Gigli, Nicola; Mosconi, Sunra J. N. A variational approach to the Navier-Stokes equations. (English. French) Zbl 1242.49063 Bull. Sci. Math. 136, No. 3, 256-276 (2012). Summary: We propose a time discretization of the Navier–Stokes equations inspired by the theory of gradient flows. This discretization produces Leray/Hopf solutions in any dimension and suitable solutions in dimension 3. We also show that in dimension 3 and for initial data in \(H^{1}\), the scheme converges to strong solutions in some interval [0,T) and, if the data satisfy the classical smallness condition, it produces the smooth solution in [\(0,\infty \)). Cited in 4 Documents MSC: 49M25 Discrete approximations in optimal control 35Q30 Navier-Stokes equations 76Y05 Quantum hydrodynamics and relativistic hydrodynamics Keywords:Navier–Stokes equations; gradient flows; discretization; Leray/Hopf solutions; smallness condition PDF BibTeX XML Cite \textit{N. Gigli} and \textit{S. J. N. Mosconi}, Bull. Sci. Math. 136, No. 3, 256--276 (2012; Zbl 1242.49063) Full Text: DOI References: [1] Blunck, S., Maximal regularity of discrete and continuous time evolution equations, Studia math., 146, 157-176, (2001) · Zbl 0981.39009 [2] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. pure appl. math., 35, 771-831, (1982) · Zbl 0509.35067 [3] DiPerna, R.J.; Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. math., 98, 511-547, (1989) · Zbl 0696.34049 [4] Galdi, G.P., An introduction to the Navier-Stokes initial boundary value problem, () · Zbl 1108.35133 [5] Giga, Y., The Stokes operator in \(L_r\) spaces, Proc. Japan acad., 57, 85-89, (1981) · Zbl 0471.35069 [6] Kunstmann, P.C.; Weis, L., Maximal \(L_p\) regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty\)-functional calculus, () · Zbl 1097.47041 [7] Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. math., 38, 3, 309-332, (1981/1982) · Zbl 0505.76100 [8] Sohr, H.; von Wahl, W., On the regularity of the pressure of weak solutions of Navier-Stokes equations, Arch. math., 46, 428-439, (1986) · Zbl 0574.35070 [9] Solonnikov, V.A., Estimates for solutions of nonstationary Navier-Stokes equations, J. soviet math., 8, 467-529, (1977) · Zbl 0404.35081 [10] Temam, R., Navier-Stokes equations. theory and numerical analysis, (1977), North-Holland Publishing Company · Zbl 0383.35057 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.