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]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[3]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[4]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[5]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[6]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[7]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[8]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[9]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[10]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[11]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[12]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[13]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[14]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[15]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322

[16]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[17]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[18]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[19]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[20]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (147)
  • HTML views (124)
  • Cited by (0)

Other articles
by authors

[Back to Top]