Global Solutions of Two-Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data

Xianpeng Hu, Fang-Hua Lin

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)372-404
Number of pages33
JournalCommunications on Pure and Applied Mathematics
Volume69
Issue number2
DOIs
StatePublished - Feb 1 2016

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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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