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), p. 899-945 / Harvested from Numdam

Résumé

We study weak solutions of the 3D Navier–Stokes equations with $ {L}^{2}$ initial data. We prove that $ {\mathrm{\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 $ {\mathrm{\nabla }}^{2}u$ was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$ {L}_{\mathit{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.

Détail

Publié le : 2014-01-01

MR 3258360 | Zbl 1297.76047

DOI : https://doi.org/10.1016/j.anihpc.2013.08.001
Classification:  76D05,  35Q30

Bibliographie

[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 763762 | Zbl 0573.76029

[2] H. Beirão Da Veiga, A new regularity class for the Navier–Stokes equations in $ {\mathbf{R}}^{n}$ , Chin. Ann. Math., Ser. B 16 no. 4 (1995), 407 -412 , Chin. Ann. Math., Ser. A 16 no. 6 (1995), 797 | MR 1380578 | Zbl 0837.35111

[3] Jöran Bergh, Jörgen Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. vol. 223 , Springer-Verlag, Berlin (1976) | MR 482275 | Zbl 0344.46071

[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 1920038 | Zbl 1075.35031

[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 2805865 | Zbl 1270.35334

[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 673830 | Zbl 0509.35067

[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 2680400 | Zbl 1204.35063

[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 2354490 | Zbl 1130.35101

[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 2735101 | Zbl 1206.35192

[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 2564471 | Zbl 1186.35137

[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 1225511 | Zbl 0864.42009

[12] Peter Constantin, Navier–Stokes equations and area of interfaces, Commun. Math. Phys. 129 no. 2 (1990), 241 -266 | MR 1048693 | Zbl 0725.35080

[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 93649 | Zbl 0084.31901

[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 2329262 | Zbl 1128.35081

[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 1049156 | Zbl 0728.35083

[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 1992563 | Zbl 1064.35134

[17] C. Fefferman, E.M. Stein, $ {H}^{p}$ spaces of several variables, Acta Math. 129 no. 3–4 (1972), 137 -193 | MR 447953 | Zbl 0257.46078

[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 607552 | Zbl 0472.35070

[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 2060233 | Zbl 1080.35072

[20] Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ (2004) | MR 2449250 | Zbl 1148.42001

[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 2308753 | Zbl 1126.35042

[22] Eberhard Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213 -231 | MR 50423 | Zbl 0042.10604

[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 2557452 | Zbl 1181.35173

[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 | MR 236541 | Zbl 0194.12805

[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 241822

[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 1938147 | Zbl 1034.35093

[27] Jean Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 no. 1 (1934), 193 -248 | MR 1555394

[28] Fanghua Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Commun. Pure Appl. Math. 51 no. 3 (1998), 241 -257 | MR 1488514 | Zbl 0958.35102

[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 0866.76002

[30] Giovanni Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. (4) 48 (1959), 173 -182 | MR 126088 | Zbl 0148.08202

[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 2312388 | Zbl 1120.35075

[32] Laurent Saloff-Coste, Aspects of Sobolev-Type Inequalities, Lond. Math. Soc. Lect. Note Ser. vol. 289 , Cambridge University Press, Cambridge (2002) | MR 1872526 | Zbl 0991.35002

[33] Vladimir Scheffer, The Navier–Stokes equations on a bounded domain, Commun. Math. Phys. 73 no. 1 (1980), 1 -42 | MR 573611 | Zbl 0451.35048

[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 1396285 | Zbl 0862.35086

[35] G.A. Seregin, A new version of the Ladyzhenskaya–Prodi–Serrin condition, Algebra Anal. 18 no. 1 (2006), 124 -143 | MR 2225215

[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 150444

[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 2270163 | Zbl 1141.49035

[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 162065 | Zbl 0168.08202

[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 1232192 | Zbl 0821.42001

[40] Michael Struwe, On partial regularity results for the Navier–Stokes equations, Commun. Pure Appl. Math. 41 no. 4 (1988), 437 -458 | MR 933230 | Zbl 0632.76034

[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 2683756 | Zbl 05800075

[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 2374209 | Zbl 1142.35066

[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 | MR 2548880 | Zbl 1154.35426