Study On The Solutions Of Euler-Poisson Equations With Rotating

We consider the following Euler-Poisson equations:where t > 0, x D, D is a bounded domain of RN (N> 3). p = p(t,x) is the density of gaseous stars, v = v(t, x) RN is the velocity.(t, x) is the energy potential of the self-gravitational force. Here (t, x) denotees the time and space variables , G is the gravitational constant.wN is the measure of the unit ball in RN , and P is the pressure satisfying the following state equation:where 7 > 1 is the adiabatic exponent, with the entropy function S=S(t,x) in The system (0.1) is compressible Euler equations; the gravitional potential is determined by the density distribution of the gas itself through the Posson equation the fourth equation of (0.1).For the case when TV = 3 Tao Luo and Joel Smoller studied the existence of solutions for (0.1) with rotating the x3-anix.In this paper, we extend the result to the high dimensions in D of Rn with given angular velocity. It is worth noticing that N is an odd number.For a ball domain and a constant angular velocity, both existence theorem and non-existence theorem are also obtained by using the time-maping method , depending on the adiabatic gas constant r.In addition we obtain the monotonicity of the radius of the star with both angular velocity and center density . We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is different to the case of the non-rotating star.When the domains is a general domain and the angular velocities are variable, we obtain existence theorems for the isentropic equations of state by monotone method .