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

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

* 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.

Citation: Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331
References:
[1]

B. BenešováJ. ForsterC. 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. CarbouM. 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. CarilloM. ChipotV. 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. ChipotI. ShafrirV. 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. KružíkU. 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. LinC. 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. RobinsonJ. 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. BenešováJ. ForsterC. 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. CarbouM. 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. CarilloM. ChipotV. 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. ChipotI. ShafrirV. 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. KružíkU. 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. LinC. 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. RobinsonJ. 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.

[1]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181

[2]

Wenjing Zhao. Weak-strong uniqueness of incompressible magneto-viscoelastic flows. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2907-2917. doi: 10.3934/cpaa.2020127

[3]

Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202

[4]

Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437

[5]

T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491

[6]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[7]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005

[8]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026

[9]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217

[10]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553

[11]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

[12]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87

[13]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure and Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465

[14]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777

[15]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure and Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011

[16]

Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371

[17]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625

[18]

Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136

[19]

Elder J. Villamizar-Roa, Elva E. Ortega-Torres. On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 825-847. doi: 10.3934/dcdsb.2011.15.825

[20]

Zhoude Shao. Existence and continuity of strong solutions of partly dissipative reaction diffusion systems. Conference Publications, 2011, 2011 (Special) : 1319-1328. doi: 10.3934/proc.2011.2011.1319

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (238)
  • HTML views (299)
  • Cited by (0)

[Back to Top]