论文标题
从统计角度来看，在中间长波方程的深水和浅水极限上
On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint
论文作者
论文摘要
（由于arxiv设置的摘要的字符数量的限制，无法在此处显示完整的摘要。）我们研究中间长波方程（ILW）的收敛问题，具有深度参数$Δ> 0 $，在深水限制中，在深层限制中（$δ\ to \ fofty $ to \ for \ frowty $ to the Ballow limitive the limitive）（$ fimit）$Δ$Δ$ $Δ特别是，我们在深水和浅水限制中建立了ILW不变的吉布斯动力学的融合。为此，我们首先构建ILW的Gibbs度量，$ 0 <Δ<\ infty $。由于它们在分布中得到支持，因此需要重新归一化。随着灯芯的重新归一化，我们进行了吉布斯措施ILW的构建。然后，我们证明，Gibbs的ILW测量结果将总变异汇合到深水极限中的本杰明·诺克方程（BO）。在浅水状态下，在应用缩放转换后，我们证明，作为$δ\ 0 $，gibbs的尺度量表量度薄弱地与korteweg-de vries方程（KDV）相关。我们指出，第二个结果特别感兴趣，因为尺度ILW和KDV的Gibbs度量是互惠的（而Gibbs的ILW和BO量度相当）。 我们还讨论了相关的动态问题的收敛性。 最后，我们指出的是，我们的结果也适用于在散落的情况下的广义ILW方程，在深水极限中融合到广义的BO，并在浅水限制中融合到广义的KDV。但是，在非排斥情况下，由于相应的Gibbs测量值的非归一化性，我们的结果无法扩展到具有较高功率的非线性。
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study convergence problems for the intermediate long wave equation (ILW), with the depth parameter $δ> 0$, in the deep-water limit ($δ\to \infty$) and the shallow-water limit ($δ\to 0$) from a statistical point of view. In particular, we establish convergence of invariant Gibbs dynamics for ILW in both the deep-water and shallow-water limits. For this purpose, we first construct the Gibbs measures for ILW, $0 < δ< \infty$. As they are supported on distributions, a renormalization is required. With the Wick renormalization, we carry out the construction of the Gibbs measures for ILW. We then prove that the Gibbs measures for ILW converge in total variation to that for the Benjamin-Ono equation (BO) in the deep-water limit. In the shallow-water regime, after applying a scaling transformation, we prove that, as $δ\to 0$, the Gibbs measures for the scaled ILW converge weakly to that for the Korteweg-de Vries equation (KdV). We point out that this second result is of particular interest since the Gibbs measures for the scaled ILW and KdV are mutually singular (whereas the Gibbs measures for ILW and BO are equivalent). We also discuss convergence of the associated dynamical problem. Lastly, we point out that our results also apply to the generalized ILW equation in the defocusing case, converging to the generalized BO in the deep-water limit and to the generalized KdV in the shallow-water limit. In the non-defocusing case, however, our results can not be extended to a nonlinearity with a higher power due to the non-normalizability of the corresponding Gibbs measures.
