Leray solutions to the Navier-Stokes equation 5 4

CHAPTER 7

THE NAVIER-STOKES EQUATION

- STILL A DRAFT -

We present some mathematical results on the Navier-Stokes equation about incom- pressible fluids in dimensiond= 2,3.

Contents

1. Introduction, a priori estimate, vortex formulation 1

2. Around the presure issue 3

3. Leray solutions to the Navier-Stokes equation 5

4. The Navier-Stokes equation in vortex formulation in 2D 9

5. Interpolation inequalities 12

6. The Hardy-Littlewood-Sobolev inequality 16

7. References 20

1. Introduction, a priori estimate, vortex formulation We consider the Navier-Stokes equation

(1.1)

∂tu−ν∆u+u· ∇u+∇p= 0 divu= 0,

on the vector (velocity) field u: (0, T)×Ω →R^d, with Ω⊂R^d. For the sake of simplicity, we will only consider the case Ω = R^d with d = 2,3, and a viscosity coefficientν= 1. The first equation is vectorial, and it is thus equivalent to

∂tui−∆ui+u· ∇ui+∂ip= 0, ∀i= 1, . . . , d.

• The pressure. The term ∇p is linked to the vanishing divergence condition and can be interpreted as a “Lagrange multiplicator” associated to this constraint.

More precisely, computing the divergence of each term involved in the first vectorial equation, we get

−∆p= div (u· ∇u) =X

ij

∂_ij²(u_iu_j),

where we have used that divu = 0 in the last equality. After having properly defined the inverse of the Laplacian operator, we may thus write

p:= (−∆)⁻¹X

ij

∂_ij²(uiuj) .

We will come back on that fundamental (but technical) point later.

1

•The energy identity. We compute 1

2 d

dtkuk²_L2 = Z

∂tu·u

= −

Z

(u· ∇u)·u+ Z

∆u·u− Z

∇p·u.

Using integrations by part, we compute each term separately. We have

− Z

∇p·u = Z

p(divu) = 0, Z

∆u·u = − Z

|∇u|²:=−X

ij

Z

(∂iuj)², and also (with Einstein’s convention of summation of repeated indices)

Z

(u· ∇u)·u= Z

(u_j∂_ju_i)u_i=1 2

Z

u_j∂_j|u|²=−1 2

Z

(divu)|u|²= 0.

All together, we have 1 2

d

dtkuk²_L2 =−k∇uk²_L2, and finally after time integration

(1.2) ku(t)k²_L2+ 2 Z t

0

k∇u(s)k²_L2ds=ku0k²_L2, ∀t≥0.

• The vortex and the dimension d = 2. In dimension d = 3, we define the vortex vector field

Ω := curlu:=∇ ∧u=





∂₂u₃−∂₃u₂

∂₃u₁−∂₁u₃

∂₁u₂−∂₂u₁



.

In dimensiond= 2, we may associated the 3dvector field ˜u:R³→R³, ˜u(x1, x2, x3) = (u1(x1, x2), u2(x1, x2),0), so that

Ω = curl ˜e u=



 0 0 ω



, ω:=∂1u2−∂2u1,

and the vortex is thus a scalar. In any dimension, we may verify (that is left as an exercise) that the vortex satisfies the evolution equation

∂tΩ +u· ∇Ω−Ω· ∇u−∆Ω = 0,

where we have just used that curl∇p= (∇ ∧ ∇)p= 0. We deduce from this that in dimensiond= 2, we have

∂_tω+u· ∇ω−∆ω= 0, where we have used

divΩ =e ∇ ·(∇ ∧u) = 0,˜ and then Ωe· ∇˜u=∇ ·(eΩ⊗u) = 0.˜

We may rather compute directly the evolution equation satisfied by the vortex functionω. We indeed have

∂tω = ∂1(∆u2−u· ∇u2−∂2p)−∂2(∆u1−u· ∇u1−∂1p)

= ∆ω−u· ∇ω+Q,

with

Q := −∂1u· ∇u2+∂2u· ∇u1

= −∂1u₁∂₁u₂−∂₁u₂∂₂u₂+∂₂u₁∂₁u₁+∂₂u₂∂₂u₁

= ∂1u1(∂2u1−∂1u2) +∂2u2(∂2u1−∂1u2)

= (divu)ω= 0.

We end with a last observation. We compute

curl ˜Ω = curl curl ˜u= (∇div −∆)˜u=−∆˜u, so that

˜

u=−∆⁻¹curl ˜Ω =−curl ∆⁻¹Ω =˜ −∇ ∧



 0 0

∆⁻¹ω



=





−∂2(∆⁻¹ω)

∂1(∆⁻¹ω) 0



.

Recalling that

∆E=δ

withE(x) = (2π)⁻¹log|x|in dimensiond= 2, we get

(1.3) u=

u˜1

˜ u2

=K∗ω with

K(x) :=∇^⊥E(x) = 1 2π

x^⊥

|x|², x^⊥= (−x2, x1).

As a conslusion, in dimensiond= 2 we obtain the following scalar equation on the vortex

∂tω+ (K∗ω)· ∇ω−∆ω= 0,

from what we may reconstruct the velocy filed thanks to the Biot-Savart equa- tion (1.3).

2. Around the presure issue

Theorem 2.1 (Hodge decomposition and pressure). For any T : Ω →R^d, there exist ϕ: Ω→Randψ: Ω→R^d such that

T = curlψ+∇ϕ,

with ψ= 0andT =∇ϕ if curlT = 0 andϕ= 0 etT = curlψ if divT = 0. As a consequence, when T ⊥ {S, divS = 0} there existsp: Ω→Rsuch thatT =∇p.

Idea of the proof. We make the following observations

curl∇= 0, div curl = 0, ∆ :=∇div −curl curl We set

w:= ∆⁻¹T, ϕ:= divw, ψ:=−curlw, so that

T = ∆w= (∇div −curl curl )w=∇ϕ+ curlψ.

On the other hand, the operator ∆⁻¹ commutes with translations and thus with both operators div and curl . To see this in another way, we may write

∆⁻¹div = ∆⁻¹div (∇div −curl curl )∆⁻¹= ∆⁻¹div∇div ∆⁻¹= div ∆⁻¹

and

∆⁻¹curl = ∆⁻¹curl (∇div −curl curl )∆⁻¹=−∆⁻¹curl curl curl ∆⁻¹

= −∆⁻¹(∇div −∆) curl ∆⁻¹= ∆⁻¹∆ curl ∆⁻¹= curl ∆⁻¹. Be careful with the fact that we have used ∆∆⁻¹ = I what is always true, but also ∆⁻¹∆ = I which is less clear, but also true in the case when Ω = R^d. As a consequence, we have

ψ= curl (∆⁻¹T) = ∆⁻¹(curlT) = 0 when curlT = 0 and

ϕ= div (∆⁻¹T) = ∆⁻¹(divT) = 0 when divT = 0.

Finally, ifT ⊥ {S, divS= 0}, we have in particular for anyφ∈ D(R^d), hcurlT, φi=hT,−curlφi= 0,

because div curlφ= 0. We deduce that curlT = 0, and thus T=∇p.

We introduce the close setsV andRof the Hilbert spaceL²(R^d) of vector field, by defining

V := {u∈L²(R^d); divu= 0}, R := {u∈L²(R^d); curlu= 0}.

Theorem 2.2(Hodge decomposition and pressure inL²(R^d)). For any vector field u∈L²(R^d), there exist a scalar functionϕ∈H˙¹(R^d)and a vector fieldψ∈H˙¹(R^d) such that

u= curlψ+∇ϕ, k∇ϕk+kDψk.kuk,

with ψ= 0 and thusu=∇ϕif u∈ R andϕ= 0and thus u= curlψif u∈ V. As a consequence, when u⊥ V there existsp∈H˙¹ such that u=∇p.

Idea of the proof. In the Fourier side, everything becomes easier.

Theorem 2.3 (pressure again). We define the projection P=I− ∇∆⁻¹div. There is equivalence between

(1) T =∇p, (2) PT = 0, (3) T ⊥ V.

Idea of the proof. (1)⇒(2) ForT =∇p, we have

PT =T − ∇∆⁻¹div∇p=T − ∇p= 0.

(2)⇒(3) Assume PT = 0. For anyv∈ V, there holdsPv=v, and therefore (T, v) = (T,Pv) = (PT, v) = 0.

That meansT ⊥ V.

(3)⇒(1) That follows immediately from the Hodge decomposition.

3. Leray solutions to the Navier-Stokes equation We consider the Navier-Stokes equation

(3.1)

∂_tu−∆u+u· ∇u+∇p= 0 divu= 0,

on the vector fieldsu: (0, T)×Ω→R^d, withd= 2,3.

Theorem 3.1. For any u₀∈ V, there exists at least one weak global solution u∈XT :=L^∞(0, T;V)∩L²(0, T;H¹), ∀T >0,

in the sense that Z

u_t·ψ_tdx+ Z t

0

Z

∇u:∇ψ dxds=

= Z

u0·ψ0dx+ Z t

0

Z

u⊗u:∇ψ dxds− Z t

0

Z

u·∂sψ dxds, for anyt∈(0, T)andψ∈C¹,divψ= 0. Moreover, the following energy inequality holds

(3.2) ku(t)k²_L2+ 2 Z t

0

k∇u(s)k²_L2ds≤ ku₀k²_L2, ∀t≥0.

We accept that the above Navier-Stokes equation is equivalent to the two following formulations

(3.3) ∂tu−∆u−P(u· ∇u) = 0

and

(3.4) ∂tu−∆u−u· ∇u⊥ D ⊗ V.

We observe that (3.1)⇒(3.3) is just a consequence of the fact thatPu=ubecause divu= 0 and the fact thatP∇p= 0.

There exist two classical strategies in order to establish the Leray Theorem 3.1 about existence of solutions.

A first way consists in considering a Friedrich discretization scheme

∂_tu_n−S_n[∆u_n−Pdiv (S_nu_n⊗S_nu_n)] = 0,

where S_n is a finite dimensional range operator for which existence of solutions is given by the Cauchy-Lipschitz on ODE, and then to pass to the limitn→ ∞.

A second way consists in considering the regularized equation (3.5) ∂tuε−∆uε+ (ρε∗uε)· ∇uε+∇pε= 0, for a sequence of mollifiers (ρ_ε) and then to pass to the limit ε→0.

We will follow that second way but we rather start by proving a stability principle.

Theorem 3.2. Consider a sequence(u0,n)of V such that u0,n→u0 in L² and a sequence of associated Leray weak global solutions(un). Then, there existsu∈XT

and a subsequence (un⁰) such that un⁰ → uand u is a Leray weak global solution associated to u0.

In the proof we use in a crucial way the following classical compactness lemma.

Lemma 3.3 (Aubin-Lions). Consider a sequence(un)which satisfies (i) (un) is bounded in L²_tx,

(ii) (∂tun) is bounded in L²_t(H_x^−s), s∈R⁺, (iii) (∇xun) is bounded in L²_tx.

Then, there exists u∈L²_tx and a subsequence (u_n⁰) such that u_n⁰ →u strongly in L²((0, T)×B_R)asn→ ∞for any R >0.

Idea of the proof. Step 1. We may write ∂tun = D^sgn with (gn) bounded in L²_tx. We introduce a sequence of mollifiers (ρ_ε), that isρ_ε(x) :=ε^−dρ(ε⁻¹x) with 0≤ρ∈ D(R^d),hρi= 1. We observe that

∂

∂t Z

R^d

un(t, y)ρε(x−y)dx= Z

R^d

gn(t, y)D^sρε(x−y)dy,

where the RHS term is bounded in L²((0, T)×R^d) uniformly inn for any fixed ε >0. We also clearly have

∇x

Z

R^d

un(t, y)ρε(x−y)dx=− Z

R^d

un∇yρε(x−y)dy,

where again the RHS term is bounded in L²((0, T)×R^d) uniformly in n for any fixedε >0. In other words,u_n∗ρ_εis bounded inH¹((0, T)×R^d). Thanks to the Rellich-Kondrachov Theorem, we get that (up to the extraction of a subsequence) (un∗ρε)n is strongly convergent inL²((0, T)×BR), for anyR >0. On the other hand, from (i) and the Banach-Alaoglu weak compacteness theorem, we know that there existsu∈L²_tx and a subsequence (un⁰) such thatun⁰* uweakly inL²_tx. All together, for any fixedε >0, we then get

un∗ρε→u∗ρε strongly inL²((0, T)×BR) asn→ ∞.

Step 2. We now observe that Z

(0,T)×R^d

|w−w∗ρε|²dxdt = Z

(0,T)×R^d

Z

R^d

(w(t, x)−w(t, x−y))ρε(y)dy

2

dxdt

= Z

(0,T)×R^d

Z

R^d

Z 1 0

∇xw(t, zs)·yρε(y)dsdy

2

dxdt, withz_s:=x+sy. From the Jensen (or Cauchy-Schwarz) inequality, we deduce Z

(0,T)×R^d

|w−w∗ρε|²dxdt ≤ ε² Z

(0,T)×R^d

Z

R^d

Z 1 0

|∇xw(t, zs)|²1 ε^d

|y|² ε² ρ y

ε

dsdydxdt

≤ ε² Z

(0,T)×R^d

|∇_xw(t, z)|²dtdz Z

R^d

|z|ρ(z)dy

≤ ε²C_ρk∇_xwk²_L2 tx. We conclude thatu_n→uinL²((0, T)×B_R) by writing

un−u= (un−un∗ρε) + (un∗ρ−u∗ρ) + (u∗ρε−u)

and using the previous convergence and estimates.

Idea of the proof of Theorem 3.2. From an interoplation theorem and Sobolev embedding, in dimensiond= 2, we have

L^∞_t (L²)∩L²_t( ˙H¹)⊂L⁴_t( ˙H^1/2)⊂L⁴_tx,

because fors= 1/2, the relation 1/s^∗= 1/2−s/dgivess^∗= 4. We deduce that u_n⊗u_n is bounded in L²((0, T)×R²).

Similarly, in dimensiond= 3, we have

L^∞_t (L²)∩L²_t( ˙H¹)⊂L⁴_t( ˙H^1/2)⊂L⁴_t(L³),

because for s = 1/2, the relation 1/s^∗ = 1/2−s/d gives s^∗ = 3. Because of the same Sobolev embedding, we haveL^3/2⊂H˙^−1/2, and then

u_n⊗u_n is bounded in L²_t(L^3/2(R³))⊂L²_t( ˙H^−1/2(R³)).

We recall the definition of the projector

P=I− ∇∆⁻¹div, which in the Fourier side also writes

Pˆ =I−ξ⊗ξ

|ξ|² . Arguing in the Fourier side, we see that

Pdiv (un⊗un) is bounded in L²_t(H^s(R^d)),

withs=−1 ifd= 2 ands=−3/2 if d= 3. As a consequence, we find

∂tun= ∆un−Pdiv (un⊗un),

which is bounded in L²(0, T;H^s(R^d)), for the same values of s as above when d = 2,3. From, Aubin-Lions’ lemma, we deduce thatun → u strongly L²_loc and thenun⊗un→u⊗ustronglyL¹_loc. As a consequence, we may pass to the limit in the weak formulation in the sense that

Z

un(t)·ψ(t)dx = − Z t

0

Z

∇un:∇ψ dxds+ Z t

0

Z

un⊗un :∇ψ dxds +

Z

u0,n·ψ0dx+ Z t

0

Z

un·∂tψ dxds,

for anyψ∈C_c¹(R^d+1), div_xψ= 0, and we get that ualso satisfies the same weak

formulation of the Navier-Stokes equation.

Idea of the proof of Theorem 3.1. We now consider the equation (3.5), and we start with considering the linear mapping which to a vector fieldu: (0, T)×R^d →R^d associates the solutionv: (0, T)×R^d→R^d to the linear equation

(3.6)

∂tv−∆v+ (u∗ρε)· ∇v+∇p= 0 divv= 0, v(0) =u0.

Let us make the problem more precise. Fixu₀∈ V and (ρ_ε) a mollifier. We define XT :=C([0, T];V)∩L²(0, T;H¹(R^d))∩H¹(0, T;H⁻¹(R^d)),

and foru, v, ψ∈ X_T, we may multiply (3.6) and integrate the resulting equation in order to get (at least formally)

Z

vt·ψtdx = − Z t

0

Z

∇v:∇ψ dxds+ Z t

0

Z

(u∗ρε)⊗v:∇ψ dxds +

Z

u0·ψ0dx+ Z t

0

Z

v·∂tψ dxds.

It is worth emphasizing that because of divu= 0, we also have div (u∗ρε) = 0, and then (u∗ρε)· ∇v = div ((u∗ρε)⊗v). That is nothing but the J.-L. Lions’

variational formulation of (3.6) when making of spaces and operator H:=V, V :=H¹, Λtv:= ∆v−(ut∗ρε)· ∇v− ∇p,

wherepstands for a Lagrange multiplicator. We have to show that−Λtis bounded and satisfiesG˚arding’s inequality. We indeed observe thatV endowed with theL² scalar product is an Hilbert space (because it is closed in L²) and next for any v, ψ∈ V ∩H¹, we have

|(Λtv, ψ)_L2|= Z

((u_t∗ρ_ε)⊗v+∇v) :∇ψ

≤C_εkvkH¹kψkH¹, using thatku(t)∗ρ_εk∞≤ ku(t)kL²kρεkL² ≤C_ε, and

(Λ_tv, v)_L2 =− Z

((u_t∗ρ_ε)⊗v+∇v) :∇v≤ −1

2kvk²_H1+C_εkvk²_L2.

From J.-L. Lions theorem, we therefore establish the existence and uniqueness of a variational solution v ∈ XT to the linear equation (3.6) and that one satisfies the energy identity

kv(t)k²_L2+ 2 Z t

0

k∇v(s)k²_L2ds=ku₀k²_L2, ∀t∈(0, T), by just repeating the arguments leading to (1.2).

Denoting v := Φ(u), we have shown Φ : XT → XT. Considering u1, u2 ∈ XT

with kui(t)k2 ≤ ku0k2 for any t ∈ (0, T) and denoting vi := Φ(ui), v =v2−v1, u=u2−u1, we have

1 2

d dt

Z

|v|² = Z

(∆v₂−(u₂∗ρ_ε)· ∇v₂− ∇p₂)·v

− Z

(∆v₁−(u₁∗ρ_ε)· ∇v1− ∇p1)·v

= Z

(∇v2+ (u2∗ρε)⊗v2) :∇v− Z

(∇v1+ (u1∗ρε)⊗v1) :∇v

= Z

|∇v|²+ Z

[(u2∗ρε)⊗v) + (u∗ρε)⊗v1)] :∇v.

As a consequence, we have 1

2 d

dtkvk²₂ ≤ Z

|(u₂∗ρ_ε)⊗v|²+ Z

|(u∗ρ_ε)⊗v₁)|²

≤ ku2∗ρ_εk²_∞ Z

|v|²+ku∗ρ_εk²_∞ Z

|v1|²

≤ C_εku₀k²₂kvk²₂+C_εkuk²₂ku₀k²₂, and from the Gronwall lemma, we deduce

kvtk²₂≤ Z t

0

Ce^C(t−s)kusk²₂ds, C:=C_εku0k²₂, and finally

sup

[0,T^∗]

kvtk²₂≤[e^CT∗−1] sup

[0,T^∗]

kutk²₂.

As a conclusion, when T^∗>0 is small enough, Φ is a contraction inXT^∗, so that Φ admits a unique fixed point which is thus the unique solution to the regularized Navier-Stokes equation (3.6).

Let us come back to the above computation in order to make it completely rigorous when we have to deal with variational solutions. With the same notations v_i :=

Φ(u_i),v=v₂−v₁,u=u₂−u₁andψ=v, we write first 1

2kv(t)k² = kv(t)k²− Z t

0

hv⁰(s), v(s)ids

= (v₂(t), ψ_t)−(u₀, ψ₀)− Z t

0

hψ_s⁰, v₁(s)ids

−(v1(t), ψt) + (u0, ψ0) + Z t

0

hψ⁰_s, v1(s)ids

= −

Z t 0

Z

∇v2:∇ψ dxds+ Z t

0

Z

(u2∗ρε)⊗v2:∇ψ dxds Z t

0

Z

∇v1:∇ψ dxds− Z t

0

Z

(u₁∗ρ_ε)⊗v₁:∇ψ dxds

= −

Z t 0

Z

[∇v+ (u2∗ρε)⊗v2−(u1∗ρε)⊗v1] :∇v dxds, and we may then estimate the RHS term similarly as we did above.

4. The Navier-Stokes equation in vortex formulation in 2D In this section, we consider the Navier-Stokes equation in dimension d= 2 in its vortex formulation, that we will simply call from now on thevortex equation, namely

(4.1) ∂tω+u· ∇ω−∆ω= 0,

whereuis given that the Biot-Savart equation (1.3).

u:=K∗ω, K(x) := 1 2π

x^⊥

|x|², x^⊥ = (−x₂, x₁).

Theorem 4.1. For any ω0 ∈ L¹(R²), there exists a unique global solution to the vortex equation

ω∈C([0,∞);L¹(R²))∩C((0,∞);L^∞(R²)) such that

kω(t)kL¹ ≤ kω0kL¹, t^1−1pkω(t)kL^p≤Ckω0kL¹,

for anyt≥0 and anyp∈(1,∞]for a universal constantC >0. The equation has to be understood in the mild sense

ωt=e^∆tω0− Z t

0

e^(t−s)∆(us· ∇ωs)ds, wheree^∆t stands for the heat semigroup.

We will not give the proof in the full generality, but rather explain how to proceed in the particuar caseω₀∈L¹∩L^ρ,ρ >1. That case yet contains the main difficulties and the proof we will see thus contains the main tools in order to overcome them.

•We start with some a priori estimates. Forp∈[1,∞), we compute d

dt Z

|ω|^pdx = p Z

ω|ω|^p−2∆ω dx−p Z

ω|ω|^p−2u· ∇ω dx

= −p(p−1) Z

|ω|^p−2|∇ω|²dx− Z

u· ∇|ω|^pdx

= −4p−1 p

Z

|∇(ω|ω|^p/2−1)|²dx+ Z

(divu)|ω|^pdx

= −4p−1 p

Z

|∇(ω|ω|^p/2−1)|²dx, by performing two integrations by parts. We deduce

(4.2) kω_tk_Lp≤ kω₀k_Lp, ∀t≥0, ∀p∈[1,∞], as well as

(4.3)

Z ∞ 0

k∇(ωs|ωs|^p/2−1)k_L2ds.kω0kL^p, ∀p∈(1,∞).

• We now briefly explain a possible strategy in order to obtain the existence of solution which consists to follow the same line of arguments as in the previous section when we have considered the NS equation in dimension d = 2,3. We introduce the kernel K_ε := K∗ρ_ε for a sequence of mollifiers (ρ_ε) and we first consider the regularized equation

(4.4) ∂_tω−∆ω+K_ε∗ω· ∇ω= 0, ω(0) =ω₀.

That equation may be tackled thanks to the classical way. We define the mapping g7→f, wheref is the solution to the linear equation

∂tf−∆f+Kε∗g· ∇f = 0, f(0) =ω0.

When furthemore, ω0 ∈ L², we prove that this mapping has a fixed point in the J.-L.-Lions space XT for T > 0 small enough, and we get then a global solution ωε ∈ XT, ∀T > 0, to the regularized equation (4.4) by repeating the argument (valid on a small intervalle (0, T), with T :=T(k ω0k_L2)). Whenp= 2, we may pass to the limitε→0 exacltly as in the previous section and in that way we obtain a solution

ω∈C(R+;L²_w)∩L^∞(R+;L²)∩L²(R+,H˙¹)

to the vortex equation (4.1). For the more general casep >1, we may also proceed similarly by adapting Aubin-Lions’ lemma to aL^p framework, we however do not developp further that issue.

• We finally concentrate on the uniqueness issue for solutions which belong to L^∞(R+;L^ρ(R²)),ρ >1. We start with several intermediate results of independant interest.

Lemma 4.2. For any 1≤r≤p≤ ∞and for anyg∈L^q(R²), there holds ke^t∆gk_Lp(R²). 1

t^1r−1p

kgk_Lr(R²)

and

k∇e^t∆gkL^p(R²). 1

t^12+1r−1p kgkL^r(R²).

The proof is straightforward by writing the kernel representation of the heat semi- group.

We also recall the following formulation of the Holder inequality (4.5) kf gkL^r ≤ kfk_LβkgkL^p, 1

r = 1 β +1

p, forβ, p≥1 such that 1/β+ 1/p≤1.

We finally accept the following consequence of the Hardy-Littlewood-Sobolev (HLS) inequality

(4.6) kK∗gk_Lβ(R²)≤Cqkgk_Lq(R²), 1 β =1

q −1 2,

which holds true for anyq∈(1,2). Recalling the Young inequality about convolu- tion product stipulating that

(4.7) kf∗gk_Lβ ≤ kfkL^pkgkL^q, ∀f ∈L^p, g∈L^q, for anyp, q∈[1,+∞] such that 1/p+ 1/q <1 and with

1 β := 1

q +1 p−1,

the HLS inequality can be seen as a critical case of the Young inequality in the case whenf =K /∈L²(R²).

Let us start the uniqueness proof and thus consider two solutions ωⁱ, i= 1,2, to the vortex equation that we write in mild form

ω_tⁱ=e^t∆ω₀− Z t

0

e^(t−s)∆(div (u_sω_s))ds.

Introducing the differenceω:=ω²−ω¹ andu=u²−u¹, we have ω_t=

Z t 0

e^(t−s)∆(div (u_sω_s²))ds+ Z t

0

e^(t−s)∆(div (u¹_sω_s))ds=:Q²_t+Q¹_t. Let us fixp∈(1,min(ρ,2)) and observe that

kωⁱk_p,t≤t^1−1/pkω₀k_Lp→0, ast→0, where we have set

kfkp,t:= sup

s∈(0,t)

[s^1−1/pkfskL^p].

We estimate

kQ¹_tkL^p ≤ Z t

0

ke^(t−s)∆(div (u¹_sωs))kL^p ds .

Z t 0

1

(t−s)^12+1r−1pku¹_sωskL^r ds .

Z t 0

1 (t−s)^1p

ku¹_sk_Lβkω_sk_Lp ds

. Z t

0

1 (t−s)^1p

kω¹_skL^qkωskL^p ds,

where we have successively used the second estimate on the heat semigroup as stated in Lemma 4.2 withrdefined by

1 r = 2

p−1 2, next the Holder inequality (4.5) with

1 β = 1

r−1 p, and finally the HLS inequality with

1 β = 1

q−1 2. From the three above definitions, we have

1 q = 1

β +1 2 = 1

r−1 p+1

2 = 1 p. As a consequenceq=p, and we may write

kQ¹_tkL^p . Z t

0

ds (t−s)^1ps^2(1−1p)

kω¹kp,tkωkp,t

. C 1 t^1−1p

kω¹kp,tkωkp,t

with the constant

C:=

Z 1 0

du (1−u)^p1u^2(1−1p)

is finite because of the choice of p∈(1,2). The same computations lead to As a consequenceq=p, and we may write

kQ²_tkL^p . C 1

t^1−1p kω²kp,tkωkp,t

Coming back to the estimate on the mild formulation, we deduc e t^1−1pkωtkL^p ≤ C[kω¹kp,t+kω¹kp,t] kωkp,t

≤ C[t^1−1/pkω0kL^p]kωkp,t, so that

kωkp,t≤1 2kωkp,t,

fort >0 small enough, and that ends the proof of the uniqueness.

5. Interpolation inequalities About interpolation inequalities :

•L^p∩L^q ⊂L^r for anyp≤r≤qand kuk_Lr ≤ kuk^θ_Lpkuk^1−θ_Lq , 1

r =θ

p+1−θ

q , ∀u∈L^p∩L^q.

We use the Holder inequality and we then write kukL^r = Z

|u|^rθ|u|^r(1−θ)1/r

≤ Z

|u|^rθs1/(rs)Z

|u|^{s0r(1−θ)1/(rs0)} ,

with 1/s+ 1/s⁰ = 1 choosen in such a way that rθs = p. As a consequence, 1/(rs) =θ/pand

1 rs⁰ = 1

r 1−1 s

= 1 r 1−θr

p =1

r −θ

p =1−θ q , and that ends the proof of the first interpolation inequality.

•L^p1(L^p2)∩L^q1(L^q2)⊂L^r1(L^r2), and more precisely kuk_L^r1(L^r2)≤ kuk^θ_Lp1(L^p2)kuk^1−θ_Lq1(L^q2), 1

ri

= θ pi

+1−θ qi

for anyu∈L^p1(L^p2)∩L^q1(L^q2) for a sameθ∈(0,1).

We proceed similarly as above. Thanks to the first interpolation inequality and the Holder inequality, we then write

kuk_Lr1(L^r2) = Z

|kuk^r_L¹r2

1/r1

≤ Z

kuk^r_L¹p^θ2kuk^r_L¹q^(1−θ)2

^1/r1

≤ Z

kuk^r_L¹p^θs2

^1/(r1s)Z

kuk^r_L¹q^(1−θ)s2 ⁰

^1/(r1s⁰)

,

with 1/s+ 1/s⁰ = 1 choosen in such a way that r₁θs = p₁. As a consequence, 1/(r₁s) =θ/p₁ and

1 r1s⁰ = 1

r1

1−1 s

= 1 r1

1−θr₁ p1

= 1 r −θ

p₁=1−θ q1

, and that ends the proof of the second interpolation inequality.

•H˙^s1∩H˙^s2 ⊂H˙^s for anys1≤s≤s2 and kukH˙^s ≤ kuk^θ_H˙s1kuk^1−θ_˙

H^s2, s=θs1+ (1−θ)s2, ∀u∈H˙^s1∩H˙^s2. On the Fourier side, we have

kuk²_H˙s = Z

|ˆu|^2θ|ξ|^2θs1|ˆu|^2(1−θ)|ξ|^2(1−θ)s2dξ

≤ Z

|ˆu|²|ξ|^2s1dξ^θZ

|ˆu|²|ξ|^2s2dξ1−θ

, by using the Holder inequality with exponentp= 1/θ andp⁰ = 1/(1−θ).

•L^p( ˙H^a)∩L^q( ˙H^b)⊂L^r( ˙H^c), and more precisely kuk_Lr( ˙H^c)≤ kuk^θ_{Lp( ˙Ha)}kuk^1−θ

L^q( ˙H^b), 1 r = θ

p+1−θ

q , c=θa+ (1−θ)b for anyu∈L^p( ˙H^a)∩L^q( ˙H^b) for a sameθ∈(0,1).

We write indeed

kuk_Lr( ˙H^c)≤ kkuk^θ_H˙akuk^1−θ_˙

H^b kL^r

and we argue exactly as in the proof of the second interpolation inequality.

•Sobolev embedding. For anys∈(0, d/2), there holds (5.1) H˙^s(R^d)⊂L^p(R^d), with 1

p= 1 2−s

d.

In particular, ˙H^1/2(R²)⊂L⁴(R²) and ˙H^1/2(R³)⊂L³(R²). The proof of (5.1) is a bit tricky and it is presented below. We start with the Cavalieri’s principle

(5.2) kfk^p_Lp=p

Z ∞ 0

t^p−1|{|f(x)|> t}|dt because

Z

R^d

|f(x)|^pdx= Z

R^d

Z ∞ 0

1t<|f(x)|pt^p−1dt dx=p Z ∞

0

nZ

R^d

1|f(x)|>tdxo t^p−1dt, by the Fubini theorem. We then introduce the splitting

f =fA+f^A, with

fA:=F⁻¹(1_|ξ|≤Afˆ), f^A:=F⁻¹(1_|ξ|≥Afˆ),

where for a function g : R^d → C we define its Fourier transform ˆg =Fg and its inverse Fourier transformF⁻¹g by

Fg(ξ) := 1 (2π)^d/2

Z

R^d

g(x)e^−ix·ξdx, F⁻¹g(x) := 1 (2π)^d/2

Z

R^d

g(ξ)e^−ix·ξdξ.

We observe that

kfAk_∞ ≤ (2π)^−d/2kfˆAk1= (2π)^−d/2 Z

1_|ξ|≤A|fˆ|dξ

≤ (2π)^−d/2Z

1_|ξ|≤A|ξ|^−2sdξ1/2Z

|ξ|^2s|fˆ|²dξ1/2

≤ (C1/2)A^d/2−skfkH˙^s,

where we have used the Cauchy-Schwartz inequality and we denote here and below byCi some constant which only depend onp, sand d. We deduce that

{|fA|> t/2}=∅ if t≥tA:=C1A^d/2−skfkH˙^s. Together with the inclusion

{|f(x)|> t} ⊂ {|f_A(x)|> t/2} ∪ {|f^A(x)|> t/2}, we get

{|f(x)|> t} ⊂ {|f^At(x)|> t/2}, where we have defined

At:= C1t/kfkH˙^s

^1/(d/2−s) . Coming back to (5.2), we then have

kfk^p_Lp≤p Z ∞

0

t^p−1|{|f^At(x)|> t/2}|dt We recall the Tchebychev inequality

|{|g|> λ}| ≤ Z |g|²

λ² 1_{|g|>λ}dx≤ 1 λ²kgkL²

and the Plancherel idenitity

kgkL² =kˆgkL².

The three last inequalities together, we have kfk^p_Lp≤p

Z ∞ 0

t^p−14

t²kfˆ^Atk²_L2dt.

We then write Z ∞

0

t^p−3kfˆ^Atk²_L2dt = Z ∞

0

t^p−3 Z

R^d

|fˆ|²1_|ξ|≥Atdξdt

= Z

R^d

|fˆ|²Z t|ξ|

0

t^p−3dt dξ

= 1

p−2 Z

R^d

|fˆ|²

C1|ξ|^d/2−skfkH˙^s

p−2

dξ,

where we have used the Fubini theorem at the second line. From the very definition p:= 2d/(d−2s), we have

d 2 −s

(p−2) = (d−2s) d d−2s−1

=d−(d−2s) = 2s.

Putting together the three last estimates and identities, we thus obtain kfk^p_Lp≤C2

Z

R^d

|fˆ|²|ξ|^2sdξkfk^p−2_˙

H^s =C2kfk^p_˙

H^s,

which is nothing but the announced estimate.

•Young inequality for convolution. For anyp, q∈[1,+∞] such that 1/p+ 1/q <1, there holds

(5.3) kf ∗gkL^r ≤ kfkL^pkgkL^q, ∀f ∈L^p, g∈L^q, wherer∈(1,∞) is defined by

1 p+1

q = 1 +1 r.

We fix 0≤f ∈L^p, 0≤g∈L^q, 0≤h∈L^r0, and observing that 1

r + 1 q⁰ =1

p, 1 r+ 1

p⁰ =1 q, 1

p⁰ + 1 q⁰ = 1

r⁰, we may write

Z

(f∗g)h dx = Z Z

(f(x−y)^prg(y)^qr)(g(y)^pq0h(y)^r

0

p0)(f(x−y)^qp0h(x)^r

0 q0)dxdy.

Because

1 p⁰ + 1

q⁰ +1 r = 1, we may use the generalized Holder inequality and we get Z

(f∗g)h dx ≤ Z Z

f^pg^qdxdy^1/rZ Z

g^qh^r0dxdy^1/p0Z Z

f^ph^r0dxdy^1/q0

= kfk^p/r_Lpkgk^q/r_Lqkgk^q/p_Lq⁰khk^r0/p0

L^r0 kfk^p/q_Lp⁰khk^r0/q0

L^r0 =kfkL^pkgkL^qkhk_Lr0. Together with the duality estimate

kf ∗gkL^r = sup

khk_Lr0≤1

Z

(f∗g)h dx,

we immedialtely conclude that (6.11) for positive functions, and then for any func-

tions.

6. The Hardy-Littlewood-Sobolev inequality

We establish now a slight but fundamental improvement of the Young inequality.

The proof uses smart and accurate arguments ofreal harmonic analysis.

6.1. The maximal function.

Lemma 6.1 (Covering lemma). Let E ⊂ R^d measurable which is covered by a familly of balls{Bj} of volume bounded by a same constant. Then we can select a (disjoint finite or denombrable) subsequence denoted byB₁, ...,B_k, ...so that

(6.1) X

k

λ(Bk)≥K⁻¹λ(E),

whereλstands for the Lebesgue measure and K:= 5^d is a convenient choice.

Proof of Lemma 6.1 . We note J the set of indices for the set of distinct balls {Bj}? As a first step, we chooseB1in the family{Bj}j∈J which satisfies

diamB₁≥ 1

2sup{diamB_j;j ∈J}.

We proceed recursively by choosingBk,k≥2, such that diamB_k ≥1

2sup{diamB_j, j∈J andB_j∩ B_i=∅ ∀i= 1, ..., k−1), and build in that way a finite or denombrable sequence B1, ...,Bk, .... We can assume

(6.2) X

k

λ(Bk)<∞,

otherwise inequality (6.1) is trivial. We defineB^∗_kas the ball having same center as B_k but whose radius is five times larger and we claim that

E⊂[

k

B^∗_k,

from what we immediately deduce (6.1) withK = 5^d. It is thus enough to prove that for anyBj in the familly{Bj} which is not in the family{Bk}, we have

B_j ⊂[

k

B_k^∗.

We fix such a ball Bj. From (6.2), we clearly have λ(Bk)→0 when k→ ∞, and we then may choosekas the first integer such that

diamB_k+1<1

2diamB_j.

From the definition of (B`) we must haveBj∩ Bi 6=∅ for somei∈ {1, ..., k} (oth- erwise we haveB_j =Bk+1 because of the diameter condition and that contradict our choice of B_j which precisely not belongs to that family). We conclude that

B_j ⊂ B_i^∗, and that ends the proof.

For a functionf ∈L¹_loc(R^d), we define the maximal function M f(x) := sup

r>0

1 λ(B(x, r))

Z

B(x,r)

|f(y)|dy.

The maximal function M f is measurable (take r ∈ Q⁺∗) and positive whenever f 6≡0.

Theorem 6.2 (Maximal function). For any p∈(1,∞) and any f ∈L^p(R^d), we have

(6.3) kM fk_Lp≤Ckfk_Lp,

for some positive constantC which only depends onpandd.

Proof of Theorem 6.2 . Step 1. We prove that forg∈L¹(R^d), there holds (6.4) λ({x∈R^d; (M g)(x)> α})≤K

α kgkL¹, ∀α >0,

for the same constantKas in the covering Lemma 6.1. Let us then fixg∈L¹(R^d).

For anyα >0, we define

Eα:={x;M g(x)> α}.

By definition of the maximal function, for anyx∈E_α, there exists a least one ball Bx with centerxsuch that

(6.5) α λ(Bx)<

Z

Bx

|g(y)|dy.

Because the right hand side is bounded by kgkL¹ the volume ofB_x is necessarily smaller than kgk_L1/α independently of x∈ E_α. Since the union of the balls B_x obviously coversE_α, we may make use of the covering Lemma 6.1 and deduce that there exists a sequence of ballsB_k (extracted from the previous family) such that they are mutually disjoint and

(6.6) λ(Eα)≤CX

k

λ(Bk)

Putting together (6.5), (6.6) and the fact that the balls (Bk) are disjoint, we get α λ(Eα)≤CX

k

αλ(Bk)≤X

k

Z

Bk

|g(y)|dy= Z

∪Bk

|g(y)|dy,

from which (6.4) immediately follows.

Step 2. We prove (6.3). Let us fixf ∈L^p(R^d). For any givenα >0, we define g(x) :=f(x)1_{|f(x)|≥α/2} and Eα:={x;M f(x)> α}.

From|f(x)| ≤ |g(x)|+α/2, we deduceM f(x)≤M g(x) +α/2, and therefore Eα⊂ {x;M g(x)> α/2}.

Applying (6.4) to the functiong (which belongs toL¹(R^d)), we have (6.7) λ(Eα)≤λ({x∈R^d; (M g)(x)> α})≤ 2K

α kgkL¹ = 2K α

Z

|f|≥α/2

|f|dx.

On the other hand, we observe that thanks to the Fubini Theorem, we have Z

R^d

(M f)^pdx = Z

R^d

Z M f(x) 0

pα^p−1dα dx= Z ∞

0

Z

R^d

1_{M f(x)>α}dx pα^p−1dα

= Z ∞

0

λ({x∈R^d; (M g)(x)> α})pα^p−1dα.

Together with (6.7) and using again the Fubini Theorem, we deduce Z

R^d

(M f)^pdx ≤ Z ∞

0

2K α

Z

|f|≥α/2

|f|dx pα^p−1dα.

= 2Kp

Z

R^d

|f(x)|Z 2|f(x)|

0

α^p−2dα dx

= K p

p−12^p Z

R^d

|f(x)|^pdx,

which ends the proof of (6.3).

6.2. Hardy-Littlewood-Sobolev inequality.

(6.8)

1

|z|∗f

_L2r/(2−r)(

R²)≤Crkfk_Lr(R²), ∀r∈(1,2), Ford≥2, we define the functionalI by

I[f](x) :=

Z

R^d

f(y)

|x−y|^d−1dy, ∀f ∈ D(R^d).

Ford≥1,α∈(0, d), we define the functionalF by F(x) :=

Z

R^d

f(y)

|x−y|^αdy, ∀f ∈ D(R^d).

We assume thatp, q >1 are such that

(6.9) 1

p+α

d = 1 +1 q. In particular, ford≥2 andα=d−1, we have

1 p−1

d = 1 q,

so that q = p^∗ is the classical Sobolev exponent associated to W^1,p(R^d) when p∈(1, d).

Theorem 6.3. For any p∈(1, d), there holds I:L^p(R^d)→L^q(R^d), 1

q := 1 p−1

d. In particular, there exists a constantC such that

(6.10)

f∗ K _L4(

R²)≤Ckfk_L4/3(R²), ∀f ∈L^4/3(R²).

•Hardy-Littlewood-Sobolev inequality. For anyp∈(1, d), there holds (6.11) kf∗Gk_Lp∗ ≤CpkfkL^p, ∀f ∈L^p,

whereGandp^∗∈(1,∞) are defined by G(x) := 1

|x|^d−1, 1 p^∗ =1

p−1 d.

It is worth emphasizing that it is a critical case of the Young inequality since G almost belongs to theL^q Lebesgue space with q:=d/(d−1) and that for such a exponentq the exponentrassociated to the Young inequality satisfies

1

p+d−1

d = 1 +1 r

which is preciselyr=p^∗.

We take 0≤f ∈L^p and we define F(x) :=

Z

R^d

f(y)

|x−y|^d−1dy.

For a fixedδ >0, we defineF =Fδ+F^δ with F_δ(x) :=

Z

|x−y|<δ

f(y)

|x−y|^d−1dy, F^δ(x) :=

Z

|x−y|≥δ

f(y)

|x−y|^d−1dy.

On the one hand, forF_δ, we write Fδ(x) :=

Z

|x−y|<δ

f(y)

|x−y|^αdy

= Z

|x−y|<δ

f(y) 1 1 +α

Z ∞

|x−y|

r^−α−1dr dy

= 1

1 +α Z δ

0

r^d−α−11 r^d

Z

|x−y|<r

f(y)dy dr

≤ λ_d(B₁) 1 +α

Z δ 0

r^d−α−1drM f(x)

= C₁δ^d−αM f(x), C₁:= λ_d(B₁) (1 +α)(d−α),

where we have used the Fubini theorem at the third line and the definition of the maximal functionM f at the fourth line.

On the other hand, forF^δ, we write F^δ(x) :=

Z

|x−y|>δ

f(y)

|x−y|^αdy

≤ kfkL^p

Z

|x−y|>δ

1

|x−y|^αp0dy^1/p0

=C2δ^d/p0−αkfkL^p, where we have used

Z

|z|>δ

dz

|z|^αp0dy1/p⁰

= λ_d−1(S1)^1/p0Z ∞ δ

r^d−1−αp0dr1/p⁰

= λ_d−1(S₁)^1/p0

(αp⁰−d)^1/p0 δ^{(d−αp0)/p0}.

It is worth emphasizing that the last integral converges because from (6.9), there holds

d−αp⁰ =dp⁰(1/p⁰−α/d) =−dp⁰/q <0.

Both estimates together imply

F(x) . δ^d−αM f(x) +δ^d/p0−αkfkL^p

. (M f(x))1−(p/d)(d−α)kfk^{(p/d)(d−α)}_Lp

= (M f(x))p(1/p−1+α/d)kfk^p(1−α/d)_Lp