Title: Dissipation concentration in two-dimensional fluids Show Chinese title

Title: 二维流体中的耗散集中

Abstract: We study the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. For Leray-Hopf solutions it is proved that the dissipation is Lebesgue in time and, for almost every time, it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is proved to be absolutely continuous with respect to a suitable "quadratic" space-time vorticity measure. This results into the trivial measure if the initial vorticity has singular part of distinguished sign, or a spatially purely atomic measure if wild oscillations in time are ruled out. In fact, the dynamics at the Kolmogorov scale is the only relevant one, in turn offering new criteria for anomalous dissipation. We provide kinematic examples highlighting the strengths and the limitations of our approach. Quantitative rates, dissipation life-span and steady fluids are also investigated. Show Chinese abstract

Abstract: 我们研究二维不可压缩流体粘性趋于零极限中出现的耗散测度。对于Leray-Hopf解，证明了耗散在时间上是勒贝格可积的，并且对于几乎所有的时刻，它相对于解的强紧性缺陷测度是绝对连续的。当初始涡量是一个测度时，证明了耗散相对于一个合适的“二次”时空涡量测度是绝对连续的。如果初始涡量具有特定符号的奇异部分，或者在时间上排除了剧烈振荡，则结果为平凡测度；如果初始涡量是纯点测度，则结果为纯空间点测度。事实上，科莫戈罗夫尺度的动力学是唯一相关的一个，从而提供了异常耗散的新标准。我们提供了运动学例子，突出了我们方法的优势和局限性。定量速率、耗散寿命和定常流也进行了研究。