Blow-up And Non-existence To The Initial Value Problem Of A Class Of Fluid Mechanics Equations

The thesis consists of two parts for studying blow-up of classical solutions to the initial value problem of Euler-Poisson equations with external electrostatic potential force and non-existence in Sobolev space to the initial value problem of the compressible Navier-Stokes equations.The first part considers the initial value problem of full Euler-Poisson equations with external electrostatic potential force and isentropic Euler-Poisson equations.We prove classical solutions to the initial value problem will blow up in finite time,if the initial data satisfy some conditions.Since Euler-Poisson equations come from Euler equations coupling gravitational field or electric field,however,gravitational field and electric field are non-local term,which destroy the finite propagation of compact support of density.Therefore,the method introduced by Sideris[39]on proving blow-up of Euler equations does not work for Euler-Poisson equations.There are two critical points in Sideris' s proof.One is the finite propagation of compact support of density,and the other is the estimate of radial component of momentum.But,the estimate of radial component of momentum can be replaced by the estimate of the upper bound of internal energy.Consequently,we need to find a method to avoid using the finite propagation of compact support of density.For our problem,we need to face two difficulties,one is to estimate the nonlocal terms,the other is to deduce blow-up without the finite propagation of compact support of density.The two difficulties are coupled together.Our method is to estimate the upper bound of the internal energy by using Hardy-Littlewood-Sobolev inequality and the lower bound of the internal energy by using Chemin's inequality,and finally compare the coefficients of the upper and lower decay rates to find the right conditions for the initial data for blow-up.The second part considers non-existence in Sobolev space to the initial value problem of the compressible full Navier-Stokes equations and one-dimensional isentropic Navier-Stokes equations.Xin[45],Cho and Jin[61]proved that any classical solution in Sobolev space C1(0,T;Hm(Rn)),m > [n2] + 2 to n-dimensional full compressible NavierStokes equations will blow-up in finite time if the initial density has compact support,this result is also true for one-dimensional isentropic Navier-Stokes equations.We prove that there is no solution in Sobolev space C1(0,T;Hm(Rn)),m > [n2] + 2 to the n-dimensional full compressible Navier-Stokes equations with positive heat conduction if the initial density has compact support,and the result holds for one-dimensional isentropic NavierStokes equations with additional assumptions on initial data.Our method is to reduce the original initial value problem to an over-determined initial boundary value problem of an integro-differential equations by an observation and then define an appropriate parabolic or integro-differential operator with degeneration on t-derivative and then establish Hopf's lemma and strong maximum principle and finally prove the desired result by contradiction.