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Time-periodic weak solutions to incompressible generalized Newtonian fluids. (English) Zbl 1472.35282

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.

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
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[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
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