Boundedness of Solution Operator Families for the Navier-Lam 𝒆́ Equations with Surface Tension in Whole Space

ABSTRACT

A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Tooth paste is an example of the non-Newtonian fluids. Since it become runnier when shaken.
In our daily lives, there are many condition related to the fluids phenomena. Therefore, studying fluid dynamics become an interesting point. The air we breathe, water flowing through the tap, blood stream flowing within our body, etc. Those are examples of fluid phenomenon. These examples plays an essential role in making life possible on out earth. Over years, scientists and researchers have contributed in this field of science to uncover the interesting phenomena and behaviour of fluids under varied condition. Moreover, the nature phenomena made it possible to be understood.
Fluid dynamics helps human to imagine the movement of fluids or gases. The fluid movement is described by conservation of mass, conservation of momentum and a transport energy. There are two critical elements of fluid dynamics that are viscosity and fluid flow regimes. One of the mechanical behaviour of the fluid is polymers. DNA is one of the biological synthesis polymers which is the important component of our body. Others example of the fluid motion are turbulent character such as river and ocean currents. Many technological applications use the phenomena of turbulence to achieve their purpose. Navier-Lame equation is one of the model fluid flow which is important. Since, the well-posedness of the model problem can be used as reference for other model such as Oldroyd-B model fluid flow.
Recently, many researchers investigated the behaviour of the fluid motion. Majority of them conducted in numerical analysis point of view, rarely of them focused on mathematical side. Therefore, this situation become motivation of author to conducted fluid dynamics in mathematical point of view. In 2012, Girault et.al using finite element methods to investigated the Navier-Stokes of the model fluid flow in theory and algorithms (Girault & Raviart, 2012).
In 20 th century, Jacques Hadamard a French mathematician was defined that well-posed as a condition that the mathematical model of physical phenomena hold three properties that are the solution of the model problem exists, unique and the behaviour of the solution changes continuously with the initial conditions. For researchers who conducting in this mathematical model the main purpose of the research is investigating not only local wellposedness but also global well-posedness with many different methods.
In this article, we consider the ℛ-boundedness of the solution operator families for Navier-Laḿ equation with surface tension in whole space. As we known, that the Navier-Laḿ equation is the fundamental equation of motion in classical linear elastodynamics (Eringen & Suhubi, 1975). Navier-Laḿ equation in cylinder coordinate has been investigated by Sakhr (Sakhr & Chronik, 2017). In 2020, (D. Liu & Li, 2020) have studied the blood flow problem in a blood vessel. This problem related to elastic Navier-Lame equations.
To Find the ℛ-sectoriality, first of all, we are applying Fourier transformation to the model problem. Then by using Weis's operator valued Fourier multiplier theorem, we estimate all multiplier. This ℛ-sectoriality introduced for first time by (Denk et al., 2005). The ℛ-bounded operator families arising from the study of Barotropic compressible flows with free surface has been studied by Zhang (Zhang, 2020). Some exact solution of Laḿ equations with = 3 by using Lie point transformations has been proved by Ozer (Özer, 2003). On the other hand, (Cao, 2009) investigated the solution of Navier equations and their representation structure. Flag partial differential equations and representation of Lie algebra studied by Xu (Xu, 2008).
Recently, there are many researchers who concern studying ℛ-boundedness. In 2014, Murata investigated the ℛ-boundedness of the Stokes operator families with slip boundary condition (Murata, 2014). On the other hand, (Maryani, 2016b) proved the maximal − regularity for Oldroyd-B model fluid flow without surface tension in bounded and unbounded domain. In the same year, (Maryani, 2016a) investigated the global well-posedness in some bounded domain case.
As we known that multiphase flows are new phenomenon in fluid motion which particularly relevant in subsurface flow. For this case, it was assumed that the fluids are well separated which mean the fluids do not mix each other and also there are no additional particles resolved in the fluids. For two-phase problem, (Maryani & Saito, 2017) studied the ℛ-boundedness of the solution operator families for Stokes equation. Furthermore, (Inna et al., 2020) investigated other fluid model i.e Korteweg equation. In that paper, they prove the ℛ-boundedness of the solution operator families for Korteweg model problem. Other researchers who concerned in Korteweg problem is (Murata & Shibata, 2020). They studied the global-well-posedness for the compressible fluid model case. Other researchers who consider two phase problem of compressible and incompressible viscous fluids motion without surface tension studied by (Kubo & Shibata, 2021). In 2020, Saito and Zhang, investigated elliptic problems with two phase incompressible flows in unbounded domain (Saito & Zhang, 2019). In one year later, (Zhang, 2020) considered The R-bounded operator families arising from the study of the barotropic compressible flows with free surface. On the other hand, the validity of the NSE for nanoscale liquid flows was investigated by (C. Liu & Li, 2011).
In 2021  investigated the Stokes equations in half-space. In the article, she showed the formula of the Stokes equations in half-space without surface tension.
One year before (Alif et al., 2021) studied Stokes equation's formula in three dimension Euclidean space by using Fourier transform. In 2020, (Oishi, 2021) investigated the solution formula and R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann Boundary condition. On the R-boundedness also studied by (Götz & Shibata, 2014) in 2014. They investigated compressible fluid flow with free surface. Before we state our main result in the next session, we introduce our notation used throughout the paper.
Notation ℕdenotes the sets of natural numbers and we set ℕ = ℕ ∪ { }. ℂandℝ denote the sets of complex numbers and real numbers, respectively. For any multi-index κ = (κ 1 , … , κ N ) ∈ ℕ , we write |κ| = κ 1 + ⋯ + κ N and = 1 1 ⋯ with x = (x 1 , … , x N ). where ∶ ≡ ∑ , =1 . The letter C denotes generic constants and the constant , ,… depends on , , …. The values of constants Cand , ,… denote a positive constant which maybe different even in a single chain of inequalities. We use small boldface letter, e.g. to denote vector-valued functions and capital boldface letters, e.g. to denote matrix-valued functions, respectively. But, we also use the Greek letters, e.g , , , , such as mass densities.
In this paper we construct the solution formula of velocity ( ) for the model problem for Navier-Laḿ equation in whole space. For this purpose, we apply Laplace transform for the resolvent problem then by using Fourier transform and inverse Fourier transform we get the operator families of the solution. In this step, we also use Weis's operator valued Fourier multiplier theorem.

B. METHODS
The research methodology which used in this paper is literature review of the related articles. In this article, we defined the solution of the velocity of the Navier-Laḿ equations in whole space case. The procedures are in the following, first of all, transforming model problem by using Laplace transform to be resolvent problem. Then applying Fourier transform and inverse Fourier transform in whole space case, we have the solution formula of velocity . The last step, we estimate the multiplier of the solution formula , by using Weis's multipliers theorem. In this paper, we study ℛ-boundedness of the solution operator families for Navier-Laḿ equation with surface tension in whole space. This ℛ-boundedness is the tools to prove further research which related to maximal − regularity class. We prove the ℛ-bounded solution operators of the generalized resolvent problem of the Navier-Laḿ equations by using Weis's operator valued Fourier multiplier.
As we know that the mathematical model of fluid motion formed by conservation of mass and conservation of momentum. These conservations guarantee the boundary condition of the model problem. To prove the existence of ℛ-bounded solution operators, first of all, we transform model problem to resolvent problem by using Laplace transform. Then, applying Fourier transform to model for getting multiplier of the operator families. This methods have been introduced by (Enomoto & Shibata, 2013). As we explained above that the research methods in this paper, we apply Fourier transform and Laplace transform to the model problem to get the solution operator families. Then by using Weis's multiplier theorem, we prove the existence of the model problem in whole space. The procedures of the research can be described in the following diagram (Figure 1 The model problem of Navier-Laḿ equation with surface tension will be explained in the following section.

C. RESULT AND DISCUSSION 1. Construction
Let and Ω be a velocity field and a bounded domain in -dimensional Euclidean space ℝ ( ≥ 2), respectively. The formula of the generalized resolvent problem of Navier-Laḿ equations in bounded domain with surface tension is described in the following where ′ = ( 1 , … , ) ∈ ℝ − and ′ ⋅ ∇ ′ η = ∑ −1
Before state our main result, first of all we introduce the definition of ℛ-boundedness and the operator valued Fourier multiplier theorem due to Weis (Weis, 2001). with some constant , depends solely on and .
The proof of Lemma 4 can be seen in (Shibata & Tanaka, 2004).
The following theorem is the main theorem of this article.
Multiplying (4) where ℱ and ℱ −1 denote the Fourier transform and inverse Fourier transform, respectively which defined in (1a). We can write the equation (6) in the following