American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances
  • DCDS-S Home
  • This Issue
  • Next Article
    Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow
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  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, 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

[1]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
[2]

Wenjing Zhao. Weak-strong uniqueness of incompressible magneto-viscoelastic flows. Communications on Pure & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491
[6]

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
[7]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[8]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
[9]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
[10]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337
[11]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87
[12]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[13]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[14]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011
[15]

Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371
[16]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[17]

Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136
[18]

Elder J. Villamizar-Roa, Elva E. Ortega-Torres. On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 825-847. doi: 10.3934/dcdsb.2011.15.825
[19]

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
[20]

Hao Yang, Fuke Wu, Peter E. Kloeden. Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5553-5567. doi: 10.3934/dcdsb.2019071

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (27)
  • HTML views (148)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences