Global existence in critical spaces for incompressible viscoelastic fluids

We investigate local and global strong solutions for the incompressible viscoelastic system of Oldroyd--B type. We obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations. More precisely, the initial velocity has the same critical regularity index as for the incompressible Navier--Stokes equations, and one more derivative is needed for the deformation tensor. We point out a smoothing effect on the velocity and a $L^1-$decay on the difference between the deformation tensor and the identity matrix. Our result implies that the deformation tensor $F$ has the same regularity as the density of the compressible Navier--Stokes equations.

[1]  Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$ , 2006, math/0611044.

[2]  R. Danchin.  Density-dependent incompressible viscous fluids in critical spaces , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  J. Chemin,et al.  Théorèmes d’unicité pour le système de navier-stokes tridimensionnel , 1999 .

[4]  Robert C. Armstrong,et al.  Dynamics of polymeric liquids: Kinetic theory , 1987 .

[5]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[6]  Yoshihiro Shibata,et al.  Global existence and exponential stability of small solutions to nonlinear viscoelasticity , 1992 .

[7]  N. Lerner,et al.  Flow of Non-Lipschitz Vector-Fields and Navier-Stokes Equations , 1995 .

[8]  R. Agemi.  Global existence of nonlinear elastic waves , 1999 .

[9]  J. Chemin,et al.  Wellposedness and stability results for the Navier-Stokes equations in R 3 , 2006 .

[10]  Thomas C. Sideris,et al.  Global existence for three‐dimensional incompressible isotropic elastodynamics via the incompressible limit , 2005 .

[11]  T. Sideris,et al.  Global existence for three‐dimensional incompressible isotropic elastodynamics , 2007 .

[12]  Jacques-Louis Lions,et al.  On Some Questions in Boundary Value Problems of Mathematical Physics , 1978 .

[13]  Zhen Lei,et al.  Global existence for a 2D incompressible viscoelastic model with small strain , 2007 .

[14]  Raphaël Danchin,et al.  On the uniqueness in critical spaces for compressible Navier-Stokes equations , 2005 .

[15]  R. Bird.  Dynamics of Polymeric Liquids , 1977 .

[16]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[17]  Raphaël Danchin,et al.  Global existence in critical spaces for compressible Navier-Stokes equations , 2000 .

[18]  R. Danchin.  A few remarks on the Camassa-Holm equation , 2001, Differential and Integral Equations.

[19]  Hiroshi Fujita,et al.  On the Navier-Stokes initial value problem. I , 1964 .

[20]  Thomas C. Sideris.  Nonresonance and global existence of prestressed nonlinear elastic waves , 2000 .

[21]  Noel J. Walkington,et al.  Digital Object Identifier (DOI) 10.1007/s002050100158 An Eulerian Description of Fluids Containing Visco-Elastic Particles , 2022 .

[22]  Yi Zhou,et al.  Global Existence of Classical Solutions for the Two-Dimensional Oldroyd Model via the Incompressible Limit , 2005, SIAM J. Math. Anal..

[23]  Herbert Koch,et al.  Well-posedness for the Navier–Stokes Equations , 2001 .

[24]  R. Larson.  The Structure and Rheology of Complex Fluids , 1998 .

[25]  Jianzhen Qian,et al.  Well-posedness in critical spaces for incompressible viscoelastic fluid system , 2010 .

[26]  C. M. Dafermos,et al.  Hyberbolic [i.e. Hyperbolic] conservation laws in continuum physics , 2005 .

[27]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[28]  Yi Zhou,et al.  Global Solutions for Incompressible Viscoelastic Fluids , 2008, 0901.3658.

[29]  Ping Zhang,et al.  On hydrodynamics of viscoelastic fluids , 2005 .