In this paper, we prove global well-posedness for compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium. This result allows us to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms.
We show a new Bernstein's inequality which generalizes the results of CannonePlanchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and supercritical dissipation for the small initial data in the critical Besov space, and local wellposedness for the large initial data. (2000): 76U05, 76B03, 35Q35
Mathematics Subject Classification
Abstract. We improve and extend some known regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony's para-product decomposition.
Abstract. We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, i. e. sup j∈Zwhere ∆ j is a frequency localization on |ξ| ≈ 2 j .
We prove the global well-posedness for the 3-D micropolar fluid system in the critical Besov spaces by making a suitable transformation to the solutions and using the Fourier localization method, especially combined with a new L p estimate for the Green matrix to the linear system of the transformed equation. This result allows to construct global solutions for a class of highly oscillating initial data of Cannone's type. Meanwhile, we analyze the long behavior of the solutions and get some decay estimates.
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