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Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity
University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany |
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.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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 and R. D. James,
Micromagnetics of very thin films, Proc. Roy. Soc. London A, 453 (1997), 213-223.
doi: 10.1098/rspa.1997.0013. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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 and R. D. James,
Micromagnetics of very thin films, Proc. Roy. Soc. London A, 453 (1997), 213-223.
doi: 10.1098/rspa.1997.0013. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
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