### 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 L^{2}∩L^{∞} and the initial velocity is small in L^{2} and bounded in L^{p} 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 L^{p} for some p>2 may due to techniques we employed. The smallness assumption on the L^{2} 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 language | English (US) |
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Pages (from-to) | 372-404 |

Number of pages | 33 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 69 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics