PHASE SPACE ANALYSIS APPLIED TO GEOPHYSICAL FLUIDS AND THERMOELASTICITY

UNIVERSIT ` A DEGLI STUDI DI TRIESTE

XXIV CICLO DEL DOTTORATO DI RICERCA IN

ENVIRONMENTAL AND INDUSTRIAL

FLUID MECHANICS

PHASE SPACE ANALYSIS APPLIED

TO GEOPHYSICAL FLUIDS AND

THERMOELASTICITY

Settore scientifico-disciplinare: MAT/05 ANALISI MATEMATICA

DOTTORANDO

MARCO PIVETTA

COORDINATORE

PROF. VINCENZO ARMENIO

SUPERVISORE

PROF. DANIELE DEL SANTO

ANNO ACCADEMICO 2011/2012

Introduction

Since it was used for the first time, the phase space analysis has become one of the most important tool in dealing with partial differential equations. The starting idea behind this theory is that there is a strong connection between space and frequency when one has to consider physical models and for this reason also mathematical analysis must take into account both variables. Fourier trasform is indeed the main tool in this context, since it is the bridge between the two visions.

One of the theories that arose from this approach and that will be widely used in this thesis thanks to its flexibility is Littlewood-Paley ([LP]) decomposition, that is based on a dyadic partition of unity in the frequency space (see e.g. [AG] or [M]).

first of all this theory allows to define Sobolev spaces in an equivalent way using the properties of the L² norms of the dyadic blocks, then it also gives an easy way to define interpolation spaces, such as Besov spaces, that will be repeatedly used in this work. This technique gives also origin to other fundamental instruments, like Bony’s paraproduct that will be crucial to prove the inequalities necessary for one of the main results (see [B]).

The thesis is devoted to prove three theorems, the first two deals with existence and uniqueness of mild and weak solutions of Navier-Stokes equations for geophys- ical incompressible fluids, while the third concerns uniqueness for thermoelastic system. Here we give an overview of the three results.

We start from the topic discussed in chapter 2 and 3, namely Navier-Stokes equa- tions for geophysical fluids, that we will refer to as Navier-Stokes-Coriolis equa- tions, which are represented by the following system







∂tu − ν∆u + (u · ∇)u + Ωe3 × u + ∇p = 0

∇ · u = 0 u(0) = u₀.

(1)

We focus our attention on mild solutions for this system, whose existence and uniqueness will be proved by using a suitable fix point theorem on the following

iii

equation, that can be obtained from system (1)

T u = G(t)u₀+ B(u, u), (2)

with

B(u, v) := − Z t

0

G(t − τ )P∇ · (u ⊗ v)dτ,

and where G(t) is the following time dependant operator

G(t)w = F⁻¹



cos

 Ωtξ3

|ξ|



I + sin

 Ωtξ3

|ξ|

 R(ξ)



e^−νt|ξ|2w(ξ)b

 ,

See chapter 1 for precise definitions. Our purpose is to deal with an initial datum belonging to a low regular space: the idea comes from the result obtained in a similar framework on classical Navier-Stokes equations. Let us briefly describe this point. It is known that Navier-Stokes equations have a scaling property, which means that if u(t, x) is a solution of N-S system then also

u_λ(t, x) = λu(λ²t, λx)

still represents a solution. Since a fixed point theorem will be used to prove exis- tence of solutions, the norm of the spaces in which this theorem will be performed must be invariant with respect to this scaling (scaling invariance is also connected with other properties of the solutions like selfsimilarity, remarked in the pioneering work [L]).

The first invariant space used in studying mild solution for three dimensional NS equations is the Sobolev space ˙H¹² and this space was used by Fujita and Kato ([FK]) in 1964, while some years later Kato ([K]) generalized the previous result considering the space L³.

Thanks to the characterization of Besov spaces, Cannone, Meyer and Planchon were able to obtain existence and uniqueness result for the space ˙B

3 p−1

p,∞, providing a result in which highly oscillating initial data can be considered, since negative index of Besov space is allowed (see [CMP] and [CP]).

The final step was obtained by Koch and Tataru, who proved the sharpest result on this field ([KT]). They were able to enlarge the result in [CMP] to the space

∇BM O. The last space that could be considered is the largest scaling invariant

v space in R³, that is ˙B_∞,∞⁻¹ , but for this space a counterexample of non existence has been proved.

We can suggestively resume all these results in the following picture, which prop- erly underline the continuous imbeddings that link these spaces.

H˙ ¹²(R³)

| {z }

F ujita−Kato

,→ L³(R³)

| {z }

Kato

,→ ˙B

3 p−1

∞,p(R³)

| {z }

Cannone M eyer P lanchon

,→ ∇BM O(R³)

| {z }

Koch−T ataru

,→ B_∞,∞⁻¹ (R³).

Before describing the technique of the proof, let us consider what can be said so far about Navier-Stokes-Coriolis equations: from [HS] we know that existence and uniqueness of mild solutions are known when the initial datum belongs to the Sobolev space ˙H¹². In chapter 2 we will obtain this result using the same technique used by Chemin in [C]. In the following chapter we try to follow the road represented above by proving two theorems, the first using L³ space, the second using the Besov space ˙B

3 p−1

p,∞, but limiting ourselves to 3 < p < 4. We state here only the theorem concerning Besov space, since the first is analougous.

Theorem 1 Let 3 < p < 4. There exists a constant c > 0, depending on Ω/ν, such that if ku₀kB

H˙ 1 2 , ˙B

3 p,∞p −1

≤ c, then there exist a unique fixed point u(t) ∈ K₄ of (2).

Moreover it holds that

i) ku(t)k_K4 ≤ 2kG(t)u₀k_K4, ii) u ∈ L^∞(0, +∞; B

H˙¹², ˙B

3p −1 p,∞

),

iii) B(u, u) ∈ C(0, +∞; ˙H¹²) and lim

t→0⁺kB(u(t), u(t))k_˙

H¹² = 0.

The space K_p is defined in this way K_p :=

(

f ∈ C(]0, +∞[; L^p)/kf k_Kp := sup

t∈[0,+∞[

t^12(1−3p)kf k_L^p < +∞

) ,

while the definition of the space B

H˙¹², ˙B

3p −1 p,∞

is the following:

let χ_B(0,1) be the characteristic function of the ball B(0, 1), we define f⁽¹⁾(t, x) = F⁻¹(χ_B(0,1)f )b

f⁽²⁾(t, x) = F⁻¹((1 − χB(0,1)) bf ),

we say that f ∈ B

H˙¹², ˙B

3p −1 p,∞

if

kf k_B

H˙1 2 , ˙B

3p −1 p,∞

= kf⁽¹⁾k_˙

H¹² + kf⁽²⁾k

B˙

3 p,∞p −1

< +∞,

in the three dimensional case it holds that

H˙ ¹² ,→ B

H˙¹², ˙B

3p −1 p,∞

,

so this theorem improves the result in [HS], moreover the choice of Besov spaces with negative index allows us to choose highly oscillating initial data. The idea of hybrid spaces is due to the work in [CMZ], where the authors obtain a result similar to ours, considering a Ω dependent space for initial datum.

The proof of this theorem consists essentially of three steps:

i) we first study the mapping property of G(t) acting on the initial datum, mainly focusing on its codomine;

ii) we prove a bilinear estimate for B(·, ·);

iii) we make use of a fix point theorem, that gives us the existence and uniqueness of a mild solution.

In chapter 4 we will briefly address the following anisotropic Navier-Stokes-Coriolis system







∂_tu − ν_h∆_hu + ∇ · (u ⊗ u) + B(t, x₁, x₂) × u +¹_ρ∇p = 0

∇ · u = 0 u(0) = u0

(AN SC) (3) where the operator ∆_h = ∂_x²₁+ ∂_x²₂ is the horizontal laplacian. The system is called anisotropic since 0 vertical viscosity is considered. For this system we study here only existence: in fact we prove the following theorem

Theorem 2 Let u₀ ∈ B^0,12 a divergence free vector fields. There exists c > 0 such that if ku₀k

B^{0, 12} ≤ cν_h then there exists a global solution u of ANSC such that u ∈ L^∞(R⁺; B^0,12) and ∇hu ∈ L²(R⁺; B^0,12).

vii This theorem is the consequence of two results obtained in [MP] and [P]: in the first the system (ANSC) is studied in the anisotropic Sobolev space H^0,s, for s > ¹₂, while in the second anisotropic Navier-Stokes system (without Coriolis term) is studied in the anisotropic Besov space B^0,12, since the case s = ¹₂ can only be treated using this space instead of Sobolev one. This Besov space is defined as follows

kukB^{0, 12} =X

q∈Z

2^q2k∆^v_quk_L² < +∞,

where ∆^v_q is the vertical Littlewood-Paley decomposition. This result does not represent a proper generalization of [MP] since

B^0,12 ,→ H^0,12, but it allows to go beyond the assumption s > ¹₂.

The proof relies on Ascoli-Arzel`a theorem: first we will write a sequence of ap- proximating systems of equations and build a sequence of solutions, then we will prove that there exists a subsequence converging to the solution of (3). Littlewood- Paley decomposition will play a crucial role in different points in this context, since it will be used to prove the boundness condition necessary to use Ascoli-Arzel`a theorem.

The theorems in [MP] and [P] also deal with continuity and uniqueness of the solution. In our work we do not describe completely these two points, but we think that one can obtain a better result than the one above, since Coriolis term does not add any real difficulty and the proof in [P] can be repeated also in this case. So the theorem one could prove is the following:

Theorem 3 Let u0 ∈ B^0,12 a divergence free vector fields. There exists c > 0 such that if ku₀k

B^{0, 12} ≤ cν_h then there exists a unique global solution u of ANSC such that

u ∈ Cb(R⁺; B^0,12) and ∇hu ∈ L²(R⁺; B^0,12).

The third result, shown in chapter 5, concerns the uniqueness of the solution for the following backward thermoelastic system











∂_t²u − ∂x(a(t, x)∂xu) + α(t, x)∂xθ(t, x) + β(t, x)θ(t, x) = f (t, x)

∂_tθ + ∂_x(b(t, x)∂_xθ) + δ(t, x)∂_x∂_tu(t, x) + ρ(t, x)∂_tu(t, x) + φ(t, x)u(t, x) = g(t, x) )

]0, T [×R u(0, x) = u₀(x), ∂_tu(0, x) = u₁(x), θ(0, x) = θ₀(x) on R,

(4) where low regularity assumptions for the coefficients a(t, x) and b(t, x) are consid- ered, i.e.

a(t, x) ∈ LL(R⁺t , LL(Rx))

b(t, x) ∈ C^µ(R⁺t , C^1,(Rx)), (5) where LL refers to log-lipschitz space, while µ is a modulus of continuity satistfying Osgood condition, namely

Z 1 0

1

µ(t)dt = +∞,

and another technical condition, that we will refer to as ? condition. These results improves the one by Koch and Lasiecka (see [KL]), where the same system was studied considering Lipschitz regularity in space and time for a and b. The com- bination of two Carleman inequalities, that we will describe below, used in [KL] is actually the same in this work. At the end of the chapter we will also introduce a non Lipschitz function space, whose modulus of continuity satisfies Osgood and ? condition. For a description of the use of Carleman inequalities see [Ca].

The system (4) is represented by two partial differential equations coupled: the coupling is represented by the term δ(t, x)∂_x∂_tu(t, x) that represents the influence of the motion on the heat tranfer process (see for example [Bi] and [ChS]).

The principal part of these two equations are composed by a hyperbolic and a backward parabolic operator, writen in divergence form. Studying the uniqueness problem for these operators when low regular coefficients are considered is not a new issue, but it is just a new episode of a long story. The first and pioneering result on this topic is contained in the work by Colombini, De Giorgi and Spag- nolo [CDGS] in 1979, where they first addressed the problem of uniqueness for hyperbolic operator with time dependent non-Lipschitz coefficient.

Being the source of a very fertile production of results, this work soon became a classical result, since even more than thirty years later some of the techniques used in [CDGS] are still performed in the result proved nowadays. All these results are

ix based on an energy estimate for the hyperbolic operator, with loss of derivatives.

Results on hyperbolic operators have been obtained by Tarama in [T], Colombini and Lerner [CL] and Colombini and Del Santo [CDS], just to give a (very) short list. The result in [CL] is in fact the one that inspired the result, since the regularity used here is just the same, but since we need a Carleman estimate for the energy, we also need to borrow some ideas from [CDS], where Log-Zygmund coefficients are considered.

For the backward parabolic operator our guide is the work by Del Santo and Prizzi ([DSP]) where a Carleman estimates for the backward parabolic operator is ob- tained when the coefficients are continuous in time with a modulus of continuity that satisfies Osgood condition. Due to the final combination of the two inequali- ties we will use in this work, we need one more hypotesis, the so called ? condition, that we will describe precisely in the first chapter.

Since the system is linear in order to prove uniqueness it is sufficient to prove that the homogeneous system associated with the one above has only the trivial solution. In order to do this we first obtain two Carleman estimates involving the operators Pu = ∂_t²u − ∂_x(a(t, x)∂_xu) and Lθ = ∂_tθ + ∂_x(b(t, x)∂_xθ) and then we combine this two inequalities to conclude the proof. The weights used in the Carleman estimates are defined using the properties of the modulus of continuity µ, that will be descibed in chapter 1.

The two inequalities are the following

Z ^T₂

0

e^{2γΦ(γ(T −t))}E(t)dt ≤ C₀ Φ^{0 γT}₂

²

Z ^T₂

0

e^{2γΦ(γ(T −t))}kPuk²_H−ω−β∗tdt

Z ^T₂

0

e^{γ2Φ(γ(T −t))}kLθk²_H−1−ω−β∗tdt ≥

≥ K Z ^T₂

0

e^{2γΦ(γ(T −t))}



γkθk²_H−1−ω−β∗t +√

γkθk²_H−ω−β∗t + 1

(Φ⁰(γT ))kθk²_H1−ω−β∗t

 dt,

where the energy E(t) is equivalent to k∂_tuk_H−ω−β∗t+kuk_H−ω−β∗t+kuk_H1−ω−β∗t. Φ(·) is a function defined using µ, while ω belongs to ]0,¹₂[. The two inequalities hold for any γ greater than a fixed γ₀. As stated above for the proof of these two inequalities a very important role will be played by Bony’s paraproduct, necessary to address difficulties arising with commutators.

The combination of the two inequalities allows us to prove that E(t) ≡ 0 for t beloging to a small time interval, this implies the following theorem

Theorem 4 Let

v ∈ H²([0, T ], L²(Rx)) ∩ H¹([0, T ], H¹(Rx)) ∩ L²([0, T ], H²(Rx)) ζ ∈ H¹([0, T ], L²(R)) ∩ L²([0, T ], H²(Rx))

solutions of (4), with f ≡ g ≡ u₀ ≡ u₁ ≡ θ₀ ≡ 0.

Then under the hypotesis (5) we have that v ≡ ζ ≡ 0.

Contents

Introduction iii

1 Background 1

1.1 Navier-Stokes-Coriolis equations . . . 1

1.1.1 Semigroup computation . . . 2

1.1.2 Mild solution . . . 4

1.1.3 Existence of mild solutions . . . 6

1.2 Littlewood-Paley theory . . . 7

1.3 Modulus of continuity . . . 12

1.4 Functional spaces . . . 14

2 Mild solutions for the Navier-Stokes-Coriolis system: the ˙H¹² case 17 2.1 Technical results . . . 18

2.2 Main theorem . . . 25

3 Mild solutions for the Navier-Stokes-Coriolis system: the L³ and Besov case 29 3.1 The ˙H¹² − L³ case . . . 29

3.2 The case ˙H¹² − ˙B 3 p−1 p,∞ . . . 42

4 Anisotropic Navier-Stokes-Coriolis system 45 4.1 Main Theorem . . . 46

5 Backward uniqueness for thermoelastic system 53 5.1 Carleman estimate for the hyperbolic operator . . . 54

5.2 Carleman Estimate for parabolic operator . . . 58

5.3 Statement and Proof of the main theorem . . . 69

5.4 Log_K-Lipschitz functions . . . 75

Appendix 79

xi

Chapter 1

Background

1.1 Navier-Stokes-Coriolis equations

In this work we will study Navier-Stokes equations for geophysical fluids, namely the following system:







∂tu − ν∆u + (u · ∇)u + Ωe3 × u + ∇p = 0

∇ · u = 0 u(0) = u₀,

(1.1)

where u is the velocity field, p the pressure, ν the viscosity and Ω the Coriolis parameter, which is proportional to the inverse of Rossby number.

This system models the behaviour of a fluid in the presence of the Coriolis force, which appears when one considers the dynamics in a rotating framework: the term Ωe₃× u represents the consequence of rotation on the velocity field u. For a detailed description of this system and all its features we refer to [CB].

Using the divergence free condition we can recast the system in the following way







∂_tu − ν∆u + ∇ · (u ⊗ u) + Ωe₃× u + ∇p = 0

∇ · u = 0 u(0) = u₀,

(1.2)

1.1.1 Semigroup computation

We start solving, in a formal way, the following linearized system.







∂_tu − ∆u + Ωe₃× u + ∇p = 0

∇ · u = 0 u(0) = u0,

(1.3)

where we set ν = 1 to fix ideas.

For convenience we now write this system component by component:















∂_tu₁− ∆u₁− Ωu₂+ ∂_xp = 0

∂_tu₂− ∆u₂+ Ωu₁ + ∂_yp = 0

∂_tu₃− ∆u₃ + ∂_zp = 0

∂xu1+ ∂yu2+ ∂zu3 = 0 u(0) = u₀.

(1.4)

We want to use Fourier transform, so we need the Fourier transform of p. To find it we take the divergence of the first three equation and we make use of the divergence free condition, thus obtaining:

∆p = Ω(∂_xu₂− ∂_yu₁) = Ωω₃,

in which we obtain the third component ω₃ of the vorticity vector field. Using Fourier transform we get:

p = −b Ωcω₃

|ξ|²,

with cω₃ = −i(ξ₁ub₂ − ξ₂ub₁). We write the system in frequency space and we obtain























∂_tub₁ + |ξ|²ub₁− Ωub₂ + iξ₁Ωcω₃

|ξ|² = 0

∂_tub₂ + |ξ|²ub₂+ Ωub₁+ iξ₂Ωcω₃

|ξ|² = 0

∂_tub₃ + |ξ|²ub₃ + iξ₃Ωcω₃

|ξ|² = 0 ξ1ub1+ ξ2ub2+ ξ3ub3 = 0.

(1.5)

1.1. NAVIER-STOKES-CORIOLIS EQUATIONS 3 We multiply the first equation by iξ₂, the second by iξ₁ and then we subtract the second from the first, obtaining

∂_tcω₃+ |ξ|²cω₃+ iΩξ₃ub₃ = 0, and so

ub₃ = i∂_tcω₃+ |ξ|²ωc₃

Ωξ₃ , (1.6)

substituting on the third equation we obtain:

∂_t(i∂_tcω₃+ |ξ|²cω₃

Ωξ₃ ) + |ξ|²i∂_tcω₃+ |ξ|²cω₃

Ωξ₃ +iΩξ₃

|ξ|² cω₃ = 0.

Or equivalently:

(∂_t+ |ξ|²)²cω₃ Ωξ₃ + ξ₃

|ξ|²Ωcω₃ = 0

Multiplying by Ωξ₃ and by e^|ξ|2t and calling f = ie^|ξ|2tcω₃ we can write f⁰⁰+ Ω² ξ₃²

|ξ|²f = 0, Writing explicitly the solution, we obtain

f (t) = A cos ξ₃

|ξ|Ωt



+ B sin ξ₃

|ξ|Ωt



and imposing the initial conditions we obtain f (t) = (ξ₁ub₂(0) − ξ₂ub₁(0)) cos ξ₃

|ξ|Ωt



+ |ξ|ub₃(0) sin ξ₃

|ξ|Ωt

 .

Returning to cω₃(t) we find

cω3(t) = −ie^−|ξ|2t



(ξ1ub2(0) − ξ2ub1(0)) cos ξ₃

|ξ|Ωt



+ |ξ|ub3(0) sin ξ₃

|ξ|Ωt



.

Returning now in (1.6) we obtain the expression for ub₃:

ub₃ = e^−|ξ|2t



ub₃(0) cos ξ₃

|ξ|Ωt



+ ξ₂

|ξ| bu₁(0) − ξ₁

|ξ| bu₂(0)



sin ξ₃

|ξ|Ωt



.

Similarly we can obtain the other components:

bu(t, ξ) = cos ξ₃

|ξ|Ωt



e^−|ξ|2tIub₀(ξ) + sin ξ₃

|ξ|Ωt



e^−|ξ|2tR(ξ)ub₀(ξ), (1.7)

where I s the identity matrix and

R(ξ) = 1

|ξ|





0 ξ₃ − ξ₂

−ξ3 0 ξ1

ξ₂ − ξ₁ 0



.

The Stokes-Coriolis semigroup can be explicitly represented by:

G(t)f = F⁻¹



cos ξ₃

|ξ|Ωt



I + sin ξ₃

|ξ|Ωt

 R



e^νt|ξ|2fb

 ,

1.1.2 Mild solution

Our goal is to study mild solution for Navier-Stokes-Coriolis, namely the fixed point of the map:

T u = G(t)u₀+ B(u, u), with

B(u, v) := − Z t

0

G(t − τ )P∇ · (u ⊗ v)dτ,

where P is the Leray projector on divergence free vector field and G(t) is the semigroup associated with the linearized Stokes - Coriolis system.

We start working on G(t) defined by the linear problem







∂_tu − ν∆u + Ωe₃× u = f − ∇p

∇ · u = 0 u(0) = u0.

(1.8)

In the last section we obtained and wrote G(t) as

G(t)w = F⁻¹



cos

 Ωtξ₃

|ξ|



I + sin

 Ωtξ₃

|ξ|

 R(ξ)



e^−νt|ξ|2w(ξ)b

 .

1.1. NAVIER-STOKES-CORIOLIS EQUATIONS 5 We obtain that

u(t) = G(t)u₀− Z t

0

G(t − s)(P(f(s)))ds

is solution of the problem (1.8), since G(t)u0 is the solution of

 ∂_tu − ν∆u + Ωe3× u = 0

u(0) = u₀, (1.9)

while Duhamel’s principle tells us that Z t

0

G(t − s)(P(f(s)))ds is the solution of

 ∂_tu − ν∆u + Ωe₃× u = Pf

u(0) = 0, (1.10)

Now let us compute the divergence of u(t), since ∇ · u₀ = 0 and ∇ · (Pw) = 0 we get

∇ · u(t) = ∇ ·



G(t)u₀+ Z t

0

G(t − s)(P(f(s)))ds



= G(t)(∇ · u₀) + Z t

0

G(t − s)(∇ · (P(f(s))))ds

= 0, so ∇ · u = 0 as we want.

Now, since

P(∂^tu − ν∆u + Ωe3× u) = ∂tu − ν∆u + Ωe3× u, we obtain that

P(∂tu − ν∆u + Ωe₃× u − f ) = 0, and so

∂_tu − ν∆u + Ωe₃× u = Pf − ∇p.

As a consequence of these computations the solutions of the problem (1.2) will be seen as the fixed point of

u(t) = G(t)u₀− Z t

0

G(t − s)(P(∇ · (u ⊗ u)))(s)ds, (1.11) with ∇ · u₀ = 0.

1.1.3 Existence of mild solutions

The strategy to show existence of mild solution is the following:

• choose a space X(R³) for the initial data u₀;

• choose a space Y ([0, T ], R³) such that G(t)u₀ ∈ Y ;

• prove that B : Y × Y → Y is a bilinear bounded operator;

• use a fixed point argument.

The last point is represented by the following lemma:

Lemma 1 Let (Y, k · k) be a Banach space and B : Y × Y → Y a bilinear operator, such that

kB(y₁, y₂)k ≤ ηky₁kky₂k

then for any y0 ∈ Y such that 4ηky0k < 1 the equation y = y0 + B(y, y) has a unique solution in the ball B(0,_2η¹ ). In particular we have

kyk ≤ 2ky₀k.

Proof. We start by defining the ball B(0, R) with

R = 1 −p1 − 4ηky₀k

2η .

We will prove that

T (y) = y₀+ B(y, y)

is a contraction in B(0, R). First of all let us prove that T : B(0, R) 7→ B(0, R),

kT (y)k ≤ ky₀k + kB(y, y)k ≤ ky₀k + ηkyk² ≤ ky₀k + ηR² = R, where the last equality comes from the definition of R.

1.2. LITTLEWOOD-PALEY THEORY 7 Now we consider y₁ and y₂ and we compute

kT (y₁) − T (y₂)k = kB(y₁, y₁− y₂) + B(y₁− y₂, y₂)k

≤ η(ky₁k + ky₂k)ky₁− y₂k ≤ 2ηRky₁− y₂k, since 2ηR < 1 then we obtain that T is a contraction.

Thanks to Banach-Caccioppoli-Picard theorem we can state that the equation

y = y₀+ B(y, y)

has a unique solution in the ball B(0, R). Moreover from this equation we get

kyk ≤ ky₀k + ηkyk² and so

kyk(1 − ηkyk) ≤ ky₀k.

Now since kyk ≤ R ≤ 1

2η, we finally obtain that kyk ≤ 2ky0k.

To prove uniqueness in the ball B(0,_2η¹ ) let y = ye ₀+ B(y,e y) be another solution,e with y ∈ B(0,e _2η¹ ). We have

ky −eyk = kB(y, y −ey) + B(y −ey,ey)k

≤ η(kyk + keyk)ky −eyk ≤ η(R + 1

2η)ky −eyk < ky −eyk.



1.2 Littlewood-Paley theory

In this paragraph we briefly introduce Littlewood-Paley theory, showing some results concerning Sobolev and H¨older spaces

Let ϕ ∈ C₀^∞(R), 0 ≤ ϕ ≤ 1, ϕ0(ξ) = 1 if |ξ| ≤ ¹¹₁₀ and ϕ₀(ξ) = 0 if |ξ| ≥ ¹⁹₁₀.

For ν ∈ N, we consider ϕν(ξ) = ϕ(2^−νξ) and we define byϕe_ν(x) the inverse Fourier transform of ϕν(ξ).

We define the following operators

S−1u = 0, S_νu =ϕe_ν ∗ u = ϕ_ν(D)u, for ν ≥ 0,

∆₀u = S₀u, ∆_νu = S_νu − S_ν−1u, for ν ≥ 1.

Remark.

The Littlewood-Paley decomposition can be defined in a “homogeneous” way as follows

∆_νu = S_νu − S_ν−1u for ν ∈ Z, (1.12) this definitionis will be used in the definition of homogenous spaces.

We now report three proposition without proof concerning characterization of Sobolev and H¨older spaces through Littlewood-Paley decomposition:

Proposition 1 Let s ∈ R. A tempered distribution u belongs to H^s if and only if the following two conditions hold:

• for every ν ≥ 0, ∆_νu ∈ L²

• the sequence δ_ν = 2^νsk∆_νuk_L² belongs to `²

Moreover there exists C_s > 1 such that, for every u ∈ H^s,

1

C_skuk_H^s ≤ X

ν

δ_ν²

!¹₂

≤ C_skuk_H^s. (1.13)

Proposition 2 Let s ∈ R and let R > 2. Let (uk)_k a sequence of L² functions such that

i) suppub0 is contained in {|ξ| ≤ R} and suppubkis contained in {_R¹2^k≤ |ξ| ≤ R2^k} for k ≥ 1.

ii) the sequence (δ_k)_k with δ_k = 2^ksku_kk_L² is in `².

1.2. LITTLEWOOD-PALEY THEORY 9 Then P

ku_k is a converging series in H^s with sum u ∈ H^s and the norm in H^s of u is equivalent to the norm in `² of (δ_k)_k (that is there exists C_s such that (1.13) holds.

If s > 0, it is sufficient to suppose that supp bu_k is contained in {|ξ| ≤ R2^k} for every k ≥ 1.

Proposition 3 A bounded function a belongs to C^1, (with 0 <  < 1), the space of H¨older functions of indices 1, , if and only if the sequence (α_k)_k, with α_k = 2^k(1+)k∆kakL^∞, is in `^∞.

Moreover there exists C_> 1, such that, for every a ∈ C^1,, 1

C_kak_C^1, ≤ sup

k

α_k ≤ C_kak_C^1,. (1.14)

Paraproduct

We now introduce Bony’s paraproduct, giving some results that can be obtained using this tool. Let’s start with a definition.

Definition. Let a ∈ L^∞ and u ∈ H^s. Bony’s paraproduct T_au is defined as follows

T_au =

+∞

X

ν=3

S_ν−3a∆_νu.

The following two propositions concern with some mapping properties of the para- product:

Proposition 4 i) Let s ∈ R and a ∈ L^∞. Then T_a: H^s → H^s and kT_auk_H^s ≤ C_skak_L^∞kuk_H^s.

ii) Let  ∈]0, 1[, a ∈ C^1,, s ∈]0, 1 + ]. Then u 7→ au − T_au maps H^−s in H^1−s and kau − Tauk_H^1−s ≤ CkakC^1,kuk_H^−s.

Proof. Since the result of i) is classical and can be found in [B] we give only the proof of ii). We have

au − T_au =

+∞

X

k=3

∆_kaS_k−3u

| {z }

(A)

+

+∞

X

k=3

( X

j ≥ 0

|j − k| ≤ 2

∆_ka∆_ju)

| {z }

(B)

,

regarding (A) we can observe that the support of the Fourier transform of ∆kaSk−3u is contained in {2^k−2 ≤ |ξ| ≤ 2^k+2} and one has

k∆_kaS_k−3uk_L² ≤ k∆_kak_L^∞kS_k−3uk_L²

≤ CkakC^1,2^−k(1+)

k−3

X

j=0

2^jsδj,

where (δ_j)_j is in `<sup>2</sup> and k(δ<sub>j</sub>)<sub>j</sub>k<sub>`</sub>² = C_skuk_H^−s. This yields

k∆kaSk−3uk_L² ≤ CkakC^1,2^{−k(1+−s)}

k−3

X

j=0

2^−(k−j)sδj.

Now, defining eδ_k =

k−3

X

j=0

2^−(k−j)sδ_j, as a consequence of Young inequality for convo- lution in `<sup>p</sup>, we have that (eδ<sub>k</sub>)<sub>k</sub>∈ `² and

k(eδk)kk_{`</sub><sup>2</sup> ≤ 2k(δ<sub>k</sub>)kk<sub>`}². We easily obtain that (A) ∈ H^1−s and

k(A)k_H^1−s ≤ C_kak_C^1,kuk_H^−s. Considering now (B) we have

(B) =

+∞

X

k=3

(∆_ka∆_k−2u + ∆_ka∆_k−1u + ∆_ka∆_ku + ∆_ka∆_k+1u + ∆_ka∆_k+2u).

1.2. LITTLEWOOD-PALEY THEORY 11

Every term can be treated analogously. We take the term

+∞

X

k=3

∆ka∆ku as an ex- ample. The support of its Fourier transform is contained in {|ξ| ≤ 2^k+2} and

k∆_ka∆_kuk_L² ≤ k∆_kak_L^∞k∆_kuk_L² C_kakC^1,2^−k(1+)2^−ksδ_k, where as before (δ_j)_j ∈ `<sup>2</sup> and k(δ<sub>j</sub>)<sub>j</sub>k<sub>`</sub>² = C_skuk_H^−s.

As a consequence

+∞

X

k=3

∆_ka∆_ku ∈ H^1−s and

k

+∞

X

k=3

∆_ka∆_kuk_H^1−s ≤ C_kakC^1,kuk_H^−s.

 Remark. In the last part, since the support of the Fourier transform of ∆ka∆ku is contained in a ball (and not in a ring) we can apply the second part of Proposition 2 and so it is necessary that 1 +  − s > 0. On the contrary it is easy to see that the result of Proposition 4 is valid also for s = 0.

The next result will be necessary in dealing with the backward parabolic operator of the thermoelastic system in Chapter 5.

Proposition 5 Let  ∈]0, 1[, s ∈]0, 1 + ], a ∈ C^1,. Then

+∞

X

ν=0

2^−2νsk∂_x([∆_ν, T_a]∂_xu)k²_L2

!¹₂

≤ C_kak_C^1,kuk_H^1−s,

for every u ∈ H^1−s.

([A, B] denotes the commutator of A and B, i.e. [A, B]u = A(B(u)) − B(A(u)).) Proof. We start by observing that

[∆_ν, T_a]w =

+∞

X

K=3

[∆_ν, S_k−3a]∆_kw,

so

∂_x([∆_ν, T_a]∂_xu) = ∂_x

+∞

X

K=3

[∆_ν, S_k−3a]∆_k(∂_xu)

! .

 From the support of the Fourier transform of [∆_ν, S_k−3]∆_kw we can obtain that [∆_ν, S_k−3]∆_kw is identically zero if |k − ν| ≥ 4.

This allows us to infer that the sum in k reduces at most to 7 terms, so

∂_x([∆_ν, T_a]∂_xu) = ∂_x([∆_ν, S_ν−6a]∆_ν−3(∂_xu)) + . . . + ∂_x([∆_ν, S_νa]∆_ν+3(∂_xu)).

The support of Fourier transform of each of these terms is contained in a ball proportional to 2^ν. We consider one of these terms: one has

k∂_x([∆_ν, S_ν−3a]∆_ν(∂_xu))k_L² ≤ C2^νk[∆_ν, S_ν−3a]∂_x(∆_νu)k_L²

≤ C2^νkak_Lipk∆_νuk_L²,

where the first inequality comes from Bernstein’s inequality, while the second de- rives from theorem 35 in Coifman and Meyer ([CM], see also [Ta]). The Proposition follows from this inequality and proposition 1.

1.3 Modulus of continuity

Let µ : [0, 1] → [0, 1] continuous, concave and strictly increasing function, with µ(0) = 0. We say thatµ is a modulus of continuity. Let I ⊆ R and f : I → R. We define f ∈ C^µ(I, R) if f ∈ L^∞(I, R) and

sup

0 < |t − s| < 1 t, s ∈ I

|f (t) − f (s)|

µ(|t − s|) < +∞.

For example if µ(s) = s then C^µ= Lip.

In the following elementary proposition we collect some useful results:

Proposition 6 Let µ be a modulus of continuity. Then

1.3. MODULUS OF CONTINUITY 13

• µ(s) ≥ sµ(1) for all s ∈ [0, 1];

• the function s 7→ µ(s)

s is decreasing in ]0, 1];

• there exists lim

s→0⁺

µ(s) s ;

• the function σ 7→ µ(¹_σ)

(_σ¹) is increasing in [1, +∞];

• the function σ 7→ 1

σ²µ(_σ¹) is decreasing in [1, +∞].

We remark that if sup

s∈]0,1]

µ(s)

s < +∞, then there exists C > 0 such that µ(s) ≤ Cs for any s ∈ [0, 1]: so C^µ = Lip. Moreover, if C^µ6= Lip, then lim

s→0⁺

µ(s)

s = +∞.

We now define the so called Osgood condition.

Definition. (Osgood condition)

Let µ be a modulus of continuity. We say that µ satisfies the Osgood condition if:

Z 1 0

1

µ(s)ds = +∞.

From the modulus of continuity we will now define a weight function, that will play an important role in the study of the thermoelastic system.

Let µ be a modulus of continuity satisfying the Osgood condition, we define φ(t) =

Z 1

1 t

1 µ(s)ds,

φ ∈ C¹ and it is strictly increasing. From the Osgood condition we obtain that φ([1, +∞[) = [0, +∞[ and φ⁰(t) = _(t2µ(¹1

t)) > 0 for any t ∈ [1, +∞[. We define:

Φ(τ ) = Z τ

0

φ⁻¹(s)ds,

so Φ⁰(τ ) = φ⁻¹(τ ), then

τ →∞lim Φ⁰(τ ) = +∞.

Moreover

Φ⁰⁰(τ ) = (Φ⁰(τ ))²µ

 1 Φ⁰(τ )



, (1.15)

for every τ ∈ [0, +∞[ and since σ 7→ σµ(_σ¹) is an increasing funtion on [1, +∞[, we obtain that:

τ →+∞lim Φ⁰⁰(τ ) = lim

τ →+∞(Φ⁰(τ ))²µ

 1 Φ⁰(τ )



= +∞. (1.16)

To conclude this section we introduce a property of µ(s) that will be fundamental in the last chapter, where we will also give an example of a class of non - Lipschitz functions whose modulus of continuity satisfies this condition

Definition. ( ? condition)

Let µ be a modulus of continuity and let Φ defined as above: we say that µ satisfies the ? condition if there exists 0 < a < 1 such that

s→+∞lim

Φ⁰(s)

(Φ⁰(as))² = 0. (?)

We remark that since Φ⁰ is an increasing function then, if µ satisfies this condition with a₀, the same condition holds for every 0 < a₀ < a < 1.

1.4 Functional spaces

In this section we introduce some functional spaces used in this thesis, focusing in particular on the less known ones.

1.4. FUNCTIONAL SPACES 15

Besov spaces

We start by introducing Besov spaces, using the definition of the operators ∆_jgiven in section 1.2. We start with the inhomogeneous version: we say that f ∈ B_p,q^s if

kf k_B^s_p,q = X

j∈N

2^qjsk∆_jf k^q_Lp

!¹_q

< +∞,

while for the homogeneous version we say that f ∈ ˙B_p,q^s if

kf kB˙^s_p,q = X

j∈Z

2^qjsk∆jf k^q_Lp

!¹_q

< +∞,

where in this second case the operators ∆_j are defined as in 1.12. We remark that in the case p = q = 2 we have that

B_2,2^s = H^s where H^s is the Sobolev space of index s.

In this work we will also use an anistropic version of Besov spaces, namely the space B^0,s whose norm is defined as follows

kf k_B^0,s =X

j∈Z

2^jsk∆^v_jf k_L² < +∞,

where the operators ∆^v_j define a monodimensional version of Littlewood-Paley decomposition.

The space fL^p

In chapter 3 we will use the space fL^∞(R⁺, B^0,12), so it is worth to define fL^p spaces and remark its connections with the space L^p, using the space B^0,12 for example.

We say that u belongs to fL^p(R⁺, B^0,12) if

kuk

Lf^p(R⁺,B^{0, 12}) =X

j∈Z

2^j2k∆^v_juk_L^p_(R⁺_,L²_(R)) < +∞.

This definition slightly differs from the usual space L^p(R⁺, B^0,12), that is defined as follows

kukL^p(R⁺,B^{0, 12})=

X

j∈Z

2^2jk∆^v_juk_L²_(R)) L^p(R⁺)

< +∞,

and it is easy to prove that

kukL^p(R⁺,B^{0, 12}) ≤ kuk

Lf^p(R⁺,B^{0, 12}).

Chapter 2

Mild solutions for the

Navier-Stokes-Coriolis system:

the ˙ H ^{1 2} case

In this section we investigate the existence and uniqueness of mild solutions for the Navier-Stokes-Coriolis equations with Sobolev initial data. We consider the problem







∂_tu − ν∆u + (u · ∇)u + Ωe₃ × u + ∇p = 0

∇ · u = 0 u(0) = u₀,

(2.1)

where u is the velocity field, p the pressure, ν the viscosity and Ω the Coriolis parameter.

Using the divergence free condition we can recast the system in the following way







∂_tu − ν∆u + ∇ · (u ⊗ u) + Ωe₃× u + ∇p = 0

∇ · u = 0 u(0) = u0,

(2.2)

2.1 Technical results

In this section we investigate some properties of the solution of the linear problem in the case of ˙H^s initial data.

Lemma 2 Let v0 ∈ ˙H^s and f ∈ L²([0, T ]; ˙H^s−1). We set v(t) = G(t)v0+

Z t 0

G(t − s)f (s)ds.

Then

v ∈

+∞

\

p=2

L^p([0, T ]; ˙H^s+2p)

!

∩ C([0, T ]; ˙H^s).

Moreover

(i) kv(t)k²_H˙s + 2ν Z t

0

k∇v(t⁰)k²_H˙sdt⁰ = kv₀k²_H˙s+ 2 Z t

0

< f (t⁰), v(t⁰) >_s dt⁰;

(ii) Z

R³

|ξ|^2s( sup

0≤t⁰≤T

|bv(t⁰)|)²dξ ≤ 9



kv₀kH˙^s + 1

√2νkf k_L2([0,T ]; ˙H^s−1)

2

;

(iii) kv(t)k

L^p([0,T ];H^{s+ 2p}) ≤ 3 (ν)^1p



kv0kH˙^s + 1

√νkf k_L2([0,T ]; ˙H^s−1)

 . Proof. One has

bv(t, ξ) =

 cos

 Ωtξ₃

|ξ|



Id + sin

 Ωtξ₃

|ξ|

 R(ξ)



e^−ν|ξ|2tbv₀(ξ)

+ Z t

0

 cos



Ω(t − s)ξ₃

|ξ|



Id + sin



Ω(t − s)ξ₃

|ξ|

 R(ξ)



e^{−ν|ξ|2(t−s)}f (s, ξ)dsb

so we can write

|v(t, ξ)| ≤ 3|b bv₀(ξ)| + Z t

0

3e^{−ν|ξ|2(t−s)}| bf (t − s, ξ)|ds

≤ 3 |bv₀(ξ)| +

Z t 0

e^{−2ν|ξ|2(t−s)}ds

¹₂ Z t 0

| bf (t − s, ξ)|²ds

¹₂!

≤ 3 |bv0(ξ)| + 1

p2ν|ξ|²k bf (·, ξ)k_L²_([0,t])

! ,

2.1. TECHNICAL RESULTS 19 and then

|ξ|^2s

 sup

0≤t⁰≤t

|bv(t⁰, ξ)|

2

≤ 9 |vb₀(ξ)| + 1

p2ν|ξ|²k bf (·, ξ)k_L²_([0,t])

!2

|ξ|^2s,

and finally we obtain (ii), in fact

Z

R³

|ξ|^2s

 sup

0≤t⁰≤t

|bv(t⁰, ξ)|

2

dξ

!¹₂

≤ 3k|bv₀(ξ)| + 1

p2ν|ξ|²k bf (·, ξ)k_L²_([0,t])kH˙^s

≤ 3kbv₀kH˙^s+ 3

√2ν

Z

R³

k bf (·, ξ)k²_L2[0,t]|ξ|^2s−2dξ

¹₂

≤ 3



kbv0kH˙^s+ 1

√2νkf k_L2([0,t], ˙H^s−1)

 .

Now we can deduce that v ∈ C([0, T ], ˙H^s). First we observe that form (ii) just obtained, one has v ∈ L^∞([0, T ], ˙H^s) in fact

Z

R³

|ξ|^2s|bv(t, ξ)|²dξ ≤ Z

R³

|ξ|^2s

 sup

0≤t≤T

|bv(t, ξ)|²

 dξ

≤ 9



kbv₀kH˙^s+ 1

√2νkf k_L2([0,t], ˙H^s−1)

2

,

the continuity is consequence of Lebesgue dominated convergence theorem, since

|ξ|^2s|bv(t, ξ)|² ≤ 18



|bv₀(ξ)|²+ 1 2ν|ξ|²

Z T 0

| bf (s, ξ)|²ds



|ξ|^2s.

Let us now prove (i). Suppose first that v ∈ L²([0, T ]; ˙H^s+1) ∩ C([0, T ], ˙H^s); since we have

( ∂_tv − ν∆v + Ωe3× v = f − ∇p v(0) = v₀,

then

∂_tbv + ν|ξ|²bv + Ωe₃×bv = bf − c∇p,