We consider a free boundary problem for the incompressible ideal magnetohydrodynamic equations that describes the motion of the plasma in vacuum. The magnetic field is tangent and the total pressure vanishes along the plasma-vacuum interface. Under the Taylor sign condition of the total pressure on the free surface, we prove the local well-posedness of the problem in Sobolev spaces.2010 Mathematics Subject Classification. 35L65, 35Q35, 76B03, 76W05.

We consider suitable weak solutions of the incompressible Navier-Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the twodimensional Hausdorff measure of the set of singular points is equal to zero in both cases.2010 Mathematics Subject Classification. 35Q30, 35B65, 76D05.

In this paper, we consider suitable weak solutions of incompressible Navier-Stokes equations in four spatial dimensions. We prove that the two-dimensional time-space Hausdorff measure of the set of singular points is equal to zero.

This paper is concerned with the three dimensional compressible Euler-Poisson equations with moving physical vacuum boundary condition. This fluid system is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local existence of classical solutions for initial data in certain weighted Sobolev spaces is established in the case that the adiabatic index satisfies 1 < γ < 3. find the solution X (n) to the linearized problem (with respect to v (n) ) of X-equation. We can also get an energy type estimate of X (n) , which will be used to construct contract map in the second step. In this process, we will make the fundamental use of the higher-order Hardy-type inequality introduced in [9]. We mention that our intermediate variable X is different from the one used in [9]. By using our intermediate variable, the improvement of the space regularity for X (n) will be easier and clearer with less computations. However, due to our choice of the intermediate variable, the symmetric structure of Xequation is different from that in [9], we will need to make use of κ to construct the contract map. In the second step, we derive a new approximate velocity field v (n+1) by a linear elliptic system of equations, which is constructed by defining the divergence, curl and vertical component on the boundary of v (n+1) by using v (n) , X (n) and their derivatives directly. The idea is that we will add a proper small perturbation to v (n) 's divergence, curl and vertical component on the boundary to define v (n+1) 's divergence, curl and vertical component on the boundary. To achieve this goal, we will make use of the evolution equations for div η v, curl η v in the domain and the evolution equation for normal component v 3 on the boundary. The first equation is derived from our definition of intermediate variable X, the other two equations are derived from κ-equations. For v (n) and X (n) , these equations do not hold, but they can be regarded as small perturbations from zero. Then combining with the energy type estimate we get in the first step, we can construct the contract map Θ : v (n) → v (n+1) and use the fixed-point scheme to get solutions to the κ-equations.Lastly, we will derive κ-independent a prior estimates for the approximated solutions. This kind a prior estimates were introduced in [7,9] for the Euler equations. We combine some ideas from [20] with [9] to carry out a little simpler proof in Section 8 than that in [9, Section 9]. The main strategy of a prior estimates is that: we first get the curl estimate and then perform the L 2 type high order energy estimates with respect to time derivative and tangential Φ = φ • η (Lagrangian potential field), F = [Dη] (Deformation tensor), J = detF (Jacobian determinant), F * = JF −1 (Cofactor matrix). (2.2) Then equation (2.1) can be rewritten in the Lagragian coordinates as v(x, t) = η t (x, t). This equation holds throughout the paper.6