Rescalings at possible singularities of Navier–Stokes equations in half-space

Seregin, G.; Šverák, V.

doi:10.1090/S1061-0022-2014-01317-9

Rescalings at possible singularities of Navier–Stokes equations in half-space
HTML articles powered by AMS MathViewer

by G. Seregin and V. Šverák

St. Petersburg Math. J. 25 (2014), 815-833

DOI: https://doi.org/10.1090/S1061-0022-2014-01317-9

Published electronically: July 18, 2014

PDF | Request permission

Abstract:

The relationship is clarified between a possible blow-up for strong solutions of the initial boundary value problem for the incompressible Navier–Stokes equations in $\{x_3>0\}$, and the Liouville theorem for mild bounded ancient solutions.

References

L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. MR 673830, DOI 10.1002/cpa.3160350604
John G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), no. 5, 639–681. MR 589434, DOI 10.1512/iumj.1980.29.29048
Hao Jia, Gregory Seregin, and Vladimír Šverák, Liouville theorems in unbounded domains for the time-dependent Stokes system, J. Math. Phys. 53 (2012), no. 11, 115604, 9. MR 3026549, DOI 10.1063/1.4738636
H. Jia, G. Seregin, and V. Sverak, A Liouville theorem for the Stokes system in half-space, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 410 (2013), no. Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 43, 25–35, 187 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 195 (2013), no. 1, 13–19. MR 3048260, DOI 10.1007/s10958-013-1561-9
A. A. Kiselev and O. A. Ladyženskaya, On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid, Izv. Akad. Nauk SSSR Ser. Mat. 21 (1957), 655–680 (Russian). MR 0100448
Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math. 203 (2009), no. 1, 83–105. MR 2545826, DOI 10.1007/s11511-009-0039-6
H. Koch and V. A. Solonnikov, $L_p$-estimates for a solution to the nonstationary Stokes equations, J. Math. Sci. (New York) 106 (2001), no. 3, 3042–3072. Function theory and phase transitions. MR 1906033, DOI 10.1023/A:1011375706754
O. A. Ladyzhenskaya, Mathematical problems of the dynamics of viscous incompressible fluids, 2nd ed., Nauka, Moscow, 1970. (Russian)
Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech. 4 (2002), no. 1, 1–29. MR 1891072, DOI 10.1007/s00021-002-8533-z
G. A. Seregin, A note on local boundary regularity for the Stokes system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370 (2009), no. Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 40, 151–159, 221–222 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 166 (2010), no. 1, 86–90. MR 2749216, DOI 10.1007/s10958-010-9847-7
G. Seregin and V. Šverák, On type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations, Comm. Partial Differential Equations 34 (2009), no. 1-3, 171–201. MR 2512858, DOI 10.1080/03605300802683687
V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153–231 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 7. MR 0415097
V. A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, J. Math. Sci. (N.Y.) 114 (2003), no. 5, 1726–1740. Function theory and applications. MR 1981302, DOI 10.1023/A:1022317029111
V. A. Solonnikov, Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 123–156 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 2, 331–365. MR 1992567, DOI 10.1070/RM2003v058n02ABEH000613
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192