We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting:
$ \partial \mathbf{u}/\partial t+$ $(\nabla f(\mathbf{u}) \cdot \nabla)$ $\mathbf{u} -νΔ\mathbf{u}=$ $\nabla η,\quad t \geq 0,\ \mathbf{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d,$ under the assumption that $\mathbf{u}$ is a gradient. Here $f$ is 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 solutions $\mathbf{u}$ of this equation, we study Sobolev norms of $\mathbf{u}$ averaged 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 $\mathbf{l}$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in $ν$. We generalise the results obtained for the one-dimensional case in \cite{BorW}, confirming the physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates does not depend on the dimension: the powers of $ν, |\mathbf{k}|, \mathbf{l}$ are the same in the one- and the multi-dimensional setting.
本文综述了A.Biryuk和作者在\cite{Bir01,BorK,BorW,BorD}中关于圆上广义Burgers方程u_t+ f '（u）u_x= v u_{xx} +η，| x \in S ^1 = \R / \Z的湍流结果.这里，f是光滑的并且是强凸的，而常数0<v << 1对应于粘性系数。我们将考虑η=0的情况和η是一个随机力的情况，该随机力在x方向上是光滑的，在t方向上是不规则的（反冲或白噪声）。在这两种情况下，都得到了u的Sobolev范数在时间上和系综上的C ν^{-δ}，δ>=0的精确界，其中δ的上界和下界具有相同的值.这些结果产生小尺度量表征湍流的急剧边界，确认物理预测\cite{BK07}。
Kolmogorov's theory of turbulence and its rigorous 1d model
本文是对A. Boritchev，S. Kuksin，\textit{One-Dimensional Turbulence and the Stochastic Burgers Equation}，AMS Publications，2021（待发表）.这本书致力于随机Burgers方程作为一个模型的一维湍流，并讨论了其内容有关的Kolmogorov理论的湍流。
Study of a model equation in detonation theory: multidimensional effects
我们扩展了Kasimov等人在Phys.Rev.Lett.中提出的反应Burgers方程，110（2013）和Faria等人，SIAM J.Appl.Maths，74（2014），以包括多维效应。此外，我们解释了如何根据Faria et al. JFM（2015）中发展的渐近理论的思想合理地证明模型。该模型是非定常小扰动跨音速流方程的强迫形式。我们表明，物理上合理的选择强迫函数，行波解类似于爆震波存在。结果表明，多维效应在行波的稳定性和动力学中起着重要的作用。数值模拟表明，该模型的解往往会形成类似于气体爆轰胞腔的多维模式。
Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进数学研究所）；Department of Mathematics, KTH Royal Institute of Technology（皇家理工学院数学系）；Department of Mathematics and Statistics, McMaster University（麦克马斯特大学数学与统计系）；Graduate School of Economics, Hitotsubashi University（一桥大学经济学研究科）
