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Global existence and regularity for the dynamics of viscous oriented fluids

  • Received: 22 July 2019 Accepted: 10 October 2019 Published: 18 October 2019
  • MSC : 35L65, 74N15, 76D03, 76D05

  • We prove global existence of weak solutions to regularized versions of balance equations representing the dynamics over a torus of complex fluids, with microstructure described by a vector field taking values in the unit ball. Regularization is offered by the presence of second-neighbor microstructural interactions and our choice of filtering the balance of macroscopic momentum by inverse Helmholtz operator with unit length scale.

    Citation: Luca Bisconti, Paolo Maria Mariano. Global existence and regularity for the dynamics of viscous oriented fluids[J]. AIMS Mathematics, 2020, 5(1): 79-95. doi: 10.3934/math.2020006

    Related Papers:

  • We prove global existence of weak solutions to regularized versions of balance equations representing the dynamics over a torus of complex fluids, with microstructure described by a vector field taking values in the unit ball. Regularization is offered by the presence of second-neighbor microstructural interactions and our choice of filtering the balance of macroscopic momentum by inverse Helmholtz operator with unit length scale.


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    [1] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2 Eds., Amsterdam: Elsevier, 2003.
    [2] S. Agmon, Lectures on Elliptic Boundary Value Problems, Providence: American Mathematical Society, 2010.
    [3] J. Benameur, R. Selmi, Long time decay to the Leray solution of the two-dimensional NavierStokes equations, Bull. London Math. Soc., 44 (2012), 1001-1019. doi: 10.1112/blms/bds033
    [4] L. C. Berselli, T. Iliescu, W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Berlin: Springer-Verlag, 2006.
    [5] L. C. Berselli, R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré-Non Linéaire Anal., 29 (2012), 171-198.
    [6] L. C. Berselli, L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130. doi: 10.1016/j.na.2011.08.011
    [7] L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations, Math. Methods Appl. Sci., 38 (2015), 1437-1450. doi: 10.1002/mma.3160
    [8] L. Bisconti, D. Catania, Remarks on global attractors for the 3D Navier-Stokes equations with horizontal filtering, Commun. Pure Appl. Anal., 16 (2015), 1861-1881.
    [9] L. Bisconti, D. Catania, Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region, Discrete Contin. Dyn. Syst. Ser. B, 20 (2017), 59-75.
    [10] L. Bisconti, D. Catania, On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations, Math. Methods Appl. Sci., 41 (2018), 4923-4935. doi: 10.1002/mma.4939
    [11] L. Bisconti, P. M. Mariano, Existence results in the linear dynamics of quasicrystals with phason diffusion and non-linear gyroscopic effects, Multiscale Mod. Sim., 15 (2017), 745-767. doi: 10.1137/15M1049580
    [12] L. Bisconti, P. M. Mariano, V. Vespri, Existence and regularity for a model of viscous oriented fluid accounting for second-neighbor spin-to-spin interactions, J. Math. Fluid Mech., 20 (2018), 655-682. doi: 10.1007/s00021-017-0339-0
    [13] S. Bosia, Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows, Commun. Pure Appl. Anal., 11 (2012), 407-441.
    [14] B. Climent-Ezquerra, F. Guillen-González, M. Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys. ZAMP, 71 (2006), 984-998.
    [15] P. Constantin, C. Foias, Navier-Stokes Equations, Chicago: University of Chicago Press, 1988.
    [16] P. D'Ancona, D. Foschi, S. Sigmund, Atlas of products for wave-Sobolev spaces on $\R^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.
    [17] L. C. Evans, Partial Differential Equations, 2 Eds., Providence: American Mathematical Society, 2010
    [18] C. Foias, O. Manley, R. Rosa, et al. Navier-Stokes Equations and Turbulence, Cambridge: Cambridge University Press, 2001.
    [19] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704
    [20] C. E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0
    [21] O. A. Ladyzhenskaya, Solution in the large of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Comm. Pure Appl. Math., 12 (1959), 427-433. doi: 10.1002/cpa.3160120303
    [22] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York: Gordon & Breach, 1969.
    [23] F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503
    [24] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéeaires, Paris: Dunod, Gauthier-Villars, 1969.
    [25] P. M. Mariano, Multifield theories in mechanics of solids, Adv. Appl. Mech., 38 (2002), 1-93. doi: 10.1016/S0065-2156(02)80102-8
    [26] P. M. Mariano, Mechanics of material mutations, Adv. Appl. Mech., 47 (2014), 1-91 doi: 10.1016/B978-0-12-800130-1.00001-1
    [27] P. M. Mariano, Trends and challenges in the mechanics of complex materials: A view, Phil. Trans. Royal Soc. A, 374 (2016), 20150341.
    [28] P. M. Mariano, F. L. Stazi, Strain localization due to crack-microcrack interactions: X-FEM for a multifield approach, Comput. Meth. Appl. Mech. Eng., 193 (2004), 5035-5062. doi: 10.1016/j.cma.2003.08.010
    [29] R. Selmi, Global well-posedness and convergence results for 3D-regularized Boussinesq system, Canad. J. Math., 64 (2012), 1415-1435. doi: 10.4153/CJM-2012-013-5
    [30] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Boston: Birkhäuser Boston Inc., 1991.
    [31] J. Wu, Inviscid limits and regularity estimates for the solutions of the 2-D dissipative quasigeostrophic equations, Indiana Univ. Math. J., 46 (1997), 1113-1124.
    [32] V. G. Zvyagin, D. A. Vorotnikov, Topological approximation methods for evolutionary problems of nonlinear hydrodynamics, Berlin: Walter de Gruyter & Co., 2008.
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