标题： 关于三维可压缩欧拉方程与非线性泊松方程耦合或非耦合的自由边界问题 显示英文标题

标题： On the Free Boundary Problems of 3-D Compressible Euler Equations Coupled or Uncoupled With a Nonlinear Poisson Equation

摘要： 对于非等熵可压缩欧拉方程与非线性泊松方程耦合的问题，在三维空间中，电势满足狄利克雷边界条件，自由边界为一般形式且不限制为图的情况下，我们确定了电势和压力的适当稳定性条件，在这些条件下，我们得到了流体和电变量的Sobolev范数的先验估计以及自由表面几何量的界。在这种情况下，对于一般的变量熵，稳定性条件是电势的外法向导数在自由表面上为正，而压力的外法向导数为负。在等熵情况下，稳定性条件简化为一个条件，即比热与电势之差的外法向导数在自由表面上为负。对于不与非线性泊松方程耦合的非等熵可压缩欧拉方程的变熵自由边界问题，也在泰勒符号条件下获得了相应的高阶估计。还发现，在电势满足狄利克雷边界条件并满足电势和压力的稳定性条件下，与非等熵可压缩欧拉方程的问题相比，对于耦合非线性泊松方程的非等熵可压缩欧拉方程问题，闭合能量估计所需的导数更少。 显示英文摘要

摘要： For the problem of the non-isentropic compressible Euler Equations coupled with a nonlinear Poisson equation with the electric potential satisfying the Dirichlet boundary condition in three spatial dimensions with a general free boundary not restricting to a graph, we identify suitable stability conditions on the electric potential and the pressure under which we obtain a priori estimates on the Sobolev norms of the fluid and electric variables and bounds for geometric quantities of free surface. The stability conditions in this case for a general variable entropy are that the outer normal derivative of the electric potential is positive on the free surface, whereas that on the pressure is negative. In the isentropic case, the stability condition reduces to a single one, the outer normal derivative of the difference of the enthalpy and the electric potential is negative on the free surface. For the free boundary problem of the non-isentropic compressible Euler equations with variable entropy without coupling with the nonlinear Poisson equation, the corresponding higher-order estimates are also obtained under the Taylor sign condition. It is also found that one less derivative is needed to close the energy estimates for the problem for the non-isentropic compressible Euler Equations coupled with a nonlinear Poisson equation when the electric potential satisfies the Dirichlet boundary condition under the stability conditions on the electric potential and the pressure, compared with the problem of the non-isentropic compressible Euler equations.