Blow Up Of The Solution Of The Three-dimensional Euler Equation With Time Decay

In this paper,we are concerned with the blow up problems of smooth solutions of the 3-D compressible Euler equation with time-dependent damping where x?R3,a(t)>0,?>0 is a constant,?0,u0?C0?(R3),(?0,u0)?0,?(0,x)>0 and ?>0 is sufficiently small.In this thesis,we will devote to proving the blow up of the solution to problem(0.0.1).In the process of proving the blow up of solution to problem(0.0.1)with time-dependent damping,the method we adopted is to construct the function F(t)and to establish the ordinary differential inequality satisfied by the function F(t),and to obtain the blow up of the solution based on integrability of a(t).We mainly refer to the method to treat(0.0.2)where x?R3,?>0,??0 and ?>0 are constants,?0,u0?C0?(R3),(?0,u0)?0,p(0,x)>0 and ?>0 is sufficiently small.At first,we give the blow up of smooth solutions of the 3-D compressible Euler equation with time-dependent damping in finite time,then we give sharp upper esti-mate of the lifespan of solutions when initial date are small.Finally,we obtain the following main conclusions.Theorem 2.0.1:We define two functions suppose q0(l)>0,q1(l)>0 hold for all l ?(M0,M)and supp?0,suppu0(?){x:|x|?M} and ?0? a(t)dt<?,where MO is a fixed constant satisfying 0 ? M0<M.Then the problem(0.0.1)exists an ?0>0 such that,for 0<?<?0,the lifespan T?of smooth solution of(0.0.1)is finite.In Chapter 1,we give some introdution to Theorem 2.0.1.The detailed proof of Theorem 2.0.1 are proved in Chapter 2.