一类阻尼二阶发展方程解的能量和弱湍流廓线的量子化
Quantization of energy and weakly turbulent profiles of the solutions to some damped second order evolution equations
摘要：
我们考虑一个二阶方程的线性“弹性”的一部分和一个非线性阻尼项取决于一个权力的范数的速度。在考虑衰减率和能量不为零的约束条件下，我们对解进行了适当的尺度变换，研究了解的渐近行为，发现了一个相当出乎意料的二分法现象。具有多个傅里叶分量的解是无阻尼线性化方程解的渐近解，并且表现出能量在分量之间的某种均分。具有无穷多个傅立叶分量的解弱趋于零，但不强趋于零。我们还证明了解的能量极限仅取决于其Fourier分量的个数，我们的结果的证明是由一个简化模型的分析所启发的，这个模型是我们通过一个平均过程设计的，它的解表现出与原方程的解相同的渐近性质。
英文摘要：
We consider a second order equation with a linear "elastic" part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero.We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fouriercomponents are asymptotic to solutions of the linearized equationwithout damping, and exhibit some sort of equipartition of theenergy among the components. Solutions with infinitely manyFourier components tend to zero weakly but not strongly. We showalso that the limit of the energy of solutions depends only on thenumber of their Fourier components.The proof of our results is inspired by the analysis of asimplified model which we devise through an averaging procedure,and whose solutions exhibit the same asymptotic properties as thesolutions to the original equation.
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