Kalousek, Martin; Kortum, Joshua; Schlömerkemper, Anja Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. (English) Zbl 1455.35195 Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 17-39 (2021). Summary: The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization.First, we show that our model possesses global in time weak solutions, thus extending work by B. Benešová et al. [SIAM J. Math. Anal. 50, No. 1, 1200–1236 (2018; Zbl 1390.74059)]. Compared to that work, we include the stray field energy and relax the assumptions on the elastic energy density. Second, we prove the local-in-time existence of strong solutions. Both existence results are based on the Galerkin method. Finally, we show a weak-strong uniqueness property. Cited in 11 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 74F15 Electromagnetic effects in solid mechanics Keywords:magnetoviscoelastic flow; existence of weak solutions; existence of strong solutions; weak-strong uniqueness; energetic variational approach Citations:Zbl 1390.74059 PDFBibTeX XMLCite \textit{M. Kalousek} et al., Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 17--39 (2021; Zbl 1455.35195) Full Text: DOI arXiv References: [1] B. Benešová; J. Forster; C. Liu; A. Schlömerkemper, Existence of weak solutions to an evolutionary model for magnetoelasticity, SIAM J. Math. Anal., 50, 1200-1236 (2018) · Zbl 1390.74059 [2] G. Carbou; M. A. Efendiev; P. Fabrie, Global weak solutions for the Landau-Lifschitz equation with magnetostriction, Math. Methods Appl. Sci., 34, 1274-1288 (2011) · Zbl 1219.35297 [3] S. Carillo; M. Chipot; V. Valente; G. Vergara Caffarelli, A magneto-viscoelasticity problem with a singular memory kernel, Nonlinear Anal. Real World Appl., 35, 200-210 (2017) · Zbl 1367.35168 [4] M. Chipot; I. Shafrir; V. Valente; G. Vergara Caffarelli, On a hyperbolic-parabolic system arising in magnetoelasticity, J. Math. Anal. Appl., 352, 120-131 (2009) · Zbl 1173.35007 [5] I. Ellahiani, E.-H. Essoufi and M. Tilioua, Global existence of weak solutions to a three-dimensional fractional model in magneto-viscoelastic interactions, Boundary Value Problems, (2017), 20 pp. · Zbl 1483.74032 [6] J. Forster, Variational Approach to the Modeling and Analysis of Magnetoelastic Materials, Ph.D thesis, University of Würzburg, (2016), urn: nbn: de: bvb: 20-opus-147226. [7] G. Gioia; R. D. James, Micromagnetics of very thin films, Proc. Roy. Soc. London A, 453, 213-223 (1997) [8] M. Kalousek, On dissipative solutions to a system arising in viscoelasticity, J. Math. Fluid Mech., 21 (2019), Art. 56, 15 pp. · Zbl 1427.35205 [9] M. Kružík; U. Stefanelli; J. Zeman, Existence results for incompressible magnetoelasticity, Discrete Contin. Dyn. Syst., 35, 2615-2623 (2015) · Zbl 1332.74017 [10] F.-H. Lin; C. Y. Wang, On the uniqueness of heat flow of farmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31, 921-938 (2010) · Zbl 1208.35002 [11] F.-H. Lin; C. Liu; P. Zhang, On hydrodynamics of viscolelastic fluids, Comm. Pure Appl. Math., 58, 1437-1471 (2005) · Zbl 1076.76006 [12] C. Liu; N. J. Walkington, An Eulerian description of fluids containing viscoelastic particles, Arch. Ration. Mech. Anal., 159, 229-252 (2001) · Zbl 1009.76093 [13] A. Schlömerkemper; J. Žabenský, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, Nonlinearity, 31, 2989-3012 (2018) · Zbl 1397.35226 [14] A. Schlömerkemper; J. Žabenský, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, Nonlinearity, 31, 2989-3012 (2018) · Zbl 1403.35249 [15] W. J. Zhao, Local well-posedness and blow-up criteria of magneto-viscoelastic flows, Discrete Contin. Dyn. Syst., 38, 4637-4655 (2018) · Zbl 1403.35249 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.