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doi: 10.3934/dcdss.2020408

Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space

Pedro Gabriel Fernández-Dalgo and  Pierre Gilles Lemarié–Rieusset ,

LaMME, Univ Evry, CNRS, Université Paris-Saclay, 91025, Evry, France

* Corresponding author: Pierre Gilles Lemarié–Rieusset

Received  January 2020 Revised  June 2020 Published  July 2020

We characterise the pressure term in the incompressible 2D and 3D Navier–Stokes equations for solutions defined on the whole space.

Keywords: Navier–Stokes equations, pressure term, Leray projection, extended Galilean invariance, suitable solutions.
Mathematics Subject Classification: Primary: 35Q30, 76D05.
Citation: Pedro Gabriel Fernández-Dalgo, Pierre Gilles Lemarié–Rieusset. Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020408
References:
[1]

A. Basson, Solutions Spatialement Homogènes Adaptées des Équations de Navier–Stokes, Thèse, Université d'Évry, 2006. Google Scholar

[2]

A. Basson, Homogeneous statistical solutions and local energy inequality for 3D Navier–Stokes equations, Commun. Math. Phys., 266 (2006), 17-35.  doi: 10.1007/s00220-006-0009-1.  Google Scholar

[3]

Z. Bradshaw and T. P. Tsai, Discretely self-similar solutions to the Navier-Stokes equations with data in $L^2_ {\rm loc} $ satisfying the local energy inequality, Analysis and PDE, 12 (2019), 1943-1962.  doi: 10.2140/apde.2019.12.1943.  Google Scholar

[4]

Z. Bradshaw and T. P. Tsai, Global existence, regularity, and uniqueness of infinite energy solutions to the Navier–Stokes equations, preprint, arXiv: 1907.00256. Google Scholar

[5]

Z. Bradshaw, I. Kukavica and T. P. Tsai, Existence of global weak solutions to the Navier-Stokes equations in weighted spaces, preprint, arXiv: 1910.06929v1. Google Scholar

[6]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.  Google Scholar

[7]

D. Chae and J. Wolf, Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $ L^2_{\rm loc }(\mathbb{R}^3)$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1019-1039.  doi: 10.1016/j.anihpc.2017.10.001.  Google Scholar

[8]

D. Chamorro, P. G. Lemarié–Rieusset and K. Mayoufi, The role of the pressure in the partial regularity theory for weak solutions of the Navier–Stokes equations, Arch. Rat. Mech. Anal., 228 (2018), 237-277.  doi: 10.1007/s00205-017-1191-3.  Google Scholar

[9]

S. Dostoglou, Homogeneous measures and spatial ergodicity of the Navier–Stokes equations, preprint, 2002. Google Scholar

[10]

S. Dubois, What is a solution to the Navier–Stokes equations?, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 27-32.  doi: 10.1016/S1631-073X(02)02419-6.  Google Scholar

[11]

E. Fabes, B. F. Jones and N. Riviere, The initial value problem for the Navier–Stokes equations with data in $L^p$, Arch. Ration. Mech. Anal., 45 (1972), 222-240.  doi: 10.1007/BF00281533.  Google Scholar

[12]

P. G. Fernández-Dalgo and P. G. Lemarié–Rieusset, Weak solutions for Navier–Stokes equations with initial data in weighted $L^2$ spaces, Arch. Ration. Mech. Anal., 237 (2020), 347-382.  doi: 10.1007/s00205-020-01510-w.  Google Scholar

[13]

H. Fujita and T. Kato, On the non-stationary Navier-Stokes system, Rendiconti Seminario Math. Univ. Padova, 32 (1962), 243-260.   Google Scholar

[14]

G. Furioli, P. G. Lemarié-Rieusset and E. Terraneo., Unicité dans $\text{L}^3({\mathbb{R}^{3}})$ et d'autres espaces limites pour Navier–Stokes, Revista Mat. Iberoamericana, 16 (2000), 605-667.  doi: 10.4171/RMI/286.  Google Scholar

[15]

N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality, in Nonlinear Equations and Spectral Theory, 141–164, Amer. Math. Soc. Transl. Ser. 2,220, Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/trans2/220/07.  Google Scholar

[16]

I. Kukavica, On local uniqueness of solutions of the Navier–Stokes equations with bounded initial data, J. Diff. Eq., 194 (2003), 39-50.  doi: 10.1016/S0022-0396(03)00153-0.  Google Scholar

[17]

I. Kukavica and V. Vicol, On local uniqueness of weak solutions to the Navier–Stokes system with $ {\rm BMO}^{-1}$ initial datum, J. Dynam. Differential Equation, 20 (2008), 719-732.  doi: 10.1007/s10884-008-9116-3.  Google Scholar

[18]

P. G. Lemarié–Rieusset, Solutions faibles d'énergie infinie pour les équations de Navier–Stokes dans $\mathbb{R}^{3}$, C. R. Acad. Sci. Paris, Ser. I, 328 (1999), 1133-1138.  doi: 10.1016/S0764-4442(99)80427-3.  Google Scholar

[19] P. G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, CRC Press, 2002.  doi: 10.1201/9781420035674.  Google Scholar
[20]

P. G. Lemarié–Rieusset, The Navier–Stokes Problem in the 21st Century, Chapman & Hall/CRC, 2016. doi: 10.1201/b19556.  Google Scholar

[21]

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

[22]

M. I. Vishik and A. V. Fursikov, Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes, Ann. Scuola Norm. Sup. Pisa, série IV, 4 (1977), 531-576.   Google Scholar

[23]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Dordrecht: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-1423-0.  Google Scholar

[24]

J. Wolf, On the local pressure of the Navier–Stokes equations and related systems, Adv. Differential Equations, 22 (2017), 305-338.   Google Scholar

show all references

References:
[1]

A. Basson, Solutions Spatialement Homogènes Adaptées des Équations de Navier–Stokes, Thèse, Université d'Évry, 2006. Google Scholar

[2]

A. Basson, Homogeneous statistical solutions and local energy inequality for 3D Navier–Stokes equations, Commun. Math. Phys., 266 (2006), 17-35.  doi: 10.1007/s00220-006-0009-1.  Google Scholar

[3]

Z. Bradshaw and T. P. Tsai, Discretely self-similar solutions to the Navier-Stokes equations with data in $L^2_ {\rm loc} $ satisfying the local energy inequality, Analysis and PDE, 12 (2019), 1943-1962.  doi: 10.2140/apde.2019.12.1943.  Google Scholar

[4]

Z. Bradshaw and T. P. Tsai, Global existence, regularity, and uniqueness of infinite energy solutions to the Navier–Stokes equations, preprint, arXiv: 1907.00256. Google Scholar

[5]

Z. Bradshaw, I. Kukavica and T. P. Tsai, Existence of global weak solutions to the Navier-Stokes equations in weighted spaces, preprint, arXiv: 1910.06929v1. Google Scholar

[6]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.  Google Scholar

[7]

D. Chae and J. Wolf, Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $ L^2_{\rm loc }(\mathbb{R}^3)$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1019-1039.  doi: 10.1016/j.anihpc.2017.10.001.  Google Scholar

[8]

D. Chamorro, P. G. Lemarié–Rieusset and K. Mayoufi, The role of the pressure in the partial regularity theory for weak solutions of the Navier–Stokes equations, Arch. Rat. Mech. Anal., 228 (2018), 237-277.  doi: 10.1007/s00205-017-1191-3.  Google Scholar

[9]

S. Dostoglou, Homogeneous measures and spatial ergodicity of the Navier–Stokes equations, preprint, 2002. Google Scholar

[10]

S. Dubois, What is a solution to the Navier–Stokes equations?, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 27-32.  doi: 10.1016/S1631-073X(02)02419-6.  Google Scholar

[11]

E. Fabes, B. F. Jones and N. Riviere, The initial value problem for the Navier–Stokes equations with data in $L^p$, Arch. Ration. Mech. Anal., 45 (1972), 222-240.  doi: 10.1007/BF00281533.  Google Scholar

[12]

P. G. Fernández-Dalgo and P. G. Lemarié–Rieusset, Weak solutions for Navier–Stokes equations with initial data in weighted $L^2$ spaces, Arch. Ration. Mech. Anal., 237 (2020), 347-382.  doi: 10.1007/s00205-020-01510-w.  Google Scholar

[13]

H. Fujita and T. Kato, On the non-stationary Navier-Stokes system, Rendiconti Seminario Math. Univ. Padova, 32 (1962), 243-260.   Google Scholar

[14]

G. Furioli, P. G. Lemarié-Rieusset and E. Terraneo., Unicité dans $\text{L}^3({\mathbb{R}^{3}})$ et d'autres espaces limites pour Navier–Stokes, Revista Mat. Iberoamericana, 16 (2000), 605-667.  doi: 10.4171/RMI/286.  Google Scholar

[15]

N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality, in Nonlinear Equations and Spectral Theory, 141–164, Amer. Math. Soc. Transl. Ser. 2,220, Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/trans2/220/07.  Google Scholar

[16]

I. Kukavica, On local uniqueness of solutions of the Navier–Stokes equations with bounded initial data, J. Diff. Eq., 194 (2003), 39-50.  doi: 10.1016/S0022-0396(03)00153-0.  Google Scholar

[17]

I. Kukavica and V. Vicol, On local uniqueness of weak solutions to the Navier–Stokes system with $ {\rm BMO}^{-1}$ initial datum, J. Dynam. Differential Equation, 20 (2008), 719-732.  doi: 10.1007/s10884-008-9116-3.  Google Scholar

[18]

P. G. Lemarié–Rieusset, Solutions faibles d'énergie infinie pour les équations de Navier–Stokes dans $\mathbb{R}^{3}$, C. R. Acad. Sci. Paris, Ser. I, 328 (1999), 1133-1138.  doi: 10.1016/S0764-4442(99)80427-3.  Google Scholar

[19] P. G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, CRC Press, 2002.  doi: 10.1201/9781420035674.  Google Scholar
[20]

P. G. Lemarié–Rieusset, The Navier–Stokes Problem in the 21st Century, Chapman & Hall/CRC, 2016. doi: 10.1201/b19556.  Google Scholar

[21]

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

[22]

M. I. Vishik and A. V. Fursikov, Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes, Ann. Scuola Norm. Sup. Pisa, série IV, 4 (1977), 531-576.   Google Scholar

[23]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Dordrecht: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-009-1423-0.  Google Scholar

[24]

J. Wolf, On the local pressure of the Navier–Stokes equations and related systems, Adv. Differential Equations, 22 (2017), 305-338.   Google Scholar

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