Localization and multiplicity for stationary Stokes systems with variable viscosity

Abstract

In this paper we discuss the localization and the multiplicity of solutions for the stationary Stokes system with variable viscosity and a reaction force term. The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems.

Authors

Renata Bunoiu
University of Lorraine, Metz, France

Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babes-Bolyai, University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Stokes system, localization and multiplicity of solutions, fixed point method, homogenization

Paper coordinates

R. Bunoiu, R. Precup, Localization and multiplicity for stationary Stokes systems with variable viscosity, Communications in Mathematical Analysis and Applications, (2025), 1-28, https://dx.doi.org/10.2139/ssrn.4887415

PDF

About this paper

Journal
Communications in Mathematical Analysis and Applications
Publisher Name
Print ISSN
Online ISSN

google scholar link

[1] Allaire, G., Ghosh, T., Vanninathan, M.: Homogenization of Stokes System using Bloch Waves. Netw. Heterog. Media 12(4), 525–550 (2017)

[2] Bunoiu, R., Cardone, G., Kengne, R., Woukeng, J.L.: Homogenization of 2D Cahn-Hilliard-Navier-Stokes system. J. Elliptic Parabol. Equ. 6, 377–408 (2020)

[3] Bunoiu, R., Precup, R.: Vectorial Approach to Coupled Nonlinear Schrödinger Systems under Nonlocal Cauchy Conditions. Appl. Anal. 95(4), 731–747 (2016)

[4] Bunoiu, R., Precup, R.: Localization and multiplicity in the homogenization of nonlinear problems. Adv. Nonlinear Anal. 9, 292–304 (2020)

[5] Colli, P., Gilardi, G., Marinoschi, G.: Global solution and optimal control of an epidemic propagation with a heterogeneous diffusion. Appl. Math. Optim. 89 (2024)

[6] Cai, Z., Wang, Y.: Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 63(3), 341–356 (2010)

[7] Camaño, J., Gatica, G.N., Oyarzúa, R., Tierra, G.: An augmented mixed finite element method for the Navier-Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54(2), 1069–1092 (2016)

[8] Cardone, G., Fares, R., Panasenko, G.P.: Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure. J. Math. Phys. 53 (2012)

[9] Codina, R., Soto, O.: Finite element solution of the Stokes problem with dominating Coriolis force. Comput. Methods Appl. Mech. Engrg. 142, 215–234 (1997)

[10] Constantin, P., Foiaş, C.: Navier-Stokes Equations (1988)

[11] Fares, R., Panasenko, G.P., Stavre, R.: A viscous fluid flow through a thin channel with mixed rigid-elastic boundary: Variational and asymptotic analysis. Abstr. Appl. Anal. 2012 (2012)

[12] Gahn, M., Neuss-Radu, M., Knabner, P.: Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface. SIAM J. Appl. Math. 76, 1819–1843 (2016)

[13] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order (1998)

[14] Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier-Stokes Equations (1981)

[15] Jost, J.: Partial Differential Equations (2007)

[16] Kohr, M., Mikhailov, S.E., Wendland, W.L.: Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with L$^\infty$ strongly elliptic coefficient tensor. Complex Var. Elliptic Equ. 66, 109–140 (2020)

[17] Lee, S., Ryi, S.-K., Lim, H.: Solutions of Navier-Stokes equation with Coriolis force. Adv. Math. Phys. 2017, Article ID 7042686, 1–9 (2017)

[18] Lukaszewicz, G., Kalita, P.: Navier-Stokes Equations (2016)

[19] Neuss-Radu, M., Jäger, W.: Effective Transmission Conditions for Reaction-Diffusion Processes in Domains Separated by an Interface. SIAM J. Math. Anal. 39, 687–720 (2007)

[20] Panasenko, G.P., Stavre, R.: Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Netw. Heterog. Media 5(4), 783–812 (2010)

[21] Perov, A.I.: On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn. 2, 115–134 (1964)

[22] Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Modell. 49, 703–708 (2009)

[23] Precup, R.: Moser-Harnack inequality, Krasnosel’skii type fixed point theorems in cones and elliptic problems. Topol. Methods Nonlinear Anal. 40, 301–313 (2012)

[24] Precup, R.: Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3, 197–207 (2014)

[25] Sohr, H.: The Navier-Stokes Equations (2001)

[26] Stan, A.: Nonlinear systems and Nash type equilibria. 66, 397–408 (2021)

[27] Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis (1995)

Related Posts

No results found.