Abstract
We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on the external forces. Weak-strong uniqueness criteria based on various growth conditions at the infinity of weak solutions are also given. This is done by employing an energy estimate and a Hardy’s inequality. Several estimates of stream functions are carried out and two density lemmas with suitable weights for the homogeneous Sobolev space on 2-dimensional space are proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albritton, D., Bruè, E., Colombo, M.: Gluing non-unique Navier-Stokes solutions. Ann. PDE 9, 17 (2023). https://doi.org/10.1007/s40818-023-00155-8
Amick, C.J.: On Leray’s problem of steady Navier-Stokes flow past a body in the plane. Acta Math. 161, 71–130 (1988)
Finn, R., Smith, D.R.: On the stationary solutions of the Navier-Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 25, 26–39 (1967)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edn. Springer Monographs in Mathematics. Springer, New York (2011)
Galdi, G.P., Sohr, H.: On the asymptotic structure of plane steady flow of a viscous fluid in exterior domains. Arch. Ration. Mech. Anal. 131, 101–119 (1995)
Galdi, G., Padula, M., Passerini, A.: Existence and asymptotic decay of plane steady flow in aperture domain. In: Concus, P., Lancaster, K. (eds.) Advances in Geometric Analysis and Continuum Mechanics, pp. 81–99. International Press, Cambridge (1995)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)
Gilbarg, D., Weinberger, H.F.: Asymptotic properties of steady plane solutions of the Navier-Stokes equation with bounded Dirichlet integral. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 5, 381–404 (1978)
Guillod, J., Wittwer, P.: On the stationary Navier-Stokes equations in the half-plane. Ann. Henri Poincaré 17(11), 3287–3319 (2016)
Guillod, J., Wittwer, P.: Existence and uniqueness of steady weak solutions to the Navier-Stokes equations in \(\mathbb{R}^{2}\). Proc. Am. Math. Soc. 146(10), 4429–4445 (2018)
Guillod, J., Korobkov, M., Ren, X.: Existence and uniqueness for plane stationary Navier-Stokes flows with compactly supported force. Commun. Math. Phys. 397(2), 729–762 (2023)
Heywood, J.: On uniqueness questions in the theory of viscous flow. Acta Math. 136, 61–102 (1976). https://doi.org/10.1007/BF02392043
Korobkov, M.M., Pileckas, K., Russo, R.: Leray’s plane steady state solutions are nontrivial. Adv. Math. 376, 107451 (2021)
Ladyzhenskaia, O.A.: The Mathematical Theory of Viscous Incompressible Fluid. Gordon & Breach, New York (1969)
Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l‘hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)
Leray, J.: Sur le mouvement d’un liquide vosqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)
Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Berlin (2011)
Nazarov, S.A.: On the two-dimensional aperture problem for Navier-Stokes equations. C. R. Acad. Sci. Sér. 1 Math. 323(6), 699–703 (1996)
Nazarov, S.A., Sequeira, A., Videman, J.H.: Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution. J. Math. Pures Appl. 80(10), 1069–1098 (2001)
Nazarov, S.A., Sequeira, A., Videman, J.H.: Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution. J. Math. Pures Appl. 81(8), 781–810 (2002). https://doi.org/10.1016/S0021-7824(01)01232-6
Smith, D.R.: Estimates at infinity for stationary solutions of the Navier-Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 20, 341–372 (1965)
Yamazaki, M.: The stationary Navier-Stokes equation on the whole plane with external force with antisymmetry. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 55(2), 407–423 (2009)
Acknowledgements
This work was done partially when Adrian Calderon was an undergraduate student at the University of Tennessee and a graduate student at Boston University. Adrian Calderon would like to thank the University of Tennessee and Boston University for the generous supports. T. Phan’s research is partially supported by Simons Foundation, grant #2769369. The authors would like to thank anonymous referees for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Calderon, A.D., Le, V. & Phan, T. On Stationary Navier-Stokes Equations in the Upper-Half Plane. Acta Appl Math 189, 7 (2024). https://doi.org/10.1007/s10440-024-00636-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10440-024-00636-3