Abbatiello, Anna Time-periodic weak solutions to incompressible generalized Newtonian fluids. (English) Zbl 1472.35282 J. Math. Fluid Mech. 23, No. 3, Paper No. 63, 20 p. (2021). Summary: In this study we are interested in the Navier-Stokes-like system for generalized viscous fluids whose viscosity has a power-structure with exponent \(q\). We develop an existence theory of time-periodic three-dimensional flows. Cited in 1 Document MSC: 35Q35 PDEs in connection with fluid mechanics 35Q30 Navier-Stokes equations 35D30 Weak solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B10 Periodic solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids Keywords:time-periodic solution; weak solution; global existence; generalized Newtonian fluid; viscous fluid PDFBibTeX XMLCite \textit{A. Abbatiello}, J. Math. Fluid Mech. 23, No. 3, Paper No. 63, 20 p. (2021; Zbl 1472.35282) Full Text: DOI arXiv References: [1] Abbatiello, A.; Feireisl, E., On a class of generalized solutions to equations describing incompressible viscous fluids, Ann. Mat. Pura Appl. (4), 199, 3, 1183-1195 (2020) · Zbl 1434.35082 · doi:10.1007/s10231-019-00917-x [2] Abbatiello, A.; Crispo, F.; Maremonti, P., Electrorheological fluids: ill posedness of uniqueness backward in time, Nonlinear Anal., 170, 47-69 (2018) · Zbl 1469.35172 · doi:10.1016/j.na.2017.12.014 [3] Abbatiello, A.; Maremonti, P., Existence of regular time-periodic solutions to shear-thinning fluids, J. Math. Fluid Mech., 21, 2, 14 (2019) · Zbl 1416.35177 · doi:10.1007/s00021-019-0435-4 [4] Axmann, Š.; Pokorný, M., Time-periodic solutions to the full Navier-Stokes-Fourier system with radiation on the boundary, J. Math. Anal. Appl., 428, 1, 414-444 (2015) · Zbl 1318.35073 · doi:10.1016/j.jmaa.2015.03.023 [5] Barhoun, A.; Lemlih, AB, A reproductive property for a class of non-Newtonian fluids, Appl. Anal., 81, 1, 13-38 (2002) · Zbl 1024.76002 · doi:10.1080/0003681021000021042 [6] Blechta, J.; Málek, J.; Rajagopal, KR, On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion, SIAM J. Math. Anal., 52, 2, 1232-1289 (2020) · Zbl 1432.76075 · doi:10.1137/19M1244895 [7] Breit, D.; Diening, L.; Schwarzacher, S., Solenoidal Lipschitz truncation for parabolic PDEs, Math. Models Methods Appl. Sci., 23, 14, 2671-2700 (2013) · Zbl 1309.76024 · doi:10.1142/S0218202513500437 [8] Burczak, J., Modena, S., Székelyhidi, L.: Non-uniqueness of power-law flows. Arxiv Preprint Series arXiv:2007.08011 (2020) [9] Crispo, F., A note on the existence and uniqueness of time-periodic electro-rheological flows, Acta Appl. Math., 132, 237-250 (2014) · Zbl 1295.76004 · doi:10.1007/s10440-014-9897-9 [10] Crispo, F.; Grisanti, C.; Maremonti, P., Singular p-Laplacian parabolic system in exterior domains: higher regularity of solutions and related properties of extinction and asymptotic behavior in time, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19, 3, 913-949 (2019) · Zbl 1428.35188 [11] Dal Maso, G.; Murat, F., Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems, Nonlinear Anal., 31, 3-4, 405-412 (1998) · Zbl 0890.35039 · doi:10.1016/S0362-546X(96)00317-3 [12] DiBenedetto, E., Degenerate parabolic equations. Universitext (1993), New York: Springer, New York · Zbl 0794.35090 · doi:10.1007/978-1-4612-0895-2 [13] Diening, L.; Růžička, M.; Wolf, J., Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9, 1, 1-46 (2010) · Zbl 1253.76017 [14] Feireisl, E.; Matušu-Nečasová, Š.; Petzeltová, H.; Straškraba, I., On the motion of a viscous compressible flow driven by a time-periodic external flow, Arch. Ration. Mech. Anal., 149, 69-96 (1999) · Zbl 0937.35131 · doi:10.1007/s002050050168 [15] Feireisl, E.; Mucha, P.; Novotný, A.; Pokorný, M., Time periodic solutions to the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 204, 745-786 (2012) · Zbl 1287.76187 · doi:10.1007/s00205-012-0492-9 [16] Galdi, GP, Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane, Discrete Contin. Dyn. Syst. Ser. S, 6, 5, 1237-1257 (2013) · Zbl 1260.35110 [17] Galdi, GP; Grisanti, CR, Womersley flow of generalized Newtonian liquid, Proc. R. Soc. Edinb. Sect. A, 146, 4, 671-692 (2016) · Zbl 1360.76023 · doi:10.1017/S0308210515000736 [18] Galdi, GP; Kyed, M.; Giga, Y.; Novotný, A., Time-periodic solutions to the Navier-Stokes equations, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 509-578 (2018), Cham: Springer, Cham · doi:10.1007/978-3-319-13344-7_10 [19] Lions, JL, Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93, 155-175 (1965) · Zbl 0132.10601 · doi:10.24033/bsmf.1620 [20] Málek, J.; Nečas, J.; Rokyta, M.; Růžička, M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation (1996), London: Chapman & Hall, London · Zbl 0851.35002 [21] Málek, J.; Rajagopal, KR; Dafermos, CM; Feireisl, E., Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Handbook of Differential Equations: Evolutionary Equations, 371-459 (2005), Amsterdam: Elsevier, Amsterdam · Zbl 1095.35027 · doi:10.1016/S1874-5717(06)80008-3 [22] Maremonti, P., Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4, 2, 503-529 (1991) · Zbl 0737.35065 · doi:10.1088/0951-7715/4/2/013 [23] Maremonti, P., Padula, M.: Existence, uniqueness and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 233 (1996), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 27, 142-182, 257 · Zbl 0930.35126 [24] Prouse, G., Soluzioni periodiche dell’equazione di Navier-Stokes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 35, 443-447 (1963) · Zbl 0128.43504 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.