New version of the Ladyzhenskaya--Prodi--Serrin condition

G. A. Seregin

Algebra i analiz, Tome 18 (2006) no. 1, pp. 124-143.

Résumé

A new local version of the Ladyzhenskaya–Prodi–Serrin regularity condition for weak solutions of the nonstationary 3-dimensional Navier-Stokes system is proved. The novelty is in that the energy of the solution is not assumed to be finite.

Bibliographie

[1] Serrin J., “The initial value problem for the Navier–Stokes equations”, 1963, Nonlinear Problems, ed. R. Langer, Univ. Wisconsin Press, Madison, 1963, 69–98 | MR

[2] Ladyzhenskaya O. A., “O edinstvennosti i gladkosti obobschennykh reshenii uravnenii Nave–Stoksa”, Zap. nauch. semin. LOMI, 5, 1967, 169–185 | MR | Zbl

[3] Giga Y., “Solutions for semilinear parabolic equations in $L_p$ and regularity of weak solutions of the Navier–Stokes system”, J. Differential Equations, 62 (1986), 186–212 | DOI | MR

[4] Sohr H., “A regularity class for the Navier–Stokes equations in Lorentz spaces”, J. Evol. Equ., 1 (2001), 441–467 | DOI | MR | Zbl

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

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

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

[8] Scheffer V., “Partial regularity of solutions to the Navier–Stokes equations”, Pacific J. Math., 66 (1976), 535–552 | MR | Zbl

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

[10] Scheffer V., “The Navier–Stokes equations in a bounded domain”, Comm. Math. Phys., 73 (1980), 1–42 | DOI | MR | Zbl

[11] Scheffer V., “Boundary regularity for the Navier–Stokes equations”, Comm. Math. Phys., 85 (1982), 275–299 | DOI | MR | Zbl

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

[13] Seregin G. A., “Differentsialnye svoistva slabykh reshenii uravnenii Nave–Stoksa”, Algebra i analiz, 14:1 (2002), 194–237 | MR | Zbl

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

[15] Takahashi S., “On interior regularity criteria for weak solutions of the Navier–Stokes equations”, Manuscripta Math., 69 (1990), 237–254 | DOI | MR | Zbl

[16] Chen Z.-M., Price W. G., “Blow-up rate estimates for weak solutions of the Navier–Stokes equations”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2625–2642 | DOI | MR | Zbl

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

[18] Escauriaza L., Seregin G., Šverák V., “On backward uniqueness for parabolic equations”, Zap. nauch. semin. POMI, 288, 2002, 100–103 | MR | Zbl

[19] Escauriaza L., Seregin G., Šverák V., “Backward uniqueness for parabolic equations”, Arch. Rational Mech. Anal., 169 (2003), 147–157 | DOI | MR | Zbl

[20] Escauriaza L., Seregin G., Šverák V., “Backward uniqueness for the heat operator in half-space”, Algebra i analiz, 15:1 (2003), 201–214 | MR | Zbl

[21] Iskauriaza L., Seregin G. A., Shverak V., “$L_{3,\infty}$-resheniya uravnenii Nave–Stoksa i obratnaya edinstvennost”, Uspekhi mat. nauk, 58:2 (2003), 3–44 | MR | Zbl

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

[23] Seregin G., “Some estimates near the boundary for solutions to the non-stationary linearized Navier–Stokes equations”, Zap. nauch. semin. POMI, 271, 2000, 204–223 | MR | Zbl

[24] Solonnikov V. A., “Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm”, Zap. nauch. semin. POMI, 288, 2002, 204–231 | MR | Zbl

[25] Takahashi S., “On a regularity criterion up to the boundary for weak solutions of the Navier–Stokes equations”, Comm. Partial Differential Equations, 17 (1992), 261–285 | MR | Zbl

[26] Takahashi S., “Erratum to: «On a regularity criterion up to the boundary for weak solutions of the Navier–Stokes equations»”, Comm. Partial Differential Equations, 19 (1994), 1015–1017 | MR | Zbl

[27] Choe H. J., “Boundary regularity of weak solutions of the Navier–Stokes equations”, J. Differential Equations, 149 (1998), 211–247 | DOI | MR | Zbl

[28] Kang K., “On boundary regularity of the Navier–Stokes equations”, Comm. Partial Differential Equations, 29 (2004), 955–987 | DOI | MR | Zbl

[29] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[30] Bugrov Ya. S., “Funktsionalnye prostranstva so smeshannoi normoi”, Izv. AN SSSR. Ser. mat., 35:5 (1971), 1137–1158 | MR | Zbl

[31] Giga Y., Sohr H., “Abstract $L^p$-estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains”, J. Funct. Anal., 102 (1991), 72–94 | DOI | MR | Zbl

[32] Maremonti P., Solonnikov V. A., “Ob otsenkakh reshenii nestatsionarnoi zadachi Stoksa v anizotropnykh prostranstvakh S. L. Soboleva so smeshannoi normoi”, Zap. nauch. semin. POMI, 222, 1995, 124–150 | MR | Zbl