REGULARITY CRITERION ON WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS

다국어 초록 (Multilingual Abstract)

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

참고문헌 (Reference)

1 K. Masuda, "Weak solutions of Navier-Stokes equations" 36 (36): 623-646, 1984

2 C. Foias, "Une remarque sur lunicite des solutions des ´equations de Navier-Stokes en dimension n" 89 : 1-8, 1961

3 E. Hopf, "Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen" 4 : 213-231, 1951

4 J. Leray, "Sur le mouvement d’un liquide visqueux emplissant l’espace" 63 (63): 193-248, 1934

5 T. Kato, "Strong solutions of the Navier-Stokes equation in Morrey spaces" 22 (22): 127-155, 1992

6 P. Constantin, "Remarks on the Navier-Stokes Equations, New perspectives in turbulence (Newport, RI, 1989)" Springer 229-261, 1991

7 H. Kozono, "Regularity criterion of weak solutions to the Navier-Stokes equations" 2 (2): 535-554, 1997

8 P. G. Lemarie-Rieusset, "Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics" Chapman & Hall/CRC 431-, 2002

9 J. Serrin, "On the interior regularity of weak solutions of the Navier-Stokes equations" 9 : 187-195, 1962

10 R. Temam, "Navier-Stokes Equations, Studies in Mathematics and its Applications, Vol. 2" North-Holland Publishing Co. 1977

11 P. G. Lemarie-Rieusset, "Multipliers between Sobolev spaces and fractional differentiation" 322 (322): 1030-1054, 2006

12 E. M. Stein, "Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32" Princeton University Press 1971

13 C. Fefferman, "Hp spaces of several variables" 129 (129): 137-193, 1972

14 E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III" Princeton University Press 1993

15 R. Coifman, "Compensated compactness and Hardy spaces" 72 (72): 247-286, 1993

16 L. Tartar, "Compensated Compactness and Applications to Partial Differential Equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV" 39 : 136-212, 1979

17 F. Murat, "Compacit´e par compensation" 5 (5): 489-507, 1978

18 H. Kozono, "Bilinear estimates in BMO and the Navier-Stokes equations" 235 (235): 173-194, 2000

19 M. E. Taylor, "Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations" 17 (17): 1407-1456, 1992

20 H. Beirao da Veiga, "A new regularity class for the Navier-Stokes equations in Rn" 16 (16): 407-412, 1995