August  2021, 14(8): 2917-2931. doi: 10.3934/dcdss.2020408

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

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

* Corresponding author: Pierre Gilles Lemarié–Rieusset

Received  January 2020 Revised  June 2020 Published  August 2021 Early access  July 2020

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

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 and Continuous Dynamical Systems - S, 2021, 14 (8) : 2917-2931. doi: 10.3934/dcdss.2020408
References:
[1]

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

[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.

[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.

[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.

[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.

[6]

L. CaffarelliR. 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.

[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.

[8]

D. ChamorroP. 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.

[9]

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

[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.

[11]

E. FabesB. 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.

[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.

[13]

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

[14]

G. FurioliP. 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.

[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.

[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.

[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.

[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.

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

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

[21]

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

[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. 

[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.

[24]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[6]

L. CaffarelliR. 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.

[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.

[8]

D. ChamorroP. 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.

[9]

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

[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.

[11]

E. FabesB. 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.

[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.

[13]

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

[14]

G. FurioliP. 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.

[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.

[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.

[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.

[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.

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

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

[21]

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

[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. 

[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.

[24]

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

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