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Global existence and regularity for the dynamics of viscous oriented fluids

1 DIMAI, Università di Firenze, viale Morgagni 67/a, I-50134 Firenze, Italy
2 DICEA, Universita di Firenze, via Santa Marta 3, I-50136 Firenze, Italy

Special Issues: Initial and Boundary Value Problems for Differential Equations

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

26. P. M. Mariano, Mechanics of material mutations, Adv. Appl. Mech., 47 (2014), 1-91    

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.    

29. R. Selmi, Global well-posedness and convergence results for 3D-regularized Boussinesq system, Canad. J. Math., 64 (2012), 1415-1435.    

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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