具有有限Weissenberg数的一般Oldroyd模型的飞溅奇点
Splash singularities for a general Oldroyd model with finite Weissenberg number
摘要：
在本文中，我们研究了一个二维Oldroyd自由边界模型，它描述了粘弹性流体的演变。我们证明存在飞溅奇点，即点的边界保持光滑，但自我相交。本文将前人对无限Weissenberg数的结果推广到任意有限Weissenberg数的情形。本文的主要困难是由于弹性张量的非线性平衡定律，不能减少，在无限Weissenberg的情况下，变形梯度的输运方程。我们的策略在准确的局部存在的结果取决于Weissenberg数和共形和拉格朗日变换的组合。飞溅奇点的存在是通过适当选择初始数据并结合稳定性估计来保证的。
英文摘要：
In this paper we study a 2D Oldroyd free-boundary model which describes the evolution of a viscoelastic fluid. We prove existence of splash singularities, namely points where the boundary remains smooth but self-intersects. This paper extends the previous results obtained for infinite Weissenberg number to the case of any finite Weissenberg number. The main difficulty of this paper is due to the non linear balance law of the elastic tensor which cannot be reduced, as in the case of infinite Weissenberg, to the transport equations for the deformation gradient. Our strategy in accurate local existence result depending on the Weissenberg number and the combination of Conformal and Lagrangian transformations. The existence of splash singularities is guarantee by a suitable choice of the initial data combined with stability estimates.
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