Some Investigation Of The PDE Related To Fluid Mechanic Models

In three dimensional space, We study some topics in fluid mechanic model.Firstly, we consider the steady problem of Euler-Poisson system in a smooth bounded domain. Assume the macroscopic entropy S is S(x)=θln|x|, hereθis a constant. Through a nonlinear transformation, the steady problem of Euler-Poisson system becomeswhereΩ(?)R3 is an open bounded domain and contains 0, h(x)∈C1(Ω) is al-lowed to change sign. Using variational method, under different restriction to the strength of velocity field, different assumptions on the isentropic function and adi-abatic exponent, we get the existence, multiplicity and uniqueness of solutions to (EPS).Then, we consider the nonlinear instability of incompressible Euler equations in a tubewhereρ=ρ(t,x,y,z), V=V(t,x,y,z) and P denote the density, velocity and pressure respectively, G=(g,0,0) is the vector along the x direction of gravity, t≥0 is the time. If a steady density is non-increasing, then the smooth steady state is nonlinear instability.The evolution of viscous heat-conducting polytropic idea fluid with self-gravitating can be formulated by the NSP systemwhereHere (?), cv, k≥0 denote temperature, specific heat at constant volume, coefficient of heat conduction respectively.▽tυis the transpose of▽υ.λandμare viscosity coefficients satisfyingμ>0,3λ+2μ≥0. We present a sufficient condition on the blowup of smooth solutions to the (NSP) system. Then we construct a family of analytical solutions which blowup in finite time.