TY  - JOUR
T1  - Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
A1  - Bathory, Michal
A1  - Bulíček, Miroslav
A1  - Málek, Josef
JA  - Advances in Nonlinear Analysis
Y1  - 2021
VL  - 10
IS  - 1
SP  - 501
EP  - 521
M2  - doi: 10.1515/anona-2020-0144
N2  - We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity $\ve$, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor $\B$. By a proper choice of the constitutive relations for the Helmholtz free energy �uchy stress tensor $\T$
	(which, however, is non-standard in the current literature  despite the fact that this choice is well motivated from the point of view o physics) and for the energy dissipation, we are able to prove that $\B$ enjoys the same regularity as $\ve$ in the classical three-dimensional Navier-Stokes equations. This enables us to handle any kind of objective derivative of $\B$, thus obtaining existence results for the class of diffusive Johnson-Segalman models as well. Moreover, using a suitable approximation scheme, we are able to show that $\B$ remains positive definite if the initial datum was a positive definite matrix (in a pointwise sense). We also show how the model we are considering can be derived from basic balance equations and thermodynamical principles in a natural way.
M1  - Preprint project = NCMM
M1  - Preprint year = 2020
M1  - Preprint number = 01
M1  - Preprint ID = NCMM/2020/01
ER  -