The well-known Constantin–Lax–Majda (CLM) equation, an important toy model of the 3D Euler equations without convection, can develop finite time singularities (Constantin et al. in Commun Pure Appl Math 38:715–724, 1985). De Gregorio modified the CLM model by adding a convective term (De Gregorio in Math Methods Appl Sci 19(15):1233–1255, 1996), which is known important for fluid dynamics (Hou and Lei in Commun Pure Appl Math 62(4):501–564, 2009; Okamoto in J Math Soc Jpn 65(4):1079–1099, 2013). Presented are two results on the De Gregorio model. The first one is the global well-posedness of such a model for general initial data with non-negative (or non-positive) vorticity which is based on a newly discovered conserved quantity. This verifies the numerical observations for such class of initial data. The second one is an exponential stability result of ground states, which is similar to the recent significant work of Jia et al. (Ration Mech Anal, 231:1269–1304, 2019), with the zero mean constraint on the initial data being removable. The novelty of the method is the introduction of the new solution space $${\mathcal {H}}_{DW}$$ H DW together with a new basis and an effective inner product of $${\mathcal {H}}_{DW}$$ H DW .

中文翻译：

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关于具有对流的 Constantin-Lax-Majda 模型

著名的 Constantin-Lax-Majda (CLM) 方程是没有对流的 3D Euler 方程的重要玩具模型，可以产生有限时间奇点（Constantin 等人，在 Commun Pure Appl Math 38:715-724, 1985 中）。De Gregorio 通过添加对流项（De Gregorio in Math Methods Appl Sci 19(15):1233–1255, 1996）修改了 CLM 模型，这对流体动力学非常重要（Hou and Lei in Commun Pure Appl Math 62(4) ):501–564, 2009；冈本在 J Math Soc Jpn 65(4):1079–1099, 2013)。展示了 De Gregorio 模型的两个结果。第一个是这种模型的全局适定性，用于具有非负（或非正）涡度的一般初始数据，该模型基于新发现的守恒量。这验证了此类初始数据的数值观察。第二个是基态的指数稳定性结果，类似于 Jia 等人最近的重要工作。(Ration Mech Anal, 231:1269–1304, 2019)，对初始数据的零均值约束是可移除的。该方法的新颖之处在于引入了新的解空间 $${\mathcal {H}}_{DW}$$ H DW 以及 $${\mathcal {H}} 的新基和有效内积_{DW}$$ H DW。