ω)

!1/2

≤ kH_z²(W(|u|²))k_L²_(Ω)× X

p≥0

E_p²kW(|u|²)_pk²_L2(Ω)

!1/2

≤Ckuk²_XkHz(W(|u|²))k_L²_(Ω) ≤Ckuk⁴_X

where we used (4.2.23), the fact that α <1/2and that Ep is increasing and goes to infinity withp. Finally :

kH_z^1+α/2(W(|u|²))k_L²_(Ω) ≤Ckuk²_X. (4.2.24) Combining (4.2.21) with (4.2.22) and (4.2.24) finally provides us with the follo-wing tame estimate, according to (4.2.3) :

∀u∈Y, kW(|u|²)k_Y ≤Ckuk_Ykuk_X

whereC > 0does not depend onu. Following the exact same lines, (4.2.16) can also easily be proved. Now, let us consider any functionu∈Yand prove estimate (4.2.17) thanks to the equivalence (4.2.6). First, we know that :

kW(|u|²)uk_H^2+α_(Ω) ≤ kW(|u|²)k_H^2+α_(Ω)kuk_L^∞_(Ω)+kuk_H^2+α_(Ω)kW(|u|²)k_L^∞_(Ω)

≤ kW(|u|²)k_Ykuk_X+kuk_YkW(|u|²)k_X

where we used the continuous embeddings X⊂L^∞(Ω) and Y⊂H^2+α(Ω). Applying (4.2.15) and (4.2.12) gives :

kW(|u|²)uk_H^2+α_(Ω) ≤Ckuk²_Xkuk_Y. (4.2.25) As far as the normH_x(W(|u|²)u) is concerned, we readily have :

H_x W(|u|²)u

=H_x W(|u|²)

u−W(|u|²)∆_xu−2∇_x W(|u|²)

· ∇_xu. (4.2.26) However, since Xis an algebra,

kHx W(|u|²)

uk_X ≤CkHx W(|u|²)

k_Xkuk_X

≤Ckuk_Ykuk²_X (4.2.27)

where we used (4.2.17) and the equivalence (4.2.6).

Moreover,

kW(|u|²)∆_xuk_X ≤CkW(|u|²)k_Xk∆_xuk_X

≤Ckuk_Ykuk²_X (4.2.28)

where we used (4.2.12), and the equivalence (4.2.6). Finally,

∇_x W(|u|²)

· ∇_xu=

2

X

i=1

∂_i W(|u|²)

u+W(|u|²)∂_iu. (4.2.29) Fixi∈ {1,2}, we readily have

∂_i W(|u|²)

=W ∂_i(|u|²)

and, applying Lemma 4.2.4 with f =∂1(|u|²)and ε= 0 yields kW ∂_i(|u|²)

k_X ≤Ck∂_i(|u|²)k_X. Therefore, combined with (4.2.29), this leads to

∇x W(|u|²)

· ∇xu X ≤C

2

X

i=1

k∂i(|u|²)k_Xk∂iuk_X ≤C

2

X

i=1

k∂iuk²_Xkuk_X. (4.2.30) Applying Lemma 4.2.3, combined with (4.2.30) allows us to conclude that

∇_x W(|u|²)

· ∇_xu

X ≤Ckuk²_Xkuk_Y. (4.2.31) Finally, combining (4.2.31) with (4.2.26), (4.2.27) and (4.2.28) yields

kHx W(|u|²)u

k_X ≤Ckuk²_Xkuk_Y,

which, combined with (4.2.25) and the equivalence (4.2.6) concludes the proof of (4.2.17). Estimates (4.2.15) and (4.2.17) are now proved. As far as (4.2.18) is concer-ned, we write :

W(|u|²)u−W(|v|²)v = W(|u|²)−W(|v|²)

u+W(|v|²)(u−v).

Then, as Y is an algebra, then

kW(|u|²)u−W(|v|²)vk_Y ≤ kW(|u|²)−W(|v|²)k_Ykuk_Y+kW(|v|²)k_Yku−vk_Y

≤ kW((|u|+|v|)|u−v|)k_Ykuk_Y+kW(|v|²)k_Yku−vk_Y

≤Ck|u|+|v|k_Yku−vk_Ykuk_Y+Ckvk_Xkvk_Yku−vk_Y where we used (4.2.16) and (4.2.15). Therefore, we get

kW(|u|²)u−W(|v|²)vk_Y ≤C kuk²_Y+kvk²_Y

ku−vk_Y which ends the proof of (4.2.18)

Now that we have obtained a priori estimates on bothV^εandW nonlinearities, we focus on the existence and uniqueness results for both initial (4.1.1)-(4.1.2)-(4.1.4)-(4.1.5) and intermediate system (4.1.16)-(4.1.17)-(4.1.13)-(4.1.14).

Dans le document Fanny Delebecque préparée à l’unité de recherche UMR 6625 IRMAR (Page 140-146)