Local boundary regularity for the Navier–Stokes equations in nonendpoint borderline Lorentz spaces

T. Barker

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 45, Tome 444 (2016), pp. 15-46

Résumé

We prove local regularity up to the flat part of the boundary, for certain classes of distributional solutions that are $L_\infty L^{3,q}$ with $q$ finite. The corresponding result, for the interior case, was proven recently by Wang and Zhang, see also work by Phuc. For local regularity, up to the flat part of the boundary, $q=3$ was established by G. A. Seregin. Our result can be viewed as an extension of this to $L^{3,q}$ with $q$ finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new $\epsilon$-regularity criterion are central themes in providing this extension.

Bibliographie

[1] J. Bergh, J. Löfström, Jrgen Interpolation spaces, An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin–New York, 1976 | DOI | MR

[2] L. Caffarelli, R.-V. Kohn, L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier–Stokes equations”, Comm. Pure Appl. Math., 35 (1982), 771–831 | DOI | MR | Zbl

[3] Russian Math. Surveys, 58:2 (2003), 211–250 | DOI | DOI | MR | Zbl

[4] Giusti Enrico, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003 | MR | Zbl

[5] Hopf Eberhard, “Über die Anfangswertaufabe für die hydrodynamischen Grundgleichungen”, Math. Nachr., 4 (1951), 213–231 (German) | DOI | MR | Zbl

[6] J. Serrin, “On the interior regularity of weak solutions of the Navier–Stokes equations”, Arch. Rational Mech. Anal., 9 (1962), 187–195 | DOI | MR | Zbl

[7] Kim Hyunseok, Kozono Hideo, “Interior regularity criteria in weak spaces for the Navier–Stokes equations”, Manuscripta Math., 115:1 (2004), 85–100 | DOI | MR | Zbl

[8] A. A. Kiselev, O. A. Ladyzhenskaya, “On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid”, Izv. Akad. Nauk SSSR. Ser. Mat., 21:5 (1957), 655–680 | MR | Zbl

[9] Kozono Hideo, “Removable singularities of weak solutions to the Navier–Stokes equations”, Comm. Partial Differential Equations, 23:5–6 (1998), 949–966 | DOI | MR | Zbl

[10] O. A. Ladyzhenskaya, “Uniqueness and smoothness of generalized solutions of Navier–Stokes equations”, Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5, 1967, 169–185 | MR | Zbl

[11] O. A. Ladyzhenskaya, Mathematical problems of the dynamics of viscous incompressible fluids, 2nd edition, Nauka, Moscow, 1970

[12] O. A. Ladyzhenskaya, G. A. Seregin, “On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations”, J. Math. Fluid Mech., 1:4 (1999), 356–387 | DOI | MR | Zbl

[13] American Math. Soc., Providence, 1968 | Zbl

[14] J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace”, Acta Math., 63 (1934), 193–248 | DOI | MR | Zbl

[15] Lin Fanghua, “A new proof of the Caffarelli–Kohn–Nirenberg theorem”, Comm. Pure Appl. Math., 51:3 (1998), 241–257 | 3.0.CO;2-A class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[16] Y. Luo, T. P. Tsai, Regularity criteria in weak $L^3$ for 3D incompressible Navier–Stokes equations, April 2014, arXiv: 1310.8307

[17] M. Struwe, “On partial regularity results for the Navier–Stokes equations”, Comm. Pure Appl. Math., 41 (1988), 437–458 | DOI | MR | Zbl

[18] J. Math. Sci. (N.Y.), 143:2 (2007), 2924–2935 | DOI | MR | Zbl

[19] N. C. Phuc, The Navier–Stokes equations in nonendpoint borderline Lorentz spaces, July 2014, arXiv: 1407.5129 | MR

[20] V. Scheffer, “Hausdorff measure and the Navier–Stokes equations”, Comm. Math. Phys., 55:2 (1977), 97–112 | DOI | MR | Zbl

[21] G. A. Seregin, “On the number of singular points of weak solutions to the Navier–Stokes equations”, Comm. Pure Appl. Math., 54:8 (2001), 1019–1028 | DOI | MR | Zbl

[22] G. Seregin, V. Sverak, “Navier–Stokes equations with lower bounds on the pressure”, Arch. Ration. Mech. Anal., 163:1 (2002), 65–86 | DOI | MR | Zbl

[23] G. Seregin, “Local regularity of suitable weak solutions to the Navier–Stokes equations near the boundary”, J. math. fluid mech., 4:1 (2002), 1–29 | DOI | MR | Zbl

[24] G. A. Seregin, “On smoothness of $L_{3,\infty}$-solutions to the Navier–Stokes equations up to boundary”, Mathematische Annalen, 332 (2005), 219–238 | DOI | MR | Zbl

[25] G. A. Seregin, “A note on local boundary regularity for the Stokes system”, Zapiski Nauchn. Seminar. POMI, 370, 2009, 151–159 | MR

[26] G. A. Seregin, “A certain necessary condition of potential blow up for Navier–Stokes equations”, Comm. Math. Phys., 312:3 (2012), 833–845 | DOI | MR | Zbl

[27] W. Wang, Z. Zhang, “On the interior regularity criteria and the number of singular points to the Navier–Stokes equations”, J. Anal. Math., 123 (2014), 139–170 | DOI | MR | Zbl