×

Characterisation of the pressure term in the incompressible Navier-Stokes equations on the whole space. (English) Zbl 1476.35171

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

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI arXiv

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, 17-35 (2006) · Zbl 1106.76017
[3] Z. Bradshaw; T. P. Tsai, Discretely self-similar solutions to the Navier-Stokes equations with data in \(\begin{document}L^2_{\rm loc} \end{document}\) satisfying the local energy inequality, Analysis and PDE, 12, 1943-1962 (2019) · Zbl 1431.35100
[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. · Zbl 1448.35360
[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. Caffarelli; R. Kohn; L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831 (1982) · Zbl 0509.35067
[7] D. Chae; J. Wolf, Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in \(\begin{document} L^2_{\rm loc }(\mathbb{R}^3)\end{document} \), Ann. Inst. H. Poincaré Anal. Non Linéaire, 35, 1019-1039 (2018) · Zbl 1391.76057
[8] D. Chamorro; P. G. Lemarié-Rieusset; 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, 237-277 (2018) · Zbl 1382.35185
[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, 27-32 (2002) · Zbl 0999.35073
[11] E. Fabes; B. F. Jones; N. Riviere, The initial value problem for the Navier-Stokes equations with data in \(L^p\), Arch. Ration. Mech. Anal., 45, 222-240 (1972) · Zbl 0254.35097
[12] P. G. Fernández-Dalgo; P. G. Lemarié-Rieusset, Weak solutions for Navier-Stokes equations with initial data in weighted \(L^2\) spaces, Arch. Ration. Mech. Anal., 237, 347-382 (2020) · Zbl 1434.35050
[13] H. Fujita; T. Kato, On the non-stationary Navier-Stokes system, Rendiconti Seminario Math. Univ. Padova, 32, 243-260 (1962) · Zbl 0114.05002
[14] G. Furioli; P. G. Lemarié-Rieusset; E. Terraneo., Unicité dans \(\text{L}^3({\mathbb{R}^3})\) et d’autres espaces limites pour Navier-Stokes, Revista Mat. Iberoamericana, 16, 605-667 (2000) · Zbl 0970.35101
[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. · Zbl 1361.35130
[16] I. Kukavica, On local uniqueness of solutions of the Navier-Stokes equations with bounded initial data, J. Diff. Eq., 194, 39-50 (2003) · Zbl 1050.35070
[17] I. Kukavica; V. Vicol, On local uniqueness of weak solutions to the Navier-Stokes system with \(\begin{document} {\rm BMO}^{-1} \end{document}\) initial datum, J. Dynam. Differential Equation, 20, 719-732 (2008) · Zbl 1148.35064
[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, 1133-1138 (1999) · Zbl 0930.35125
[19] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, Chapman & Hall/CRC, 2016. · Zbl 1342.76029
[20] J. Leray, Essai sur le mouvement d’un fluide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05
[21] M. I. Vishik; A. V. Fursikov, Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes, Ann. Scuola Norm. Sup. Pisa, série IV, 4, 531-576 (1977) · Zbl 0373.76055
[22] M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Dordrecht: Kluwer Academic Publishers, 1988.
[23] J. Wolf, On the local pressure of the Navier-Stokes equations and related systems, Adv. Differential Equations, 22, 305-338 (2017) · Zbl 1457.76056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.