| Title: | Remark on regularity criterion for weak solutions to the shear thinning fluids (English) | 
| Author: | Kim, Jae-Myoung | 
| Language: | English | 
| Journal: | Mathematica Bohemica | 
| ISSN: | 0862-7959 (print) | 
| ISSN: | 2464-7136 (online) | 
| Volume: | 149 | 
| Issue: | 3 | 
| Year: | 2024 | 
| Pages: | 287-294 | 
| Summary lang: | English | 
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| Category: | math | 
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| Summary: | J. Q. Yang (2019) established a regularity criterion for the 3D shear thinning fluids in the whole space $\Bbb {R}^3$ via two velocity components. The goal of this short note is to extend this result in viewpoint of Lorentz space. (English) | 
| Keyword: | shear thinning fluids | 
| Keyword: | regularity criterion | 
| MSC: | 35Q35 | 
| MSC: | 76D05 | 
| idZBL: | Zbl 07953703 | 
| idMR: | MR4801102 | 
| DOI: | 10.21136/MB.2023.0024-23 | 
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| Date available: | 2024-09-11T13:43:47Z | 
| Last updated: | 2024-12-13 | 
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152533 | 
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| Reference: | [1] Alghamdi, A. M., Gala, S., Ragusa, M. A., Yang, J. Q.: Regularity criterion via two components of velocity on weak solutions to the shear thinning fluids in $\Bbb R^3$.Comput. Appl. Math. 39 (2020), Article ID 234, 9 pages. Zbl 1463.35138, MR 4132926, 10.1007/s40314-020-01281-w | 
| Reference: | [2] Astarita, G., Marrucci, G.: Principles of Non-Newtonian Fluid Mechanics.McGraw-Hill, London (1974). | 
| Reference: | [3] Bae, H.-O., Choe, H. J., Kim, D. W.: Regularity and singularity of weak solutions to Ostwald-de Waele flows.J. Korean Math. Soc. 37 (2000), 957-975. Zbl 0977.76005, MR 1803282 | 
| Reference: | [4] Berselli, L. C., Diening, L., Růžička, M.: Existence of strong solutions for incompressible fluids with shear-dependent viscosities.J. Math. Fluid Mech. 12 (2010), 101-132. Zbl 1261.35118, MR 2602916, 10.1007/s00021-008-0277-y | 
| Reference: | [5] Böhme, G.: Non-Newtonian Fluid Mechanics.North-Holland Series in Applied Mathematics and Mechanics 31. North-Holland, Amsterdam (1987). Zbl 0713.76004, MR 0882542 | 
| Reference: | [6] Bosia, S., Pata, V., Robinson, J. C.: A weak-$L^p$ Prodi-Serrin type regularity criterion for the Navier-Stokes equations.J. Math. Fluid Mech. 16 (2014), 721-725. Zbl 1307.35186, MR 3267544, 10.1007/s00021-014-0182-5 | 
| Reference: | [7] 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 (2010), 1-46. Zbl 1253.76017, MR 2668872, 10.2422/2036-2145.2010.1.01 | 
| Reference: | [8] Krylov, N. V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces.Graduate Studies in Mathematics 96. AMS, Providence (2008). Zbl 1147.35001, MR 2435520, 10.1090/gsm/096 | 
| Reference: | [9] Ladyzhenskaya, O. A.: New equations for the description of the motion of viscous incompressible fluids and solvability in the large of boundary value problems for them.Tr. Mat. Inst. Steklova 102 (1967), 85-104 Russian. Zbl 0202.37802, MR 0226907 | 
| Reference: | [10] Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow.Gordon and Breach, New York (1969). Zbl 0184.52603, MR 0254401 | 
| Reference: | [11] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Etudes mathematiques. Dunod, Paris (1969), French. Zbl 0189.40603, MR 0259693 | 
| Reference: | [12] Loayza, M., Rojas-Medar, M. A.: A weak-$L^p$ Prodi-Serrin type regularity criterion for the micropolar fluid equations.J. Math. Phys. 57 (2016), Article ID 021512, 6 pages. Zbl 1342.35223, MR 3462971, 10.1063/1.4942047 | 
| Reference: | [13] Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs.Applied Mathematics and Mathematical Computation 13. Chapman & Hall, London (1996). Zbl 0851.35002, MR 1409366, 10.1007/978-1-4899-6824-1 | 
| Reference: | [14] Málek, J., Rajagopal, K. R.: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations.Evolutionary Equations. Volume II Handbook of Differential Equations. Elsevier, Amsterdam (2005), 371-459. Zbl 1095.35027, MR 2182831, 10.1016/S1874-5717(06)80008-3 | 
| Reference: | [15] O'Neil, R.: Convolution operators and $L(p,q)$ spaces.Duke Math. J. 30 (1963), 129-142. Zbl 0178.47701, MR 0146673, 10.1215/S0012-7094-63-03015-1 | 
| Reference: | [16] Pineau, B., Yu, X.: A new Prodi-Serrin type regularity criterion in velocity directions.J. Math. Fluid Mech. 20 (2018), 1737-1744. Zbl 1419.35153, MR 3877494, 10.1007/s00021-018-0388-z | 
| Reference: | [17] Pokorný, M.: Cauchy problem for the non-Newtonian viscous incompressible fluid.Appl. Math., Praha 41 (1996), 169-201. Zbl 0863.76003, MR 1382464, 10.21136/AM.1996.134320 | 
| Reference: | [18] Triebel, H.: Theory of Function Spaces.Monographs in Mathematics 78. Birkhäuser, Basel (1983). Zbl 0546.46027, MR 0781540, 10.1007/978-3-0346-0416-1 | 
| Reference: | [19] Yang, J.: Regularity criteria for 3D shear thinning fluids via two velocity components.Comput. Math. Appl. 77 (2019), 2854-2858. Zbl 1442.76011, MR 3945092, 10.1016/j.camwa.2019.01.017 | 
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