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arXiv 1004.2192math.AP
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全耦合Navier-Stokes和Q-张量系统的整体存在性和正则性

Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system

Marius Paicu, Arghir Zarnescu

摘要:

本文研究了一类非线性液晶流体演化方程组解的整体存在性和正则性。流体被描述为一个系统,耦合强迫Navier-Stokes系统与抛物型系统。在我们以前的工作,\cite{pz1},我们假设某个参数,$ξ$,是零,这有取消某些条款的效果。在目前的工作中,我们不做这个假设,并研究整个系统,观察到这些额外的条款的存在有一个非平凡的效果,即四倍指数增长的高范数。我们还不同地估计某些条款已经存在于简化系统和改进的估计\cite{pz1}。我们证明了在二维和三维中整体弱解的存在性。在第二个维度,我们证明了更高的正则性的解决方案,并表明,高规格的时间增加最多四倍指数。我们还证明了在二维中的弱-强唯一性。

英文摘要:

We study the global existence and regularity of solutions for a system describing the evolution of a nematic liquid crystal fluid. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. In our previous work, \cite{pz1}, we assumed that a certain parameter, $ξ$, is zero, which had the effect of cancelling certain terms. In the current work we do not make this assumption and study the full system, observing that the presence of these additional terms has a non-trivial effect, namely the quadruply exponential increase of the high norms. We also estimate differently certain terms already existent in the simplified system and improve the estimates in \cite{pz1}. We prove the existence of global weak solutions in dimensions two and three. In dimension two we prove the higher regularity of solutions and show that the high norms increase in time at most quadruply exponential. We also show the weak-strong uniqueness in dimension two.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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