The Study About The Time-periodic Solutions Of One Dimensional Viscous Burgers Equation Driving By Periodic Boundary Conditions

In this paper,we study the time periodic solution of the one-dimensional viscous Burgers equation driving by the periodic Dirichlet boundary conditions on a finite interval.It is a nonlinear parabolic equation.Firstly,we homogenize the boundary con-ditions,transform it into a nonlinear parabolic problem with periodic force,and get the weak form of the equation.To prove the existence,we use the Galerkin method doing the projection,getting the sufficient conditions of the linearized approximate problem and the Schaefer fixed point method obtaining the existence of nonlinear approximate problem.Then we combine some priori estimates and use the Ascoli compactness theorem to prove the approximate nonlinear solutions can converge to the weak solution,so the solution of nonlinear parabolic problem with periodic force exists.Finally,we show the time asymptotically stable in the H~1sense under some extra conditions on the periodic boundary.In summary when the periodic boundary is sufficiently small,and the substraction of the two boundary function~?s integration in a period is 0,the viscous Burgers equation~?s solution is not only existent,but also stable.