Abstract. In this paper, we consider the global wellposedness of 2-D incompressible magnetohydrodynamical system with smooth initial data which is close to some non-trivial steady state. It is a coupled system between the Navier-Stokes equations and a free transport equation with an universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system (1.1) in the Lagrangian coordinates (2.19). Then we employ anisotropic Littlewood-Paley analysis to establish the key a priori L 1 (R + ; Lip(R 2 )) estimate for the Lagrangian velocity field Yt. With this estimate, we can prove the global wellposedness of (2.19) with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of (2.19) is crucial for the proofs to work. The global wellposedness of the original system (1.1) then follows by a suitable change of variables.

In this paper, we investigate the large-time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous Navier-Stokes equations. In particular, we prove that given any global smooth solution .a; u/ of (1.2), the velocity field u decays to 0 with an explicit rate, which coincides with the L 2 norm decay for the weak solutions of the 3D classical Navier-Stokes system [26,29] as t goes to 1. Moreover, a small perturbation to the initial data of .a; u/ still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution .a; u/ for t > 0. We should point out that the main results in this paper work for large solutions of (1.2).

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Let us consider an initial data v0 for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to L 3 2 ∩L 2 . We prove that if the solution associated with v0 blows up at a finite time T ⋆ , then for any p in ]4, ∞[, and any unit vector e of R 3 , the L p norm in time with value inḢ 1 2 + 2 p of (v|e) R 3 blows up at T ⋆ .

In this paper, we prove the global existence of smooth solutions to the three‐dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, (e3,0). Compared with previous works by Lin, Xu, and Zhang and by Xu and Zhang, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood‐Paley theory for the Lagrangian formulation of the system, we achieve the global L1‐in‐time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large‐time decay rate of the solution is also obtained.© 2016 Wiley Periodicals, Inc.

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