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The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support., 31 pages

In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-\Delta)^{\alpha}$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $\alpha$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $\alpha=0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global regularity issue on the 2D resistive MHD equations, namely, the case when the MHD equations only have standard Laplacian magnetic diffusion., Comment: 31 pages. We have added some materials in this updated version. arXiv admin note: text overlap with arXiv:2205.00688

In this paper, we consider the Cauchy problem of the two-dimensional (2D) magnetic Bénard system with partial dissipation. On the one hand, we obtain the global regularity of the 2D magnetic Bénard system with zero thermal conductivity. The main difficulty is the zero thermal conductivity. To bypass this difficulty, we exploit the structure of the coupling system about the vorticity and the temperature and use the Maximal $L_t^{p}L_x^{q}$ regularity for the heat kernel. On the other hand, we also establish the global regularity of the 2D magnetic Bénard system with horizontal dissipation, horizontal magnetic diffusion and with either horizontal or vertical thermal diffusivity. This settles the global regularity issue unsolved in the previous works. Additionally, in the Appendix, we also show that with a full Laplacian for the diffusive term of the magnetic field and half of the full Laplacian for the temperature field, the global regularity result holds true as long as the power $\alpha$ of the fractional Laplacian dissipation for the velocity field is positive.

We investigate the differentiability issue of the drift-diffusion equation with nonlocal L��vy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial H��lder regularity, we prove that the vanishing viscosity solution is differentiable with some H��lder continuous derivatives for any positive time., 24 pages. Submitted