We derive a criterion for the breakdown of solutions to the Oldroyd-B model in $\R^3$ in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space $H^m$, $m> 5/2$, then either a unique solution exists within this space indefinitely, or, at the time where the solution breaks down, the time integral of the $L^\infty$-norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Eluer equations of incompressible fluids.
A Beale-Kato-Majda criterion with optimal frequency and temporal localization
我们得到了三维Navier-斯托克斯方程具有最佳频率和时间局部化的Beale-Kato-Majda型准则。与以前的结果相比，我们的条件只需要控制傅立叶模式低于临界频率，其值是明确的时间尺度。作为应用，它给出了空间$B^{-1}_{\infty,\infty}$中正则性的强频率局部化条件以及可能爆破解的$L^p$范数$2\leq p <3$衰减率的下界.证明依赖于新的估计，在小的时间尺度，这可能是独立的利益的截止耗散和能量。
Global well-posedness of strong solutions to the initial-boundary value problem for a two-dimensional stress-diffusive Oldroyd-B model in the creeping flow regime
MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）
Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；School of Mathematics, Statistics and Mechanics, Beijing University of Technology（北京工业大学数学、统计与力学学院）
