Abstract
In the present paper we consider mild bounded ancient (backward) solutions to the Navier–Stokes equations in the half plane. We give two different definitions, prove their equivalence and prove smoothness up to the boundary. Such solutions appear as a result of rescaling around a singular point of the initial boundary value problem for the Navier–Stokes equations in the half-plane.
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References
Escauriaza, L., Seregin, G. Šverák, V.: \({L_{3, \infty}}\)-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk 58 2(350), 3–44(2003); translation in Russian Math. Surveys 58(2), 211–250 (Russian) (2003)
Giga Y., Hsu P.-Y., Maekawa Y.: A Liouville theorem for the planer Navier–Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion. Commun. Partial Differ. Equ. 39(10), 19061935 (2014)
Jia H., Seregin G., Seregin G.: Liouville theorems in unbounded domains for the time-dependent Stokes system. J. Math. Phys. 53, 115604 (2012)
Jia, H., Seregin, G., Sverak, V.: A Liouville theorem for the Stokes system in half-space. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 410 (2013), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 43(187), 25–35; translation in J. Math. Sci. (N. Y.) 195(1), 13–19 (2013)
Koch G., Nadirashvili N., Seregin G., Šverák V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203, 83–105 (2009)
Koch H., Solonnikov V.A.: \({L_p}\)-estimates for a solution to the nonstationary Stokes equations. Function theory and phase transitions. J. Math. Sci. (N. Y.) 106(3), 3042–3072 (2001)
Ladyzhenskaya O.A.: Mathematical problems of the dynamics of viscous incompressible fluids. 2nd edn. Nauka, Moscow (1970)
Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Nečas J., Ružička M., Šverák V.: On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176(2), 283–294 (1996)
Seregin G.A.: Local regularity of suitable weak solutions to the Navier–Stokes equations near the boundary. J. Math. Fluid Mech. 4(1), 1–29 (2002)
Seregin G.: A note on local boundary regularity for the Stokes system. Zapiski Nauchn. Semin. POMI 370, 151–159 (2009)
Seregin G.: A certain necessary condition of potential blow up for Navier–Stokes equations. Commun. Math. Phys. 312(3), 833–845 (2012)
Seregin, G.: Lecture Notes on regularity theory for the Navier–Stokes equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2015)
Seregin G.: Liouville theorem for 2D Navier–Stokes equations in half space. Zapiski Nauchn. Semin. POMI 425, 137–148 (2014)
Seregin G., Šverák V.: On Type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations. Commun. PDE’s 34, 171–201 (2009)
Seregin, G., Šverák V.: Rescalings at possible singularities of the Navier–Stokes equations in half-space. Algebra i Analiz 25(5), 146–172 (2013); translation in St. Petersburg Math. J. 25(5), 815–833 (2014)
Solonnikov V.A.: Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 70, 213–317 (1964) (Russian)
Solonnikov V.A.: Estimates of solutions to the non-stationary Navier–Stokes system. Zapiski Nauchn. Semin. LOMI 28, 153–231 (1973)
Solonnikov V.A.: On nonstationary Stokes problem and Navier–Stokes problem in a half space with initial data nondecreasing at infinity. J. Math. Sci. 114(5), 1726–1740 (2003)
Solonnikov, V.A.: Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator. (Russian) Uspekhi Mat. Nauk 58, 2(350), 123–156 (2003); translation in Russian Math. Surveys 58(2), 331–365 (Russian) (2003)
Stein E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)
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Communicated by V. A. Solonnikov
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Barker, T., Seregin, G. Ancient Solutions to Navier–Stokes Equations in Half Space. J. Math. Fluid Mech. 17, 551–575 (2015). https://doi.org/10.1007/s00021-015-0211-z
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DOI: https://doi.org/10.1007/s00021-015-0211-z