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The stability of multi-dimensional shock waves is an important issue in thetheory of gas dynamics. This paper studies a Cauchy problem for the 3Dcompressible Navier-Stokes equations, concerning the nonlinear time-asymptoticstability of the planar shock profiles. We show that if the initialperturbation is small and of zero mass, then the shock profile with weakstrength is asymptotically stable in the $ L^\infty(\mathbb{R}^3) $ space up toa constant shift. Moreover, we can obtain the exponential decay rate of thenon-zero mode associated with the perturbed profile. One novelty of this paperis that the initial perturbation can be periodic at the spatial infinity whilefor positive time the perturbation is no longer periodic, which is obviouslynot integrable, along the direction of the wave propagation. To deal with theseoscillations, we construct a suitable ansatz that oscillates together with thesolution, which makes the $ L^2 $-energy method available. A key point in thestability analysis is the use of the anti-derivative technique and Galileaninvariance to estimate the zero mode of the perturbation.

Nonlinear asymptotic stability of planar shock profiles for 3D compressible Navier-Stokes equations with spatially periodic perturbations