Asymptotic stability of planar rarefaction waves under periodic perturbations for 3-d Navier-Stokes equations

In this talk, we study a Cauchy problem for the 3-d compressible isentropic Navier-Stokes equations, in which the initial data is a 3-d periodic perturbation around a planar rarefaction wave. We prove that the solution of the Cauchy problem exists globally in time and tends to the background rarefaction wave in the $ L^\infty(\R^3) $ space as $ t\to +\infty. $ The result reveals that even though the initial perturbation has infinite oscillations at the far field and is not integrable along any direction of space, the planar rarefaction wave is nonlinearly stable for the 3-d N-S equations. The key point is to construct a suitable ansatz $ (\rhot, \uvt) $ to carry the same oscillations as those of the solution $ (\rho, \uv) $ at the far field in the normal direction of the rarefaction wave, so that the difference $ (\rho-\rhot, \uv-\uvt) $ belongs to some Sobolev space and the energy method is available.