2013/12/24 by Xianpeng Hu, Fanghua Lin, Hu, Xianpeng +1 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1312.6749
openalex publication_date 2013/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a long standing open problem, and it is studied in this paper. We show the 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 the global well-posedness . One of the key observations in the paper is that the velocity and the \textquotedblleft effective viscous flux\textquotedblright 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.