Existence and blow up criterion for strong solutions to the compressible biaxial nematic liquid crystal flow. (English) Zbl 1530.35217
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A15 | Liquid crystals |
| 35D35 | Strong solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
References:
| [1] | GoversE, VertogenG. Elastic continuum theory of biaxial nematics. Phys Rev A. 1984;1984:30. |
| [2] | GoversE, VertogenG. Erratum: slastic continuum theory of biaxial nematics [Phys. Rev. A, 1984, 30]. Phys Rev A. 1985;1985:31. |
| [3] | HuangJ, LinJ. Orientability and asymptotic convergence of Q‐tensor flow of biaxial nematic liquid crystals. Cal Var PDE. 2022;61:173. · Zbl 1503.35164 |
| [4] | HuangJ, DingS. Compressible hydrodynamic flow of nematic liquid crystals with vacuum. J Differ Equ. 2015;258:1653‐1684. · Zbl 1446.76070 |
| [5] | HuangT, WangC, WenH. Strong solutions of the compressible nematic liquid crystal flow. J Differ Equ. 2012;252:2222‐2265. · Zbl 1233.35168 |
| [6] | DuH, LiY, WangC. Weak solutions of non‐isothermal nematic liquid crystal flow in dimension three. J Elliptic Parabolic Equ. 2020;6:71‐98. · Zbl 1442.35334 |
| [7] | DingS, WangC, WenH. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete Contin Dyn Syst Ser B. 2011;15(2):357‐371. · Zbl 1217.35099 |
| [8] | JiangF, JiangS, WangD. Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions. Arch Rational Mech Anal. 2014;214:40‐451. · Zbl 1307.35225 |
| [9] | LinJ, LaiB, WangC. Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three. SIAM J Math Anal. 2015;47:2952‐2983. · Zbl 1321.35167 |
| [10] | MaS. Classical solutions for the compressible liquid crystal flows with nonnegative initial densities. Anal Appl. 2013;397:595‐618. · Zbl 1256.35088 |
| [11] | YangY, DouC, JuQ. Weak‐strong uniqueness property for the compressible flow of liquid crystals. J Differ Equ. 2013;25:1233‐1253. · Zbl 1441.76020 |
| [12] | HuangT, WangC, WenH. Blow up criterion for compressible nematic liquid crystal flows in dimension three. Arch Rational Mech Anal. 2012;204:285‐311. · Zbl 1314.76010 |
| [13] | LiuL, LiuX. A blow up criterion of strong solutions to the compressible liquid crystals system. Chinese Ann Math Ser A. 2011;32(4):393‐406. · Zbl 1249.82030 |
| [14] | LiuS, ZhaoX, ChenY. A new blow up criterion for strong solutions of the compressible nematic liquid crystal flow. Disc Cont Dyn Syst Ser B. 2020;25:4515‐4533. · Zbl 1451.35132 |
| [15] | ZhuW, ChenX, FanJ. A blow up criterion for compressible nematic liquid crystal flows. Appl Math Comput. 2013;219:7365‐7368. · Zbl 1453.76191 |
| [16] | SeveringK, SaalwachterK. Biaxial nematic phase in a thermotropic liquid‐crystalline side‐chain polymer. Phys Rev Lett. 2004;92:125501. |
| [17] | LinJ, LiY, WangC. On static and hydrodynamic biaxial nematic liquid crystals. arXiv:2006.04207; 2020. |
| [18] | DuH, HuangT, WangC. Weak compactness of simplified nematic liquid flows in 2D. arXiv:2006.04210v1; 2020. |
| [19] | GongW, LinJ. Existence of solutions to incompressible biaxial nematic liquid crystals flows. Applicable Analysis, 2021, doi:10.1080/00036811.2021.1909723 · Zbl 1504.35324 |
| [20] | LiS, WangC, XuJ. Well‐posedness of frame hydrodynamics for biaxial nematic liquid crystals. arXiv:2202.08998v1. |
| [21] | LiS, XuJ. Frame hydrodynamics of biaxial nematics from molecular‐theory‐based tensor models. arXiv:2110.12137v1, in revision at SIAM J. Appl. Math. |
| [22] | ZhuL, LinJ. Existence and uniqueness of solution to one‐dimensional compressible biaxial nematic liquid crystal flows. Z Angew Math Phys. 2022;2022:73. doi:10.1007/s00033‐021‐01670‐z · Zbl 1514.35370 |
| [23] | WahlW. Estimating [( \nabla \]\) u by divu and curlu. Math Methods Appl Sci. 1992;15:123‐143. · Zbl 0747.31006 |
| [24] | ChoY, ChoeH, KimH. Unique solvability of the initial boundary value problems for compressible viscous fluids. J Math Pures Appl. 2004;83:243‐275. · Zbl 1080.35066 |
| [25] | SunY, WangC, ZhangZ. A Beale‐Kato‐Majda blow‐up criterion for the 3‐D compressible Navier‐Stokes equations. J Math Pures Appl. 2011;95:36‐47. · Zbl 1205.35212 |
| [26] | BourguignonJ, BrezisH. Remarks on the Euler equation. J Funct Anal. 1974;15:341‐363. · Zbl 0279.58005 |
| [27] | PadulaM. Existence of global solutions for 2‐dimensional viscous compressible flows. J Fuct Anal. 1986;69:1‐20. · Zbl 0633.76072 |
| [28] | SimonJ. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J Math Anal. 1990;21(5):1093‐1117. · Zbl 0702.76039 |
| [29] | ChoeH, KimH. Strong solutions of the Navier‐Stokes equations for isentropic compressible fluids. J Differ Equ. 2003;190:504‐523. · Zbl 1022.35037 |
| [30] | AcquistapaceP. On BMO regularity for linear elliptic systems. Ann Math Pur Appl. 1992;1992:231‐269. · Zbl 0802.35015 |
| [31] | BrézisH, WaingerS. A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm PDE. 1980;5:773‐789. · Zbl 0437.35071 |
| [32] | KozonoH, TaniuchiY. Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Comm Math Phys. 2000;214:191‐200. · Zbl 0985.46015 |
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.
