Multidimensional Potential Burgers Turbulence
Multidimensional Potential Burgers Turbulence
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DOI：
10.1007/s00220-015-2521-7
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发表时间：
2013-12
影响因子：
2.4
通讯作者：
A. Boritchev
中科院分区：
文献类型：
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作者：
A. Boritchev
我们考虑多维广义随机Burgers方程的空间周期设置： ∂ u ∂ 不 + ( ∇ F ( u ) · ∇ ) u - ν Δ u = ∇ η , 不 ≥ 0 , X ∈ 不 D = ( R / Z ) D , 假设u是一个梯度其中，ν是强凸的，满足增长条件，ν是小的，正的，而η是一个随机强迫项，空间光滑，时间白色。为了求解此方程，我们研究了在时间和系综上的Sobolev范数：这些范数中的每一个都表现为ν的给定负幂。这些结果产生尖锐的上限和下限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。这些量表现为相关参数的范数的幂，这些参数分别是物理空间中的分离度和傅立叶空间中的波数。我们的界不依赖于初始条件，并且一致成立。我们推广了[10]中一维情况下的结果，证实了[4，30]中的物理预测。注意，估计的形式不依赖于维度：在一维和多维设置中的幂是相同的。
We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: ∂ u ∂ t + ( ∇ f ( u ) · ∇ ) u - ν Δ u = ∇ η , t ≥ 0 , x ∈ T d = ( R / Z ) d , under the assumption thatuis a gradient. Herefis strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For solutionsuof this equation, we study Sobolev norms ofuaveraged in time and in ensemble: each of these norms behaves as a given negative power of ν. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation ℓ in the physical space and the wavenumberkin the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in. We generalise the results obtained for the one-dimensional case in [10], confirming the physical predictions in [4, 30]. Note that the form of the estimates does not depend on the dimension: the powers ofare the same in the one- and the multi-dimensional setting.