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Stationary Solutions to the Navier–Stokes System in an Exterior Plane Domain: 90 Years of Search, Mysteries and Insights

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Abstract

In this survey, we study the boundary value problem for the stationary Navier–Stokes system in planar exterior domains. With no-slip boundary condition and a prescribed constant limit velocity at infinity, this problem describes stationary Navier–Stokes flows around cylindrical obstacles. Leray’s invading domains method is presented as a starting point. Then we discuss the boundedness and convergence of general D-solutions (solutions with finite Dirichlet integrals) in exterior domains. For the Leray solutions of the flow around an obstacle problem, we study the nontriviality, and the justification of the limit velocity at small Reynolds numbers. Further, under the same assumption of small Reynolds numbers the global uniqueness theorem for the problem is established in the class of D-solutions, its proof deals with the accurate perturbative analysis based on the linear Oseen system, inspired by classical Finn-Smith technique; the classical Amick and Gilbarg–Weinberger papers are involved here as well. The forced Navier–Stokes system in the whole plane is also presented as a closely related problem. A list of unsolved problems is given at the end of the paper.

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Notes

  1. Stokes himself gave the following explanation: The pressure of the cylinder on the fluid continuously tends to increase the quantity of fluid which it carries with it, while the friction of the fluid at a distance from the cylinder continually tends to diminish it. In the case of a sphere, these two causes eventually counteract each other, and the motion becomes uniform. But in the case of a cylinder, the increase in the quantity of the fluid carried continually gains on the decrease due to the friction of the surrounding fluid, and the quantity carried increases indefinitely as the cylinder moves on [37, p. 65].

  2. This convergence is uniform on every bounded set.

  3. It is natural to restrict our attention to the class of D-solutions since the Dirichlet integral is the natural energy integral for the stationary Navier–Stokes system. Physically, finiteness of the Dirichlet integral means that the total energy dissipation rate in the fluid is finite.

  4. The irony of fate—precisely because of the embedding theorems, the Millennium problem for the dynamical NS-system is easily solved in the two-dimensional case, and remains unapproachable for the three-dimensional case.

References

  1. Amick, Ch.J.: On Leray’s problem of steady Navier–Stokes flow past a body in the plane. Acta Math. 161, 71–130 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amick, Ch.J.: On the asymptotic form of Navier–Stokes flow past a body in the plane. J. Differ. Equ. 91, 149–167 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. Babenko, K.I.: The asymptotic behavior of a vortex far away from a body in a plane flow of viscous fluid. J. Appl. Math. Mech. USSR 34, 869–881 (1970)

    Article  MATH  ADS  Google Scholar 

  4. Chang, D.-C., Krantz, S.G., El Stein, M.: \(H^p\) theory on a smooth domain in \(\textbf{R}^N\) and elliptic boundary value problems. J. Funct. Anal. 114, 286–347 (1993)

  5. Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. IX Sér. 72, 247–286 (1993)

    MathSciNet  MATH  Google Scholar 

  6. Ferone, A., Russo, R., Tartaglione, A.: The Stokes paradox in inhomogeneous elastostatics. J. Elast. 142, 35–52 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Finn, R., Smith, D.R.: On the linearized hydrodynamical equations in two dimensions. Arch. Ration. Mech. Anal. 25, 1–25 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fujita, H.: On the existence and regularity of the steady-state solutions of the Navier–Stokes equation. J. Fac. Sci. Univ. Tokyo Sect. I(9), 59–102 (1961)

    MATH  Google Scholar 

  10. Galdi, G.P.: On the existence of symmetric steady-state solutions to the plane exterior Navier–Stokes problem for arbitrary large Reynolds number. Adv. Fluid Dyn. 4, 1–25 (1999)

    MathSciNet  MATH  ADS  Google Scholar 

  11. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problems. Springer, Berlin (2011)

    MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gilbarg, D., Weinberger, H.F.: Asymptotic properties of Leray’s solutions of the stationary two-dimensional Navier–Stokes equations. Russ. Math. Surv. 29, 109–123 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gilbarg, D., Weinberger, H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equation with bounded Dirichlet integral. Ann. Scuola Norm. Sup. Pisa Cl. Sci 4(5), 381–404 (1978)

  15. Guillod, J., Wittwer, P.: Optimal asymptotic behavior of the vorticity of a viscous flow past a two-dimensional body. J. Math. Pures Appl. 108, 481–499 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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, 4429–4445 (2018)

  17. Guillod, J., Korobkov, M., Ren, X.: Existence and uniqueness for plane stationary Navier–Stokes flows with compactly supported force. Commun. Math. Phys. 397, 729–762 (2023)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  18. Jia, H., Šverák, V.l.: Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions. Invent. Math. 196, 233–265 (2014)

  19. Kapitanskiĭ, L.V., Piletskas, K.I.: On spaces of solenoidal vector fields and boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries. Trudy Mat. Inst. Steklov. 159, 5–36 (1983). English Transl.: Proc. Math. Inst. Steklov. 159, 3–34 (1984)

  20. Korobkov, M.V., Ren, X.: Uniqueness of plane stationary Navier–Stokes flow past an obstacle. Arch. Ration. Mech. Anal. 240, 1487–1519 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Korobkov, M.V., Ren, X.: Leray’s plane stationary solutions at small Reynolds numbers. J. Math. Pures Appl. 158, 71–89 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Korobkov, M.V., Tsai, T.-P.: Forward self-similar solutions of the Navier–Stokes equations in the half space. Anal. PDE 9, 1811–1827 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Korobkov, M.V., Pileckas, K., Russo, R.: The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain. J. Math. Pures Appl. IX Sér. 101, 257–274 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Korobkov, M.V., Pileckas, K., Russo, R.: Solution of Leray’s problem for stationary Navier–Stokes equations in plane and axially symmetric spatial domains. Ann. Math. 2(181), 769–807 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Korobkov, M.V., Pileckas, K., Russo, R.: On convergence of arbitrary D-solution of steady Navier–Stokes system in 2D exterior domains. Arch. Ration. Mech. Anal. 233, 385–407 (2019)

  26. Korobkov, M.V., Pileckas, K., Russo, R.: On the steady Navier–Stokes equations in 2D exterior domains. J. Differ. Equ. 269(3), 1796–1828 (2020)

  27. Korobkov, M.V., Pileckas, K., Russo, R.: Leray’s plane steady state solutions are nontrivial. Adv. Math. 376, Paper No.107451 (2021)

  28. Ladyženskaja, O.A., Solonnikov, V.A.: Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier–Stokes equation. Zapiski Nauchn. Sem. LOMI, 59, 81–116 (1976) (in Russian); English translation in J. of Soviet Mathematics, 10(2), 257–286 (1978)

  29. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Mathematics and its Applications, vol. 2. Gordon and Breach Science Publishers, New York (1969)

    MATH  Google Scholar 

  30. Leray, J.: Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

    MATH  Google Scholar 

  31. Oseen, C.W.: Neuere methoden und ergebnisse in der hydrodynamik. Akademische Verlagsgesellschaft m.b.H, Leipzig (1927)

    MATH  Google Scholar 

  32. Russo, A.: A note on the exterior two-dimensional steady-state Navier–Stokes problem. J. Math. Fluid Mech. 11(3), 407–414 (2009)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  33. Russo, R.: On Stokes’ Problem. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, pp. 473–511. Springer, Berlin (2010)

    Chapter  Google Scholar 

  34. Sazonov, L.I.: On the asymptotic behavior of the solution of the two-dimensional stationary problem of the flow past a body far from it. Math. Notes 65(2), 202–207 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Seregin, G.: Lecture Notes on Regularity Theory for the Navier–Stokes Equations. World Scientific Publishing Co. Pte. Ltd., Hackensack (2015)

    MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Stokes, G.-G.: On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Philos. Soc. 9, 8–106 (1851)

    ADS  Google Scholar 

  38. Vorovich, I.I., Yudovich, V.I.: Stationary flow of a viscous incompressible fluid. Mat. Sb. 53(4), 393–428 (1961) (in Russian)

  39. Yudovich, V.I.: Eleven great problems of mathematical hydrodynamics. Mosc. Math. J. 3, 711–737 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Mikhail Korobkov & Xiao Ren

  2. Sobolev Institute of Mathematics, pr-t Ac. Koptyug, 4, Novosibirsk, 630090, Russia

    Mikhail Korobkov

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  1. Mikhail Korobkov
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In honor of Professor Olga Ladyzhenskaya. This article is part of the Topical collection Ladyzhenskaya Centennial Anniversary edited by Gregory Seregin, Konstantinas Pileckas and Lev Kapitanski.

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Korobkov, M., Ren, X. Stationary Solutions to the Navier–Stokes System in an Exterior Plane Domain: 90 Years of Search, Mysteries and Insights. J. Math. Fluid Mech. 25, 55 (2023). https://doi.org/10.1007/s00021-023-00792-w

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