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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

Francis Hounkpe 1,, and  Gregory Seregin 1,2,

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.

Keywords: Singular limits, self-similar solutions, incompressible Navier-Stokes system, global convergence, Lamé system.
Mathematics Subject Classification: Primary: 35K55, 35J60, 35C06, 35A35; Secondary: 35A01.
Citation: Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249. Springer-Verlag New York, 2008.  Google Scholar

[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. Google Scholar

[7]

F. Hounkpe, Decay estimate for some toy-models related to the Navier-Stokes system, Preprint (2020), arXiv:2008.08712. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249. Springer-Verlag New York, 2008.  Google Scholar

[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. Google Scholar

[7]

F. Hounkpe, Decay estimate for some toy-models related to the Navier-Stokes system, Preprint (2020), arXiv:2008.08712. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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