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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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