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We obtain the global large solutions to the compressible Navier–Stokes equations in $${\mathbb {R}}^2$$ . The solution is large in the sense that there is no smallness assumption applied to one component of the initial incompressible velocity.

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.

In the present paper, we consider the global well-posedness of the density-dependent incompressible flow of liquid crystals in $\mathbb{R}^{2}$ . The local existence and uniqueness of the system are obtained without the assumption of small density variation. The global well-posedness is proved when the initial density and liquid crystal orientation are small. However, the initial velocity field is allowed to be arbitrarily large.

In this paper, we consider the global well-posedness of the incompressible magnetohydrodynamic system with anisotropic Besov initial data in the critical Besov spaces. Under the nonlinear smallness condition on the critical anisotropic Besov norm of the horizontal components of the initial velocity field and magnetic field, we get the system has a unique global solution. Our result allows to construct global solutions for a class of highly oscillating initial data of Cannone's type.

In this paper, we study the global well posedness of the 3D incompressible magnetohydrodynamic system with horizontal dissipation and horizontal magnetic diffusion in the scaling invariant Besov–Sobolev-type spaces. We first get a unique global solution to this system with small initial data by the classical Friedrich’s regularization method. Then using a weighted Chemin–Lerner-type norm, we prove the system also can generate a global solution if the horizontal components of the initial data are small enough compared to the vertical components. In particular, our results imply the global large solutions with highly oscillating initial data.