Lax Pairs For The Perturbed Euler Equations

In this paper, we consider Cauchy problem for the Navier-Stokes equation:where u : R×R~n→Rnis presented as velocity field, p : R×R~n→R~n is the pressure.The equation shows the motion of incompressible flow. Whenν= 0, The equation canbe writted into the former of Euler equation:The equation presents the motion of incompressible flow. As is known to all , theequation is local-posedness in many standar spaces[10, 11, 12], when n = 2 has the globalsolution. However, when n = 3, it is an open problem in the ?eld of mathematical physicshow to extend the local solution to global solution. Recently, Charles has reasearchedon the Lax pairs and their isospectral theory for the Euler equation of 2D and 3D[13, 14].Besides, Charles researched the Darboux transformations for Euler equation[15]. It is ofgreat significance to the well- posedness of Euler equation.Although Lax pairs plays an important role in the study of PDE. It is very difficultto find a Lax pairs of nonlinear PDE. Now the integrabal theory is facing a basic problem,which is how to judge whether a PDE is Lax integrabal. In other words, it is how to?nd two operators so that a PDE can be presented in Lax former. It is quite difficult tosolve the problem. So far, the better method is to continue structure. However, morePDE will depend on our intuition and experience[6, 20].In this article , we will find Lax pairs for three types of perturbation of Euler equa-tion for inviscid incompressible fluid flows. Moreover, we will investigate the isospectraltheorem of the perturbed Euler equations. These Lax pairs have the greatest potential that they can be very useful in establishing global well-posedness of 3D Euler equationand Navier-Stokes equation.