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May  2020, 19(5): 2907-2917. doi: 10.3934/cpaa.2020127

Weak-strong uniqueness of incompressible magneto-viscoelastic flows

Department of Mathematics, College of Sciences, Northeastern University, Shenyang 110819, China

Received  August 2019 Revised  September 2019 Published  March 2020

Our aim in this paper is to prove the weak-strong uniqueness property of solutions to a hydrodynamic system that models the dynamics of incompressible magneto-viscoelastic flows. The proof is based on the relative energy approach for the compressible Navier-Stokes system.

Citation: 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
References:
[1]

B. BenešováJ. ForsterC. García-CerveraC. Liu and A. Schlömerkemper, Analysis of the flow of magnetoelastic materials, Proc. Appl. Meth. Mech., 16 (2016), 663-664.  doi: 10.1002/pamm.201610320.  Google Scholar

[2]

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.  Google Scholar

[3]

C. CavaterraE. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differ. Equ., 255 (2013), 24-57.  doi: 10.1016/j.jde.2013.03.009.  Google Scholar

[4]

Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differ. Equ., 31 (2006), 1793-1810.  doi: 10.1080/03605300600858960.  Google Scholar

[5]

E. Emmrich and R. Lasarzik, Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions, Discrete Contin. Dyn. Syst., 38 (2018), 4617-4635.  doi: 10.3934/dcds.2018202.  Google Scholar

[6]

E. FeireislY. Lu and A. Novotný, Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law, Sci. China Math., 61 (2018), 2003-2016.  doi: 10.1007/s11425-017-9272-7.  Google Scholar

[7]

E. FeireislB. J. Jin and A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9.  Google Scholar

[8]

J. Forster, Variational Approach to the Modeling and Analysis of Magnetoelastic Materials, Ph.D thesis, University of Würzburg, 2016. Google Scholar

[9]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1.  Google Scholar

[10]

M. Grasselli and H. Wu, Long-time behavior for a nematic liquid crystal model with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002.  doi: 10.1137/120866476.  Google Scholar

[11]

X. P. Hu and F. H. Lin, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Commun. Pure Appl. Math., 69 (2016), 372-404.  doi: 10.1002/cpa.21561.  Google Scholar

[12]

X. P. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 3437-3461.  doi: 10.3934/dcds.2015.35.3437.  Google Scholar

[13]

Y. HyonD. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: maximum dissipation principle, Discrete Contin. Dyn. Syst., 26 (2010), 1291-1304.  doi: 10.3934/dcds.2010.26.1291.  Google Scholar

[14]

N. Jing, H. Liu, and Y. L. Luo, Global classical solutions to an evolutionary model for magnetoelasticity, preprint, arXiv: 1904.09531v1. Google Scholar

[15]

M. Kalousek, On dissipative solutions to a system arising in viscoelasticity, preprint, arXiv: 1903.03635. doi: 10.1007/s00021-019-0459-9.  Google Scholar

[16]

M. Kalousek, J. Kortum and A. Schlömerkemper, Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity, preprint, arXiv: 1904.07179. Google Scholar

[17]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[18]

Z. Lei and Y. Zhou, Global existence of classical solutions for 2D Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.  doi: 10.1137/040618813.  Google Scholar

[19]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.  Google Scholar

[20]

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

[21]

F. H. Lin and P. Zhang, On the initial boundary value problem of the incompressible viscoelastic fluid system, Commun. Pure Appl. Math., 61 (2008), 539-558.  doi: 10.1002/cpa.20219.  Google Scholar

[22]

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

[23]

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

[24]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.  Google Scholar

[25]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396.  doi: 10.3934/dcds.2010.26.379.  Google Scholar

[26]

H. WuX. Xu and C. Liu, On the general Ericksen-Leslie system: Parodi's relation, wellposedness and stability, Arch. Ration. Mech. Anal., 208 (2013), 59-107.  doi: 10.1007/s00205-012-0588-2.  Google Scholar

[27]

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. BenešováJ. ForsterC. García-CerveraC. Liu and A. Schlömerkemper, Analysis of the flow of magnetoelastic materials, Proc. Appl. Meth. Mech., 16 (2016), 663-664.  doi: 10.1002/pamm.201610320.  Google Scholar

[2]

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.  Google Scholar

[3]

C. CavaterraE. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differ. Equ., 255 (2013), 24-57.  doi: 10.1016/j.jde.2013.03.009.  Google Scholar

[4]

Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differ. Equ., 31 (2006), 1793-1810.  doi: 10.1080/03605300600858960.  Google Scholar

[5]

E. Emmrich and R. Lasarzik, Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions, Discrete Contin. Dyn. Syst., 38 (2018), 4617-4635.  doi: 10.3934/dcds.2018202.  Google Scholar

[6]

E. FeireislY. Lu and A. Novotný, Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law, Sci. China Math., 61 (2018), 2003-2016.  doi: 10.1007/s11425-017-9272-7.  Google Scholar

[7]

E. FeireislB. J. Jin and A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9.  Google Scholar

[8]

J. Forster, Variational Approach to the Modeling and Analysis of Magnetoelastic Materials, Ph.D thesis, University of Würzburg, 2016. Google Scholar

[9]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1.  Google Scholar

[10]

M. Grasselli and H. Wu, Long-time behavior for a nematic liquid crystal model with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002.  doi: 10.1137/120866476.  Google Scholar

[11]

X. P. Hu and F. H. Lin, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Commun. Pure Appl. Math., 69 (2016), 372-404.  doi: 10.1002/cpa.21561.  Google Scholar

[12]

X. P. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 3437-3461.  doi: 10.3934/dcds.2015.35.3437.  Google Scholar

[13]

Y. HyonD. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: maximum dissipation principle, Discrete Contin. Dyn. Syst., 26 (2010), 1291-1304.  doi: 10.3934/dcds.2010.26.1291.  Google Scholar

[14]

N. Jing, H. Liu, and Y. L. Luo, Global classical solutions to an evolutionary model for magnetoelasticity, preprint, arXiv: 1904.09531v1. Google Scholar

[15]

M. Kalousek, On dissipative solutions to a system arising in viscoelasticity, preprint, arXiv: 1903.03635. doi: 10.1007/s00021-019-0459-9.  Google Scholar

[16]

M. Kalousek, J. Kortum and A. Schlömerkemper, Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity, preprint, arXiv: 1904.07179. Google Scholar

[17]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398.  doi: 10.1007/s00205-007-0089-x.  Google Scholar

[18]

Z. Lei and Y. Zhou, Global existence of classical solutions for 2D Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.  doi: 10.1137/040618813.  Google Scholar

[19]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.  Google Scholar

[20]

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

[21]

F. H. Lin and P. Zhang, On the initial boundary value problem of the incompressible viscoelastic fluid system, Commun. Pure Appl. Math., 61 (2008), 539-558.  doi: 10.1002/cpa.20219.  Google Scholar

[22]

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

[23]

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

[24]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.  Google Scholar

[25]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396.  doi: 10.3934/dcds.2010.26.379.  Google Scholar

[26]

H. WuX. Xu and C. Liu, On the general Ericksen-Leslie system: Parodi's relation, wellposedness and stability, Arch. Ration. Mech. Anal., 208 (2013), 59-107.  doi: 10.1007/s00205-012-0588-2.  Google Scholar

[27]

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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