The Existence And Analyticity Of The Euler Equation In The Besov Space

The Euler equation describes the law of motion of gas or fluid.It's widely used in a-coustics,mechanics,hydraulics and other fields.In this paper,we study the existence and analyticity of the weak solution to Euler equation.By a priori estimates and compact-ness method,we show that there exists unique weak solution in u ? C([0,T];B1,1s(Rn))if u0?B1,1s(Rn),s>n+1.The result about the existence of the weak solution of the Euler equation for Pak and Park[22]in B1,1n+1(Rn)is generalized;Sawada O,Takada R[11]showed the analyticity of the weak solution to Euler equation in B?,1 1.We get the ana-lyticity of the weak solution to Euler equation in B1,1n+1(Rn).By using a priori estimates we derive the suitable estimates for the higher order derivatives of the nonlinear term of Euler equation.It is showed that the solution is analytic if the initial velocity is analytic by the inductive method.We show that the analyticity of the weak solution to Euler equation in B1,1n+1(Rn).