September  2018, 38(9): 4637-4655. doi: 10.3934/dcds.2018203

Local well-posedness and blow-up criteria of magneto-viscoelastic flows

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

* Corresponding author: Wenjing Zhao

Received  December 2017 Published  June 2018

In this paper, we investigate a hydrodynamic system that models the dynamics of incompressible magneto-viscoelastic flows. First, we prove the local well-posedness of the initial boundary value problem in the periodic domain. Then we establish a blow-up criterion in terms of the temporal integral of the maximum norm of the velocity gradient. Finally, an analog of the Beale-Kato-Majda criterion is derived.

Citation: Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
References:
[1]

J. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349. Google Scholar

[2]

B. Benešová, J. Forster, C. Liu and A. Schlömerkemper, Existence of weak solutions to an evolutionary model for magnetoelasticity, arXiv: 1608.02992.Google Scholar

[3]

B. BenešováJ. ForsterC. García-CerveraC. Liu and A. Schlömerkemper, Analysis of the flow of magnetoelastic materials, PAMM, 16 (2016), 663-664. Google Scholar

[4]

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. Differential Equations, 255 (2013), 24-57. doi: 10.1016/j.jde.2013.03.009. Google Scholar

[5]

J. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317. Google Scholar

[6]

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 Differential Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960. Google Scholar

[7]

Z.-F. FengC.-J. Zhu and R.-Z. Zi, Blow-up criterion for the incompressible viscoelastic flows, J. Funct. Anal., 272 (2017), 3742-3762. doi: 10.1016/j.jfa.2016.10.024. 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]

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

[10]

X.-P. Hu and R. Hynd, A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15 (2013), 431-437. doi: 10.1007/s00021-012-0124-z. Google Scholar

[11]

X.-P. Hu and F.-H. Lin, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Comm. 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]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704. Google Scholar

[15]

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

[16]

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

[17]

T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, RAM: Research in Applied Mathematics, 32. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar

[18]

F.-H. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919. doi: 10.1002/cpa.21402. Google Scholar

[19]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. 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, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. Google Scholar

[21]

F.-H. Lin and C.-Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130361, 18pp. Google Scholar

[22]

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

[23]

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

[24]

Q. Liu and S.-B. Cui, Regularity of solutions to 3-D nematic liquid crystal flows, Electron. J. Differential Equations, 173 (2010), 1-5. Google Scholar

[25]

Q. LiuJ.-H. Zhao and S.-B. Cui, Logarithmically improved BKM's criterion for the 3D nematic liquid crystal flows, Nonlinear Anal., 75 (2012), 4942-4949. doi: 10.1016/j.na.2012.04.009. Google Scholar

[26]

N. MasmoudiP. Zhang and Z.-F. Zhang, Global well-posedness for 2D polymeric fluid models and growth estimate, Phys. D, 237 (2008), 1663-1675. doi: 10.1016/j.physd.2008.03.020. Google Scholar

[27]

A. Schlömerkemper and J. Žabensky, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, arXiv: 1703.07858.Google Scholar

[28]

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

[29]

B.-Q. Yuan, Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow, Discrete Contin. Dyn. Syst., 33 (2013), 2211-2219. doi: 10.3934/dcds.2013.33.2211. Google Scholar

[30]

B.-Q. Yuan and R. Li, The blowup criterion of a smooth solution to the incompressible viscoelastic flow, J. Math. Anal. Appl., 406 (2013), 158-164. doi: 10.1016/j.jmaa.2013.04.055. Google Scholar

[31]

S. Zheng, Nonlinear Evolution Equations, Pitman Ser. Monogr. and Surv. on Pure and Appl. Math., 133, Chapman & Hall/CRC, Boca Raton, FL 2004. doi: 10.1201/9780203492222. Google Scholar

show all references

References:
[1]

J. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349. Google Scholar

[2]

B. Benešová, J. Forster, C. Liu and A. Schlömerkemper, Existence of weak solutions to an evolutionary model for magnetoelasticity, arXiv: 1608.02992.Google Scholar

[3]

B. BenešováJ. ForsterC. García-CerveraC. Liu and A. Schlömerkemper, Analysis of the flow of magnetoelastic materials, PAMM, 16 (2016), 663-664. Google Scholar

[4]

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. Differential Equations, 255 (2013), 24-57. doi: 10.1016/j.jde.2013.03.009. Google Scholar

[5]

J. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317. Google Scholar

[6]

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 Differential Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960. Google Scholar

[7]

Z.-F. FengC.-J. Zhu and R.-Z. Zi, Blow-up criterion for the incompressible viscoelastic flows, J. Funct. Anal., 272 (2017), 3742-3762. doi: 10.1016/j.jfa.2016.10.024. 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]

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

[10]

X.-P. Hu and R. Hynd, A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15 (2013), 431-437. doi: 10.1007/s00021-012-0124-z. Google Scholar

[11]

X.-P. Hu and F.-H. Lin, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Comm. 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]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704. Google Scholar

[15]

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

[16]

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

[17]

T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, RAM: Research in Applied Mathematics, 32. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar

[18]

F.-H. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919. doi: 10.1002/cpa.21402. Google Scholar

[19]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. 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, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. Google Scholar

[21]

F.-H. Lin and C.-Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130361, 18pp. Google Scholar

[22]

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

[23]

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

[24]

Q. Liu and S.-B. Cui, Regularity of solutions to 3-D nematic liquid crystal flows, Electron. J. Differential Equations, 173 (2010), 1-5. Google Scholar

[25]

Q. LiuJ.-H. Zhao and S.-B. Cui, Logarithmically improved BKM's criterion for the 3D nematic liquid crystal flows, Nonlinear Anal., 75 (2012), 4942-4949. doi: 10.1016/j.na.2012.04.009. Google Scholar

[26]

N. MasmoudiP. Zhang and Z.-F. Zhang, Global well-posedness for 2D polymeric fluid models and growth estimate, Phys. D, 237 (2008), 1663-1675. doi: 10.1016/j.physd.2008.03.020. Google Scholar

[27]

A. Schlömerkemper and J. Žabensky, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, arXiv: 1703.07858.Google Scholar

[28]

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

[29]

B.-Q. Yuan, Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow, Discrete Contin. Dyn. Syst., 33 (2013), 2211-2219. doi: 10.3934/dcds.2013.33.2211. Google Scholar

[30]

B.-Q. Yuan and R. Li, The blowup criterion of a smooth solution to the incompressible viscoelastic flow, J. Math. Anal. Appl., 406 (2013), 158-164. doi: 10.1016/j.jmaa.2013.04.055. Google Scholar

[31]

S. Zheng, Nonlinear Evolution Equations, Pitman Ser. Monogr. and Surv. on Pure and Appl. Math., 133, Chapman & Hall/CRC, Boca Raton, FL 2004. doi: 10.1201/9780203492222. Google Scholar

[1]

Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332

[2]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112

[3]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[4]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

[5]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[6]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025

[7]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

[8]

Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. An existence theorem for the magneto-viscoelastic problem. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 435-447. doi: 10.3934/dcdss.2012.5.435

[9]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917

[10]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

[11]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[12]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

[13]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[14]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[15]

Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263

[16]

Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147

[17]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[18]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[19]

Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072

[20]

Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (83)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]