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January  2021, 14(1): 17-39. doi: 10.3934/dcdss.2020331

Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

Martin Kalousek , Joshua Kortum and  Anja Schlömerkemper ,

University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

* Corresponding author: Anja Schlömerkemper

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  June 2019 Revised  September 2019 Published  January 2021 Early access  April 2020

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 Benešová et al. 2018. 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.

Keywords: Magnetoviscoelastic flow, existence of weak solutions, existence of strong solutions, weak-strong uniqueness, energetic variational approach.
Mathematics Subject Classification: Primary: 35Q35, 35Q74; Secondary: 74F15.
Citation: Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331
References:
[1]

B. Benešová, J. Forster, C. Liu and A. Schlömerkemper, Existence of weak solutions to an evolutionary model for magnetoelasticity, SIAM J. Math. Anal., 50 (2018), 1200-1236.  doi: 10.1137/17M1111486.  Google Scholar

[2]

G. Carbou, M. A. Efendiev and P. Fabrie, Global weak solutions for the Landau-Lifschitz equation with magnetostriction, Math. Methods Appl. Sci., 34 (2011), 1274-1288.  doi: 10.1002/mma.1440.  Google Scholar

[3]

S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, A magneto-viscoelasticity problem with a singular memory kernel, Nonlinear Anal. Real World Appl., 35 (2017), 200-210.  doi: 10.1016/j.nonrwa.2016.10.014.  Google Scholar

[4]

M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, On a hyperbolic-parabolic system arising in magnetoelasticity, J. Math. Anal. Appl., 352 (2009), 120-131.  doi: 10.1016/j.jmaa.2008.04.013.  Google Scholar

[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. doi: 10.1186/s13661-017-0852-3.  Google Scholar

[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. Google Scholar

[7]

G. Gioia and R. D. James, Micromagnetics of very thin films, Proc. Roy. Soc. London A, 453 (1997), 213-223.  doi: 10.1098/rspa.1997.0013.  Google Scholar

[8]

M. Kalousek, On dissipative solutions to a system arising in viscoelasticity, J. Math. Fluid Mech., 21 (2019), Art. 56, 15 pp. doi: 10.1007/s00021-019-0459-9.  Google Scholar

[9]

M. Kružík, U. Stefanelli and J. Zeman, Existence results for incompressible magnetoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2615-2623.  doi: 10.3934/dcds.2015.35.2615.  Google Scholar

[10]

F.-H. Lin and 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 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[11]

F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscolelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.  Google Scholar

[12]

C. Liu and N. J. Walkington, An Eulerian description of fluids containing viscoelastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252.  doi: 10.1007/s002050100158.  Google Scholar

[13] J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-dimensional Navier-Stokes Equations. Classical Theory, Cambridge Studies in Advanced Mathematics, 157. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781139095143.  Google Scholar
[14]

A. Schlömerkemper and J. Žabenský, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, Nonlinearity, 31 (2018), 2989-3012.  doi: 10.1088/1361-6544/aaba36.  Google Scholar

[15]

W. J. Zhao, Local well-posedness and blow-up criteria of magneto-viscoelastic flows, Discrete Contin. Dyn. Syst., 38 (2018), 4637-4655.  doi: 10.3934/dcds.2018203.  Google Scholar

show all references

References:
[1]

B. Benešová, J. Forster, C. Liu and A. Schlömerkemper, Existence of weak solutions to an evolutionary model for magnetoelasticity, SIAM J. Math. Anal., 50 (2018), 1200-1236.  doi: 10.1137/17M1111486.  Google Scholar

[2]

G. Carbou, M. A. Efendiev and P. Fabrie, Global weak solutions for the Landau-Lifschitz equation with magnetostriction, Math. Methods Appl. Sci., 34 (2011), 1274-1288.  doi: 10.1002/mma.1440.  Google Scholar

[3]

S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, A magneto-viscoelasticity problem with a singular memory kernel, Nonlinear Anal. Real World Appl., 35 (2017), 200-210.  doi: 10.1016/j.nonrwa.2016.10.014.  Google Scholar

[4]

M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, On a hyperbolic-parabolic system arising in magnetoelasticity, J. Math. Anal. Appl., 352 (2009), 120-131.  doi: 10.1016/j.jmaa.2008.04.013.  Google Scholar

[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. doi: 10.1186/s13661-017-0852-3.  Google Scholar

[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. Google Scholar

[7]

G. Gioia and R. D. James, Micromagnetics of very thin films, Proc. Roy. Soc. London A, 453 (1997), 213-223.  doi: 10.1098/rspa.1997.0013.  Google Scholar

[8]

M. Kalousek, On dissipative solutions to a system arising in viscoelasticity, J. Math. Fluid Mech., 21 (2019), Art. 56, 15 pp. doi: 10.1007/s00021-019-0459-9.  Google Scholar

[9]

M. Kružík, U. Stefanelli and J. Zeman, Existence results for incompressible magnetoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2615-2623.  doi: 10.3934/dcds.2015.35.2615.  Google Scholar

[10]

F.-H. Lin and 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 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[11]

F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscolelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.  Google Scholar

[12]

C. Liu and N. J. Walkington, An Eulerian description of fluids containing viscoelastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252.  doi: 10.1007/s002050100158.  Google Scholar

[13] J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-dimensional Navier-Stokes Equations. Classical Theory, Cambridge Studies in Advanced Mathematics, 157. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781139095143.  Google Scholar
[14]

A. Schlömerkemper and J. Žabenský, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, Nonlinearity, 31 (2018), 2989-3012.  doi: 10.1088/1361-6544/aaba36.  Google Scholar

[15]

W. J. Zhao, Local well-posedness and blow-up criteria of magneto-viscoelastic flows, Discrete Contin. Dyn. Syst., 38 (2018), 4637-4655.  doi: 10.3934/dcds.2018203.  Google Scholar

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