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Global Solutions of Two-Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data
- Source :
- Communications on Pure and Applied Mathematics. 69:372-404
- Publication Year :
- 2015
- Publisher :
- Wiley, 2015.
-
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. © 2015 Wiley Periodicals, Inc.
Details
- ISSN :
- 00103640
- Volume :
- 69
- Database :
- OpenAIRE
- Journal :
- Communications on Pure and Applied Mathematics
- Accession number :
- edsair.doi...........eca0b2233a5eb7eee83ecd4f7f2c5c99