Global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity |
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Authors: | Hammadi Abidi Ping Zhang |
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Institution: | 1.Département de Mathématiques,Faculté des Sciences de Tunis,Tunis,Tunisia;2.Academy of Mathematics and Systems Science and HUA Loo-Keng Key Laboratory of Mathematics,Chinese Academy of Sciences,Beijing,China |
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Abstract: | Given initial data (ρ 0, u 0) satisfying 0 < m ? ρ 0 ? M, \(\rho _0 - 1 \in L^2 \cap \dot W^{1,r} (R^3 )\) and \(u_0 \in \dot H^{ - 2\delta } \cap H^1 (\mathbb{R}^3 )\) for δ ∈]1/4, 1/2 and r ∈]6, 3/1 ? 2δ, we prove that: there exists a small positive constant ? 1, which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution (ρ, u) whenever \(\left\| {u_0 } \right\|_{L^2 } \left\| {\nabla u_0 } \right\|_{L^2 } \) and \(\left\| {\mu (\rho _0 ) - 1} \right\|_{L^\infty } \leqslant \varepsilon _0 \) for some uniform small constant ? 0. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution. |
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