论文标题
在周期性扰动下，平面稀疏波的渐近稳定性
Asymptotic stability of planar rarefaction waves for 3-d isentropic Navier-Stokes equations under periodic perturbations
论文作者
论文摘要
我们研究了3-D Isentropic Navier-Stokes方程的平面稀疏波（$ x_1 $ - 方向）的渐近稳定性，其中初始扰动是定期在圆环上的$ \ mathbb {t}^3 $，平均平均值为零。为了解决这个cauchy问题，其中初始数据仅相对于$ x_2 $和$ x_3 $，但不是$ x_1，我们构建了一个合适的ANSATZ，该ANSATZ在$ x_1 $方向上携带解决方案的振荡，但仍是横向$ x_2 $ x_2 $ - x_3 $ - 和$ x_3 $ - directions的定期。以这种方式，ANSATZ和解决方案之间的区别可以在$ \ mathbb {r} \ times \ times \ mathbb {t}^2中，$可以集成，从而使我们能够在$ \ \ mathbb {r} r} $ nirequality上使用gagliardo-nirenberg type nirenberg type nirenberg type nirenberg type nirenberg type nirenberg。
We study the asymptotic stability of a planar rarefaction wave (in the $ x_1 $- direction) for the 3-d isentropic Navier-Stokes equations, where the initial perturbation is periodic on the torus $ \mathbb{T}^3 $ with zero average. To solve this Cauchy problem in which the initial data is periodic with respect to only $ x_2 $ and $ x_3 $ but not to $ x_1, $ we construct a suitable ansatz carrying the oscillations of the solution in the $ x_1 $- direction, but remaining to be periodic in the transverse $ x_2 $- and $ x_3 $- directions. In such a way, the difference between the ansatz and the solution can be integrable on the region $ \mathbb{R}\times\mathbb{T}^2, $ which allows us to utilize the energy method with the aid of a Gagliardo-Nirenberg type inequality on $ \mathbb{R}\times\mathbb{T}^2 $ to prove the result.
