Approches numeriques par des volumes finis du modele de keller-segel

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APPROCHES NUMERIQUES PAR DES VOLUMES

FINIS DU MODELE DE KELLER-SEGEL

FINITE VOLUME METHODTITRE

ةرازو

ميلعتلا

لياعلا

ثحبلاو

يملعلا

BADJI MOKHTAR –ANNABA

UNIVERSITY

UNIVERSITE BADJI MOKHTAR

ANNABA

ةعماج

يجاب

راتخم

-ةبانع

-Faculté des Sciences

Année : 2018

Département de Mathématiques

THESE

Présentée en vue de l’obtention du diplôme de Doctorat en Sciences

Option

Mathématiques appliquées

Par

MESSIKH CHAHRAZED

DIRECTEUR DE THÈSE : GUESMIA AMAR M.C. U20 Août SKIKDA

CO-ENCADREUR: SAADI SAMIRA prof. U.B.M. ANNABA

Devant le jury

PRESIDENT :

Nouri Fatma Zohra

Prof

U.B.M. ANNABA

EXAMINATEUR :

Haiour Mohamed Prof

U.B.M. ANNABA

EXAMINATEUR :

Daili Nourreddine

Prof

U.F.A. SETIF

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ةرازو

ميلعتلا

لياعلا

ثحبلاو

يملعلا

BADJI MOKHTAR -ANNABA

UNIVERSITY

UNIVERSITE BADJI MOKHTAR

ANNABA

ةعماج

يجاب

راتخم

-ةبانع

Faculté des Sciences

Année : 2018

Département de Mathématiques

THÈSE

Présentée en vue de l’obtention du diplôme de Doctorat en sciences

APPROCHES NUMERIQUES PAR DES VOLUMES

FINIS DU MODELE DE KELLER-SEGEL

Option

Par

MESSIKH CHAHRAZED

Devant le jury

EXAMINATEUR :

Maoumi Messaoud

M.C.A

U.20 Août SKIKDA

Mathématiques appliquées

PRESIDENT:

Nouri Fatma Zohra Prof

U.B.M. ANNABA

EXAMINATEUR :

Haiour Mohamed Prof

U.B.M. ANNABA

EXAMINATEUR :

Daili Nourreddine

Prof

U.F.A. SETIF

DIRECTEUR DE THÈSE : GUESMIA AMAR M.C.A U. 20 Août SKIKDA

CO-ENCADREUR : SAADI SAMIRA Prof U.B.M ANNABA

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ءادهإ

،ةبحملاو نانحلاب ينترمغ يتلا يمأ ىلإ

،ءيشب ااموي يلع لخبي مل يذلا يبأ ىلإ عضاوتملا لمعلا اذه يدهأو

،ةفرعملاو علطلا فغش ىلع ةأشنلاو لملاو ةايحلا ينومتبهو متنأ امهل لوقأ

:

،ااعيمج يترسأو يتوخإ ىلإو

،ةبيبحلا يتنبإ ىلإ

،يتمع و يمع حور ىلإ

،يتذتاسأ ىلإ

،يتليمز و يائلمز ىلإ

يمامأ قيرطلا ءيضي هقرب انس حبصأ اافرح ينملع نم لك ىلإ مث

.

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Acknowledgments

I thank all who in one way or another contributed in the completion of this thesis. First of all and foremost, I would like to thank Allah (swt) for giving me the strength to finish this study.

Furthermore, I would like to express my gratitude to my principle supervisor, Dr Guesmia Amar and Co-supervisor, Professor Saadi Samira for their constant guidance and encouragement.

I would also like to show gratitude to my committee including Professor Ms Nouri Fathima Zhora, Professor Mr Haiour Mohamed, Professor Mr Daily Noured-dine, and Professor Mr Maouni Messaoued, for their very helpful comments and suggestions.

I would also like to thank all those who participated closely or remotely to my research and to the preparation of this thesis.

Finally, I extend my sincere thanks to my family: My parents, My brothers, my little sister, and my daughter, who have accompanied, helped, supported and encouraged me throughout this thesis.

I would also like to thank my friends and colleagues for their encouragement and moral support.

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Abstract

The Keller-Segel chemotaxis model is described by a system of nonlinear PDEs. The system under consideration represents a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration.

In this thesis, the global and local solutions of both Keller-Segel model and its space fractional model were studied using certain assumptions.The proof is based on the Lax-Miligram Theorem, Galerkin’s method and the principle of the Maximum. For the Keller-Segel model, the finite volume method is utilized under certain hypotheses to prove the existence and unique-ness of an approximate positive solution. Besides, under adequate regularity conditions on the exact solution of this problem, the finite volume scheme is the first order accurate in both time and space.

Regarding the space fractional Keller-Segel model, the finite volume meth-od is used to discretise the space fractional mmeth-odel. By using the Grünwald formula the discretization of the fractional derivative term is obtained eas-ily. Moreover, the study demonstrates the stability and the convergence of finite volume method. To test the effectiveness of the proposed method, a comparison with the finite difference method is carried out.

Several examples and numerical experiments are provided. A good agree-ment between the numerical simulation and the theoretical results are ob-tained. Furthermore, the results reveal that the finite volume method is effi-cient compare to the finite difference method.

Keywords: Grünwald formula; Finite volume method; Keller-segel;

Frac-tional differential equation; Chemotaxis; Principe maximum discret; Galerkin method

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Résumé

Le modèle de chimiotaxie de Keller-Segel est décrit par un système d’EDP non linéaire. Le système considéré représente une équation de convection-diffusion pour la densité cellulaire couplée d’une équation de réaction-convection-diffusion pour concentration de chimioattractif.

Dans cette thèse, les solutions locale et globales du modèle de Keller-Segel et son modèle fractionnaire spatial ont été étudiées. La démonstration est basée sur le Théoreme de Lax-Miligram, la méthode de Galerkin et le principe du Maximum. Pour le modèle de Keller-Segel, la méthode des vol-umes finis est utilisée sous certaines hypothèses pour prouver l’existence et l’unicité d’une solution positive approximative. En outre, sous certaine conditions adéquate de la régularité de la solution exacte de ce problème, le schéma obtenu par cette méthode est de premier ordre en espace et en temps.

En ce qui concerne le modèle fractionnel de Keller-Segel, la méthode des volumes finis est utilisée pour discrétiser le modèle fractionnel spatial. En utilisant la formule de Grünwald, la discrétisation du terme de dérivée frac-tionnaire est facile à obtenir. De plus, l’étude démontre la stabilité et la con-vergence de la méthode des volumes finis. Pour tester l’efficacité de la méth-ode proposée, une comparaison avec la méthméth-ode des différences finies est effectuée.

Plusieurs exemples et expériences numériques sont fournis. Un bon ac-cord entre la simulation numérique et les résultats théoriques sont obtenus. De plus, les résultats révèlent que la méthode des volumes finis est efficace par rapport à la méthode des différences finies.

Mots clés: Formule de Grünwald; Méthode des volumes finis; Modèle de

Keller-Segel; Equation différentielle fractionnaire; Chimiotaxie; Principe du Maximum discret; Méthode de Galerkin.

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صخلم

ة بجوملا لو لحلا ة ينادحو و دو جو ة ساردب ا نمق ة حورطلأا هذه يف

جدو من يهف ة يناتلا ا مأو رجل سرليك جدو من يه ىلولأا :نيتلك شمل

أد بملاو مار غليم سكل ة يرظن مادخت سإب رجلس رليكل يرسكلا قتشملا

.يمظعلأا

ة قيرط لامعت سإب نيتلك شملا نيتا هل ة يددعلا ليلحتلا ةساردب انمق امك

جئا تنلا هذ ه نا اندجو دقل و ،دودحملا قرفلا ةقيرط و دودحملا مجحلا

.يرظنلا رابتخلإا جئاتن عم ةقفاوتم

ةيحاتفملا تاملكلا

:

لكش

درليك

،

ةقيرط

مجحلا

دودحملا

،

جدومن

رليك

لغي س

،

ةلداعملا

ةير سكلا ةيل ضافتلا

،

باد جنا

يئا يميك

،

اد بم

دحلا

لصفنملا ىصقلاا

،

ةقيرط

.يكلاق

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List of Figures

1.1 Example chemotaxis. . . 22

4.1 Two control volumes of an admissible mesh. . . 55

4.2 Concentration of chemical signal substance for example 1 . . . 65

4.3 Temporal distribution of bacteria of example 1 for x = y = 0.5. 66

4.4 Maximum of the temporal distribution of bacteria for example 1. 67

4.5 Contour plots for example 1.. . . 68

4.6 Concentration of chemical signal substance for example 2. . . 69

4.7 Temporal distribution of bacteria of example 2 for x=y= 0.5. . 69

4.8 Maximum of the temporal distribution of bacteria for example 2. 70

4.9 Contour plots for example 2.. . . 71

4.10 Comparison between FVM and FEM. . . 72

5.1 FDM and the FVM for α = 1.5, γ = 2, β = 1 . . . 86

5.2 Exacte and approximate solution for α = 1.5, γ = 1, β = 2 . . . 86

5.3 Error between exact and approximate solution for γ = 1, β = 2 87

5.4 Error between exact and approximate solution for γ = 2, β = 1 87

6.1 FDM and FVM for different values of α. . . 97

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Chapter 1 Introduction

1.1 Background

Taxis refers to the collective motion of cells or an organism in response to an attractant gradient. The nature of the attractant stimulus can be of chemical (chemotaxis), physical (baro-, electro-, magneto-, phono-, photo-, and ther-motaxis), or mechanical (gyro-, hapto-, and rheotaxis) origins. Chemotaxis is the oriented movement of cells along concentration gradients of chemicals produced by the cells themselves or in their environment, and is a signifi-cant mechanism of directional migration of cells. The movement towards a higher concentration of the chemical substance is called positive chemotaxis, whereas the movement towards regions of lower chemical concentration is called negative chemotaxis.

Many types of cells use chemotaxis to actively move to specific loca-tions. The inflammatory process provides an excellent example of chemo-taxis, wherein immune cells respond to a gradient of chemokines or chemoat-tractants, and move up the gradient to reach the site of infection. Once the immune cells “sense” the gradient, they extravasate from vascular vessels and move toward the infection site within the adjacent tissue to destroy bac-teria, remove dead cells, and heal the wound area. In case of the multi-cellular organisms, chemotaxis of cell populations plays also a crucial role throughout the life cycle: sperm cells are attracted to chemical substances released from the outer coating of the egg; during embryonic development it plays a role in organising cell positioning, for example during gastrulation

and patterning of the nervous system [19]; in the adult, it directs immune

cell migration to sites of inflammation [93] and fibroblasts into wounded

re-gions to initiate healing. Similar process can be also repeated seen cancer

growth, allowing tumour cells to invade the surrounding environment [83]

or stimulate new blood vessel growth (angiogenesis) [70]. Extensive research

has been conducted into the mechanistic and signalling processes regulating

chemotaxis in bacteria, particularly in E. coli [24], and in the life cycle of

cell slimemolds such as Dictyostelium discoideum [90]. While the

biochem-ical and physiologbiochem-ical bases are less well understood, chemotaxis also plays a crucial role in the navigation of multicellular organisms. The nematode worm C. elegans undergoes chemotaxis in response to a variety of external

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signals [44] while in insects, the fruit fly Drosophila melanogaster navigates

up gradients of attractive odours during food location [91] and male moths

follow pheromone gradients released by the female during mate location

[2]. We give the following example of chemotaxis to better understand this

phenomenon:

FIGURE1.1: Example chemotaxis.

(http://www.csi.uoregon.edu/projects/celegans/talks/figures/ nips97/1.ce-ctx.gif)

Theoretical and mathematical modelling of chemotaxis dates to the

pio-neering works of Patlak in the 1950s [72] and Keller and Segel in the 1970s

[49]. The review article by Horstmann [46] provides a detailed introduction

into the Keller–Segel mathematics model for chemotaxis. In its original form this model consists of four coupled reaction-advection-diffusion equations, which can be reduced under quasi-steady-state assumptions to a model of two unknown functions u and c. This model is the basis of our study in this research work. The general form of this model is given by the following set of partial differential equations:

u_t_{− ∇(m∇u) + ∇(ξu∇c) = 0, (x, t) ∈ R}d

× R+_,

δct− ∆c + τ c + ρu = 0, (x, t) ∈ Rd× R+, (1.1)

where u(x, t) denotes the density of bacteria in the position x ∈ Rd _{at time}

t, c is the concentration of chemical signal substance, δ ≥ 0 represents the relaxation time, the parameter ξ is the sensitivity of cells to the chemoattrac-tant, m, τ and ρ are given smooth functions.

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Referring to (1.1), the model can be either Parabolic-Parabolic model when

δ 6= 0or Elliptic-Parabolic model when δ = 0. In case when δ = ρ = 0, this

model (1.1) exhibits a profound mathematical structure and mostly only

di-mension 2 is understood, especially chemotactic collapse. The KS model has

been extensively studied in the last few years (see [43, 44,92, 48] for recent

survey articles).

The Keller-Segel model is a system of partial differential equations which models chemotactic aggregation in cellular systems. It is the diffusive char-acter of the cellular motion known to be false is many situations, (for exam-ple in the case when the interactions within comexam-plex and non homogeneous media), because diffusion is not adequately described by the classical theory

of Brownian motion and Fick ‘s Law [41, 42]. We called this phenomena the

anomalous transport.

Recently, the process of anomalous transport begun receiving attention in the mathematical and scientific community discovery of real-world processes that exhibit anomalous transport. These processes cannot be adequately modelled using the classical theory of diffusion, and hence there is a pressing need to develop new mathematical and computational techniques based on the correct anomalous transport models. Indeed, the application to anoma-lous transport has been the most significant driving force behind the rapid growth and expansion of the literature in the area of fractional calculus. Re-cently, one can find numerous textbooks published on fractional calculus

and its applications, we can cite [75,66,51,23,77].

In general the field of fractional differential equation and fractional

cal-culus provide a mean for modelling such anomalous transport [41]. The

space fractional diffusion equations are obtained from the classical diffusion equations by replacing the space derivative by a generalized derivative of fractional order. The Keller-Segel system involving in their linear parts

frac-tional powers Dα _{:= −(−∆)}α/2 _{of the Laplacian with α ∈ (0, 2]. Non-local}

operators, and in particular the fractional Laplacian, have received a lot of

attention recently [36, 35,55, 57]. In biology the motivation comes from the

fact that in many cases organisms adopt L´evy-flight search strategies and

therefore dispersal is better modeled by non-local operators [8,26].

1.2 Literature Review

Chemotaxis has attracted significant interest due to its critical role in a wide range of biological phenomena. Recently, There is an increase in the num-ber of researches that use Keller-Segel and fractional Keller-Segel models to

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1.2.1 Applications

There are many applications for chemotaxie among them:

1. Elementary living beings, like bacteria, are able to reach food sources

using only limited and very noisy sensory information. In [21], the

au-thors describe a very simple algorithm inspired from bacteria chemo-taxis. they presented a Markov chain model for studying the effect of noise on the behavior of an agent that moves according to this algo-rithm, and they showed that the application of noise can increase the expected average performance over a fixed available time. After this theoretical analysis, experiments on reworld application of this al-gorithm are carried out. In particular, they showed that the alal-gorithm is able to control a complex robot arm, actuated by 17 McKibben pneu-matic artificial muscles, without the need of any model of the robot or of its environment.

2. Bacterial Foraging Optimization (BFO) is a recent developed nature-inspired optimization algorithm, which is based on the foraging be-havior of E. coli bacteria. Due to its biological motivation and graceful structure, this algorithm has drawn the attention of many researchers

from diverse fields [67, 38, 33, 85]. Although the algorithm has

suc-cessfully been applied to many kinds of real word optimization prob-lems, experimentation with complex problems reports that the basic BFO algorithm possesses a poor performance. Improvement of this method is obtained using the Self-tuned Bacterial Foraging

Optimiza-tion (STBFO) [34].

3. Contaminated soil and ground water persistently threatens drinking-water supplies, and is difficult and expensive to remediate. In situ bioremediation is an effective remediation strategy, but is often lim-ited by inadequately distribution of bacteria throughout a contami-nated region. Bacterial chemotaxis describes the ability of bacteria to sense chemical concentration gradients in their environment, and preferentially swim toward optimal concentrations of chemicals that are beneficial to their survival. This mechanism may greatly increase the efficiency of ground-water remediation technologies by enhancing bacterial mixing within contaminated zones. Many of the native soil-inhabiting bacteria that degrade common environmental pollutants also

exhibit chemotaxis toward these compounds. In [82], authors

pre-sented a review of bacterial chemotaxis to recalcitrant ground-water contaminants, including relevant techniques for mathematically quan-tifying chemotaxis, and proposed improvements to field-scale biore-mediation methods using chemotactic bacteria. By exploiting the degrada-tive and chemotactic properties of bacteria, we can potentially improve both the economics and the efficiency of in situ bioremediation.

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1.2.2 Analytical and numerical solutions

The system1.1was extensively studied by many authors and a lot of

mathe-matical results on the existence of global in -time -solutions Authors and on

the blow-up of local in time solutions have been processed. We refer to [99,

100,26] for a quite complete bibliography.

In most cases, it is not possible to find analytical solutions to Partial differ-ential equation. It is in order to be able to calculate approximate solutions that the numerical methods have been developed. There are several numer-ical methods for the discretization of a partial differential equation. Among these methods, the most used are the finite element method (FEM), the finite difference method (FDM) and the finite volume method (VFM). The finite volume method is a well-adapted discretization method for the simulation of the conservation laws compared to with the other methods (FDM and FEM).

In the literature, there are several approaches to solve the classical Keller-Segel system numerically. The parabolic-elliptic model was approximated

by using finite difference [80,87] or finite element methods [59,78,84]. Also

a dynamic moving-mesh method [9], a variational steepest descent

approxi-mation scheme [7], and a stochastic particle approximation [40,39,65] were

developed.

Concerning numerical schemes for the parabolic-parabolic model , we

men-tion the positivity- preserving second-order finite volume method of [18],

the discontinuous Galerkin approach of [25, 84], and the conservative finite

element scheme of [79]. We also cite the paper [10] for a mixed finite

ele-ment discretization of a Keller-Segel model with nonlinear diffusion. There are only a few works in which a numerical analysis of the scheme was per-formed. Filbet proved the existence and numerical convergence of finite

vol-ume solutions [30]. Error estimates for a conservative finite element

approxi-mation were shown by Saito [78,79]. Epshteyn and Izmirlioglu proved error

estimates for a fully discrete discontinuous finite element method [25].

Con-vergence proofs for other schemes can be found in [7,39].

The numerical solution of the fractional differential equation was discussed by several authors using different methods. For example, the finite

differ-ence method was discussed in [60, 61] and the finite volume method in

[41]. In these references the standard and Shifted Grünwaled formula for

discretize the Liouville derivative fractional term was used.

1.3 Research objectives

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(P)           

ut− ∆u + div (u∇v) = 0, (t, x) ∈ [0, T ] × Ω,

−∆v + τ v = 0, x ∈ Ω, u = 0, Γ × [0, T ] , u (0, x) = u0, x ∈ I, v = g, Γ. and (FP)            ut+ Dαu + div (u∇v) = 0, (t, x) ∈ [0, T ] × Γ, −∆v + τ v = 0, x ∈ Ω, u = 0, Γ × [0, T ] , u (0, x) = u0, x ∈ I, v = g, Γ.

Where Ω is a bounded open convex domain in Rd_{, (d ≤ 3 for the}

prob-lem (P ) and d=1 for the probprob-lem (FP)), with smooth boundary Γ, u0 ∈

H1_(Ω)_{, g ∈ H}1/2_(∂Ω)_{and τ is a positive constant. Here we define D}α_{u (x) =}

F−1_{(F (D}α_{v)) (x) = F}−1_(|ξ|α

F v (ξ)) (x), where F denotes the Fourier

trans-form and F−1 its inverse.

The work aims to achieve the following objectives:

1. To study the existence and uniqueness of the solutions for problems (P) and (FP) via the Lax-Milgran theorem and Galerkin method. 2. To develop a numerical solutions of the problem (P) using FVM and

compared them with other software results.

3. To solve numerically the problem (FP) using the FVM and study its the convergence and stability.

4. To compare between FVM and FDM approximate solution for the (FP) model.

1.4 Thesis outline

This thesis is composed of two parts. The first one focus the existence and uniqueness of solutions for Keller-Segel and fractional derivative Keller-Segel models using the Galerkin method, Lax-Milgran Theorem and Maximun principle while the second focus on the study of these numerical problems using some numerical methods such as finite volume method and finite dif-ference. Following is a brief description of each chapter for this thesis is arranged as follows:

Chapter 2

In this chapter, we consider the elliptic-parabolic model’s problem (P ). The purpose of this work is to derive the formulation of classic Keller-Segel Model

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and also its minimal models that was obtained by proposing assumptions on the data. We prove the uniqueness, existence and positivity of solutions for

problem (P ) on bounded convex domains in R2 _{or R}3 _{using Lax-Milgran ’s}

Theorem, Maximum principle and Galerkin method. We introduce the en-ergy associated to problem (P ) and prove uniform decay for solutions for this problem. Finally, we draw our conclusion.

Chapter 3

In this chapter, the study of local and global solutions of a following frac-tional Keller-Segel problem (F P ) are discussed. Moreover, the uniform de-cay rates of the energy are discussed as time goes to infinity. The standard technique is used to employ a proof based on the Galerkin method and en-ergy estimates.

chapter 4

In this chapter, we are interested on the numerical simulation of the Keller-Segel Elliptic-Parabolic problem model (P ) using an implicit finite volume scheme. We have showed, under certain assumptions, the existence of a

unique and positive approximation solution. The L∞ estimate of the

ap-proximate solutions for the problem (P ) as well as the estimate error are established. Moreover, under adequate regularity assumption of the exact solution, the finite volume scheme is the first order accurate. A good agree-ment between our numerical simulations and the theoretical results has been discussed.

Chapter 5

In this chapter, we present an implicit finite volume method for the numeri-cal solution of the one- dimensional space fractional Keller-Segel system (P ) with the source on a finite domain: Our method is based on the fractionally-shifted Grünwald formulas which helped us in the discretisation of the frac-tional derivative terms. We also prove that the method is stable and con-vergent of order 1 in space and time. Finally, we report several numerical experiments illustrating the efficiency of our methods.

Chapter 6

We derive the finite difference discretization for the problem (F P ) in one di-mension domain. The main difference between the two methods is that the

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finite volume method deals directly with the differential equation in conser-vative form and eliminating the need for product rule expansions in vari-able coefficient problems. In addition to that, we compare the numerical solutions obtained with the two methods for several test problems.

Chapter 7

Finally, in this chapter, some concluding remarks are given and some possi-ble future works are suggested.

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Chapter 2 Existence and uniqueness of

solutions for the KS problem

2.1 Chemotaxis

Chemotaxis (from chemo + taxis) is the movement of an organism in re-sponse to a chemical stimulus. Somatic cells, bacteria, and other single-cell or multisingle-cellular organisms direct their movements according to certain chemicals in their environment. In case when multicellular living, chemo-taxis plays an important role in the development and physiological func-tioning of the organism. For example the animal moves to seek food or runs away in case there is danger. The mathematical analysis of chemotaxis mod-els shows a plenitude of spatial patterns such as the chemotaxis modmod-els

ap-plied to skin pigmentation patterns [69, 71, 93] that lead to aggregations of

one type of pigment cell into a striped spatial pattern. And other models

ap-plied to the aggregation patterns in an epidemic disease [93], tumor growth

( these mechanisms allow chemotaxis in animals can be subverted during

cancer metastasis [15]), angiogenesis in tumor progression [12] , and many

other examples (we can see the thesis of Moustafa Ibrahim [68] ).

Theoretical and mathematical modeling of chemotaxis dates to the

pio-neering works of Patlak in the 1950s [72] and Keller and Segel in the 1970s

[49, 50]. The review article by Horstmann [46] provides a detailed

intro-duction into the mathematics of the Keller–Segel (KS) model for chemotaxis. In its original form, this model consists of four coupled reaction-advection-diffusion equations. These can be reduced under quasi-steady-state

assump-tions to a model for two unknown funcassump-tions u and v (we can see [76]). The

general form of the Keller–Segel model in which we are interested is given by the following system

ut = ∇(K1(u, v)∇u − K2(u, v)u∇v) + K3(u, v), (2.1)

vt = Dv∆v + K4(u, v) + K5(u, v)v. (2.2)

Where

1. u denotes the cell density.

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3. K1 is the diffusivity of the cells

4. K2 is the chemotactic sensitivity.

5. K3 describes the cell growth and death.

6. K4 and K5 describe the production and degradation of the chemical

signal, in signal concentration model.

Remark 2.1.1. The cell density refers to the number of cells per unit volume. Often

cell density is denoted as viable cell density which is the number of living cells per unit volume.

Remark 2.1.2. The concentration of a chemical in a solution refers to how many of

the chemical’s molecules are sitting in a small volume of the solution. Concentration could be measured in molecules per liter, although molecules are so small compared to a liter that we usually use different units . A gradient is a measurement of how much something changes as you move from one region to another. So a concentration gradient is a measurement of how the concentration of something changes from one place to another.

Note that cell migration is dependent on the signal gradient. The Keller–Segel

model (2.1-2.2) is studied by many authors, for example we can cite the

re-view articles of Horstmann [46] and the paper of Bendahmane et al. [3].

2.2 Formulation of classic Keller-Segel Model

We will derive the formulation of classic Keller-Segel Model and minimal models that was obtained by proposing assumptions on the data.

By posing D1 = K1(u, v), χ = K2(u, v), f = K3, D2 = K4(u, v), g =

K5(u, v), and h = K6(u, v). The Keller-Segel model defined by equations (2.1

and (2.2) is simplified to

ut = ∇(D1∇v − χu∇v) + f (2.3)

vt = D2∆v + g − h (2.4)

Although the parameters in classical KS model are straight forward, it is very important to understand the formulation steps of this classical model. Generally, the classical KS model is derived from a basic assumption of an

arbitrary surface S enclosing a volume V [44]. According to the general

conservation equation, the change rate of the material amount u in V equals to the flux rate of u across S out of V plus the u created/disappeared in V . Thus, ∂ ∂t Z V udV = − Z S ϕ.ηdS + Z V f dV, (2.5)

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where ϕ is the flux of material u and f is the source term of u. According to the Divergence Theorem,we can write that

Z S ϕ.ηdS = Z V ∇.ϕdV. (2.6)

Since the function of the cell density u is continuous, and the volume V is

arbitrary, the integrand have be zero. Thus, equation (2.6) can be rewritten

as _Z

V

(ut+ ∇.ϕ − f )dV = 0. (2.7)

So, we obtain

ut+ ∇.ϕ − f = 0. (2.8)

This equation describe the general flux transport ϕ whether by diffusion or by some other processes. Since the flux in our chemotaxic model is con-tribute by two different terms, which are cell diffusion flux and chemotaxis flux. The flux is defined as

ϕtotal= ϕdif f + ϕchemo, (2.9)

where we consider Fick’s law as the process of cell diffusion flux ϕdif f is

given by

ϕdif f = −D1∇u, (2.10)

ϕchemo is the chemotaxis flux defined as

ϕchemo = χu∇v, (2.11)

and χ is chemotactic coefficient.

The analysis of χ in various forms has been carried out by different

re-searchers. Now, putting the ϕtotalinto equation (2.8) yields

ut= ∇(D1∇u − χu∇v) + f, (2.12)

the cell density part of the classical KS model.

By repeating the same process above, for one chemical attractant, we yield the chemical attractant concentration part of the classical KS model.

vt= D2∆v + g(u) − h(v). (2.13)

Yet, the classical KS model is still too complicated for us to solve and to simulate the cell behaviour. Some more assumption needs to be made to simplify our model. Thus, we come up with Minimal Model of classical KS model. The necessity assumptions are as follow:

1. Individual cells undergo a combination of random motion and chemo-taxis towards chemical attractant.

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3. The attractant is produced at constant rate.

4. The degradation rate of attractant is linearly dependent on its concen-tration.

5. The attractant diffuses passively over the field.

Using these assumptions, the cell proliferation/death term f (u, v) of

equa-tion (2.12) is now 0, the term g(u, v) in the equation (2.13) is now only the

function of u, and the term h(u, v) in the equation (2.13) is now only the

func-tion of v. Taking D1, D2, and χ also be positive constant, thus the parabolic

quasi-linear equation of minimal model of KS model can be noted as

ut = ∇(D1∇v − χu∇v), (2.14)

vt = D2∆v + au − bv. (2.15)

Remark 2.2.1. Under the assumptions (1-5) and if the system (2.14-2.15) is assume

that there isn’t a variation in the concentration of the chemical signal a long as the

time, then (vt) is equal to zero. Therefore, we obtain the parabolic-elliptic

Keller-Segel system.

2.3 Existence of weak solution of the problem (P)

For the Keller-Segel system defined in (2.14-2.15), putting K1 = K2 = 1,

K3 = 0, K4 = τ v(τ is a is positive constant), vt = 0and Dv = −1a new form

can be obtained as follows;

(P )            (P1)   

ut− ∆u + div (u∇v) = 0, (t, x) ∈ R+× Ω,

u = 0, Γ,

u (0, x) = u0, x ∈ Ω,

(P2)

−∆v + τ v = 0, x ∈ Ω,

v = g, Γ.

Where Ω is a bounded convex domain in R2 _{or R}3 _{with smooth}

bound-ary Γ. The first equation of problem P1 expresses the movement of bacteria

(representing a random distribution side and a deterministic drift in the

di-rection of high concentrations) and first equation of problem P2 describes

the diffusion degradation of v.

To demonstrate an existence and uniqueness of weak solutions for the system (P), we follow the next steps: Firstly, we have to demonstrate a

existence and uniqueness of weak solution for the system (P2) using

Lax-Milgran Theorem. The end, we will prove the same result of (P1) to (P2)

using Galerkin method. As (P) is formulated by two systems (P1)and (P2).

The existence and the uniqueness result are proved. Based on that, the

fol-lowing initial-boundary conditions on u0and g assumptions are set to prove

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H1 g ∈ L 1 2 (Γ) . H2 g ∈ L 3 2 (Γ) . H3 u0 ∈ L2(Ω) . H4 u0 ≥ 0 and g ≥ 0.

2.3.1 Variational formulation of the Keller-Segel problem

We will derive a variational (or weak) formulation of problems (P1)and (P2).

We will discuss all fundamental theoretical results that provide a rigorous understanding of how to solve these problems using the Lax-Milgran

The-orem and Galerkin method. If the hypothesis H1 is satisfies and using the

Trace Theorem, then there exists ˜g ∈ H₀1(Ω)such that γ0(˜g) = g.

Now looking for v having the form v = ev + ˜g ∈ H

1_(Ω) _{reduces the}

prob-lem (P2)toev ∈ H 1 0(Ω). (P2) −∆ e v + τ_ev − ∆˜g + τ ˜g = 0, in x ∈ Ω, e v = 0, on Γ.

Definition 2.3.1. We say (u,ev) ∈ L

2_{(0, T ; H}1

0 (Ω)) × H01(Ω)with

ut∈ L2(0, T ; H−1(Ω))is a weak solution of the problem (P) if and only if

hut, wi + B (u, w, t) = 0, (2.16)

a (_ev, q) = l (q) , (2.17)

where _





B (u, w, t) =R_Ω(∇u∇w + ∇v∇uw + τ vuw) dx ,

a (_ev, q) =R_Ω(∇_ev∇q + τ_evq) dx , l (q) = −R_Ω(∇˜g∇q + τ ˜gq) dx, for all (w, q) ∈ (H1 0(Ω)) 2 , 0 ≤ t ≤ T, and u (0, x) = u0 ∈ L2(Ω) . (2.18)

Remark 2.3.1. Note that u ∈ C ([0, T ] ; L2_(Ω)) _{as u ∈ L}2_{(0, T ; H}1

0(Ω)) and

ut∈ L2(0, T ; H−1(Ω)). Then, equality (2.18) makes sense.

2.3.2 Existence, uniqueness and positivity of a weak

solu-tion for the problem (P

2

)

In this subsection, we state and prove the existence, uniqueness and the

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1. Existence and uniqueness result:

Theorem 2.3.1. (Existence and uniqueness of a weak solution) If the

hypoth-esis H1 holds. Then, the problem (P2)has only one solution c ∈ H1(Ω) for

any q ∈ H1_{(Ω) .}

Proof. By applying the Lax-Milgran Theorem, the solution ev of the

problem (2.17) exists and it is unique. So, (P2)has unique solution.

Remark 2.3.2. Elliptic regularity Theorem remains valid provided that the

boundary condition g is in the space L32 (Γ)which is the space image H2(Ω)

by the operator trace γ because γ : H2_{(Ω) ,→ L}3₂ _(Ω)_.

2. Positivity a weak solution: Using the Maximum principle one can show

that the solution of the problem (P2)is positive as follows.

Multiplying the first equation of (P2) by q ∈ H01(Ω), we obtain other

variational formula for problem (P2)

e P3 Z Ω (∇v∇q + τ vq) dx = 0.

Proposition 2.3.1. If g ∈ L12 (Γ)and τ > 0, then the problem

e P3 have a positive solution v∈ H1_(Ω)_.

Proof. As Γ is smooth enough and g ∈ L12 (Γ), then from Theorem2.3.1

v ∈ H1(Ω). If v = g ≥ 0 on Γ, then v− = v ∈ (v, 0) ∈ H₀1(Ω). So, we have Z Ω vv−dx = Z Ω v−2 dx, Z Ω ∇v∇v−dx = Z Ω ∇v−2 dx.

This implies that 0 = Z ∇v−2 + τ v−2dx ≥ min (1, τ ) v− 2 H1 0(Ω) . So v− _{= 0.} _{As v = v}

−+ v+ with v+ = max(v, 0), then we conclut that

v = v+is a positive function.

Remark 2.3.3. (a) If g ∈ L32(Γ), then the solution v ∈ H2(Ω).

(b) If v ∈ H2_(Ω) _{and v is a solution of problem (P}

2)this implies that v ∈

W1,q(Ω) because H2(Ω) ,→ W1,q(Ω)for 1 ≤ q ≤ 2∗_.

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2.3.3 Existence, uniqueness and positivity of a weak

solu-tion of (P

1

)

Before proving the existence and uniqueness of weak solution of problem

(P1), we need the following Lemma:

Lemma 2.3.1. 1. For all v ∈ H1

0(Ω)and if H2 holds, then B (., ., t) is

continu-ous in H1

0 (Ω) × H01(Ω) ,there exists a constant positive C such that

|B (u, v, t)| ≤ C kuk_H1

0(Ω)kvkH01(Ω). (2.19)

2. For any u ∈ H1

0 (Ω) and H2is hold. Then there exists a constant positive β

such that

β kuk2_H1

0(Ω) ≤ B (u, u, t) . (2.20)

Proof. 1. We use the Cauchy-Schwarz inequality and from Remark2.3.3

it follows that v ∈ H2_{(Ω) ,→ L}q_(Ω)_{for any q ∈} _1, 2n

n−2 with n = 2 or

n = 3, we obtain the relation (2.20) as follows

B (u, v, t) ≤ k∇uk_L2_(Ω)k∇vk_L2_(Ω)+ k∇vk_L4_(Ω)kuk_L2_(Ω)kvk_L4_(Ω)

+τ kvk_L4_(Ω)kuk_L2_(Ω)kvk_L4_(Ω)

≤ C kuk_H1_(Ω)kvk_H1_(Ω).

2. Making use of −∆v + τ v = 0 the expression of B (u, u, t) becomes

B (u, u, t) = Z (∇u)2+∇v 2 ∇u 2_{+ τ vu}2_dx = Z (∇u)2+ τ v −∆v 2 u2dx = Z (∇u)2+1 2τ vu 2 dx ≥ k∇uk2_L2_(Ω).

Finally, by Poincare inequality yields

B (u, u, t) ≥ β kuk2_H1

0(Ω).

1. Galerkin method

To demonstrate the existence of weak solution of (P1)via the method

of Galerkin, we assume wk= wk(x)are smooth functions verifying

{wk}

∞

k=1 is an orthogonal basis of H

1

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and {wk} ∞ k=1 is an orthonormal basis of L 2 (Ω) . (2.22)

Consider a positive integer m and we will look for a function

um : [0 T ] → H01(Ω)of the form um(t) := m X k=1 dk_m(t) wk (2.23) which satisfies dk_m(0) = (u0, wk) , (2.24) and hu0_m, wki + B (um, wk, t) = 0, 0 ≤ t ≤ T and k = 1, ..., m (2.25)

where u0 = u_tand here (., .) denotes the scalar product in L2_(Ω)_.

Theorem 2.3.2. (Construction of the approximate solution) For each integer

m, there exists a unique function um of the form (2.23) satisfying (2.24) and

(2.25).

Proof. Assuming um has the structure (2.23). Substituting (2.23) into

(2.24) and using (2.22) we obtained

d0k_m(t) +

m X

l=1

dl_mB (wl, wk, t) = 0 0 ≤ t ≤ T and k = 1, ..., m. (2.26)

According to standard existence theory for ordinary differential

equa-tions, there exists a unique absolutely continuous functions dm(t) =

(d1_m, d2_m, ..., dm_m)satisfying (2.24) and (2.26). So, um stated in (2.23)

satis-fies (2.24) and (2.25) for all t ∈ [0 T ].

2. Existence result:

We will introduce the energy associated to problem (P1)and prove

uni-form decay of solutions for this problem. We propose now to send m

to infinity and show a subsequence of our solutions um of problems

(2.24) and (2.25) converges to a weak solution of (P1). For this, we will

need some uniform estimates.

Theorem 2.3.3. (Energy estimates). There exists a constant C, depending

only on Ω, T and v, such that max 0≤t≤TkumkL2(Ω)+ kumkL2(0,T ; H01(Ω)) + ku 0 mkL2_{(0,T ; H}−1_(Ω))≤ C ku0k_L2_(Ω). (2.27) for m = 1, 2, . . .

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Step1. Multiplying equation (2.25) by dk

m(t), summing for k = 1, ..., m,

and then recalling (2.23) we find

(´um, um) + B (um, um, t) = 0for all 0 ≤ t ≤ T. (2.28)

From Lemma2.3.1, there exists constant β > 0 such that

β kumk2_H1

0(Ω) ≤ B (um, um, t) for all 0 ≤ t ≤ T. (2.29)

Consequently (2.29) yields the inequality

d dt kumk2_L2_(Ω) + β kumk2_H1 0(Ω) ≤ 0 for all 0 ≤ t ≤ T. (2.30)

This implies that

kumk2_L2_(Ω)≤ kum(0)k_L22_(Ω)≤ ku0k2_L2_(Ω) for all 0 ≤ t ≤ T. (2.31)

So, we have

max

0≤t≤TkumkL2(Ω) ≤ ku0kL2(Ω). (2.32)

Step2. Integrate inequality (2.30) from 0 to T and employing the

in-equality (2.32) yield kumk 2 L2₍_{0,T ; H}1 0(Ω)) = T Z 0 kumk 2 H1 0(Ω)dt ≤ C ku0k 2 L2_(Ω).

Step3. Fix any w ∈ H1

0(Ω), with kwk 2 H10(Ω) ≤ 1, and write w = w 1_{+ w}2_, where w1 _{∈ span (w} k) k=m k=1 ,and (w 2_{, w} k) = 0 (k = 1, ..., m).

Using (2.25), we deduce that

u0_m, w1 + B um, w1, t = 0 for all 0 ≤ t ≤ T. Then, (2.23) implies hu0_m, vi = (u0_m, w) = u0_m, w1 = −B um, w1, t , consequently |hu0_m, vi| ≤ C kumk_H1 0(Ω). Since kv1_k2 H1 0(Ω)≤ kvk 2 H1 0(Ω) ≤ 1, thus ku0_mk_H−1_(Ω) ≤ C kumk_H1 0(Ω), and therefore ku0 mk 2 L2_{(0,T ; H}−1_(Ω))= Z T 0 ku0 mk 2 H−1_(Ω)dt ≤ C Z T 0 kumk2_H1 0(Ω)dt ≤ C ku0k 2 L2_(Ω).

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Now, we are in a position to state ours results. Next, we pass to lim-its as m → ∞, to build a weak solution of our initial boundary-value

problem (P1).

Theorem 2.3.4. (Existence of weak solution). Under hypothesis H2and H3,

there exists a weak solution of (P1).

Proof. The proof of this Theorem do in three steps:

Step 1. According to the energy estimates (2.27), we see that the

se-quence {um} ∞ m=1is bounded in L 2_{(0, T ; H}1 0(Ω))and {u0m} ∞ m=1is bounded in L2_{(0, T ; H}−1_(Ω)) .

Consequently, there exists a subsequence which is also noted by {um}

∞ m=1 and a function u ∈ L2_{(0, T ; H}1 0(Ω)), with u0 ∈ L2(0, T ; H−1(Ω)), such that u_m * u weakly in L2_{(0, T ; H}1 0(Ω)) , u0_m * u0 weakly in L2_{(0, T ; H}−1_{(Ω)) .} (2.33)

Step 2. Next, fix an integer N and choose a function v ∈ C1_{(0, T ; H}1

0(Ω))

having the form

v (t) = N X k=1 dk(t) wk, (2.34) wheredk N

k=1are given smooth functions.

We choose m ≥ N , multiply equation (2.25) by dk_{(t) ,} _{sum for k =}

1, . . . , N, and then integrate with respect to t to find

Z t

0

hu0_m, wi + B (um, w, t) dt = 0. (2.35)

We recall (2.33) to find upon passing to weak limits that

Z t

0

hu0, wi + B (u, w, t) dt = 0 ∀w ∈ L2 0, T ; H₀1(Ω) . (2.36)

As functions of the form (2.34) are dense in L2_{(0, T ; H}1

0(Ω)).

Hence, in particular

hu0, wi + B (u, w, t) = 0 ∀w ∈ H₀1(Ω) and ∀t ∈ [0 T ] , (2.37)

and from Remark2.3.1we have u ∈ C (0, T ; L2_{(Ω)) .}

Step3. In order to prove u (0) = u0,we first note from (2.36) that

Z T

0

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for each w ∈ C1_{(0, T ; H}1

0(Ω))with w (T ) = 0.

Similary, from (2.35) we deduce

Z T

0

− hum, w0i + B (um, w, t) dt = (um(0) , w (0)) . (2.39)

We use again (2.38), we obtain

Z T

0

− hu, w0i + B (u, w, t) dt = (u0, w (0)) , (2.40)

since um(0) → u0 in L2(Ω). Comparing (2.38) and (2.40), we conclude

u (0) = u0.

3. Uniqueness result

Theorem 2.3.5. (Uniqueness of weak solutions ) A weak solution of (P1)is

unique.

Proof. We suppose there exists two weak solution u1and u2. We put

U = u2− u1,

then U is also a solution of (P1)with

U0 = (u2− u1) (0) ≡ 0.

Setting v = U in identity (2.37), we have

d dt 1 2kU k 2 L2_{(U )} + B (U, U, t) = 0.

From Lemma2.3.1, we have B (U, U, t) ≥ β kU k2H1

0(U ) ≥ 0, so d dt 1 2kU k 2 L2_{(U )}

≤ 0, then integrate with respect to t to find

kU k2_L2_(Ω)≤ kU0k2_L2_(Ω) = 0,

thus U ≡ 0.

4. Positivity solutions

Proposition 2.3.2. (Positivity solutions) Under assumption H2− H4. Then,

the weak solution of (P1) u ∈ C ([0, T ] ; L2(Ω))is positive on ]0, T [ × Ω.

Proof. If u0 ≥ 0 on Γ. Therefore u−= min (u, 0) ∈ L2(]0, T [ ; H01(Ω)).

A reasoning similar to the Proposition2.3.1, we obtain for all 0 ≤ t ≤ T

1 2 d dt Z Ω u−2dx + Z Ω B u−, u−, t dx = 0.

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Using the Lemma2.3.1and integrating with respect to τ from 0 to t, we get 1 2 Z Ω u−2dx + β Z t 0 ku (s)k2_H1 0(U )ds ≤ 1 2 Z Ω u−(0)2dx = 0. Since u−_{(0) = (u} 0) − = 0.So u− _{= 0.}

2.4 Uniform decayof solutions

In this section, we prove the exponential decay for solutions of problem ( P1)

we also prove the positivity of solutions for problem ( P).

Theorem 2.4.1. 1. if H2, H3and g > 0 are satisfied. Then the solution (u, v) of

problem (P) is global. Furthermore it holds that there exists τ0 > 0such that

kuk_L2 ≤ e−τ0tku0kL2.

2. Assume H2− H4. Then, problem (P) has a unique positive solution

(u, v) ∈ L2_{(0, T ; H}1 0(Ω)) × H01(Ω) . Proof. We put E (t) = 1 2 Z Ω u2dx. (2.41)

We derivate the equation (2.41) and we use first equations of (P1)and (P2)

1. We have

dE

dt = −B (u, u, t) ≤ 0,

therefore

E (t) ≤ E (0) . This mean that the solution of (P) is global. Next, we have from poincaré’s inequality

dE dt = −B (u, u, t) = − Z (∇u)2+1 2τ u 2 dx ≤ −τ0kuk2_L2_(Ω) = −τ0E (t) .

This implies that

E (t) ≤ E (0) e−τ0t.

2. It follows from Theorem 2.3.1, Theorem 2.3.4, Theorem 2.3.5,

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2.5 Summary

This study deals with the global existence, uniqueness and boundedness of the weak solution for the chemotaxis system (P) defined as

(P)        ut− ∆u + ∇ (u∇v) = 0, (t, x) ∈ R+× Ω, −∆v + τ v = 0, x ∈ Ω, u = 0, v = g, Γ, u (0, x) = u0, x ∈ Ω.

The system (P) is under Dirichlet boundary conditions in a convex bounded

domain Ω ∈ Rn_{with smooth boundary Γ, g and u}

0 are two given functions.

Based on Galerkin’s method, Lax-Milgran’s Theorem and Maximum princi-ple, a proof of the existence and uniqueness of a global solution for the sys-tem (P ) is determined for some properties adequate on initial and boundary

conditions. The exponential decay of L2_(Ω) _{norm of bacteria density u has}

also been shown. Moreover that, we show that the unique solution is posi-tive if the initial and boundary conditions are posiposi-tive.

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Chapter 3 Existence and uniqueness for a

fractional derivative KSM

3.1 Anomalous transports

Nature as well as modern technology presents us a variety of disordered ma-terials ranging from composites over gels to the inner structure of biological cells. We have shown that the transport properties of microscopic particles in such materials are directly connected to strong structural heterogeneities resulting from the presence of a broad range of length scales. It is known that these heterogeneities lead to a dramatic slowing down of transport pro-cesses due to a fractal dynamic behavior which has to be contrasted to nor-mal diffusion. The latter being tightly bound to Brownian motion, is the dominant transport process in homogeneous materials and can be charac-terized by single length and time scales. Heterogeneous materials, however, lack such a single length scale, and the new fractal transport law depends on non-integer, i.e. fractal, powers of time and length. This phenomena is called anomalous transports.

3.2 Fractional derivative Keller-Segel problem

In this chapter, the study of local and global solutions of a following spatial fractional derivative Keller-Segel model(KSM):

(P)            (P1)    ut+ Dαu + div (u∇v) = 0, (t, x) (t, x) ∈ [0, T ] × I, u = 0, ∂I × [0, T ] , u (0, x) = u0, x ∈ I, (P2) −∆v + τ v = 0, x ∈ I, v = g, ∂I,

where I is a bounded open domain with the smooth boundary ∂I ∈ C1_,

u0 ∈ H1(I), g ∈ H1/2(∂I) ,and τ is a positive constant. Here we define

Dα_{u (x) = F}−1_{(F (D}α_{u)) (x) = F}−1_(|ξ|α

F u (ξ)) (x), where F denotes the

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the energy are discussed as time goes to infinity. As noted in the introduc-tion, existence results for the fractional derivatives Keller-Segel equations have received much attention in recent years, the standard technique being to employ a proof based on the Galerkin method and relatively elementary energy estimates.

3.3 Direct Approach to Weak Solutions

In this section, we study existence and uniqueness solution of the fractional Keller-Segel problem (P) in one-dimensional and multidimensional cases by using the Galerkin method and some prior estimates. We used the same

technique in [35, 5] which allows us to show that the dissipative operator

Dα_{can control the nonlinearity div (u∇v) .}

3.3.1 One-dimensional case

We see that the solution of the problem (P2)is given by

v (x) = Ae √

τ x_{+ Be}−√τ x _{∈ C}∞ _{I ,}

where I = ]a, b[ and the boundary conditions are v(a) = α and v(b) = β. So

Aand B verified

A = αe−

√

τ b_{− βe}−√τ a_/_αe√τ (a−b)_{− βe}−√τ (a−b)_,

B = βe

√

τ a_{− αe}√τ b_/_αe√τ (a−b)_{− βe}−√τ (a−b)_.

The problem (P) consists of two problem (P1)and (P2). Since the solution of

(P2) exists and belongs to C∞ I, we need only to study the problem (P1)

and look for its weak solutions with initial data u(x, 0) = u0(x)the function

u in V1 such that V1 = L∞(]0, T [ ; L2(I)) ∩ L2(]0, T [ ; H01(I)) satisfying the

following identity: Z I uφdx− Z t 0 Z I uφsdxds+ Z t 0 Z I Dα2uD α 2φ s− uvxφx dxds = Z I u0(x) φt(t, 0) dx,

for a.e. t ∈ ]0, T [ and φ (t, x) ∈ H1_{(]0, T [ × I).}

For simplicity we denote u(t, x) by u and φ(t, x) by φ. In order to simplify

our construction, we suppose u(t) ∈ H1

0(I)for t ∈]0, T [ instead of

u(t) ∈ Hα/2(I) for t ∈ ]0, T [ which could be easily generalized from the

definition of weak solution of a parabolic second equation [5].

Theorem 3.3.1. Let 0 < α ≤ 2, T > 0, and u0(x) ∈ H1(I). Then, Cauchy’s

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regularity properties:

u ∈ L∞ ]0, T [ ; H₀1(I) ∩ L2 _{]0, T [ ; H}1+α/2_{(I) ,}

and

ut ∈ L∞ ]0, T [ ; L2(I) ∩ L2 ]0, T [ ; Hα/2(I) .

For t → ∞, α > 1 and g > 0 this solution decays so that lim

t→∞

Dα/2u

2 = lim_t→∞|u|∞= 0.

Proof. Suppose u is a weak solution of (P1)and let Sn be a truncation

oper-ator such that un=Snu,(we denote un(x, t)by un),then we can consider the

following approximate problem:

(un)t+ D α_u n+ ∂ ∂x un ∂v ∂x = 0, 0 < α ≤ 2, (3.1)

with initial data un|t=0= Snu0.

Let us multiply (3.1) by unthen

1 2 d dt Z I u2_ndx + Z I Dα/2un 2 dx − Z I ∂un ∂x un ∂v ∂xdx = 0, 0 < α ≤ 2, which implies d dt|un| 2 2+ 2 Dα/2un 2 2 = −τ Z I u2_nv dx. (3.2)

Then, it holds that d dt |un| 2 2+ 2 Dα/2un 2 2 ≤ τ |un| 2 2|v|L∞. (3.3)

Likewise, upon differentiating equation (3.1) with respect to x and

multiply-ing by (un)x we obtain d dt|(un)x| 2 2+ 2 Dα/2+1_u n 2 2 = τ R Iv−3 (un) 2 x+ τ u2n dx ≤ τ max (τ, 3) |v|_L∞kunk2₁. (3.4)

From equations (3.3) and (3.4), we conclude that

kunk2₁+

Z t

0

kunk2_1+α/2dt ≤ M = M (T, τ, k(un)₀k) . (3.5)

Now, differentiate equation (3.1) with respect to t and multiply it by (un)t

d dt|(un)t| 2 2 + 2 Dα/2(un)_t 2 2 = −τ Z I v ((un)_t)2dx ≤ τ |v|_L∞|(un)_t|2₂.

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The classical Gronwall inequality gives

|(un)_t|2₂+

Z t

0

k(un)_t(s)k2_α/2ds ≤ M (T, τ ) . (3.6)

Now, we prove the uniqueness of the solution, for this, consider two weak

solutions u1 and u2of (P1). Then, their difference w = u1 − u2 satisfies

d dt|w| 2 2+ 2 Dα/2w 2 2 = −τ Z I w2vdx ≤ τ |v|_L∞kwk 2 1. (3.7)

By Gronwall’s Lemma, it holds that w(t) = 0 on [0, T ].

It holds, from (3.5) and (3.6), that a solution unis bounded.

Now, We will prove the asymptotic estimates lim

t→∞

Dα/2u

2 = lim_t→∞|u|∞= 0.

The idea of this prove is to verify the hypothesis of the following Lemma:

Lemma 3.3.1. let f be the positive function such that ftand

R∞ 0 f (τ )dτare bounded then lim t→∞f (t) = 0. Indeed, multiplying (3.1) by Dα_u d dt Dα/2u 2 2+ 2 |D α_u|2 2 = 2 Z I ∂v ∂x ∂u ∂x + τ vu Dαudx ≤ 4 max(τ |v|_L∞, ∂v ∂x L∞ ) kuk₁|Dα_u| 2 ≤ C + |Dαu|2₂. Therefore, we have d dt Dα/2u 2 2 ≤ C,

If g > 0 this implies that v > 0 and from (3.3.1) we get

|u|2₂+ 2 Z t 0 Dα/2u(s) 2 2ds + τ Z t 0 Z I u2v dx ≤ |u0|2₂ = C, 0 < α ≤ 2,

for all t. Thus, lim t→∞

Dα/2_u

2 = 0. The second asymptotic relation follows

from the Sobolev embedding Hα/2 _{⊂ L}∞

valid for α > 1.

3.3.2 Multidimensional case

Before dealing with the existence and uniqueness results solutions for the

problem (P1). We recall some known results on elliptic problem for the

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Lemma 3.3.2. If g ∈ H1/2(∂I), then there exists ˜g ∈ H1(Ω)such that

σ(˜g) = gwhere σ design the trace operator.

Definition 3.3.1. We define the space K = {v ∈ H1_{(Ω) /v − ˜}_{g ∈ H}1

0(I)}. We

call weak solution of (P1)all functions v ∈ K satisfying the following relation

∀w ∈ K, Z

I

(∇v∇w + τ vw)dx = 0.

Theorem 3.3.2. Let I be an open domain in Rn _{of C}1 _{class and g ∈ H}1/2_(∂I).

Then, the problem (P2) possesses a unique weak solution v ∈ H1(I)satisfying the

following inequalities:

min(inf

∂I g, 0) ≤ v (x) ≤ max(sup_∂I g , 0).

Now, we look for weak solutions of problem (P1)with its initial data

u(x, 0) = u0(x) ∈ H01(I),

thus u ∈ V1 such that

V1 = L∞ ]0, T [ ; L2(I) ∩ L2 ]0, T [ ; H01(I)

satisfying the following identity (for simplify the notations, we denote

u (t, x) (resp φ (t, x))by u (resp φ) ): Z I uφdx − Z t 0 Z I uφsdxds + Z t 0 Z I Dα2uD α 2φdxds − Z t 0 Z I u∇v.∇φdxds = Z I u0φ (0, x) dx,

for a.e. t ∈ ]0, T [ and φ (t, x) ∈ H1_{(]0, T [ × I)}_.

In this case and as in the one-dimensional, we also suppose u(t) ∈ H1

0(I)

for t ∈]0, T [ instead of u(t) ∈ Hα/2_(I) _{for t ∈ ]0, T [ just to simplify our}

construction. Finally, we arrive at our main result.

Theorem 3.3.3. (Existence and uniqueness result for (P1)) Let 0 < α ≤ 2, 2 ≤

n ≤ 6, and T > 0. If g ∈ C1_(∂I)_{or g ∈ H}1/2_(∂I)_{and u}

0(x) ∈ H01(I). Then,

Cauchy’s problem (P1) has a unique weak solution u ∈ V1 for α > n/3. Moreover,

usatisfies the following regularity properties:

u ∈ L∞ ]0, T [ ; H₀1(I) ∩ L2 _{]0, T [ ; H}1+α/2_{(I) ,}

and for a.e. T > 0, we have

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For t → ∞, n ≤ 3, α > 1 and g > 0 this solution decays so that lim t→∞ Dα/2u 2 = lim_t→∞|u|∞= 0.

Proof. Suppose u is a weak solution of (P1)and let Sn be a truncation

oper-ator such that un = Snu and we denote un(t, x)by un,then we can consider

the following approximate problem:

(un)t+ D

α

un+ div (un∇v) = 0, 0 < α ≤ 2, (3.8)

with initial data un|_t=0= Snu0.

Let us multiply (3.8) by un, then

Z I 1 2 d dtu 2 n+ D α/2_u n 2 + div (un∇v) un dx = 0, 0 < α ≤ 2, (3.9) which implies d dt Z I u2_ndx + 2 Z I Dα/2un 2 dx + τ Z I u2_nv dx = 0. (3.10) As H1

0 (I)and H1(I)injected in L3(I)for all n ≥ 6, then it holds that

d dt Z I u2_ndx + 2 Z I Dα/2un 2 dx = −τ Z I u2_nvdx (3.11) ≤ τ |un| 2 3|v|3 ≤ C kunk 2 1 for n ≤ 6.

Likewise, differentiating (3.8) with respect to x and multiplying it by ∇un

we obtain d dt|∇un| 2 2+ 2 Dα/2+1un 2 2 = −3 Z I τ v (∇un) 2 dx + τ2 Z I vu2_ndx. (3.12)

Applying the Cauchy-Schwarz to right member of (3.12) to obtain

−3 Z I τ v (∇un) 2 dx + τ2 Z I vu2_ndx ≤ τ max (3, τ ) |v|₃(∇un) 2 + u2_n3 2 ≤ C1kunk2_1,3 for n ≤ 6.

The assumption α > n/3 has been used in the interpolation of W1,3_-norm

of u by the norms of its fractional derivative to have 2 (6 + n) /3 (2 + α) < 2.

Indeed, that one follows from [11]. Then, we obtain

C1kunk2_1,3 ≤ C1kunk 2(6+n)/3(2+α) 1+α/2 |un| (2−2(6+n)/3(2+α)) 2 ≤ kunk2₁₊α 2 + C2|un|m₂ ,

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for any m > 0.

Devising this with (3.11-3.12) we obtain

d dtkunk 2 1+ kunk 2 1+α/2 ≤ C3 |un| m 2 + kunk 2 1 ≤ C, hence, we obtain kunk 2 1+ Z t 0 kunk 2 1+α/2dt ≤ M = M (T, τ, k(un)0k) . (3.13)

Now, differentiating equation (3.8) with respect to the time and multiplying

it by (un)twe obtain d dt|(un)t| 2 2+ 2 Dα/2(un)_t 2 2 = −τ Z I v ((un)_t)2dx. (3.14)

Using the Young’s inequality, the right-hand side of (3.14) is estimated by

τR_Iv ((un)t) 2 dx ≤ τ |v|₃|(un)t| 2 3 ≤ C1k(un)tk 2 1,3 ≤ C1k(un)_tk 2n/3α α/2 |(un)t| (2−2n)/3α 2 ≤ k(un)tk 2 α/2+ C |(un)t| m 2 . (3.15)

The crucial estimate of (3.15) only requires that α > n/3 with n ≤ 6.

The classical Gronwall inequality gives

|(un)_t|2₂+

Z t

0

k(un)_t(s)k2_α/2ds ≤ M (T, τ ) . (3.16)

It holds, from (3.13and (3.16), that a solution unis bounded. Then it is

suffi-cient in order to apply approximation Galerkin’s procedure . Hence, we can extract a subsequence which converges to a limit

u ∈ L∞(]0, T [; H₀1(I)) ∩ L2(]0, T [; H1+α/2(I)),

and

ut ∈ L∞ ]0, T [ ; L2(I) ∩ L2 ]0, T [ ; Hα/2(I) .

To finish, it remains to know if u is a solution of problem. Since injection

of H1

0(I)into L2(I)is compact, we can apply Ascoli Theorem and conclude

a strongly convergence of (un)n∈Nto u in L2(]0, T [; L2(I)).

Now, in order to conclude, it is enough to prove that (Dα_u

n) converges

strongly to (Dα_u) _{in L}1_{(]0, T [; L}2_{(I)). We denote V}

2 = L1(]0, T [; H1+α/2(I))

and note that

kDα_u

n− DαukL1_{(]0,T [;L}2_(I)) ≤ k∆un− ∆ukV2 k∆unkV2 + k∆ukV2 ,

and since the term ∆ is linear, approach problem converges weakly to a limit point, then the existence holds.

Now, we prove the uniqueness of the solution. Consider two weak solutions

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d dt|w| 2 2 + 2 Dα/2w 2 2 = −τ Z I w2vdx. (3.17)

By Generalized Gagliardo-Nirenberg inequality a right member of (3.17) can

be estimated as follows: τ Z I w2vdx ≤ τ |v|₃|w|2₃ ≤ C1kwk 2n/3α α/2 |w| (2−2n)/3α 2 ≤ kwk 2 α/2+ C|w| 2 2, (3.18) for α > n

3. By Gronwall’s Lemma it holds that w(t) on [0, T ] but for α > n/3.

Now, the proof of asymptotic estimates lim

t→∞

Dα/2u

2 = lim_t→∞|u|∞= 0

can be accomplished as follows:

Multiplying equation (3.8) by Dα_u_{we obtain}

d dt Dα/2u 2 2+ 2 |D α_u|2 2 = 2 Z I

(∇v.∇u + τ vu) Dαudx

≤ C1kuk_1,3kvk_1,6|Dαu|₂

≤ C + |Dαu|2₂.

Because kuk_1,3 is bounded for α > n/3 and as H2,2_{injected in W}1,6_for

n ≤ 3. So kvk_1,6is also bounded. Therefore, we have

d dt Dα/2u 2 2 ≤ C.

If g > 0 and from (3.10), then we get for all t

|u|2₂+ 2 Z t 0 Dα/2u(s) 2 2ds + τ Z t 0 Z I u2v dx ≤ |u0|2₂ = C, 0 < α ≤ 2.

Thus, using Lemma3.3.1, we obtain lim

t→∞

Dα/2u

2 = 0. The second

asymp-totic relation follows from the Sobolev embedding Hα/2 _{⊂ L}∞

valid for α > 1.

Remark 3.3.1. A rigorous proof is obtained by rewriting the previous differential

inequalities as integral inequalities, like (3.13)-(3.16), which are direct consequences

of the definition of the weak solution for all α ∈ ]0, 2]. We have the existence and

uniqueness global solutions of problem (P1)for α > n/3. Indeed, the crucial

esti-mate of (3.18) only requires that α > n/3.

Remark 3.3.2. In [35], the approach by prior estimates to the Cauchy problem of

type fractional Burger equation in case one-dimensional produces, in fact, regular-ity of the solutions; for a power fractional α > 3/2. When α ≤ 3/2, a direct construction of weak global in time solutions is no longer possible for initial data of this problem of arbitrary size, and the authors employed another technique to

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obtain candidate weak solutions. The used method is the method of parabolic reg-ularization. Applying the same technical method in the case one-dimensional and multidimensional to our problem help us to obtain the regularity solutions for all

α ∈ ]0 2]for n = 1 and α > n/3 for 2 ≤ n ≤ 6. Then, these regularly solutions are

global in time.

3.4 Summuray

This chapter studies the local and global in time solutions to a class of mul-tidimensional generalized Keller-Segel model with a fractional power of the Laplacian in the principal part and with algebraic nonlinearity. The obtained results include existence, uniqueness and regularity of solutions of Cauchy problem as well as the explanation of the critical exponents role. The proof is based on Galerkin method and on some Generalized inequalities of Sobolev space.

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Chapter 4 Finite volume method for a

Keller-Segel problem

4.1 Keller-Segel model

The aim of this chapter is to study the finite volume schemes applied to the elliptic-parabolic model (P) defined as:

(P)                (P1)   

ut− ∆u + div (u∇v) = 0, (t, x) ∈ R+× Ω,

u = 0, ∂Ω, u (0, x) = u0, x ∈ Ω, (P2) −∆v + τ v = 0, x ∈ Ω, v = g, ∂Ω. (4.1) Where Assumption 1

1. Ω is an open bounded polygonal subset of R2 _{or R}3_.

2. τ ≥ 0.

3. g ∈ C (∂Ω, R).

4. u0 ∈ C2(Ω, R).

The chapter is organised as follows: In section 4.3, we give some results

related to the existence and the uniqueness of the approximate solution of

problem (P2). Under adequate regularity condition on the exact solution

of this problem, the finite volume scheme is the first order accurate. In

section 4.4, we prove the same results as in section 4.3 (existence and the

uniqueness, and C2_{error estimate) for problem (P}

1). The proved is based on

the proof of Theorem 3.3 and Theorem 4.1 in [27]. We conclude in section4.5

the same results obtained in the previous sections for problem (P). Finally,

in Section 4.6, we illustrate the method’s performance against test problems

from the literature, and verify that the results are in agreement with our nu-merical analysis.