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In this paper, we consider the 2D incompressible MHD equations with horizontal dissipation and horizontal magnetic diffusion. We establish local existence and uniqueness of solutions for initial data in $H^s$($s\geq 1$). The proof relays heavily on new estimates of the nonlinear terms., Comment: The result is trival, therefore we want to withdraw this paper

In this paper, we consider the 2D second grade fluid past an obstacle satisfying the standard non-slip boundary condition at the surface of the obstacle. Second grade fluid model is a well-known non-Newtonian model, with two parameters: $\alpha$ representing length-scale, while $\nu > 0$ corresponding to viscosity. We prove that, under the constraint condition $\nu = {o}(\alpha^\frac{4}{3})$, the second grade fluid with a suitable initial velocity converges to the Euler fluid as $\alpha$ tends to zero. Moreover, we estimate the convergence rate of the solution of second grade fluid equations to the one of Euler fluid equations as $\nu$ and $\alpha$ approach zero.

In this article, we consider 2D second grade fluid equations in exterior domain with Dirichlet boundary conditions. For initial data $\boldsymbol{u}_0 \in \boldsymbol{H}^3(\Omega)$, the system is shown to be global well-posed. Furthermore, for arbitrary $T > 0$ and $s \geq 3$, we prove that the solution belongs to $L^\infty([0, T]; \boldsymbol{H}^s(\Omega))$ provided that $\boldsymbol{u}_0$ is in $\boldsymbol{H}^s(\Omega)$.

After casting Euler-$\alpha$ equations into vorticity-stream function formula, we obtain some very useful estimates from the properties of the vorticity formula in exterior domain. Basing on these estimates, one can have got the global existence and uniqueness of the solutions to Euler-$\alpha$ equations in 2D exterior domain provided that the initial data is regular enough.