Compressible micropolar equations model a class of fluids with microstructure. In this paper we establish the dissipative measure-valued solution to the micropolar fluids. We also give the weak-strong uniqueness principle to this system which means its dissipative measure-valued solution is the same as the classical solution, provided they emanate from the same initial data.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Amirat, Y., Hamdache, K.: Weak solutions to the equations of motion for compressible magnetic fluids. J. Math. Pures Appl. 91, 433–467 (2009)
Brenner, H.: Navier–Stokes revisited. Physica A 349, 60–132 (2005)
Chen, M., Xu, X., Zhang, J.: Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum. Commun. Math. Sci. 13, 225–247 (2015)
DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667–689 (1987)
Dong, B.-Q., Li, J., Wu, J.: Global well-posedness and large-time decay for the 2D micropolar equations. J. Differ. Equ. 262, 3488–3523 (2017)
Dražić, I., Mujaković, N.: 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: large time behavior of the solution. J. Math. Anal. Appl. 431, 545–568 (2015)
Duan, R.: Global strong solution for initial-boundary value problem of one-dimensional compressible micropolar fluids with density dependent viscosity and temperature dependent heat conductivity. Nonlinear Anal. Real World Appl. 42, 71–92 (2018)
Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)
Feireisl, E., Gwiazda, P., Świerczewska Gwiazda, A., Wiedemann, E.: Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. Partial Differ. Equ. 55, pp. Art. 141, 20 (2016)
Galdi, G.P., Rionero, S.: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int. J. Eng. Sci. 15, 105–108 (1977)
Germain, P.: Weak-strong uniqueness for the isentropic compressible Navier–Stokes system. J. Math. Fluid Mech. 13, 137–146 (2011)
Liu, Q., Zhang, P.: Optimal time decay of the compressible micropolar fluids. J. Differ. Equ. 260, 7634–7661 (2016)
Łukaszewicz, G.: Micropolar Fluids. Modeling and Simulation in Science, Engineering and Technology. Theory and Applications. Birkhäuser Boston, Inc., Boston (1999)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Springer Basel AG, Basel (1995)
Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. of Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1996)
Mellet, A., Vasseur, A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier–Stokes equations. SIAM J. Math. Anal. 39, 1344–1365 (2008)
Mujaković, N.: One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution. Rad. Mat. 10, 181–193 (2001)
Neustupa, J.: Measure-valued solutions of the Euler and Navier–Stokes equations for compressible barotropic fluids. Math. Nachr. 163, 217–227 (1993)
Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow, vol. 27 of Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2004)
Nowakowski, B.: Large time existence of strong solutions to micropolar equations in cylindrical domains. Nonlinear Anal. Real World Appl. 14, 635–660 (2013)
Pedregal, P.: Parametrized Measures and Variational Principles, vol. 30 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Verlag, Basel (1997)
Rojas-Medar, M.A., Boldrini, J.L.: Magneto-micropolar fluid motion: existence of weak solutions. Rev. Mat. Complut. 11, 443–460 (1998)
Wu, Z., Wang, W.: The pointwise estimates of diffusion wave of the compressible micropolar fluids. J. Differ. Equ. 265, 2544–2576 (2018)
Yamaguchi, N.: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math. Methods Appl. Sci. 28, 1507–1526 (2005)
Yamazaki, K.: Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete Contin. Dyn. Syst. 35, 2193–2207 (2015)
The work of B.-K. Huang is supported by the grant from NNSFC under the contract 11901148 and ”the Fundamental Research Funds for the Central Universities”. B.-K. Huang is grateful to Prof. Eduard Feireisl for his hospitality during the his visit to Institute of Mathematics of the Academy of the Czech Republic. He also express his gartitude to the anonymous referee for his/her suggestions that considerably have improved the presentation of the results
Conflict of interest
The authors declare that they have no conflict of interest.
Communicated by E. Feireisl
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original version of this article was revised due to a retrospective Open Access cancellation.
Rights and permissions
About this article
Cite this article
Huang, B. On the Existence of Dissipative Measure-Valued Solutions to the Compressible Micropolar System. J. Math. Fluid Mech. 22, 59 (2020). https://doi.org/10.1007/s00021-020-00529-z
- Micropolar fluid flow
- Measure-valued solution
- Weak-strong uniqueness