Free Boundary And Partial Regularity Of Partial Differential Equations Arising From Fluid Mechanics

Partial differential equations arising from fluid mechanics are important research objects in the theory of partial differential equations. In this dissertation, we consider two problems in the partial differential equations arising from fluid mechanics. The first one is the free boundary problem, we study the local wellposedness of compressible Euler-Poisson equations with physical vacuum in both one and three dimension. The second one is the partial regularity theory, we consider the partial regularity of suitable weak solutions to four dimensional incompressible Navier-Stokes equations and MHD equations.First, we consider the compressible Euler-Poisson equations with physical vacuum. Well-posedness of ideal complex fluid with physical vacuum is a long-standing problem concerned by many mathematicians and make wonderful success only for Euler equations [15,16,40,41]. In chapter three and four, we establish the local wellposedness for one dimensional and three dimensional compressible Euler-Poisson equations. We use La-grange coordinates to transform the free boundary problem to a fixed domain problem, which is similar to the Euler equations [15,16,40,41]. And then we give an explicit formula for the potential force term to decouple the Euler-Poisson equations to Euler equations with source term. The main difficulty of our system is the estimate of poten-tial force, especially in three dimension. In three dimensional case, the potential force is formulated by a convolution. During the high order derivative estimates, the singularity of convolution kernel will increase. We shall use Taylor’s formula and Sobolev Cα esti-mates to balance the singularity and get the estimates for the potential force in the end. Then we introduce a new artificial viscosity to construct κ-approximate equations and use new median variable and fixed point scheme to get the solution to approximate equations. Finally, we can get the local existence by doing κ-independent a prior estimates.We also study the partial regularity of suitable weak solutions to the four dimensional incompressible Navier-Stokes equations in chapter five. We extend Caffarelli, Kohn and Nirenberg’s famous partial regularity result to four dimension. Compared to three dimensional case, the main difficulty is the lack of compactness. We choose a different test function from that in [59] in the local energy inequality to get an estimate with positive decay term and establish a new iteration scheme based on this estimate. In this way, we can get a weak decay estimate and improve the decay rate by bootstrap argument and parabolic regularity. In this way, we can obtain the Holder’s continuity and partial regularity for both interior case and boundary case without the compactness argument.Finally, we study the boundary partial regularity of suitable weak solutions to the four dimensional incompressible MHD equations in chapter six. We give two different kind ε-regularity criteria by following a similar argument we used for Navier-Stokes equations. One only requires the smallness of scaling Lp,q norm of u, another requires the smallness of scaling space time L2 norm of Vu and boundedness of scaling norm of H or ▽H. We can use the second kind ε-regularity to get the boundary partial regularity.