Higher derivatives estimate for the 3D Navier–Stokes equation
Vasseur, Alexis
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 1189-1204 / Harvested from Numdam
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In this article, a nonlinear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier–Stokes equation) to a critical space (invariant through the canonical scaling of the Navier–Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier–Stokes equations. Those estimates are uniform, up to the possible blowing-up time. The proof uses blow-up techniques. Estimates can be obtained by this means thanks to the galilean invariance of the transport part of the equation.

Publié le : 2010-01-01
DOI : https://doi.org/10.1016/j.anihpc.2010.05.002
Classification:  76D05,  35Q30 
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     author = {Vasseur, Alexis},
     title = {Higher derivatives estimate for the 3D Navier--Stokes equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {1189-1204},
     doi = {10.1016/j.anihpc.2010.05.002},
     zbl = {05800075},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_5_1189_0}
}

Vasseur, Alexis. Higher derivatives estimate for the 3D Navier–Stokes equation. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 1189-1204. doi : 10.1016/j.anihpc.2010.05.002. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_5_1189_0/

[1] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften vol. 223, Springer-Verlag, Berlin (1976) | Zbl 0344.46071

[2] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 no. 6 (1982), 771-831 | Zbl 0509.35067

[3] R. Coifman, P.-L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 no. 3 (1993), 247-286 | Zbl 0864.42009

[4] P. Constantin, Navier–Stokes equations and area of interfaces, Comm. Math. Phys. 129 no. 2 (1990), 241-266 | Zbl 0725.35080

[5] E.B. Fabes, B.F. Jones, N.M. Rivière, The initial value problem for the Navier–Stokes equations with data in L p , Arch. Rational Mech. Anal. 45 (1972), 222-240 | Zbl 0254.35097

[6] C. Foiaş, C. Guillopé, R. Temam, New a priori estimates for Navier–Stokes equations in dimension 3, Comm. Partial Differential Equations 6 no. 3 (1981), 329-359 | Zbl 0472.35070

[7] Y. Giga, O. Sawada, On regularizing-decay rate estimates for solutions to the Navier–Stokes initial value problem, preprint.

[8] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231 | Zbl 0042.10604

[9] H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 no. 1 (2001), 22-35 | Zbl 0972.35084

[10] P.G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics vol. 431, Chapman & Hall/CRC, Boca Raton, FL (2002) | Zbl 1034.35093

[11] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math. 63 (1934), 183-248

[12] F. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Comm. Pure Appl. Math. 51 no. 3 (1998), 241-257 | Zbl 0958.35102

[13] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications vol. 3, The Clarendon Press, Oxford University Press, New York (1996) | Zbl 0866.76002

[14] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pacific J. Math. 66 no. 2 (1976), 535-552 | Zbl 0325.35064

[15] V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Comm. Math. Phys. 55 no. 2 (1977), 97-112 | Zbl 0357.35071

[16] V. Scheffer, The Navier–Stokes equations in space dimension four, Comm. Math. Phys. 61 no. 1 (1978), 41-68 | Zbl 0403.35088

[17] V. Scheffer, The Navier–Stokes equations on a bounded domain, Comm. Math. Phys. 73 no. 1 (1980), 1-42 | Zbl 0451.35048

[18] J. Serrin, The initial value problem for the Navier–Stokes equations, Nonlinear Problems, Proc. Sympos. Madison Wis., Univ. of Wisconsin Press, Madison, WI (1963), 69-98 | Zbl 0115.08502

[19] V.A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type, Trudy Mat. Inst. Steklov. 70 (1964), 133-212 | Zbl 0168.08202

[20] M. Struwe, On partial regularity results for the Navier–Stokes equations, Comm. Pure Appl. Math. 41 no. 4 (1988), 437-458 | Zbl 0632.76034

[21] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI (2001) | Zbl 0981.35001

[22] A. Vasseur, A new proof of partial regularity of solutions to Navier–Stokes equations, NoDEA Nonlinear Differential Equations Appl. 14 no. 5–6 (2007), 753-785 | Zbl 1142.35066