论文标题
$ l^2 $ - 稳定性在$ 4 $ WAVES动力学方程的均衡附近
$L^2$-stability near equilibrium for the $4$ waves kinetic equation
论文作者
论文摘要
我们考虑波湍流理论中产生的四个波空间均匀动力学方程。我们研究了雷利 - 吉恩均衡解决方案周围的长期行为和解决方案的存在。对于截止频率，我们表明，对于围绕二次情况的分散关系而言，雷利 - 吉恩斯均衡的线性化操作员是强制性的。然后，我们将其传递给完全非线性操作员，显示$ l^2 $ - 稳定性的稳定性，用于接近Rayleigh -Jeans的初始数据。
We consider the four waves spatial homogeneous kinetic equation arising in wave turbulence theory. We study the long-time behaviour and existence of solutions around the Rayleigh-Jeans equilibrium solutions. For cut-off'd frequencies, we show that for dispersion relations weakly perturbed around the quadratic case, the linearized operator around the Rayleigh-Jeans equilibria is coercive. We then pass to the fully nonlinear operator, showing an $L^2$ - stability for initial data close to Rayleigh-Jeans.
