Critical functional framework and maximal regularity in action on systems of incompressible flows
Critical functional framework and maximal regularity in action on systems of incompressible flows
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DOI：
10.24033/msmf.451
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
R. Danchin;P. Mucha
中科院分区：
文献类型：
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作者：
R. Danchin;P. Mucha
本文主要研究R的有界或外区域上的发展Stokes系统的Besov空间的极大正则性结果的陈述和证明。我们努力与时间无关的先验估计L时间可积性和齐次Besov空间中的值。最后通过应用，我们求解了两个流体力学系统。这种类型的结果是已知的整个空间的情况下，并已被证明是非常强大的调查在临界空间的一些与流体力学有关的偏微分方程的适定性问题。在[15]中，它们最近被扩展到半空间设置。在我们这里研究的更复杂的外部域情况下，我们缺乏对速度的“低频”的控制。为了克服这一点，我们采用了P. Maremonti和V.A. Solonnikov在[43]中，适当修改以匹配Besov空间框架。因此，我们的时间无关估计只在Besov空间的某些合适的交集中成立。让我们强调一下，由于使用了L1空间，所以不可能应用对偶技巧。作为我们工作的第一个重要的应用，我们解决本地大数据或全球小数据，（略）非齐次不可压缩Navier-Stokes方程在临界Besov空间，在外部域。在观察到L1时间可积性允许全局确定流动的流线之后，整个系统在拉格朗日坐标设置中被重铸。特别是，这使我们能够考虑不连续密度，如[17]，[20]。第二个应用涉及证明一个低马赫数系统的整体存在性结果（对于小的临界数据），该系统最近由第一作者和X. Liao in [14].这里的系统是在欧拉坐标系下研究的。
This memoir is mainly devoted to the statement and the proof of new maximal regularity results involving Besov spaces for the evolutionary Stokes system in bounded or exterior domains of R. We strive for time independent a priori estimates with L time integrability and values in homogeneous Besov spaces. By way of application at the end of the memoir, we solve two systems of fluid mechanics. Results of this type are known for the whole space case and have proved to be spectacularly powerful to investigate the well-posedness issue in critical spaces for a number of PDEs related to fluid mechanics. They have been extended recently to the half-space setting in [15]. In the more involved exterior domain case that we investigate here, we lack a control on the ‘low frequencies’ of the velocity. To overcome this, we adopt the method introduced by P. Maremonti and V.A. Solonnikov in [43], suitably modified to match the Besov space framework. As a consequence, our time-independent estimates hold true only in some suitable intersection of Besov spaces. Let us emphasize that no duality tricks are likely to be applied, because of the use of L1 space. As a first and important application of our work, we solve locally for large data or globally for small data, the (slightly) inhomogeneous incompressible Navier-Stokes equations in critical Besov spaces, in an exterior domain. After observing that the L1 time integrability allows to determine globally the stream lines of the flow, the whole system is recast in the Lagrangian coordinates setting. This, in particular, enables us to consider discontinuous densities, as in [17], [20]. The second application concerns the proof of a global existence result (for small critical data) for a low Mach number system that has been studied recently in the whole space setting by the first author and X. Liao in [14]. The system here is investigated in the Eulerian coordinates.