Global Solutions of Two‐Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data
Global Solutions of Two‐Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data
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DOI：
10.1002/cpa.21561
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发表时间：
2013-12
影响因子：
3
通讯作者：
Xianpeng Hu;F. Lin
中科院分区：
文献类型：
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作者：
Xianpeng Hu;F. Lin
二维不可压缩粘弹性流弱解的整体存在性是一个长期存在的开放问题，本文对其进行了研究。如果初始变形梯度在L2∩L∞上接近单位矩阵，并且初始速度在L2上很小，在Lp上有界，我们证明了全局存在性。虽然经典的Oldroyd - B模型自动满足了初始变形梯度的假设，但由于我们采用的技术，我们还附加了初始速度以Lp为界的假设。然而，初始速度L2范数的小假设对于全局适定性来说是很自然的。本文的关键观测之一是速度和“有效粘性通量”G对于正时间是足够规则的。G的正则性为变形梯度的点向估计提供了一种新的方法，无需在空间变量的速度梯度上使用L∞界。©2015 Wiley期刊公司
The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem, and it is studied in this paper. We show global existence if the initial deformation gradient is close to the identity matrix in L2 ∩ L∞ and the initial velocity is small in L2 and bounded in Lp for some p > 2. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd‐B model, the additional assumption on the initial velocity being bounded in Lp for some p > 2 may due to techniques we employed. The smallness assumption on the L2 norm of the initial velocity is, however, natural for global well‐posedness. One of the key observations in the paper is that the velocity and the “ effective viscous flux” G are sufficiently regular for positive time. The regularity of G leads to a new approach for the pointwise estimate for the deformation gradient without using L∞ bounds on the velocity gradients in spatial variables. © 2015 Wiley Periodicals, Inc.