arXiv 1608.014602016-08-04math.AP

分数阶亚临界Burgers方程的衰减湍流

Decaying turbulence for the fractional subcritical Burgers equation

Alexandre Boritchev

备注：: arXiv admin note: substantial text overlap with arXiv:1208.5241

摘要：

我们考虑一维空间周期情形下的分数阶非强迫Burgers方程：$$\partial u/\partial t+(f(u))_x +νΛ^α u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$这里$f$是强凸的，并且满足一个附加的增长条件$Λ=\sqrt{-Δ}$，$ν$是小的正的，而$α\in (1,\ 2)$是亚临界范围内的常数.对于这个方程的解$u$，我们推广了文献[10]中对于$α=2$情形（即当$-Λ^α$是拉普拉斯算子时）得到的结果。我们得到尖锐的估计的时间平均Sobolev规范的$u$作为一个功能的$ν$。这些结果产生尖锐的估计自然类似物的数量表征的流体动力学湍流，即平均值的增量和能量谱。在惯性范围内，这些量表现为相关参数的范数的幂，相关参数分别是物理空间中的间隔$\ell$和傅立叶空间中的波数$\mathbf{k}$。所有估计值的形式与$α=2$的情况相同;唯一的变化（除了隐式常数）是$ν$被$ν^{1/(α-1)}$取代。

英文摘要：

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +νΛ^α u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$ Here $f$ is strongly convex and satisfies an additional growth condition, $Λ=\sqrt{-Δ}$, $ν$ is small and positive, while $α\in (1,\ 2)$ is a constant in the subcritical range. For solutions $u$ of this equation, we generalise the results obtained for the case $α=2$ (i.e. when $-Λ^α$ is the Laplacian) in [10]. We obtain sharp estimates for the time-averaged Sobolev norms of $u$ as a function of $ν$. These results yield sharp estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation $\ell$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. The form of all estimates is the same as in the case $α=2$; the only thing that changes (except implicit constants) is that $ν$ is replaced by $ν^{1/(α-1)}$.

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摘要：

英文摘要：

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