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Asymptotic stability for the 3D Navier-Stokes equations. (English) Zbl 1142.35548
Summary: We consider the 3D Navier-Stokes equations in \(\Omega\subset \mathbb{R}^3\), not necessarily bounded. We prove the asymptotic stability for weak solutions in the class \(\nabla u \in L^{\alpha}(0, \infty;L_{\gamma}(\Omega))\) for \(2/\alpha+3/\gamma=2\) with arbitrary initial and external perturbations.

MSC:
35Q30 Navier-Stokes equations
35B35 Stability in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76D55 Flow control and optimization for incompressible viscous fluids
93D20 Asymptotic stability in control theory
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References:
[1] Beirão da Veiga H., Chin. Ann. Math. 16 pp 407– (1995)
[2] DOI: 10.1007/BF00279962 · Zbl 0678.35076
[3] DOI: 10.1007/BF00387899 · Zbl 0756.76018
[4] DOI: 10.1002/cpa.3160350604 · Zbl 0509.35067
[5] Constantin P., Navier–Stokes Equations (1988) · Zbl 0687.35071
[6] Hopf E., Math. Nach. 4 pp 213– (1951) · Zbl 0042.10604
[7] Kawanago T., Electron. J. Diff. Eqs. 15 pp 23– (1998)
[8] DOI: 10.1006/jfan.2000.3625 · Zbl 0970.35106
[9] DOI: 10.1007/BF02572306 · Zbl 0798.35127
[10] Leray J., J. Math. Pures. Appl. 12 pp 1– (1933)
[11] Lions J. L., Quelques méthodes de resolution des problémes aux limites non linéaires (1969)
[12] Lions P. L., Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models (1996) · Zbl 0866.76002
[13] DOI: 10.1081/PDE-100107819 · Zbl 1086.35077
[14] DOI: 10.2748/tmj/1178228767 · Zbl 0568.35077
[15] DOI: 10.1007/BF02102642 · Zbl 0795.35082
[16] Scheffer V., Pacific J. Math. 66 pp 535– (1976) · Zbl 0325.35064
[17] DOI: 10.1007/BF00253344 · Zbl 0106.18302
[18] Tanabe H., Equations of Evolution (1979) · Zbl 0417.35003
[19] Tian G., Comm. Anal. Geo. 7 pp 221– (1999)
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