Existence , Uniqueness and C − Di ff erentiability of Solutions in a Non-linear Model of Cancerous Tumor

In this paper, we prove the existence and uniqueness of the weak solution of a system of nonlinear equations involved in the mathematical modeling of cancer tumor growth with a non homogeneous divergence condition. We also present a new concept of generalized differentiation of non linear operators : C−differentiability. Through this notion, we also prove the uniqueness and the C−differentiability of the solution when the system is perturbed by a certain number of parameters. Two results have been established. In the first one, differentiability is according to Fréchet. The proof is given uses the theorem of reciprocal functions in Banach spaces. First of all, we give the proof of strict differentiability of a direct mapping, according to Fréchet. In the second result, differentiability is understood in a weaker sense than that of Fréchet. For the proof we use Hadamard’s theorem of small perturbations of Banach isomorphism of spaces as well as the notion of strict differentiability.


Introduction
Cancer (Marcotte, 2008) is a serious genetic disease that results in an imbalance between cell division and death, leading to cells disequilibrium.The balance between these two processes regulates the number of cells in the tissues, and the breakdown of this equilibrium leads to the development of clusters of cancer cells (called tumors) irrespective of the normal functioning of the body.The cancer cell is a wanton cell that multiplies itself in an uncontrolled and excessive manner within a normal tissue of the body.This anarchic proliferation gives rise to increasingly large tumors that grow up and then destroy the surrounding organs.The cancer cells can also swarm away from a body to form a new tumor, or circulate in a free form.By destroying its environment, the cancer can become a real danger to the survival of human being.The fight against this disease is an important field of medical research.The need to adapt various types and forms of cancers as well as the understanding of complex phenomena involved in its growth has led to the development of many mathematical models (Patrick, 2013) (Tracqui, 1995) in recent decades.Mathematical modeling of cancer evolution is a rapidly developing field.Their interest lies in their ability to gather large quantity of information accumulated by biologists.
Indeed, it is important to understand that the mathematical complexity of a model is not a sufficient criterion to judge its relevance.Thus, the nature of this phenomenon (the cancer cells have a fluid movement) motivated us to use the non-stationary compressible Navier-Stokes model, which can describe the disease.These equations do not address the tumor environment and its interactions directly, but present measurable magnitudes such as the volume density denoted by ρ = ρ(x, t) and the density of the outer forces denoted by ξ e = ξ e (x, t), which models environmental factors.Furthermore, the cells are considered to be transported by a velocity field ν = ν(x, t), with the corresponding pressure π = π(x, t).
The choice of the Navier Stokes system as working equations permits to tackle problems like unknown coupling, nonlinearity, and time dependence.The nonlinear nature of the convection term (ν • ∇ν) that appears in these equations is the source of difficulties in solving this problem.To overcome this difficulty, we use the method consisting of low estimates and convergences in regular spaces like L 2 ( (0, T ); L 1 (I, R dim ) ) . Note that in this paper the goal is to obtain the existence, uniqueness and C −differentiability of a nonlinear dynamic system solution with (ν, ρ) ≡ ℜ ε (V), where V = (ν 0 ρ 0 , ξ e ), in which ν 0 , ρ 0 and ξ e are respectively the velocity, initial density and function that models the membrane surrounding the tumor and R ε the satisfactory operator where Ψ is a continuous invertible operator and η a positive constant.Our approach is therefore to perturb the system, using measurable functions and operators, twice continuously differentiable in Banach spaces in order to obtain the proof of the differentiability of the solution (ν, ρ).We end the introduction with a brief description of the content of the document.
In section 2, we formulate the problem and give some preliminary notations that will be used in the following.Section 3 introduces the definition of a weak solution by rewriting the system equations in a particular frame, then we present a result of the existence and a criterion of weak uniqueness of the weak solution for the case of constant viscosity coefficients.It generalizes classical criteria like (Solonnikov's, 1978) criterion for the case of the initial-value problem through an additional property of the weak solution.Then we give approximate estimates (see Varga, 1971;Aubin, 1972) used during the process of passing limit in the solution.Finally, section 4 is about uniqueness and generalized differentiability of the solution when the system is perturbed by a number of parameters.

Formulation of the Problem
Cancer is an important area of research in medicine, but also the suject of applied mathematics research.Mathematics is used in particular to model the growth of cancerous tumors, with the main goal of optimizing treatment by increasing antitumor efficacy and decreasing toxicity on healthy cells.In this paper, we present a model of differential equations modeling the tumor across a given area.We consider a non-homogeneous region (variable density) as a function of time I t = I × (0, T ) occupied by the tumor, where I is a lipchitz bounded open set of R 3 and let ∂I be its border.Let x ∈ I, the size of the tumor and t ∈ (0, T ), the time parameter.At the initial time t = 0, the tumor has a size x 0 in the I domain.The non-stationary model is then described by the following differential equations where ξ e denotes the density of the external forces and L λ,µ (ν) an operator formally defined by with λ and µ respectively representing the volumetric and dynamic viscosity coefficients supposed to be constant.The system is supplemented by initial conditions on density and proliferation rate It is assumed that on the ∂I border of the domain I, the velocity checks the boundary conditions (2. 3) It should be mentioned that ρν ⊗ ν ∈ R 3 in (2.1) 1 is a tensor product of ρν and ν, and that Before announcing the results, it is necessary to define the domains in which we work.In this sub section, we introduce the notation that will be used throughout this document.

General Framework and Preliminaries
Let's give here some notations.The following function spaces provide a standard framework for obtaining the unique results of overall and differentiability of the solution of system (2.1) − (2.3).
The underlying domain.Let I ⊂ R 3 , a delimited domain ∂I its sufficiently smooth border.For T > 0, the interval (0, T ) defines the considered time interval and I t = I × (0, T ) a space-time domain with boundary ∂I t = ∂I × (0, T ).
Standard Lebesgue spaces.Let m be a non-negative integer.We denote by H m (I; R 3 ) the usual Sobolev space W m,2 (I; R 3 ) as defined in (Lions and Magenes, 1972).
We note by D(I), the space of infinitely differentiable functions with compact support.Its closure in the norm W m,p (I; An alternate characteristic in the case where m = 1 and p = 2 is where γ 0 is the ν trace operator.We also note by L p (I) 3 = L p (I; R 3 ), the lebesgue space on I with the norm ∥.∥ p and by ∥.∥ E the norm associated to a space E. If E is a Banach space, L p (0, T ; E) is the Banach space composed of functions, measurable on (0, T ) which values in E. For details concerning these spaces, see (Adams, 1945) or (Girault ,1986).Let introduce the solenoidal spaces.We consider zero divergence spaces introduced for the problem (2.1) − (2.3).
where K 1 div and K 0 div are the respective closure of C ∞ 0,σ (I; R 3 ) in L 2 (I; R 3 ) and H 1 0 (I; R 3 ).

Let us define the Stokes operator
where P : L 2 (I; R 3 ) −→ K 1 div is the orthogonal projection.Note also that, we have It should also be noted that A −1 : K 1 div −→ K 1 div is a self-adjoint compact operator on K 1 div and by the classical spectral theorems, there exists a sequence ℓ j > 0 and a sequence function {ϕ j (x)} ∞ j=1 ∈ D[A] such as Aϕ j = ℓ j ϕ j (for the existence and regularity of these functions see for example (Layzhenskaya, 1969) and (Temam, 1977).
Let us now give the definition of a weak solution for the system (2.1) − (2.3).

Weak Solution : Existence and Uniqueness
First, let's give the definition of the weak solution.
Definition 1 Let I be a bounded domain in R 3 with smooth boundary, and assume that the data ν 0 (x), ρ 0 (x), ξ e (x, t) satisfy the regularity conditions 3) on (0, T ) corresponding to the initial conditions ν 0 and ρ 0 if the following conditions are met : i) ν and ρ satisfy where C ( (0, T ); X w ) is the space of continuous functions of (0, T ) with values in a closed ball of X equipped with the weak topology of the separable Banach space X.
The next section, we discuss the existence and uniqueness of weak solution results for the system (2.1) − (2.3).

Existence
In this section, we are interested in a result of existence of a weak solution for the model (2.1) − (2.3) modeling the tumor in a three-dimensional I domain with volumetric viscosity and dynamic coefficients supposed to be constant that satisfy the following conditions : µ ≥ 0, 2µ + λ > 0. (3.4) The unknowns are the volume density ρ(x, t), the tumor cell velocity field ν(x, t) and the pressure π that appears under the effect of tumor cell movements.The first result of this document is the following lemma on the existence of weak solutions to the (2.1) system, subject to (2.2) − (2.3).
Theorem 2 Let I ⊂ R 3 be a Lipschitz bounded domain with a regular border ∂I.Let ρ 0 ∈ W 1,2 (I; R) and ξ e ∈ L 1 ( (0, T ); Then, for a given T > 0, there exists a unique weak solution (ν , ρ) of the problem (2.1) Proof.For proof, we establish the Galerkin approximation to the (2.1) − (2.3) system.We first present the approximation scheme, then we estimate a priori the approximate solution, and finally we perform the process of passing to the limit to approximate solutions.
To implement it, we take a basic functional subset of K 1 div as follows We denote by {ϕ k (x)} m k=1 linearly independent generalized eigenfunctions corresponding to each distinct unstable eigenvalue ℓ k of the operator A = −P∆ defined on D[A] ∩ K 1 div −→ K 1 div , and P the orthogonal projection of L 2 (I; R 3 ) on K 1 div .consider the eigenvalue problem It is well known that {ϕ j (x)} ∞ j=1 forms a complete orthogonal system in the space K 0 div .For a detailed analysis of the convergence of expansions of eigenfunctions and the regularity of eigenfunctions, see (Ladyzhenskaya, 1969).Suppose that ρ 0 , and ξ e satisfy the assumptions of theoreme 2. By regularizing the initial density, we choose ρ 0m so that (3.7) We define approximate solutions for the formulation (2.1) − (2.3) as follows : We say that where (3.10) In order to resolve system (3.8),we use the classical method of characteristics to construct a solution.We thus have the following result.
Proof.The proof is standard and we can refer to (Kim, 1987) with a small change in the volumic density.To simplify the mathematical formulations of the system (3.10),we introduce the following notations So we can rewrite the above system of differential equations in matrix form as follows where , it is clear that the matrices defined above belong to C(0, T ).The matrix (a m kp ) m×m is symmetric positive definite , thanks to the orthogonality of (ϕ k ) k=1,..,m in K 0 div .In particular, the matrix (a m kp ) 1≤k,p≤m is non-singular.Then (3.15) can be written as , the resolution of the initial value problem of the above system follows from the classical theory of ordinary differential equations, so we are assured of the existence and uniqueness of the solution of (3.10) and therefore, the one of the problem (3.9).
Step 2 (Parametric Sensitivity of Solutions).Let's give a prior estimate of the solutions of (3.8) − (3.9) with variable density and constant viscosity.For tumor-related reasons, we consider that the proliferation rate fields disappear on ∂I (see the condition (2.3)), and for this reason, we only consider the limit data from Dirichlet for ν.
Lemma 4 (estimated solution with low velocity assumption).Let I be a bounded domain in R 3 with smooth boundary, and assume that the data ν m (0), ρ m (0), ξ em satisfy the regularity conditions ν m (0 (3.17) Then, there is a solution (ν , ρ) of the system (3.8)− (3.9) satisfying the initial conditions (2.2) − (2.3) and the following inequality : If ρ m has the additional regularity Proof.By combining (2.1) 1 and (2.1) 2 and multiplying by ν m , we then obtain, by integration on the volume I, the following variational formulation Applying the derivation theorem, the first term on the left gives the following estimate The slow mode reaction-diffusion equations allows as to write that the integral on the I volume of the term ∇ : ρ m ∂ν m j ∂x j ν mi ν mi dx.
Since ν m = 0 on ∂I, the boundary terms disappear.Further, thanks to the hypothesis of small speeds we have ρ m ∂ν m j ∂x j = 0, and so we find Integrating by parts (Green's formula), we get (where y 0 is the unique continuous linear map defined from W 1,2 0 (I; R 3 ) → L 2 (I; R 3 ) such that y 0 ν m = 0, where − → n is the normal at the border of I, denoted ∂I and ds its surface element).It follows that ) , Finally, the force provided by the membrane : Applying Young's inequality, the estimate (3.27) becomes 1 2 If we integrate the inequality (3.28) on (0, t) we get Applying the inequality of the Gronwall Lemma (Lions, 1972), the inequality (3.29) becomes, for T > 0, fixed ) exp(T ). (3.30) On the other hand, it is easy to see that (2.1) 2 can be in the form By integrating on the I domain, we obtain 1 2 Integrating on (0 , t) and using Gronwall lemma inequality Theorem 5 Let I be a bounded domain in R 3 with smooth boundary, and assume that the data ν m (0), ρ m (0), ξ em satisfy the regularity conditions So for 2 ≤ p < s < ∞ and 2µ + λ > 0, there is a solution (ν m , ρ m ) of the system (3.8)− (3.9) satisfying the initial conditions (2.1) − (2.3) and the following inequality : Proof.According to (3.26), we have the following inequality 1 2 By integrating on (0 , T ), we get the estimate 1 2 Which completes the proof of theorem 5.
Proof.To explain the ideas clearly, we present a formal argument.Let's start by using the following equations (3.40) (3.41) By summing the two systems (3.40) and (3.41) we get the following system : (3.42) Multiplying equation (3.42) 1 by (ρ − ρ) and integrating on I we get d dt On the other hand, by multiplying equation (3.42) 2 by 2(ν − ν) and integrating on I we have the next estimate : Applying the inequality of H ölder and Sobolev we have the following estimate : Summing the equations (3.46) and (3.47) we get : Thus, the application of the Gronwall inequality completes the proof of theoreme 6.

Linearization of the 3D Dynamic System
We consider a bounded domain I with the same initial conditions.In this paragraph, we construct a linear functional perturbation that linearizes the equation (2.1) 1 .However, let's look at the term (ν • ∇ν) that appears in the equation (2.1) 1 .It is at the root of the difficulties encountered in solving this problem.We will therefore linearize the system by substituting this term with the following perturbation : whese φ is a measurable function with respect to (x, t), twice continuously differentiable with respect to (v , w)∈ R 3 × R 9 , and H p = Pϑ a continuous integral operator (see Silvia, 2014) which, at any function ϑ, matches H p .That is written in expanded form : where the P(x − y, t − t ′ ) operator is a linear and continuous application in I × (0, T ).Using the new functions introduced, the initial value problem (2.1) (4.3)Note that this system is a simpler version of the (2.1) system since the term (ν • ∇ν) has been replaced by F(H, φ).This approach introduced new variables v, w which are considered respectively as an argument of the ν(x, t) field and its divergence.We will then make hypotheses about the functions φ(x, t, v, w) and ϑ(y, t ′ , v, w) defined on

Assumptions and Definition
(H-1) : Let β, β > 0 and T > 0 (fixed).For every (v, w) ∈ R 3 × R 9 , the functions (x, t, v, w) −→ φ(x, t, v, w) and (y, t ′ , v, w) −→ ϑ(y, t ′ , v, w) are measurable and verify the following conditions are twice continuously differentiable with respect to couple (v, w).Moreover : (H-3) : Let A ϵ and B ϵ be two nonlinear F-differentiable and G-differentiable operators.We note by A ′′ ϵ and B ′′ ϵ , the respective second differential of A ϵ and B ϵ defined as follows (for these notations see Trenoguine, 1985), the second derivative of A ϵ (ν) and B ϵ (ν) in ν with A ′ ϵ (ν)g = dA ϵ (ν, g).For an increase of h, independent of g, we have For h = g we deduce the following formulas Let us now give the definition of the generalized solution of the perturbed problem (4.3).

Generalized Differentiability of Non Linear Operators
In this section we propose a new concept of generalized differentiability.This concept encompasses the standard notions of Fréchet differentiability, strict differentiability, according to Gateaux and Lipschitz continuity.We now begin with our first definition of generalized differentiability.
where d H) is a linear and continuous operator, and R(ν, gh) satisfies the following condition: Proposition 10 Let I ⊂ R 3 be a bounded Lipschitz domain with a regular border ∂I.For all (x, t) ∈ I × (0, T ), let's define the operator A ϵ by If φ and A ϵ satisfy the assumptions (H-1)-(H-3), then, dA ϵ (v, g) is C −continuous and C −differentiable on W(0, T ).On the other hand, A ′′ ϵ (v)g 2 is continuous and is defined by the following operator Proof.Suppose (H-1)-(H-3) are checked, and that the operator A ϵ (ν) checks the equality (a), we have for all small enough g and h Let's introduce the following reflexive space W(0, T ) =: } .
Let's divide the last equality by τ and taking the norm on W(0, T ), we have In order to make writing easier, we often omit the variables x and t.