Vanishing viscosity limit of the 3D incompressible Oldroyd-B model (Q2235193)

The paper establishes the existence and uniqueness of solutions to Cauchy problems for systems of PDEs modelling the Eulerian mechanics of an incompressible elastic material that occupies the whole Euclidean space \(\mathbb{R}^3\) and that is homogeneous (i.e., with constant mechanical properties). The PDEs are Navier-Stokes equations complemented by an evolution equation for a viscoelastic component of the Cauchy stress, the so-called Oldroyd-B model with 3 parameters. In addition to a (constant) elastic modulus and a (constant) time-scale, which together define the first viscosity \(\lambda_1\), there is the second viscosity \(\lambda_2\) as parameter of the PDEs. The author shows that under given small analytic initial conditions, global-in-time solutions exist and are unique uniformly for all small values of the second viscosity parameter. It allows to define a limit when the second viscosity vanishes, which formally corresponds to a global-in-time solution of the Cauchy problem for the incompressible elastodynamic system with small analytic initial conditions. Note that the paper heavily relies on specialized analysis in Besov spaces. Global-in-time solutions to the Cauchy problem for the incompressible elastodynamic system in \(\mathbb{R}^3\) are known to exist only since the recent papers by \textit{T. C. Sideris} and \textit{B. Thomases} [Commun. Pure Appl. Math. 58, No. 6, 750--788 (2005; Zbl 1079.74028); Commun. Pure Appl. Math. 60, No. 12, 1707--1730 (2007; Zbl 1127.74016)]. Analyticity of the initial conditions is required specifically here only to handle the additional nonlinear terms that appear in the evolution equation for a tensor (Cauchy stress), in comparison with the usual nonlinear term that models the material advection of a scalar quantity, for instance.