Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows

Abstract

We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation gradient and for the magnetization obtained from a special case of the micromagnetic energy. It turns out that the conditions on uniqueness coincide with those for the well-known Navier-Stokes equations in bounded domains: weak solutions are unique in two spatial dimensions, and weak solutions satisfying the Prodi-Serrin conditions are unique among all weak solutions in three dimensions. That is, we obtain the so-called weak-strong uniqueness result in three spatial dimensions.

Weak–strong uniqueness of renormalized solutions to reaction–cross-diffusion systems

“…Weak-strong uniqueness was established by Leray [20] for incompressible Navier-Stokes equations and by Dafermos [8] for conservation laws; see the review by Wiedemann [24] for more details. Later this concept has been applied to other fluid models, including measurevalued solutions [15,19]; to magneto-viscoelastic flow equations [22]; and to gradient flows based on optimal transport [3]. As far as we know, there are very few works on the weak-strong uniqueness involving renormalized solutions.…”

Weak-strong uniqueness of incompressible magneto-viscoelastic flows

“…proved the existence of global weak solutions to the system (1) − (4) with an additional regularizing term κ∆F (κ > 0) in equation 3for the deformation tensor F . Later, uniqueness of solutions to the same regularized problem with slightly different boundary conditions was studied in [23]. The authors proved the uniqueness of global weak solutions when the spatial dimension is two and also a weak-strong type of uniqueness result in three dimensions.…”

Local well-posedness and blow-up criteria of magneto-viscoelastic flows

“…In [8], the author proved the existence of global weak solutions to the magneto-viscoelastic system (1) − (4) with an additional regularizing term κ∆F (κ > 0) in the deformation equation (3). Later, Schlömerkemper andŽabensky [27] investigated the uniqueness of global weak solutions for the same problem with slightly different boundary conditions. It is worth mentioning that the artificial regularizing term κ∆F plays an essential role in their analysis.…”

“…In this paper, we aim to establish the local well-posedness and some blow-up criteria of the magneto-viscoelastic system (1) − (4), in particular, without the regularizing term κ∆F as in [8,27]. Following the notation in [22], we define the usual strain tensor in the form of…”