Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Navier-Stokes equations; solution uniqueness; weak Leray-Hopf solution; multiplier space
Navier-Stokes equations; solution uniqueness; weak Leray-Hopf solution; multiplier space
Summary:
Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2( \Bbb R^d) $. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2(( 0,T) ;\dot X _1(\Bbb R^d)^d)$ where $\dot X_1(\Bbb R^d)$ is the multiplier space, then we have $u=v$.
References:
[1] Coifman, R., Lions, P. -L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72 (1993), 247-286. MR 1225511 | Zbl 0864.42009
[2] Constantin, P.: Remarks on the Navier-Stokes equations. In: New Perspectives in Turbulence Springer New York (1991), 229-261. MR 1126937
[3] Foias, C.: Une remarque sur l'unicité des solutions des équations de Navier-Stokes en dimension $n$. Bull. Soc. Math. Fr. 89 (1961), 1-8 French. MR 0141902 | Zbl 0107.07602
[4] Fefferman, C., Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. DOI 10.1007/BF02392215 | MR 0447953
[5] Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), German 213-231. DOI 10.1002/mana.3210040121 | MR 0050423
[6] Kato, T.: Strong $L^p$ solutions of the Navier-Stokes equation in Morrey spaces. Bol. Soc. Bras. Mat. 22 (1992), 127-155. DOI 10.1007/BF01232939 | MR 1179482
[7] Kozono, H., Sohr, H.: Remark on uniqueness of weak solutions to the Navier-Stokes equations. Analysis 16 (1996), 255-271. DOI 10.1524/anly.1996.16.3.255 | MR 1403221 | Zbl 0864.35082
[8] Kozono, H., Taniuchi, Y.: Bilinear estimates in $BMO$ and the Navier-Stokes equations. Math. Z. 235 (2000), 173-194. DOI 10.1007/s002090000130 | MR 1785078 | Zbl 0970.35099
[9] Lemarié-Rieusset, P. G.: Recent Developments in the Navier-Stokes Problem. Chapman & Hall/CRC Boca Raton (2002). MR 1938147 | Zbl 1034.35093
[10] Lemarié-Rieusset, P. G., Gala, S.: Multipliers between Sobolev spaces and fractional differentiation. J. Math. Anal. Appl. 322 (2006), 1030-1054. DOI 10.1016/j.jmaa.2005.07.043 | MR 2250634 | Zbl 1109.46039
[11] Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta. Math. 63 (1934), 193-248 French. DOI 10.1007/BF02547354 | MR 1555394
[12] Murat, F.: Compacité par compensation. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 5 (1978), 489-507 French. MR 0506997 | Zbl 0399.46022
[13] Masuda, K.: Weak solutions of Navier-Stokes equations. Tôhoku Math. J. 36 (1984), 623-646. DOI 10.2748/tmj/1178228767 | MR 0767409 | Zbl 0568.35077
[14] Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9 (1962), 187-195. DOI 10.1007/BF00253344 | MR 0136885 | Zbl 0106.18302
[15] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press Princeton (1993). MR 1232192 | Zbl 0821.42001
[16] Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton Mathematical series. Princeton University Press Princeton (1971). MR 0304972
[17] Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4, Edinburgh 1979. Res. Notes Math. 39 (1979), 136-212. MR 0584398
[18] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Amsterdam (1977). MR 0609732 | Zbl 0383.35057
[19] Taylor, M. E.: Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Commun. Partial Differ. Equations 17 (1992), 1407-1456. DOI 10.1080/03605309208820892 | MR 1187618 | Zbl 0771.35047
Search
Partner of
