Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations

Choi, Kyudong; Vasseur, Alexis F.

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 899-945.

We study weak solutions of the 3D Navier–Stokes equations with $L^2$ initial data. We prove that ${\nabla }^{\alpha }u$ is locally integrable in space–time for any real α such that $1<\alpha <3$. Up to now, only the second derivative ${\nabla }^{2}u$ was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$L_{\mathrm{𝑙𝑜𝑐}}^{4/(\alpha +1)}$. These estimates depend only on the $L^2$-norm of the initial data and on the domain of integration. Moreover, they are valid even for $\alpha ⩾3$ as long as u is smooth. The proof uses a standard approximation of Navier–Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced.

Classification : 76D05, 35Q30
Mots clés : Navier–Stokes equations, Fluid mechanics, Blow-up techniques, Weak solutions, Higher derivatives, Fractional derivatives

[1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94 no. 1 (1984), 61 -66 | MR | Zbl

[2] H. Beirão Da Veiga, A new regularity class for the Navier–Stokes equations in ${𝐑}^n$ , Chin. Ann. Math., Ser. B 16 no. 4 (1995), 407 -412 , Chin. Ann. Math., Ser. A 16 no. 6 (1995), 797 | MR | Zbl

[3] Jöran Bergh, Jörgen Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. vol. 223 , Springer-Verlag, Berlin (1976) | MR | Zbl

[4] Luigi C. Berselli, Giovanni P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations, Proc. Am. Math. Soc. 130 no. 12 (2002), 3585 -3595 | MR | Zbl

[5] Clayton Bjorland, Alexis Vasseur, Weak in space, log in time improvement of the Ladyženskaja–Prodi–Serrin criteria, J. Math. Fluid Mech. 13 no. 2 (2011), 259 -269 | MR | Zbl

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

[7] Luis A. Caffarelli, Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math. 171 no. 3 (2010), 1903 -1930 | MR | Zbl

[8] Dongho Chae, Kyungkeun Kang, Jihoon Lee, On the interior regularity of suitable weak solutions to the Navier–Stokes equations, Commun. Partial Differ. Equ. 32 no. 7–9 (2007), 1189 -1207 | MR | Zbl

[9] Chi Hin Chan, Smoothness criterion for Navier–Stokes equations in terms of regularity along the streamlines, Methods Appl. Anal. 17 no. 1 (2010), 81 -103 | MR | Zbl

[10] A. Cheskidov, R. Shvydkoy, The regularity of weak solutions of the 3D Navier–Stokes equations in $B_{\infty ,\infty }^{-1}$ , Arch. Ration. Mech. Anal. 195 (2010), 159 -169 | MR | Zbl

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

[12] Peter Constantin, Navier–Stokes equations and area of interfaces, Commun. Math. Phys. 129 no. 2 (1990), 241 -266 | MR | Zbl

[13] Ennio De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 3 (1957), 25 -43 | MR | Zbl

[14] Hongjie Dong, Dapeng Du, On the local smoothness of solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 9 no. 2 (2007), 139 -152 | MR | Zbl

[15] G.F.D. Duff, Derivative estimates for the Navier–Stokes equations in a three-dimensional region, Acta Math. 164 no. 3–4 (1990), 145 -210 | MR | Zbl

[16] L. Escauriaza, G.A. Seregin, V. Šverák, $L_{3,\infty }$-solutions of Navier–Stokes equations and backward uniqueness, Usp. Mat. Nauk 58 no. 2(350) (2003), 3 -44 | MR | Zbl

[17] C. Fefferman, E.M. Stein, $H^p$ spaces of several variables, Acta Math. 129 no. 3–4 (1972), 137 -193 | MR | Zbl

[18] Ciprian Foiaş, Colette Guillopé, Roger Temam, New a priori estimates for Navier–Stokes equations in dimension 3, Commun. Partial Differ. Equ. 6 no. 3 (1981), 329 -359 | MR | Zbl

[19] Yoshikazu Giga, Okihiro Sawada, On regularizing-decay rate estimates for solutions to the Navier–Stokes initial value problem, Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th birthday. vols. 1, 2, Kluwer Acad. Publ., Dordrecht (2003), 549 -562 | MR | Zbl

[20] Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ (2004) | MR | Zbl

[21] Stephen Gustafson, Kyungkeun Kang, Tai-Peng Tsai, Interior regularity criteria for suitable weak solutions of the Navier–Stokes equations, Commun. Math. Phys. 273 no. 1 (2007), 161 -176 | MR | Zbl

[22] Eberhard Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213 -231 | MR | Zbl

[23] Igor Kukavica, The fractal dimension of the singular set for solutions of the Navier–Stokes system, Nonlinearity 22 no. 12 (2009), 2889 -2900 | MR | Zbl

[24] O.A. Ladyženskaja, Uniqueness and smoothness of generalized solutions of Navier–Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 169 -185 | EuDML | MR | Zbl

[25] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. vol. 23 , American Mathematical Society, Providence, RI (1967) | MR

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

[27] Jean Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 no. 1 (1934), 193 -248 | MR

[28] Fanghua Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Commun. Pure Appl. Math. 51 no. 3 (1998), 241 -257 | MR | Zbl

[29] Pierre-Louis Lions, Mathematical topics in fluid mechanics, vol. 1, Incompressible Models, Oxford Sci. Publ., Oxford Lecture Ser. Math. Appl. vol. 3 , The Clarendon Press Oxford University Press, New York (1996) | Zbl

[30] Giovanni Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. (4) 48 (1959), 173 -182 | MR | Zbl

[31] James C. Robinson, Witold Sadowski, Decay of weak solutions and the singular set of the three-dimensional Navier–Stokes equations, Nonlinearity 20 no. 5 (2007), 1185 -1191 | MR | Zbl

[32] Laurent Saloff-Coste, Aspects of Sobolev-Type Inequalities, Lond. Math. Soc. Lect. Note Ser. vol. 289 , Cambridge University Press, Cambridge (2002) | MR | Zbl

[33] Vladimir Scheffer, The Navier–Stokes equations on a bounded domain, Commun. Math. Phys. 73 no. 1 (1980), 1 -42 | MR | Zbl

[34] Maria E. Schonbek, Michael Wiegner, On the decay of higher-order norms of the solutions of Navier–Stokes equations, Proc. R. Soc. Edinb., Sect. A 126 no. 3 (1996), 677 -685 | MR | Zbl

[35] G.A. Seregin, A new version of the Ladyzhenskaya–Prodi–Serrin condition, Algebra Anal. 18 no. 1 (2006), 124 -143 | MR

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

[37] Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math. 60 no. 1 (2007), 67 -112 | MR | Zbl

[38] V.A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type, Tr. Mat. Inst. Steklova 70 (1964), 133 -212 | MR | Zbl

[39] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Monographs in Harmonic Analysis, III, Princeton Math. Ser. vol. 43 , Princeton University Press, Princeton, NJ (1993) | MR | Zbl

[40] Michael Struwe, On partial regularity results for the Navier–Stokes equations, Commun. Pure Appl. Math. 41 no. 4 (1988), 437 -458 | MR | Zbl

[41] Alexis Vasseur, Higher derivatives estimate for the 3D Navier–Stokes equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27 no. 5 (2010), 1189 -1204 | Numdam | MR | Zbl

[42] Alexis F. Vasseur, A new proof of partial regularity of solutions to Navier–Stokes equations, Nonlinear Differ. Equ. Appl. 14 no. 5–6 (2007), 753 -785 | MR | Zbl

[43] Jörg Wolf, A direct proof of the Caffarelli–Kohn–Nirenberg theorem, Parabolic and Navier–Stokes Equations. Part 2, Banach Cent. Publ. vol. 81 , Polish Acad. Sci. Inst. Math, Warsaw (2008), 533 -552 | EuDML | MR | Zbl