2008/03/03 by Luc Molinet, Molinet, Luc, Raafat Talhouk +1
Engineering · Mathematics · #35B05 #76D03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #math.AP #msc:35B05 #msc:76D03
paper · pdf · doi:10.48550/arxiv.0803.0228
arxiv created 2008/03/03 · openalex publication_date 2008/03/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with regular flows of incompressible weakly viscoelastic fluids which obey a differential constitutive law of Oldroyd type. We study the newtonian limit for weakly viscoelastic fluid flows in \RN or \TN for N=2, 3, when the Weissenberg number (relaxation time measuring the elasticity effect in the fluid) tends to zero. More precisely, we prove that the velocity field and the extra-stress tensor converge in their existence spaces (we examine the Sobolev-Hs theory and the Besov-Bs,12 theory to reach the critical case s= N/2) to the corresponding newtonian quantities. These convergence results are established in the case of "ill-prepared"' data.We deduce, in the two-dimensional case, a new result concerning the global existence of weakly viscoelastic fluids flow. Our approach makes use of essentially two ingredients : the stability of the null solution of the viscoelastic fluids flow and the damping effect,on the difference between the extra-stress tensor and the tensor of rate of deformation, induced by the constitutive law of the fluid.