Abstract
Extending the work of Jia and Šverák on self-similar solutions of the Navier–Stokes equations, we show the existence of large, forward, discretely self-similar solutions.
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Brandolese L.: Fine properties of self-similar solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 192(3), 375–401 (2009)
Caffarelli L., Kohn R.-V., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982)
Cannone, M., Meyer, Y., Planchon, F.: Solutions auto-simlaires des equations de Navier-Stokes. In: Seminaire X-EDP, Centre de Mathematiques, Ecole polytechnique (1993–1994)
Cannone M., Planchon F.: Self-similar solutions for the Navier-Stokes equations in R 3. Comm. Part. Diff. Equ. 21(1–2), 179–193 (1996)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized steady problems, Volume I. Springer, Berlin (1994)
Giga Y., Miyakawa T.: Navier-Stokes flow in R 3 with measures as initial vorticity and morrey spaces. Comm. Part. Differ. Equ. 14(5), 577–618 (1989)
Jia, H, Šverák, V.: Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math. Online doi:10.1007/s00222-013-0468-x
Ladyzenskaja, O.A.: Unique global solvability of the three-dimensional Cauchy problem for the NavierStokes equations in the presence of axial symmetry. Zapiski Nauchnykh Seminarov Leningrad Otdelenie. Matematicheski. Institut im. V. A. Steklova (LOMI) 7, 15577 (1968) (Russian)
Landau L.D.: A new exact solution of the Navier-Stokes equations. Dokl Akad. Nauk SSSR 43, 299 (1944)
Landau L.D., Lifshitz E.M.: Fluid Mechanics, 2nd edn. Butterworth-Heinemann, Paperback reprinting (2000)
Leonardi S., Malek J., Nečas J., Pokorny M.: On axially symmetric flows in R 3. Zeitschrift fur Analysis und ihre Anwendungen 18, 639–49 (1999)
Lemarié-Rieusset, P.G.: Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics 431. Chapman & Hall/CRC, Boca Raton (2002)
Mawhin J.: Leray-Schauder degree: a half century of extensions and applications. Topol. Methods Nonlinear Anal. 14(2), 195–228 (1999)
Nečas J., Růžička M., Šverák V.: On Lerays self-similar solutions of the Navier-Stokes equations. Acta Math. 176, 283–294 (1996)
Oseen, C.W.: Hydrodynamik, Leipzig (1927)
Solonnikov V.A.: Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations (Russian). Trudy Mat. Inst. Steklov. 70, 213–317 (1964)
Šverák, V.: On Landau’s solutions of the Navier-Stokes equations. Problems in Mathematical Analysis 61, 175–190 (2011). Translation in Journal of Mathematical Sciences 179(1), 208–228 (2011)
Šverák, V.: Lecture in the workshop “Evolution equations of physics, fluids, and geometry: asymptotics and singularities”. Banff International Research Station (BIRS) (2012). http://www.birs.ca/events/2012/5-day-workshops/12w5137/videos
Tsai T.-P.: On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Ration. Mech. Anal. 143, 29–51 (1998)
Ukhovskii M.R., Yudovich V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 5261 (1968)
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Tsai, TP. Forward Discretely Self-Similar Solutions of the Navier–Stokes Equations. Commun. Math. Phys. 328, 29–44 (2014). https://doi.org/10.1007/s00220-014-1984-2
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DOI: https://doi.org/10.1007/s00220-014-1984-2