On the modified Foppl - Von Karman Model

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UNIVERSITÀ DEGLI STUDI DI PISA

Dipartimento di Matematica Corso di Laurea Magistrale in Matematica - Percorso Modellistico

Tesi di Laurea

ON THE MODIFIED FVK MODEL

Relatore: Candidato:

Prof. VLADIMIR GEORGIEV GIULIO DEL CORSO

Controrelatore:

Prof. LUIGI CARLO BERSELLI

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What is the dierence between method and device? A method is a device which you used twice.

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Introduction

This thesis study a problem of PDE linked to blood ows. As shown in Chapter 1 we started from an old model developed by Foppl and Von Kar-man (FvK hereafter) in 1898 and adapt it to our problem. Using the FvK model we constructed a model for the membrane that bound the inner uid on an idea proposed by Prof. Y. Shibata (Waseda University) and Prof. V. Georgiev (University of Pisa). The Cauchy Problem associated with this model has a nonlinear not easy term that is needed to be studied before starting numerical simulations or even use this particular PDE to model the membrane.

Thus the main question of the thesis is: can we have some GWP (global well posedness) results for this model in the case of small initial data (but without imposing too high regularity on the conditions)?

The answer is positive for the two modied version of the FvK model used. Obtaining this solution involves many Harmonic Analysis instruments as Strichartz Estimates, Yukawa Potential and Riesz Trasnform. We also have needed to use the particular symmetry and structure of the nonlinear term to control the behaviour of the solution.

The original version of this thesis was accepted as article in the Proceed-ings of the SIA Conference of NTADES 2018 - Soa and here is given the same results enclosed with all the theorical instruments and conclusions that are not contained in the original article. The initial idea was to study this problem from a numerical point of view, but the theorical part (i.e. the proof of GWP for this model) needed enough works and mathematical instruments to justify a master thesis in harmonic analysis. Of course the point of arrival is still the numerical model, for this reason, as shown in the section Further development in Chapter 5, the PHD's project of reaserch submitted to the GSSI is on the link between the theorical part and the numerical one.

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8 CONTENTS The thesis is structured as follow. In the rst chapter are given motiva-tions and structure of the model proposed (i.e. Foppl - Von Karman, FvK hereafter) and also a brief historical digression of the FvK model. Indeed the original model was from 1898 and was used to study a two-dimensional simplied version of the much more dicult Navier - Stokes Equations. Only in the last few years some reaserch groups start to study blood model and the interaction between the inner uid and the membrane (for example the University of Milano Bicocca and the Waseda University).

The second chapter recall all the theorical instruments needed to com-plete the proofs of the two main results ot this master thesis. The theorical instruments needed are: Sobolev (homogeneous and inhomogenous) Spaces, Riesz Transform, Schrodinger equations, Yukawa potential and the Strichartz estimate in two cases, the well known Strichartz estimate for Schrodinger equation and a version of Strichartz estimate adapted to our plate mode.

The third and the fourth chapets contain the two main results of the thesis that are two global well posedness (GWP) results in two dierent versions of the original FvK model. The rst one

The fth chapter contains a brief overview of the results obtained in the previous chapters and some conclusion. This work was originaly thought as a starting point for a PHD work, for this reason this last chapter contain also a section, called further developments, that describe some ideas and possible future works that use the results contained in this thesis.

The chapter ends with a brief section caled Two pages Thesis that is an extremly brief description of the formal steps needed to obtain the two re-sults.

In the appendices there are some useful instruments not directly related with the results but theoretically close.

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Chapter 1

Introduction to FvK model

Our starting point is elastic plate model discussed in [17]. The model is based on two works [9] and [25] and for this is named Föppl - von Kärmán (FvK hereafter). We begin with a slightly more general variational formulation than the one proposed in [7], so we can introduce the following modied action functional involving vertical amplitude of the deformation v(t, x) and the Airy stress function u(t, x), where x ∈ Ω with Ω being open domain in R2 with suciently regular boundary ∂Ω.

A(u, v) = 1 2 Z Z I×Ω |∂_tv|2− |∆v|2_{+ |(−∆)}σ/2_u|2_{+ u{v, v}}_dxdt _(1.1)

where {f, g} is the quadratic form

{f, g} = Q(∇2f, ∇2g) = X

|α|=|β|=2

qα,β∂xαf ∂xβg. (1.2)

A typical choice of the quadratic form Q is the following one

Q(∇2f, ∇2g) = ∂_x2₁f ∂_x2₂g + ∂_x2₂f ∂_x2₁− 2∂_x₁_x₂f ∂x1x2g. (1.3)

The corresponding Euler - Lagrange equations give the system (

∂_t2v + ∆2v = {v, u}

(−∆)σu = 1₂ {v, v} (1.4)

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10 1. Introduction to FvK model Remark 1.0.1 (Energy). Multiplying the rst equation of (1.4) by ∂tv and

the second by ∂tu, after integration we use the relations

Z R2 w(t, x){u, v}(t, x)dx = Z R2 u(t, x){w, v}(t, x)dx as well as Z R2 ∂tv(t, x){u, v}(t, x)dx = 1 2 Z R2 u(t, x)∂t{v, v}(t, x)dx = d 2dt Z R2 u(t, x){v, v}(t, x)dx −1 2 Z R2 {v, v}(t, x)∂tu(t, x)dx = 1 2 d dt Z R2 u(t, x){v, v}(t, x)dx +1 2 d dt Z R2 |(−∆)σ/2u(t, x)|2dx obtain that the energy given by

E(t) = E(u, v)(t) = 1 2k∂tv(t)k 2 L2_(R2₎+ 1 2k∆v(t)k 2 L2_(R2₎− (1.5) −1 2 Z R2 u(t, x){v, v}(t, x)dx −1 2k(−∆) σ/2_u(t)k2 L2_(R2₎ is a conserved quantity.

In the rst part of this work we study the case σ = 1 obtaining a result of global well posedness for small initial data via Strichartz estimates for vibrating plate and the properties of Riesz Transform.

In the second part we generalize the result to a perturbed version of the case σ = 2, the original problem in [25], obtaining again a result of global well posedness for small initial data.

Starting with the case σ = 2 we want prove a result of global well-posedness for small initial data using a contraction theorem argument. We rst reduce the system (1.4) to a nonlocal scalar plate equation and introduce the Strichartz estimate for plate equation that we will use in the proof. Reduction to nonlocal scalar plate equation

Using the second equation in the FvK system (1.4) we nd u = 1

2(−∆)

−1_{{v, v}}

so the system (1.4) is reduced to the scalar FvK equation ∂2_tv + ∆2v = 1

2{v, (−∆)

−1_{{v, v}}.} _(1.6)

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11 Remark 1.0.2. We can further obtain the conservation of the energy ex-pressed only in the term of v, i.e.

E(v)(t) = E 1 2(−∆)

−1_{{v, v}, v}

(t) so that after integration by parts we get

Z R2 u(t, x){v, v}(t, x)dx = − Z R2 ∆u(t, x)(−∆)−1{v, v}(t, x)dx = (1.7) −2 Z R2 |(−∆)1/2u(t, x)|2dx = −1 2 Z R2 |(−∆)−1/2{v, v}(t, x)|2dx and hence E(v)(t) = 1 2k∂tv(t)k 2 L2_(R2₎+ 1 2k∆v(t)k 2 L2_(R2₎+ (1.8) +1 8 Z R2 |(−∆)−1/2{v, v}(t, x)|2dx.

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Chapter 2

Elements of Harmonic Analysis

2.1 Sobolev Spaces

Sobolev spaces, [13, 14] and [20], are useful space to measure the smoothness of functions and decay properties at innity.

Denition 2.1.1 (Classical Denition). Let k a nonnegative integer and let 1 < p ≤ ∞; the Sobolev Space Wk,p(Rn) is dened as the space of functions f ∈ Lp(Rn) for all multi-indices α that satisfy |α| ≤ k.

The space is normed by the expression kf k_Wk,p :=

X

|α|≤k

k∂αf k_Lp

where ∂(0,··· ,0)_{f = f}_.

Observation 2.1.1. The k indicates the degree of smoothness of a given function. As k increases the functions become smoother, i.e.

Lp ⊃ W1,p ⊃ W2,p⊃ W3,p ⊃ · · · Sobolev Spaces are also complete.

The denition can be extended for not real k as follows.

Denition 2.1.2 (Modern Denition). s ∈ R, 1 < p < ∞, the Inhomoge-neous Sobolev Space Hs

p(Rn) is dened as the space of all tempered

distribu-tions u in S0

(Rn) with the property that ((1 + |ξ|2)s2u)ˆ V

is an element of Lp_(Rn₎ _{for such distributions u we dene}

kuk_Hs p = ((1 + | · | 2₎₂s_u_ˆ Lp_(Rn₎ 13

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14 2. Elements of Harmonic Analysis Observation 2.1.2. The space Ws,p _{is B}s

p,2.

Theorem 2.1.3 (Sobolev Embedding Theorem). [14] 1. Let 0 < s < n

p and 1 < p < ∞; then the Sobolev Space Ws,p(Rn)

continuosly embeds in Lq_(Rn₎ _when

1 p − 1 q = s n 2. Let 0 < s = n p and 1 < p < ∞; then W s,p_(Rd₎ _{continuously embeds in} Lq_(Rn₎ _{for any} n s < q < ∞. 3. Let n

p < s < ∞ and 1 < p < ∞; then every element of W

s,p_(Rn₎ _can

be modied on a set of measure zero so that the resulting function is bounded and uniformly continuous.

Observation 2.1.3. The Sobolev Embedding Theorem is a powerful instru-ment that we use implicitly in the proof of Theorem 3.2.1. Indeed is the theorem that let us avoid the problem with the endpoints and study condition quite near the the endpoint (and the theorem guarantees the result for all the values between the right endpoint and the one we proved).

In the results of Chapter 3 and Chapter 4 we talk also of homogeneous Sobolev space. The main dierence between the inhomogeneous one is tha elements of ˙Ws,p may not themselves be elements of Lp. Another relevant dierence is that in the homogeneous Sobolev space two elements whose dierence are polynomials are identied.

Denition 2.1.4. Let s be a real number and 1 < p < ∞; the homogeneous Sobolev Space ˙Ws,p(Rn) is dened as the space of all u in S0(Rn) \ P(Rn) for which the well-dened distribution

(|ξ|su)ˆ V

coincides with a function in ˙W1,p(Rd). For distributions u ∈ ˙Ws,p(Rd) we dene

kuk_W_˙s,p =

(| · |ˆu)V _˙

W1,p_(Rn₎

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2.2 Contraction Theorem 15

2.2 Contraction Theorem

A standard and powerful tool to deal with PDE is the abstract iteration argument [24, 3]. Let us work abstractly, consider the general solution

u = ul+ DN (u)

where ul is the linear solution (ul = eit∆u0 in the case of the Schr'odinger

equation), N is the nonlinearity and D is the Duhamel operator DF (t, x) :=

Z t

0

ei(t−s)∆F (s, ·)ds

Proposition 2.2.1 (Abstract iteration argument). Let N , T be two Banach spaces, let D : N → T be a bounded linear oeprator with the bound

kDF k_T ≤ C₀kF k_N (2.1) for all F ∈ N and some constant C0 > 0, and let N : S → N , with N(0) = 0,

be a nonlinear operator which is Lipschitz continuous and obeys the bounds kN (u) − N (v)k_N ≤ 1

2C0

ku − vk_T (2.2) for all u, v in the ball Bε := {u ∈ S : kukT ≤ ε} for some ε > 0. Then for

all ul ∈ Bε/2, there exists a unique solution u ∈ Bε to the equation

u = ul+ DN (u)

with Lipschitz map ul→ u with constant at most 2. That is, we have

kuk_T ≤ 2 ku_lk_T Proof. Observe that, for v = 0, the estimate

kN (u) − N (v)k_N ≤ 1 2C0 ku − vk_T becomes kN (u)k_N ≤ 1 2C0 kuk_T (2.3)

since N(0) = 0 by hypothesis. Then, x ul∈ Bε/2and consider the map

ϕ(u) := ul+ DN (u)

Using (2.1) and (2.3) one has

kϕ(u)k_T = kul+ DN (u)kT ≤ ε 2 + C0 2C0 ε = ε

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16 2. Elements of Harmonic Analysis for all u ∈ Bε(i.e. ϕ maps the ball Bεinto Bε). Moreover ϕ is a contraction

on Bε, indeed by (2.1) and (2.2) one has

kϕ(u) − ϕ(v)k_T = kDN (u) − DN (v)k_T ≤ C0kN (u) − N (v)kN ≤ C₀ 1 2C0 ku − vk_T = 1 2ku − vkT

for all u, v ∈ Bε. Then, the contraction theorem asserts that there exists a

unique xed point for u for ϕ and moreover the map ul → u is Lipschitz

with constant at most 2.

This proposition can be extended to dierent vector space and given in dierent forms, as showed in the proof of the main results of this thesis.

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2.3 Schrödinger Equations 17

2.3 Schrödinger Equations

As we seen in previous chapter, the operator that describes our model could be see as a product of two Schrödinger operators. The Cauchy problem for the free Schrödinger equation reads as follows

(

i∂tu + ∆u = 0

u(0, x) = u0(x)

with t ∈ R and x ∈ Rd_{, d ≥ 1. We can write the solution in terms of the}

Fourier transform as follows

u(t, x) = eit∆u0 (x) :=

Z

Rd

e2πix·ξe−4π2it|ξ|2uˆ0(ξ)dξ

and eit∆ _{is known as Schrodinger propagator. The corresponding}

inho-mogeneus equation is (

i∂tu + ∆u = F (t, x).

u(0, x) = u0(x)

By Duhamel's principle, the integral form of the solution is u(t, x) = eit∆u0(·) +

Z t

0

ei(t−s)∆F (s, ·)ds.

In this case is important to study the space-time integrability properties of the solution and, for this reason, we can use the Strichartz estimates.

Strichartz estimates work in dierent function/distribution spaces like Lebesgue, Sobolev, Wiener amalgam and modulation spaces and have found applica-tions to well posedness and scattering theory for nonlinear Schrödinger equa-tions [1, 2].

These estimates could also be adapted for operator obtained from the Schrödinger operator.

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18 2. Elements of Harmonic Analysis

2.4 Strichartz Estimates

2.4.1 Schrödinger equations

While exist many estimates with xed time, like Lp_{-dispersive estimate or}

Ws,r_{-xed time estimate [4, 3], often for the study of global well posedness it}

is useful to have estimates for the solution both in time and space variables. In this direction the mani result is represented by the Strichartz estimates. Denition 2.4.1. An exponent pair (q, r) is Schrödinger admissible if d ≥ 1 and 2 ≤ q, r ≤ ∞ ; 1 q = d 2 1 2− 1 r , (q, r, d) 6= (2, ∞, 2) Denition 2.4.2. An exponent pair (q, r) is Schrödinger accetable if

1 ≤ q ≤ ∞, 2 ≤ r ≤ ∞ ; 1 q < d 1 2 − 1 r , (q, r) 6= (∞, 2)

Observation 2.4.1. The original version in Lp _{of Strichartz estimates was}

elaborated by R. Strichartz in 1977. The following theorem is an extension developed by Ginibre and Velo that used the T∗_T _{method to detach the couple}

(q, r)to (q0, r0).

The endpoint case (i.e. (q, r) = 2,_d−22d ) is treated in Tao's work [15] in which they prove that the estimates work also for the endpoint in case d ≥ 3 while for d = 2 (i.e. (q, r) = (2, ∞)) the estimates are false (see Smith [18]). Notice that the new results of this thesis are exactly in the case d = 2 and, for this reason, they require an explicit use of the structure of the non linear term to obtain results for values of (q, r) near to the endpoint.

Lemma 2.4.3. [3] [Lp_{-Fixed Time Estimate]}

eit∆u0 _L_r (Rd₎. |t| −d₍1 2− 1 r) ku₀k Lr0_(Rd₎

Proof. Given a solution u(t, x) of the Cauchy problem of the nonlinear Schrödinger equation we want to study some estimates for xed t.

Since multiplication on the Fourier transform side intertwines with convolu-tion on the space side, formula

u(t, x) = eit∆u0 (x)

can be rewritten as

u(t, x) = (Kt∗ u0)(x)

where Kt is the inverse Fourier transform of the multiplier e−4π

2_it|ξ|2 , given by Kt(x) = 1 (4πit)d2 ei|x|24t

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2.4 Strichartz Estimates 19 Since eit∆ _{is a unitary operator, we obtain L}2_{-conservation law}

eit∆u₀

L2_(Rd₎ = ku0kL2_(Rd₎. (2.4)

Furthemore, since Kt ∈ L∞ and kKtk∞ ∼ t−

d

2, applying Young inequality

to the fundamental solution we obtain the L1_{-dispersive estimate}

eit∆u0 _L∞_(Rd₎. |t| −d 2 ku₀k L1_(Rd₎ (2.5)

This shows that if the initial data u0 ha a suitable integrability in space,

then the evolution will have a power type decay in time.

Using the Riesz-Thorin theorem we can interpolate (2.4) and (2.5) to obtain the Lp_{-xed time estimates.}

Lemma 2.4.4 (Fixed Time Estimate). Let d ≥ 1, 0 < α < d and 1 < p < q < ∞ such that 1 q = 1 p − d − α d . Then the following estimate

| · |−α∗ f

q . kf kp

holds for all f ∈ Lp_(Rd_).

Theorem 2.4.5. [3] For any Schrödinger admissible couples (q, r) and (˜q, ˜r) one has the homogeneus Strichartz estimates

eit∆u0

_Lq

tLrx(R×Rd). ku0kL2x(Rd)

the dual homogeneous Strichartz estimates Z R e−is∆F (s, ·)ds _L₂ x(Rd) . kF k_L˜q0 t Lr0x˜(R×Rd)

and the inhomogeneous (retarded) Strichartz estimates Z s<t ei(t−s)∆F (s, ·)ds _Lq tLrx(R×Rd) . kF k_Lq0˜ t L˜r0x(R×Rd) .

Proof. We shall only prove the homogeneous case in the non-endpoint case as an example of use of the T∗_T_-method.

Let (q, r) be Schrödinger admissible and consider the linear operator T : L1 tL2x→ L2x dened as T (F ) = Z R e−is∆F (s, ·)ds.

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20 2. Elements of Harmonic Analysis Its adjoint T∗_{: L}2

x→ L∞t L2x is the Schrödinger propagator

T∗(u) = eit∆u

Applying Minkowski'sinequality and the xed estimate 2.4.3 2.4.4 we obtain the diagonal untruncated estimates

Z R ei(t−s)∆F (s, ·)ds _Lq tLrx(R×Rd) ≤ Z R e i(t−s)∆_{F (s, ·)} _Lr x(Rd) ds _Lq t(R) . kF k_L˜r0 x(Rd)∗ 1 |t|d(12− 1 r) _Lq t(R) . kF k_L˜q0 t L˜r0x(R×Rd)

whenever 2 < q, r ≤ ∞ are such that 2 q +

d r =

d

2 and for any Schwartz

function F ∈ F(R × Rd_{. Then, using Lemma A.0.2 one obtains the}

ho-mogeneus Strichartz estimatees and the corrisponding dual homogeneous Strichartz estimates.

Observation 2.4.2. Assuming spherical simmetry [22] the Strichartz es-timates hold also for endpoint. In a preliminary work we tried to assume spherical simmetry to obtain a better estimate and don't have to do explicit work near the endpoint. A direct approach, instead, bring better results and avoid to assume tronger ipothesis.

To obtain Strichartz estimates for potential and Sobolev spaces it is enough to notice that the Schrödinger operator eit∆ _{commutes with Fourier}

multipliers like |∆|s _{or h∆i}s_{. In particular, if u : I × R}d_{→ C is a solution to}

the inhomogeneous equation with initial data u0 ∈ ˙Hxs(Rd), then, applying

|∆|s _{to both sides and using the previous estimates, we obtain}

kuk_Lq tW˙ s,r x (I×Rd). ku0kH˙xs(Rd)+ kF kLq0t˜W˙ s,˜r0 x (I×Rd)

2.4.2 Specialization to Plate Model

The FvK model is a particular case of the general vibrating plate equation, also know under the name of Germain - Lagrange equation by the ones that rst discovered the correct dierential equation as a model for the vibration of an elastic surface      ∂2_tu + ∆2u = F (t, x) u(0, x) = u0(x) ∂tu(0, x) = u1(x) (2.6) where t ∈ R, x ∈ Rd_{, F : R × R}d_{→ C, ∆ =}Pd

j=1∂x2i the Laplace Operator

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2.4 Strichartz Estimates 21 case of the Germain - Lagrange equation in the case of d = 2 and where F (t, x)is the nonlinear operator described in [6]. In this case we can use the structure of the nonlinear term to obtain the results showed in Chapter 3 and Chapter 4 [6].

To obtain the needed result, i.e. the Strichartz estimate, we take the free vibrating plate equation

∂_t2u(t, x) + ∆2u(t, x) = 0 and note that can be factorized as the following product

(∂_t2+ ∆2)u = (i∂t+ ∆)(−i∂t+ ∆)u

which displays the relation with the Schrödinger equation. Formally the vibrating plate operator

P = ∂_t2+ ∆2

can be recovered by composing two Schrödinger-type operators S1 = i∂t+ ∆ ; S2= −i∂t+ ∆

We compute the Fourier transform of the problem (2.6) in the homoge-neous case (i.e. F (t, x) = 0), obtaining the following dierential equation

     ∂_t2u + |ξ|ˆ 4u = 0ˆ ˆ u(0, ξ) = ˆu0(ξ) ∂tu(0, ξ) = ˆˆ u1(ξ)

Solving it and taking the inverse Fourier transform of the solution lead to the following

u(t, ·) = K0(t)u0+ K(t)u1

where

K0(t) = cos(t∆) ; K(t) = sin(t∆) ∆

For every xed t, the propagators K(t), K0_(t)_{are Fourier multiplies with}

symbols cos(t|ξ|2_{), sin(t|ξ|}2_)/|ξ|2_{, ξ ∈ R}d_{. To move from the homogeneous}

case to the inhomogeneous one we can use Duhamel's formula and write u(t, ·) = cos(t∆)u0+ sin(t∆) ∆ u1+ Z t 0 sin((t − s)∆) ∆ F (s)ds. Then is enough to use the simply formulas

cos(t∆) = e it∆_{+ e}−it∆ 2 ; sin(t∆) ∆ = eit∆_{− e}−it∆ 2i∆

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22 2. Elements of Harmonic Analysis to show that estimates for K0_{follow directly from the ones for the Schrödinger.}

Estimates on K follow instead from the continuity properties of the propa-gator eit∆

∆ .

The commutativity property of Fourier multipliers eit∆ _{and ∆ can be}

combined with the xed time estimates for the Schrödinger equation eit∆u0 _Lr . |t| −d₍1 2− 1 r) ku₀k Lr0, 2 ≤ r ≤ ∞ (2.7) lead to eit∆u₀ _˙ Ws,r . |t| −d₍1 2− 1 r) ku₀k_˙ Ws,r0 ; eit∆ ∆ u1 _Lq IW˙s,r . ku1k_H˙s−2

with 2 ≤ r ≤ ∞, s ∈ R. The sharpness of these estimates follow from the optimality of (2.7) then, using the Strichartz estimates for Schrödinger equation and the commutativity of Fourier multipliers we obtain the follow-ing estimates.

Observation 2.4.3. Admissible pairs for the (2.6) equation are the admis-sible Schrödinger couples.

Theorem 2.4.6. [4] Let d ≥ 1 and s ∈ R, then the following homogeneous Strichartz estimates eit∆u₀ Lq_IW˙s,r . ku0k_H˙s ; eit∆ ∆ u1 _Lq I, ˙Ws,r . ku1k_H˙s−2

and inhomogeneous Strichartz estimates Z t 0 ei(t−s)∆ ∆ F (s)ds Lq_IW˙s,r . kF k_Lq0˜ I W˙s−2,˜r0

hold for any admissible pairs (q, r) and (˜q, ˜r).

Proof. The two previous estimates follow directly from the commutativity property of the Fourier multipliers eit∆ _{and ∆ and the Strichartz estimates}

for the Schrödinger equation.

Corollary 2.4.7 (Strichartz estimates for the vibrating plate). Let d ≥ 1, s ∈ R and (q, r), (˜q, ˜r)be admissible pairs. If u is a solution to the Cauchy problem (2.6), then

kuk_Lq

IW˙s,r . ku0

k_H_˙s+ ku1k_H˙s−2+ kF k_Lq0˜

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2.5 Riesz Transform 23

2.5 Riesz Transform

During the proof of 3.2.1 we recall the Riesz transform. To give a formal background to our work we write here a very brief introduction to the Riesz transform, pointing out the main property that we've used [20],[21].

Denition 2.5.1. A Riesz transform Rj for j = 1, · · · , n is an operator

such that Rj(f ) = f ∗ Kj where ˆ Kj(ξ) = ξj i|ξ| ; Kj(x) = cnxj |x|n+1

and cn are the following constants

cn=

Γ n+1₂ πn+12

For our work is important to recall the relation between the Riesz trans-form and the Laplacian ∆, indeed the Riesz transtrans-form of f can be written as [21]

Rf = ∇(−∆)12f

and also, given u a Schwartz function, then RiRj(∆u) = −

∂2u ∂xi∂xj

Riesz transform have also many other good properties, indeed can be characterized (in the same way as Hilbert transform) with its property of invariance.

Proposition 2.5.2. [16] A family of translation invariant operators ˆT = (T1, · · · , Tn) bounded on L2(Rn) and commutating with positive dilations,

satises the identity

l_g−1◦ T ◦ l_g = π(g) ◦ T ; g ∈ O(n)

if and only if, up to a constant multiple, it is the family of Riesz transforms. Where in the previous statement lg(f )(x) := f (g−1x)and π(g) the standard

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24 2. Elements of Harmonic Analysis

2.6 Bessel functions

During the study of Yukawa potential in Section 4.2, we want to solve a particular integral equation (i.e. Y (x) = R_R2 e

iξx

x+ξ2dξ). Solve this integral is

the core of the proof and, as expected, can't be done in a direct way. We need then to write in a dierent way this integral to point out some particular structure that we could use to obtain some kind of upper bounds. The idea is to link the previous integral with a particular Bessel function [?], then use asymptotic behaviour of Bessel function to obtain the upper bound we need [8].

Denition 2.6.1. [26] A Bessel function of argument z and order v is Kv(z) := 1 2 v Γ v + 1₂ Γ 1₂ Z π 0

cos(z cos(θ)) sin2v(θ)dθ

and the integral is convergent for general complex values of v for which R(v) exceeds −1

2.

The previous integral could then be written as a Bessel function in the following way Yξ(x) = Z R2 eixη ξ + η2dη = Z ∞ 0 Z π 0 eiρ|x| cos(θ)dθ ρ ξ + ρ2dρ := Z ∞ 0 πJ0(ρ|x|) ρ ξ + ρ2dρ

Where J0(ρ|x|) is a Bessel function [26] because Bessel functions Jn admit

an explicit integral form [8] πJn(z) = i−n Z π 0 eiz cos(θ)cos(nθ)dθ and so J0(ρ|x|) = 1 π Z π 0 eiρ|x| cos(θ)dθ

Observation 2.6.1. Obviously the previous integral is far from being solved. In [6] we tried to solve this with dierent approaches, the rst one is to directly substitutes asymptotically expansions of Bessel functions subdividing low and high values of the variable. This way was too complicated and later we decided to use a dierent approach as shown in Chapter 4.

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Chapter 3

Global Well Posedness for σ = 1

We follow in this section the original structure of the proceeding associated to this thesis. Every theoretical instruments needed are contained not only in the cited articles but also in the previous chapters.

3.1 Strichartz estimates for vibrating plate

Denition 3.1.1. [4] The pair (q, r) is admissible if 2 ≤ q, r ≤ ∞,1 q + 1 r = 1 2, (q, r) 6= (2, ∞)

Proposition 3.1.2. [4] Let I ⊆ R, (q, r) and (˜q, ˜r) admissible pairs and s ∈ R, then: eit∆v0 _Lq IW˙s,r . kv0 k_H_˙s eit∆ ∆ v1 _Lq IW˙s,r . kv1k_H˙s−2 Z t 0 ei(t−s)∆F (s)ds _Lq IW˙s,r . kF k_Lq0˜ IW˙s−2,˜r0

The two formulas cos(t∆) = e it∆_{+ e}−it∆ 2 , sin(t∆) ∆ = eit∆− e−it∆ 2i∆

show that proposition (3.1.2) could be applied to obtain the following esti-mate Z t 0 sin((t − s)∆) ∆ F (u)(s)ds _Lq IW˙s,r . kF k_Lq0˜ I W˙s−2,˜r0 (3.1) 25

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26 3. Global Well Posedness for σ = 1 In the proof of (3.2.1) we will also use the estimate

Z t 0 sin((t − s)∆) ∆ F (s)ds L∞ I W˙s,2 + Z t 0 sin((t − s)∆) ∆ F (s)ds Lq_I˜W˙s,˜r . kF k_Lq0˜ I W˙s−2,˜r0 (3.2) that follows directly from proposition 3.1.2 observing that both (r, q) and (˜r, ˜q) are admissible and that the right side of (3.2) does not depend from the left side, so both terms can be upper bounded by the same value.

3.2 GWP for FvK with small initial data

Theorem 3.2.1. [6] Given (v0, v1) ∈ ˙H2 × L2 small enough, the Cauchy

problem      ∂2 tv + ∆2v = 12v, (−∆) −1_{{v, v}} v(0, x) = v0(x) ∂tv(0, x) = v1(x)

with (t, x) ∈ I × Ω where Ω being open domain in R2 _{with suciently regular}

boundary ∂Ω has an unique solution for every I ⊆ R.

Proof. In this proof we use a contraction theorem argument.

By Duhamel's formula [7] we can rewrite the problem in the following integral form v = cos(t∆)v0+ sin(t∆) ∆ v1+ Z t 0 sin((t − s)∆) ∆ F (u)(s)ds (3.3) and dene the map

T (v)(t) = cos(t∆)v0+ sin(t∆) ∆ v1+ Z t 0 sin((t − s)∆) ∆ F (u)(s)ds (3.4) We want to control the norm of T (v), for the rst two terms on the right side of (3.3) follow directly from Strichartz estimates for elastic plate (proposition 3.1.2) for small initial data:

kcos(t∆)v₀k_Lq IW˙s,r . kv0 k_H_˙s, sin(t∆) ∆ v1 _Lq IW˙s,r . kv1k_H˙s−2

For the nonlinear term we can still use the Strichartz estimate but we could choose admissible parameters ˜q, ˜r and s to obtain a good estimate. We put s = 2and: ₍ ˜ q = 2 + α ˜ r = 2(2+α)_α =⇒ ( ˜ q0 = 2+α_1+α ˜ r0 = 2(2+α)_4+α (3.5)

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3.2 GWP for FvK with small initial data 27 with α ∈ (0, ∞) to preserve Strichartz admissibility condition in proposition 3.1.2, obtaining the following inequality

Z t 0 sin((t − s)∆) ∆ F (s)ds Lq_IH˙r . kF k L 2+α 1+α I L 2(2+α) 4+α

and, for the equation (3.2), Z t 0 sin((t − s)∆) ∆ F (s)ds _L∞ I H˙2 + Z t 0 sin((t − s)∆) ∆ F (s)ds _L2+α I H˙ 2(2+α) α . kF k L 2+α 1+α I L 2(2+α) 4+α (3.6) Thanks to the Strichartz estimate we just need to study the norm of the nonlinear term F instead the whole integral.

The reason of this preliminary result with σ = 1 is that we can use Riesz Transform to simplify the problem to study this integral. From [13], [14], we could dene the Riesz Transform as Rj with the property that ∂xu = RxDu

and ∂yu = RyDu, where D :=

√

−∆. These transforms are invariant in the norm k·kLq_Lr (with (q, r) 6= (2, ∞)) so we could write the nonlinear part as

v, (−∆)−1_{{v, v} =} = ∂_x2v∂_y2(−∆)−1{v, v} + ∂_y2v∂_x2(−∆)−1{v, v} − 2∂xyv∂xy(−∆)−1{v, v} = R_x2D2vR_y2{v, v} + R2 yD2vR2x{v, v} − 2RxRyD2vRxRy{v, v} (3.7) but {v, v} = ∂_x2v∂_y2v + ∂_y2v∂_x2v − 2∂xyv∂xyv = R_x2D2vR_y2D2v + R2_yD2vR_x2D2v − 2RxRyD2vRxRyD2v (3.8) and, by substituting (3.8) in (3.7), we obtain:

v, (−∆)−1_{{v, v} = R}2 xD2vRy2Rx2D2vR2yD2v + R2xD2vR2yRy2D2vR2xD2v − 2R2_xD2vR2_yRxRyD2vRxRyD2v + R2yD2vR2xRxD2vR2yD2v + R_y2D2vR2_xR_y2D2vR2_xD2v − 2R2_yD2vR2_xRxRyD2vRxRyD2v − 2RxRyD2vRxRyR2xD2vR2yD2v − 2RxRyD2vRxRyRy2D2vR2xD2v + 4RxRyD2vRxRyRxRyD2vRxRyD2v (3.9) By the property of subadditivity of norms, we could also write

v, (−∆)−1_{{v, v}} Lq0_I˜Lr0˜ . R2_xD2vR2_yR_x2D2vR2_yD2v Lq0_I˜L˜r0 + · · ·

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28 3. Global Well Posedness for σ = 1 where on the right of this inequality there are all the terms of (3.9).

To study each terms on the right we use the generalized Hölder inequality and obtain (here in case of the rst term)

R2_xD2vR2_yRxD2vR2yD2v Lq0_I˜L˜r0 (3.10) ≤ R2_xD2v Lq1Lr1 R2_yR2_xD2v Lq2Lr2 R2_yD2v Lq3Lr3 . D2v _L_q1_L_r1 D2v _L_q2_L_r2 D2v _L_q3_L_r3

where the last inequality is given by Riesz Transform's property [14]. Notice that every terms on the right of (3.9) have the same structure, this lead to the following kF (v)k Lq0_I˜Lr0˜ . D2v _L_q1_L_r1 D2v _L_q2_L_r2 D2v _L_q3_L_r3 (3.11) The last problem that must be solved to complete the proof is to control if exists (qi, ri) that satisfy admissibility request. We could easily give an

example of admissible quadruple but we want nd the more general set of all admissible elements with the assumption that (q1, r1) = (∞, 2) and

(q3, r3) = (q2, r2): kF (v)k Lq0_I˜L˜r0 . D2v _L∞_L2 D2v 2 Lq2Lr2

that must satisfy the following                1 ˜ q0 = _q1 1 + 2 q2 1 ˜ r0 = _r1 1 + 2 r2 1 ˜ q0 + _˜_r10 = 3₂ 1 q1 + 1 r1 = 1 2 1 q2 + 1 r2 = 1 2 =⇒ (₁ ˜ q0 = _q2 2 1 ˜ r0 = 1₂ +_r2 2 (3.12)

where the rst two rows follow from Hölder's condition, the third is the condition of admissibility for Strichartz estimate and the last two are chosen to obtain also on the right side the same norm. Notice that the fourth equa-tion is always fullled with the choice (q1, r1) = (∞, 2) and that the third

could be obtained by the rst summed to the second.

We've already given in (3.5) a description of ˜q0_{, ˜}_r0 _{in function of a}

param-eter k ∈ (0, 2), we want to write also q2, r2 with that parameter to obtain

the explicit range of k that could be used. Solving (3.12) we obtain (₁ q2 = 1+α 2(2+α) 1 r2 = 4+α 4(2+α)− 1 4

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3.2 GWP for FvK with small initial data 29 Notice that we want to keep valid the admissibility condition, so 1

q2,

1 r2 <

1 2

and this leads to α > −1 ; α > −2 that is less restrictive then α > 0. This mean that for every α ∈ (0, ∞) we could nd a solution of the system in the form that we want.

To complete the proof we can dene the norm kvk_X := 3 X j=1 D2v _L_qj_L_rj = D2 _L∞_L2+ 2 D2 _L_q2_L_r2

and the problem could be seen as a x point one T (v) = v with T dened in (3.4). From the previous steps we obtain:

kT (v)k_X ≤ Cε |{z}

small initial data

+C kvk3_X and then, if T (vk) = vk+1, kT (v_k+1) − T (vk)kX ≤ C max n kv_k+1k2_X, kvkk2X o · kv_k+1− v_kk_X and then T : B(C1ε) → B(C1ε)with C1 = 2C and ε small enough. For

con-traction theorem, then ∃!v ∈ B(C1ε)such that v = T (v) and this complete

the proof of GWP for our Cauchy problem.

As said during the proof, another way to solve the problem of the pa-rameters in Hölder inequality (3.12) is to give an explicit solution. We have to remember that, for inequality (3.6), a good choice of α is really near to 0, so we can obtain a control over the norm on the interval 1

˜ q + 1 ˜ r = 1 2 for ˜

q ∈ (2 + ε, ∞). Choosing ˜q = 2 + ε and ˜r = 2(2+ε)_ε we obtain that, for ε small enough

(

q1 = q2= q3= 3(2+ε)_1+ε

r1 = r2= r3 = 6(2+ε)_4+ε

is an admissible choice for (3.11). Notice that this solution is not con-tained in the open set described above.

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Chapter 4

Global Well Posedness for

perturbated version of σ = 2

4.1 A generalized result

We want to generalize the result obtained in previous sections. To do that we take the problem described by the following equation

(ξ − ∆)(−∆)u = 1

2{v, v} (4.1) that is the one obtained from (1.1) choosing σ = 2 and perturbing the rst term with a l that grants regularity.

By formally inverting the operator (ξ − ∆) we obtain (−∆)u = (ξ − ∆)−1 1

2{v, v}

= Y ∗ 1

2{v, v} (4.2) With Y the Yukawa (or Bessel) potential [14].

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32 4. Global Well Posedness for perturbated version of σ = 2

4.2 Study of Yukawa potential

We need to study the term on the right of (4.2) and so we write the explicit form of Y , that is a radial function, and move to polar coordinate

Yξ(x) = Z R2 eixη ξ + η2dη = Z ∞ 0 Z π 0 eiρ|x| cos(θ)dθ ρ ξ + ρ2dρ := Z ∞ 0 πJ0(ρ|x|) ρ ξ + ρ2dρ (4.3)

Where J0(ρ|x|) is a Bessel function [26] because Bessel functions Jn admit

an explicit integral form [8] πJn(z) = i−n Z π 0 eiz cos(θ)cos(nθ)dθ and so J0(ρ|x|) = 1 π Z π 0 eiρ|x| cos(θ)dθ

We want to obtain some upper bounds to Y (x) in Lp_{with appropriate p,}

to do that we use an explicit way to write our integral in terms of modied Bessel's functions.

From [8] (pag. 95, rel. (51)) we can use the following formula Z ∞

0

Jµ(bt)(t2+ z2)−νtµ+1dt = (b/2)ν−1z1+µ−νKν−µ−1(bz)/Γ(ν)

if Re(2ν − 1/2) > Re(µ) > −1 ; Re(z) > 0

(4.4) If we specialize 4.4 in our case (i.e. µ = 0 and ν = 1) we obtain

Z ∞ 0 J0(|x|ρ) ρ ρ2_{+ x}dρ = K0(|x| p ξ) (4.5)

and the conditions 4.4 are always fullled because Re(3/2) > Re(0) > −1 ; Re(z) > 0

this means that we could study K0(x), the modied Bessel function,

instead of Y (x).

We subdivide the problem in two parts, x < 1 and x > 1. From [8] (pag. 86 rel. 7), specialized per ν = 0 we obtain for large variable

K0(z) = e−z π 2z 1₂ "_{M −1} X m=0 cm(2z)−2m+ O(|z|−M) # . |z|−12e−z (4.6)

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4.2 Study of Yukawa potential 33 with

cm=

Γ(1/2 + m) Γ(1/2 − m)

For the neighborhood of the origin, instead, we obtain ([8], pag. 9 rel. (38)) K0(z) = I0(z) ln z 2 + ∞ X m=0 z 2 2ψ(m + 1) (m!)2 (4.7)

with ([8], pag. 5 rel. (13)) I0 = ∞ X n=0 z 2 2m 1 m!Γ(m + 1) and then K0(z) . ln(z/2) for z < 1.

By the combination of the two previous results (4.6) and (4.7) we obtain Yξ(x) = K0(|x| p ξ) . ( ln(|x|√ξ/2) = ln(|x|) + ln(√ξ/2) if |x| < √1 ξ |√ξ|−12e−|x| √ ξ _{if |x| >} _√1 ξ (4.8) and then Y (x) ∈ L1 _{(i.e. kY k}

L∞_L1 < CY).

Remark 4.2.1. We could obtain the same result of (4.8) in another less evident way. We could use the result of [12] to link the study of K0 with the

study of the free covariance that in our case (i.e. d = 2, m = 1 and y = 0, see rel. 7.2.2 pag. 162 [12]) is

C(x) = K0(|x|)

Notice that for the free covariance we have the following propositions that lead again to (4.8).

Proposition 4.2.1. [12] For x bounded away from 0 C(x) . |x|−12e−|x|

For |x| in a neighborhood of zero

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34 4. Global Well Posedness for perturbated version of σ = 2

4.3 GWP for generalized case

With the result of the previous section we're ready to prove the GWP theo-rem for the perturbed σ = 2 problem.

Theorem 4.3.1. [6] Given (v0, v1) ∈ ˙H2 × L2 small enough, the Cauchy

problem            ∂_t2v + ∆2v = {v, u} (ξ − ∆)(−∆)u = 1₂{v, v} v(0, x) = v0(x) ∂tv(0, x) = v1(x) (4.9) with (t, x) ∈ I × Ω where Ω being open domain in R2 _{with suciently}

reg-ular boundary ∂Ω has an unique solution for every I ⊆ R and for every ξ 6= 0. Proof. This proof is similar to the one of the previous GWP theorem 3.2.1. As seen in (4.2) we take the second equation and obtain the formal solution

u = (−∆)−1

Y ∗1 2{v, v}

and substituting this in the rst one of (4.9) we obtain ∂2_tu + ∆2u = F (u) := 1

2v, (−∆)

−1

(Y ∗ {v, v}) We can write again the iteration with the Duhamel's formula

T (v)(t) = cos(t∆)v0+ sin(t∆) ∆ v1+ Z t 0 sin((t − s)∆) ∆ F (u)(s)ds And for Strichartz estimates for VP (3.6) we obtain the usual upper bound of the three terms, in particular we have to control the norm of kF kL˜q0_L˜r0

This is the only dierence between the two proofs, notice that as in (3.7) we can write v, (−∆)−1_{(Y ∗ {v, v}) = ∂}2 xv∂2y(−∆) −1 (Y ∗ {v, v})+ + ∂_y2v∂_x2(−∆)−1(Y ∗ {v, v}) − 2∂xyv∂xy(−∆)−1(Y ∗ {v, v}) = R_x2D2vR2_y(Y ∗ {v, v}) + R2_yD2vR2_x(Y ∗ {v, v})− − 2R_xRyD2vRxRy(Y ∗ {v, v}) (4.10)

We can study this norm with the generalized Hölder inequality obtaining (in the case of the rst term, the others are the same)

R2_xD2vR2_y(Y ∗ {v, v}) Lq0˜_Lr0˜ ≤ R_x2D2v _L_q1_L_r1 R2_y(Y ∗ {v, v}) Lq2Lr2 . D2v Lq1Lr1k(Y ∗ {v, v})kLq2Lr2

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4.3 GWP for generalized case 35 with 1 ˜ q0 = _q1 1 + 1 q2 and 1 ˜ r0 = _r1 1 + 1 r2.

Now we can apply Young's inequality obtaining

k(Y ∗ {v, v})k_Lq2Lr2 ≤ kY kL∞_L1 · k{v, v}k_Lq2Lr2

but we proved in (4.8) that kY kL∞_L1 ≤ CY constant. Notice also that, for

(3.8), k{v, v}k_L_q2_L_r2 ≤ D2v _L_q3_L_r3 D2v _L_q4_L_r4 with 1 q2 = 1 q3 + 1 q4, 1 r2 = 1 r3 + 1 r4 , and so R2_xD2vR2_y(Y ∗ {v, v}) L˜q0_Lr0˜ . CY D2v _L_q1_L_r1 D2v _L_q3_L_r3 D2v _L_q4_L_r4 like before we have to control that the norms found on the right side are still admissible _                     1 ˜ q0 = _q1 1 + 1 q2 = 1 q1 + 1 q3 + 1 q4 1 ˜ r0 = _r1 1 + 1 r2 = 1 r1 + 1 r3 + 1 r4 1 ˜ q0 +_r_˜10 = 3₂ 1 q1 + 1 r1 = 1 2 1 q3 + 1 r3 = 1 2 1 q4 + 1 r4 = 1 2

but if we choose r3 = r4 ; q3 = q4 this is exactly the same as the condition

in (3.12) and we have already proved that exists a quadruple that solve this problem.

For subadditivity of norm and because all the terms in (4.10) have the same structure we obtain again

kF k_Lq0˜_Lr0˜ . D2v _L_q1_L_r1 D2v 2 Lq3Lr3

With this upper bound for kF k we can nish the proof in the same way as the theorem 3.2.1.

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Chapter 5

Conclusion and further

development

5.1 Conclusion

As seen in Chapter 3 and Chapter 4 the two main results of the thesis are the global well posedness for two dierent modications of the original Foppl -Von Karman model. The rst one, the case σ = 1 is a toy model, a problem shaped to study the particular structure of the nonlinear term and under-stand how we can use it to approach the general mode. The second result is the general case, i.e. σ = 2. This version of the problem is quite hard to study and so we reach a result on a perturbated version that allow us to use again the structure of the problem and also the Yukawa potential, as seen in Chapter 4.

Note that the result is nice for three reasons:

• Global Well Posedness is guaranteed for small initial data but for low regularity initial condition. Indeed, as seen in Chapter 3 and Chapter 4 it's enough to have (u0, u1) ∈ ˙H × L2.

• As said during the introduction, the idea of the thesis is to reach some numerical results on a model of human blood. The Global Well Posed-ness is then a starting point to construct the numerical model.

• The result of the perturbated case could be used to extend the result to the non-perturbated case (i.e. σ = 2 without the ε term).

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38 5. Conclusion and further development

5.2 Further development

This thesis, as said in the introduction, starts as a proceedings presented at the conference NTADES 2018 (New Trends in the Applications of Dieren-tial Equations in Science) in Soa. Previously the starting point was some works of Y. Shibata about modelling of the blood ow and the interaction between some membrane models, like the Foppl - Von Kármán one, and the inner uid.

For this reason further developments are of two dierent types, the rst one is more linked to Harmonic Analysys, the second one to Numerical Anal-ysis.

Harmonic Analysis and PDE

On the theorical side the rst step is to complete the proceeding [6]. This means obtain a GWP result for the problem with σ = 2, i.e.

(

∂_t2v + ∆2v = {v, u} (−∆)2u = 1₂ {v, v}

indeed this case is the one more important from a physical point of view. This could be done in two dierent ways, the rst one is to use something similar to the R-method proposed by Y. Shibata in his works on Stokes' equations [19]. The second approach instead is more direct, and study the simmetry property of our nonlinear term to obtain directly a Global Well Posedness result for case σ = 2. This second approach is the one that we hope lead to a brief and nice proof of our problem.

The second step is to try to study the mathematical properties of this model with particular attention to turbolence problem (in the membrane). The idea is to use the modied energy method (proposed by T. Tao in vari-ous articles and well described in [23]) to obtain a lower bound for Sobolev norm that could be used to obtain some good inequalities on this norm that could lead to some control over turbolence.

The third step is to approach scattering phenomena, a starting point are some works of V. Georgiev and M. Tarulli on similar models (i.e. [10]). Note that the idea to approach scattering phenomena was born during the confer-ence of NTADES 2018 where M. Tarulli presented a work about scattering on very similar model and that could be adapted to FvK model using the particoular simmetry of the nonlinear term.

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5.2 Further development 39 At last but not least it is fundamental to complete the model of the vein. This means that we want to add the inner uid to our model and study the interaction between the usual and well known Navier - Stokes equations and the FvK model developed for the membrane. Knowing the inner diculties of models that involve Navier - Stokes equation this part will be also approached in a numerical way.

Numerical Analysis

On the numerical part, the one that will be mostly developed during the PHD at GSSI, we can study two linked problems.

The rst one is to study a numerical model for the FvK model. There are several researchers at the University of Milano Bicocca that study simi-lar problems, for example D L. Beirao da Veiga developed some interesting methods [5] to approach problem like the FvK.

The second one is to study methods for the Navier - Stokes equations and the interaction between the inner uid and the membrane. These topics are exactly one of the research eld of the GSSI and the core of the numerical part of PHD.

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40 5. Conclusion and further development

5.3 Two pages Thesis

• We started from the model of a vein obtained adapting an old model of plate equation (Foppl - Von Karman).

• To obtain a solid starting point for numerical experimentation and theoretical study of this model we need to prove global well posedness for the associated Cauchy problem

(

∂_t2v + ∆2v = {v, u} (−∆)σu = 1₂ {v, v}

• Solving formally the second one and substituting in the rst one we obtain the equation that we want to study

∂_t2v + ∆2v = 1

2{v, (−∆)

−σ_{{v, v}}.}

• Main result 1: Global Well Posedness for σ = 1. Rewrite the problem using Duhamel's formula.

Dene the iteration map to use a contraction theorem argument. Using Strichartz estimates for the plate equation we obtain an easy control in terms of initial data of two terms of Duhamel's formula. The third term can be written as a function of the nonlinear function.

(First Problem) To use Strichartz estimates avoiding endpoint we need to explicitly compute possible values of the parameters. Using the known form of the nonlinear term and invariance

prop-erty of Riesz transforms we control the norm of the nonlinear part.

(Second Problem) Existence of admissible parameters to use both Strichartz estimates and Hölder inequality.

Dene the right norm that alows to use contraction theorem. • Main result 2: Global Well Posedness for a perturbated version of

σ = 2.

Using the same structure of the previous proof, is enough to write the solution as a convolution between previous one and a term that is the Yukawa potential.

Yukawa potential can be write as an integral involving Bessel function.

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5.3 Two pages Thesis 41 (Third Problem) We need an upper bound of the Yukawa

po-tential to control the solution of the general problem. By using the properties of Bessel functions and free covariance operator we obtain that result.

Use Young's inequality to control the solution expressed in terms of convolution.

(Fourth Problem) Explicit control that the parameters could be chosen to be admissible and satisfy Hölder inequatiliy.

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Appendix A

T

∗

T

-method

Discovered by Tomas in 1975 [11, 3], the T∗_T _{method is a powerful abstract}

tool of Harmonic Analysis.

This method allows to know the continuity of a linear operator T (thus also of its adjoint T∗_{) by considering the boundedness of the composition operator}

T∗T.

Denition A.0.1. For any vector space D, D∗

ais its algebraic dual, La(D, X)

is the space of linear maps from D to X (another vector space) and hϕ, fi_D is the pairing between D∗

a and D (f ∈ D, ϕ ∈ Da∗) taken to be linear in f

and antilinear in ϕ.

Lemma A.0.2. Let H be a Hilbert space, X a Banach space, X∗ _{the dual}

space of X and D a vector space densely contained in X. Let T ∈ La(D, H)

and T∗ _{∈ L}

a(H, Da∗) its adjoint, dened by

hT∗h, f i_D) hh, T f i ∀f ∈ D, ∀h ∈ H

where h·, ·i is the inner product in H (thus antilinear in the rst argument). Then the following three conditions are equivalent:

1. There exists α, 0 ≤ α < ∞ such that ∀f ∈ D kT f k_H≤ α kf k_X

2. Let h ∈ H, then T∗_h _{can be extended to a continuous linear functional}

on X, and there exists α, 0 ≤ α < ∞, such that ∀h ∈ H kT∗hk_X∗≤ α khk_H

3. Let f ∈ X, then T∗_{T f} _{can be extended to a continuous linear functional}

on X, and there exists α, 0 ≤ α < ∞, such that ∀f ∈ D kT∗T f k_X∗≤ α2kf k_X

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44 A. T∗_T_-method

The constant α is the same in all the three cases. If one of (all) those conditios is (are) satised, the operators T and T∗_T _{extend by continuity to}

bounded operators form X to H and from X to X∗_{, respectively.}

Proof. D is densely contained in X, thus it follows that X∗ _{is a subspace of}

D_a∗.

• (1 =⇒ 2) Let h ∈ H, then, ∀f ∈ D

|hT∗h, f i_D| = |hh, T f i| ≤ khk_HkT f k_H≤ α khk_Hkf k_X • (2 =⇒ 1) Let f ∈ D, then, ∀h ∈ H

|hh, T f i| = |hT∗h, f i_D| ≤ kT∗hk_X∗kf k_X ≤ α khk_Hkf k_X

• (1 + 2 =⇒ 3) obvious, and thus 1 or 2 =⇒ 3. • (3 =⇒ 1) Let f ∈ D, then

kT f k2 = |hT f, T f i| = |hT∗T f, f i_D| ≤ kT∗T f k_X∗kf k_X ≤ α2kf k2_X.

Since D is a dense subspace of X, we see that T can be extended to a bounded linear functional from X to H.

Corollary A.0.3. Let H, D ad two triplets (Xi, Ti, αi), i = 1, 2, satisfy the

conditions of Lemma A.0.2. Then for all choices of i, j = 1, 2, R(T∗ i Tj) ⊂ X_i∗ and ∀f ∈ D kT_i∗Tjf k_X∗ i ≤ α_iαjkf kXj. (A.1) In particular T∗

i Tj extends by continuity to a bounded operator from Xj to

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Bibliography

[1] F. A. Berezin and M. Shubin. The Schrödinger Equation, volume 66. Springer Science & Business Media, 2012.

[2] E. Cordero and F. Nicola. Some new strichartz estimates for the schrödinger equation. Journal of Dierential Equations, 245(7):1945 1974, 2008.

[3] E. Cordero and D. Zucco. Strichartz estimates for the schrödinger equa-tion. Cubo (Temuco), 12(3):213239, 2010.

[4] E. Cordero and D. Zucco. Strichartz estimates for the vibrating plate equation. Journal of Evolution Equations, 11(4):827845, 2011.

[5] L. B. da Veiga, K. Lipnikov, and G. Manzini. The mimetic nite dif-ference method for elliptic problems, volume 11. Springer, 2014.

[6] G. Del Corso and V. Georgiev. On the modied fvk model (proceeding). Pliska Studia Mathematica, 30, 2018.

[7] G. Düring, C. Josserand, and S. Rica. Wave turbulence theory of elastic plates. Physica D: Nonlinear Phenomena, 347:4273, 2017.

[8] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, et al. Higher transcendental functions, vol. 1, 1953.

[9] A. Föppl. Vorlesungen über technische Mechanik: Graphische Statik. 7, volume 2. BG Teubner, 1912.

[10] V. Georgiev, A. Stefanov, and M. Tarulli. Smoothing-strichartz esti-mates for the schrodinger equation with small magnetic potential. arXiv preprint math/0509416, 2005.

[11] J. Ginibre and G. Velo. Generalized strichartz inequalities for the wave equation. In Partial Dierential Operators and Mathematical Physics, pages 153160. Springer, 1995.

[12] J. Glimm and A. Jae. Quantum physics: a functional integral point of view. Springer Science & Business Media, 2012.

(46)

46 BIBLIOGRAPHY [13] L. Grafakos. Classical fourier analysis, volume 2. Springer, 2008. [14] L. Grafakos. Modern fourier analysis, volume 250. Springer, 2009. [15] M. Keel and T. Tao. Endpoint strichartz estimates. American Journal

of Mathematics, 120(5):955980, 1998.

[16] T. Kobayashi and A. Nilsson. Group invariance and l p-bounded oper-ators. Mathematische Zeitschrift, 260(2):335354, 2008.

[17] L. D. Landau and E. M. Lifshitz. Theory of elasticity. 1965.

[18] S. J. Montgomery-Smith. Time decay for the bounded mean oscillation of solutions of the schr\" odinger and wave equations. arXiv preprint math/9704212, 1997.

[19] Y. Shibata et al. On the r-boundedness of solution operators for the stokes equations with free boundary condition. Dierential and Integral Equations, 27(3/4):313368, 2014.

[20] E. M. Stein. Harmonic Analysis (PMS-43), Volume 43: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.(PMS-43), volume 43. Princeton University Press, 2016.

[21] E. M. Stein and G. Weiss. Introduction to Fourier analysis on Euclidean spaces (PMS-32), volume 32. Princeton university press, 2016.

[22] T. Tao. Spherically averaged endpoint strichartz estimates for the twodi-mensional schrödinger equation. Communications in Partial Dierential Equations, 25(7-8):14711485, 2000.

[23] T. Tao. Nonlinear dispersive equations: local and global analysis. Num-ber 106. American Mathematical Soc., 2006.

[24] P. Villaggio. Salsa, sandro: Equazioni a derivate parziali. metodi, mod-elli e applicazioni (partial dierential equations. methods, models and applications). Meccanica, 46(2):477478, 2011.

[25] T. Von Kármán. Festigkeitsprobleme im maschinenbau. Teubner, 1910. [26] G. N. Watson. A treatise on the theory of Bessel functions. Cambridge