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

Sri Maryani; Ari Wardayani; Bambang Hendriya Guswanto

doi:10.31764/jtam.v6i1.6217

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

Sri Maryani, Ari Wardayani, Bambang Hendriya Guswanto

Abstract

In this paper, we consider the boundedness of the operator families in whole space for Navier-Lame model problem in bounded domain of N dimensional Euclidean space (N≥2). To find the boundedness of the operator families, first of all we construct model problem in the form of the resolvent problem by using Laplace transform. Then, using Fourier transform, we get the solution formula of the model problem. In this paper, we use the qualitative methods to construct solution formula of velocity (u). This step is fundamental stage to find the well-posedness of the model problem. As we known that fluid motion can be described in partial differential equation (PDE). Essential point in PDE are finding existence and uniqueness of the model problem. One methods of investigating the well-posedness is R-boundedness of the solution operator families of the model problem. We can find the R-boundedness of the solution operator families not only in whole-space, half-space, bent-half space and in general domain. In this paper we investigate the R-boundedness of the solution operator families only in whole space. By using this R-boundedness, we can find that the multipliers which form of the operator families are bounded with some positive constant.

Keywords

R-boundedness; Whole-space; Navier-lame Equations; Fluid Motion; Well-posedness.

Full Text:

DOWNLOAD [PDF]

References

Alif, A. H., Maryani, S., & Nurshiami, S. R. (2021). Solution Formula of the Compressible Fluid Motion in Three Dimension Euclidean Space using Fourier Transform. Journal of Physics: Conference Series, 1751(1), 12006.

Cao, B. (2009). Solutions of Navier equations and their representation structure. Advances in Applied Mathematics, 43(4), 331–374.

Denk, R., Hieber, M., & Prüss, J. (2005). Optimal Lp-Lq-regularity for parabolic problems with inhomogeneous boundary data.

Enomoto, Y., & Shibata, Y. (2013). On the $mathcal {R} $-sectoriality and the initial boundary value problem for the viscous compressible fluid flow. Funkcialaj Ekvacioj, 56(3), 441–505.

Eringen, A. C., & Suhubi, E. S. (1975). Elastodynamics. Volume 2- Linear theory(Book). New York, Academic Press, Inc., 1975. 671 P.

Girault, V., & Raviart, P.-A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.

Götz, D., & Shibata, Y. (2014). On the ℛ-boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions. Asymptotic Analysis, 90(3–4), 207–236.

Inna, S., Maryani, S., & Saito, H. (2020). Half-space model problem for a compressible fluid model of Korteweg type with slip boundary condition. Journal of Physics: Conference Series, 1494(1), 12014.

Kubo, T., & Shibata, Y. (2021). On the evolution of compressible and incompressible viscous fluids with a sharp interface. Mathematics, 9(6), 621.

Liu, C., & Li, Z. (2011). On the validity of the Navier-Stokes equations for nanoscale liquid flows: The role of channel size. AIP Advances, 1(3), 32108.

Liu, D., & Li, L. (2020). The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel. Numerical Methods for Partial Differential Equations, 36(5), 1044–1073.

Maryani, S. (2016a). Global well‐posedness for free boundary problem of the Oldroyd‐B model fluid flow. Mathematical Methods in the Applied Sciences, 39(9), 2202–2219.

Maryani, S. (2016b). On the free boundary problem for the Oldroyd-B Model in the maximal Lp–Lq regularity class. Nonlinear Analysis: Theory, Methods & Applications, 141, 109–129.

Maryani, S., Nugroho, L. B., Sugandha, A., & Guswanto, B. H. (2021). The half-Space Model Problem for Compressible Fluid Flow. Limits: Journal of Mathematics and Its Applications, 18(1), 37–44.

Maryani, S., & Saito, H. (2017). On the $mathcal {R} $-boundedness of solution operator families for two-phase Stokes resolvent equations. Differential and Integral Equations, 30(1/2), 1–52.

Murata, M. (2014). On a maximal Lp–Lq approach to the compressible viscous fluid flow with slip boundary condition. Nonlinear Analysis: Theory, Methods & Applications, 106, 86–109.

Murata, M., & Shibata, Y. (2020). The global well-posedness for the compressible fluid model of Korteweg type. SIAM Journal on Mathematical Analysis, 52(6), 6313–6337.

Oishi, K. (2021). A solution formula and the ℛ‐boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition. Mathematical Methods in the Applied Sciences, 44(5), 3925–3959.

Özer, T. (2003). The solution of Navier equations of classical elasticity using Lie symmetry groups. Mechanics Research Communications, 30(2), 193–201.

Panton, R. L. (2013). Incompressible flow. John Wiley & Sons.

Saito, H., & Zhang, X. (2019). Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains. ArXiv Preprint ArXiv:1912.00135.

Sakhr, J., & Chronik, B. A. (2017). Solving the Navier-Lame equation in cylindrical coordinates using the Buchwald representation: some parametric solutions with applications. ArXiv Preprint ArXiv:1704.06669.

Shibata, Y., & Tanaka, K. (2004). On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion. Mathematical Methods in the Applied Sciences, 27(13), 1579–1606.

Watanabe, K. (2018). Compressible–Incompressible Two-Phase Flows with Phase Transition: Model Problem. Journal of Mathematical Fluid Mechanics, 20(3), 969–1011.

Weis, L. (2001). Operator–valued Fourier multiplier theorems and maximal $ L_p $-regularity. Mathematische Annalen, 319(4), 735–758.

Xu, X. (2008). Flag partial differential equations and representations of Lie algebras. Acta Applicandae Mathematicae, 102(2–3), 249–280.

Zhang, X. (2020). The R-bounded operator families arising from the study of the barotropic compressible flows with free surface. Journal of Differential Equations, 269(9), 7059–7105.

DOI: https://doi.org/10.31764/jtam.v6i1.6217