Results on existence for generalized nD Navier-Stokes equations

Abstract In this paper we consider a class of nD Navier-Stokes equations of Kirchhoff type and prove the global existence of solutions by using a new approach introduced in [Jday R., Zennir Kh., Georgiev S.G., Existence and smoothness for new class of n-dimentional Navier-Stokes equations, Rocky Mountain J. Math., 2019, 49(5), 1595–1615].


Introduction
In this article we investigate a Navier-Stokes equations of Kirchho type In the case a = , system (1.1) has been treated in [3]. (See [4][5][6][7].) In this paper we extend the previous works to nd for any n ≥ a new class of smooth initial data u satisfying (1.4) and a new class of functions f i , including f i = , i = , . . . , n, such that the problem (1.1), (1.2) has a solution p, u ∈ C ∞ R n × [ , ∞) .
This kind of systems appears in the models of nonlinear Kirchho -type. It is a generalization of a model introduced by Kirchho [8] in the case n = this type of problem describes a small amplitude vibration of an elastic string. The original equation is: where ≤ x ≤ L, t > and u is the lateral de ection x is the space coordenate variable while t denotes the time variable E is the Young modulus ρ is the mass density L is the lenght h is the cross section area P is the initial axial tension τ is the resistence modulus f is the external force (1.6) (For more see [9]). Here we assume that the initial data u and the force term f are as follows for any multi-index α and for any positive constant K, where C α,K is a positive constant depending on α and K, Q is a positive constant such that for any α, m, K, where B is a compact subset of R n , respectively. Several numerical methods for solving of (1.1) are used. In [10] Lagrangian and semi-Lagrangian velocity and displacement methods are introduced for the numerical solution of (1.1). In the scalar case, methods of characteristics for time discretization of convention di usion problems are extensively used (see [11] and references therein). These methods are based on time discretization of the material time derivative combined with nite di erences or nite elements for space discretization. When the characteristic methods are formulated in a xed reference domain they are called pure Lagrangian methods. The classical methods of characteristics are semi-Lagrangian and rst-order in time. There exists an extensive literature for these methods (see [12,13] and references therein). The error estimates of the norm O(h k ) + O(∆t) + O(h k+ /δt) in l ∞ (L (Ω))-norm are obtained under the assumption that the normal velocity vanishes on the boundary Ω, where h is the space step and δt is the time step (See [14][15][16]).
In order to increase the order of time and space approximations, higher order schemes for the discretization of the material derivative and higher order nite element spaces are used(see [17][18][19] and references therein). Second order characteristic method for solving of (1.1) is used in [20,21].
In this paper we propose a new method for investigation of the Cauchy problem (1.1), (1.2) which is di erent than the well-known methods. In Section 2 we give some auxiliary results. In Section 3 we proof the main result introduced in Theorem (3.1)

Preliminaries
We will start with the following useful Lemma. Here Proof. We di erentiate once in t and twice in x , . . ., xn, all equations of the system (2.1) and we see that the function u satis es the system (1.3). Now we put t = a in the rst n equations of the system (2.1) and we get We di erentiate the last system twice in x , . . ., xn, and we obtain i.e., the function u i , i ∈ { , . . . , n}, satis es the initial condition (2.2). This completes the proof.
The proof of the existence result is based on a xed point theorem for sum of two operators one of which is expansive. for any x, y ∈ M.
Next result we will use to prove our xed point theorem. Then there exists an x * ∈ X such that Tx * + Sx * = x * .
Proof. Since Y is compact and S(X) resides in Y, we have that the rst condition of Theorem 2.3 holds. Because T : X −→ E is expansive, we have that the second condition of Theorem 2.3 holds. Note that T − : Y → E exists, it is linear and contractive with a constant l ∈ ( , ). Let z ∈ S(X) be arbitrarily chosen and xed. Set Take y ∈ Y arbitrarily. De ne the sequence {yn} n∈N as follows. Then Using the principle of the mathematical induction, we get Therefore {yn} n∈N is a Cauchy sequence of elements of Y ⊂ E. Since E is a Banach space, it follows that the sequence {yn} n∈N is convergent to an element Because z ∈ S(X) was arbitrarily chosen, we conclude that S(X) ⊂ (I − T)(X), i.e., the third condition of Theorem 2.2 holds. Hence and Theorem 2.3, it follows that there exists an x * ∈ X such that This completes the proof.
Below we will suppose that x = x , . . . , x n ∈ ∂B is arbitrarily chosen and xed. Also, we will use the following notations.

Main result -proof
Our main result is as follows.
Firstly, we will prove that the Cauchy problem for i ∈ { , . . . , n} We choose the constants γ, δ, l > as follows WithK we denote the set of all equi-continuous families in E , i.e., if F ⊂K is a family of elements of E , then for every ϵ > there exists a δ = δ(ϵ) > such that Note that K is a compact subset of Q . For (u, p) ∈ Q we de ne the operators. Proof.
We choose the constant l > such that  Note that K is a compact subset of Q . For (u, p) ∈ Q we de ne the operators.