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Regularity criteria for the Navier-Stokes-Landau-Lifshitz system. (English) Zbl 1180.35406

Summary: We study regularity criteria for the Navier-Stokes-Landau-Lifshitz system. Using delicate estimates, the regularity criteria for smooth solution of Navier-Stokes-Landau-Lifshitz system in Besov spaces and the multiplier spaces are obtained. The Navier-Stokes-Landau-Lifshitz system is coupled system of the Navier-Stokes equation and Landau-Lifshitz system, our results generalize the related results for Navier-Stokes equation and Landau-Lifshitz system to our system.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B35 Stability in context of PDEs
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[1] Kim, H., A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37, 1417-1434 (2006) · Zbl 1141.35432
[2] Kozono, H.; Ogawa, T.; Taniuchi, Y., The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242, 251-278 (2002) · Zbl 1055.35087
[3] Ogawa, T., Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34, 6, 1318-1330 (2003) · Zbl 1036.35082
[4] Maz’ya, V. G., On the theory of the \(n\)-dimensional Schrödinger operator, Izv. Akad. Nauk SSSR (Ser. Mat.), 28, 1145-1172 (1964) · Zbl 0148.35602
[5] Lemarié-Rieusset, P. G., Recent Developments in the Navier-Stokes Problem (2002), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 1034.35093
[6] Maz’ya, V. G.; Shaposhnikova, T. O., Theory of Multipliers in Spaces of Differentiable Functions, Monogr. Stud. Math., vol. 23 (1985), Pitman: Pitman Boston, MA · Zbl 0645.46031
[7] Lin, F. H.; Liu, C., Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14, 289-330 (2001) · Zbl 1433.82014
[8] Triebel, H., Theory of Function Spaces II (1992), Birkhäuser Basel · Zbl 0778.46022
[9] Machihara, S.; Ozawa, T., Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc., 131, 5, 1553-1556 (2002) · Zbl 1022.46018
[10] Meyer, Y., Oscillating Patterns in Some Nonlinear Evolution Equations, Lecture Notes in Math., vol. 1871 (2006), pp. 101-187 · Zbl 1358.35096
[11] Temam, R., Navier-Stokes Equations (1977), North-Holland: North-Holland Amsterdam · Zbl 0335.35077
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