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October  2021, 41(10): 4823-4846. doi: 10.3934/dcds.2021059

## An approximation of forward self-similar solutions to the 3D Navier-Stokes system

 1 Mathematical Institute, OxPDE, University of Oxford, Oxford, UK 2 St Petersburg Department of Steklov Mathematical Institute, RAS, RUSSIA

* Corresponding author

Received  October 2020 Published  October 2021 Early access  March 2021

In this paper, we present two constructions of forward self-similar solutions to the $3$D incompressible Navier-Stokes system, as the singular limit of forward self-similar solutions to certain parabolic systems.

Citation: Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4823-4846. doi: 10.3934/dcds.2021059
##### References:
 [1] T. Barker, G. Seregin and V. Šverák, On stability of weak Navier-Stokes solutions with large $L^{3, \infty}$ initial data, Commun. Partial Differ. Equ, 43, (2018), 628–651. doi: 10.1080/03605302.2018.1449219. [2] Z. Bradshaw and T-P. Tsai, Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations Ⅱ, Ann. Henri Poincaré, 18 (2017), 1095–1119. doi: 10.1007/s00023-016-0519-0. [3] D. Chae and J. Wolf, Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $L^2_loc(\mathbb{R}^3)$, arXiv: 1610.01386. [4] Y. Giga and T. Miyakawa, Navier-Stokes flow in $\mathbb{R}^3$ with measure as initial vorticity and Morrey spaces, Commun. Partial Differ. Equ, 14, (1989), 577–618. doi: 10.1080/03605308908820621. [5] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249. Springer-Verlag New York, 2008. [6] J. Guillod and V. Šverák, Numerical investigations of non-uniqueness for the Naier-Stokes intial value problem in borderline spaces, Preprint (2017), arXiv:1704.00560. [7] F. Hounkpe, Decay estimate for some toy-models related to the Navier-Stokes system, Preprint (2020), arXiv:2008.08712. [8] H. Jia and V. Šverák, Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions, Invent math, 196 (2014), 233–265. doi: 10.1007/s00222-013-0468-x. [9] N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2,220, Adv. Math. Sci., 59, Amer. Math. Soc., Providence, RI, 2007,141–164. doi: 10.1090/trans2/220/07. [10] M. Korobkov and T-P. Tsai, Forward self-similar solutions of the Navier-Stokes equations in the half space, Analysis & PDE, 9 (2016), 1811–1827. doi: 10.2140/apde.2016.9.1811. [11] O. A. Ladyzhenskaya and G. A. Seregin, A method for the approximate solution of initial-boundary value problems for Navier-Stokes equations, J. Math Sci, 75 (1995), 2038-2057.  doi: 10.1007/BF02362945. [12] P. G. Lemarie-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC Research Notes in Mathematics, 431. Chapman and Hall/CRC, Boca Raton, FL 2002. doi: 10.1201/9781420035674. [13] P. G. Lemarie-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL (2016). xxii+718 pp. doi: 10.1201/b19556. [14] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [15] J. Mawhin, Leray-Schauder degree: a half-century of extensions and applications, Topol. Methods Nonlinear Anal., 14 (1999), 195–228. doi: 10.12775/TMNA.1999.029. [16] D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Generalised Gagliardo-Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO, Milan J. Math., 81 (2013), 265–289. doi: 10.1007/s00032-013-0202-6.

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##### References:
 [1] T. Barker, G. Seregin and V. Šverák, On stability of weak Navier-Stokes solutions with large $L^{3, \infty}$ initial data, Commun. Partial Differ. Equ, 43, (2018), 628–651. doi: 10.1080/03605302.2018.1449219. [2] Z. Bradshaw and T-P. Tsai, Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations Ⅱ, Ann. Henri Poincaré, 18 (2017), 1095–1119. doi: 10.1007/s00023-016-0519-0. [3] D. Chae and J. Wolf, Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $L^2_loc(\mathbb{R}^3)$, arXiv: 1610.01386. [4] Y. Giga and T. Miyakawa, Navier-Stokes flow in $\mathbb{R}^3$ with measure as initial vorticity and Morrey spaces, Commun. Partial Differ. Equ, 14, (1989), 577–618. doi: 10.1080/03605308908820621. [5] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249. Springer-Verlag New York, 2008. [6] J. Guillod and V. Šverák, Numerical investigations of non-uniqueness for the Naier-Stokes intial value problem in borderline spaces, Preprint (2017), arXiv:1704.00560. [7] F. Hounkpe, Decay estimate for some toy-models related to the Navier-Stokes system, Preprint (2020), arXiv:2008.08712. [8] H. Jia and V. Šverák, Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions, Invent math, 196 (2014), 233–265. doi: 10.1007/s00222-013-0468-x. [9] N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2,220, Adv. Math. Sci., 59, Amer. Math. Soc., Providence, RI, 2007,141–164. doi: 10.1090/trans2/220/07. [10] M. Korobkov and T-P. Tsai, Forward self-similar solutions of the Navier-Stokes equations in the half space, Analysis & PDE, 9 (2016), 1811–1827. doi: 10.2140/apde.2016.9.1811. [11] O. A. Ladyzhenskaya and G. A. Seregin, A method for the approximate solution of initial-boundary value problems for Navier-Stokes equations, J. Math Sci, 75 (1995), 2038-2057.  doi: 10.1007/BF02362945. [12] P. G. Lemarie-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC Research Notes in Mathematics, 431. Chapman and Hall/CRC, Boca Raton, FL 2002. doi: 10.1201/9781420035674. [13] P. G. Lemarie-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL (2016). xxii+718 pp. doi: 10.1201/b19556. [14] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [15] J. Mawhin, Leray-Schauder degree: a half-century of extensions and applications, Topol. Methods Nonlinear Anal., 14 (1999), 195–228. doi: 10.12775/TMNA.1999.029. [16] D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Generalised Gagliardo-Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO, Milan J. Math., 81 (2013), 265–289. doi: 10.1007/s00032-013-0202-6.
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