WEAK-STRONG UNIQUENESS FOR THREE DIMENSIONAL INCOMPRESSIBLE ACTIVE LIQUID CRYSTALS

  • Fan YANG ,
  • Congming LI
Expand
  • School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, China
E-mail: fanyang-m@sjtu.edu.cn

Received date: 2023-07-05

  Online published: 2024-08-30

Supported by

This work was partially supported by NSFC (11831003, 12031012) and the Institute of Modern Analysis-A Frontier Research Center of Shanghai.

Abstract

The hydrodynamics of active liquid crystal models has attracted much attention in recent years due to many applications of these models. In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the incompressible active liquid crystals in $\mathbb{R}^3$. Our results yield that if there exists a strong solution, then it is unique among the Leray-Hopf type weak solutions associated with the same initial data.

Cite this article

Fan YANG , Congming LI . WEAK-STRONG UNIQUENESS FOR THREE DIMENSIONAL INCOMPRESSIBLE ACTIVE LIQUID CRYSTALS[J]. Acta mathematica scientia, Series B, 2024 , 44(4) : 1415 -1440 . DOI: 10.1007/s10473-024-0413-7

References

[1] Albritton D, Brué E, Colombo M. Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. Annals of Mathematics, 2022, 196(1): 415-455
[2] De Anna F, Zarnescu A. Uniqueness of weak solutions of the full coupled Navier-Stokes and $Q$-tensor system in 2D. Communications in Mathematical Sciences, 2016, 14(8): 2127-2178
[3] Buckmaste T, Vicol V. Convex integration constructions in hydrodynamics. Bulletin of the American Mathematical Society, 2021, 58(1): 1-44
[4] Berselli L C. On a regularity criterion for the solutions to the 3D Navier-Stokes equations. Differential Integral Equations, 2002, 15: 1129-1137
[5] Buckmaster T, Vicol V. Nonuniqueness of weak solutions to the Navier-Stokes equation. Annals of Mathematics, 2019, 189(1): 101-144
[6] Buckmaster T, Vicol V. Convex integration and phenomenologies in turbulence. EMS Surveys in Mathematical Sciences, 2020, 6(1): 173-263
[7] Cheskidov A, Luo X. Sharp nonuniqueness for the Navier-Stokes equations. Inventiones Mathematicae, 2022, 229(3): 987-1054
[8] Cheskidov A, Luo X. $L_2$-critical nonuniqueness for the 2D Navier-Stokes equations. Annals of PDE, 2023, 9(2): Art 13
[9] Chen G Q, Majumdar A, Wang D, Zhang R. Global weak solutions for the compressible active liquid crystal system. SIAM Journal on Mathematical Analysis, 2018, 50(4): 3632-3675
[10] Chen Y, Wang D, Zhang R. On mathematical analysis of complex fluids in active hydrodynamics. Electronic Research Archive, 2021, 29(6): 3817-3832
[11] Chen G Q, Majumdar A, Wang D, Zhang R. Global existence and regularity of solutions for active liquid crystals. Journal of Differential Equations, 2017, 263(1): 202-239
[12] Dong H, Du D. The Navier-Stokes equations in the critical Lebesgue space. Communications in Mathematical Physics, 2009, 292(3): 811-827
[13] Du H, Hu X, Wang C. Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three. Archive for Rational Mechanics and Analysis, 2020, 238(2): 749-803
[14] Escauriaza L, Seregin G A, Šverák V. $L_{3,\infty}$-solutions of the Navier-Stokes equations and backward uniqueness. Russian Mathematical Surveys, 2003, 58(2): 211-250
[15] Fabes E B, Jones B F, Rivière N M. The initial value problem for the Navier-Stokes equations with data in $L^p$. Archive for Rational Mechanics and Analysis, 1972, 45(3): 222-240
[16] Giomi L, Bowick M J, Ma X, Marchetti M C. Defect annihilation and proliferation in active nematics. Physical Review Letters, 2013, 110(22): 228101
[17] Giomi L, Mahadevan L, Chakraborty B, Hagan M F. Excitable patterns in active nematics. Physical Review Letters, 2011, 106(21): 218101
[18] Giga Y. Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system. Journal of Differential Equations, 1986, 62(2): 186-212
[19] Galdi G P, Heywood J G, Rannacher R.Fundamental Directions in Mathematical Fluid Mechanics. Basle: Birkhäuser, 2012
[20] Guillén-González F, Rodríguez-Bellido M Á. Weak solutions for an initial-boundary $Q$-tensor problem related to liquid crystals. Nonlinear Analysis: Theory, Methods & Applications, 2015, 112: 84-104
[21] De Gennes P G, Prost J. The Physics of Liquid Crystals. Oxford: Oxford University Press, 1995
[22] Hopf E. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet. Mathematische Nachrichten, 1950, 4: 213-231
[23] Huang J R, Ding S J. Global well-posedness for the dynamical $Q$-tensor model of liquid crystals. Science China Mathematics, 2015, 58: 1349-1366
[24] Huang T. Regularity and uniqueness for a class of solutions to the hydrodynamic flow of nematic liquid crystals. Analysis and Applications, 2016, 14(4): 523-536
[25] Jia H, Sverak V. Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?. Journal of Functional Analysis, 2015, 268(12): 3734-3766
[26] Kozono H, Sohr H. Remark on uniqueness of weak solutions to the Navier-Stokes equations. Analysis, 1996, 16: 255-271
[27] Kato T. Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions. Mathematische Zeitschrift, 1984, 187(4): 471-480
[28] Leray J. Sur le mouvement d'un liquide visqueux remplissant l'espace. Acta Mathematica, 1934, 63: 193-248
[29] Ladyzhenskaya O A. On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations. Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory, 1969, 5: 60-66
[30] Lian W, Zhang R. Global weak solutions to the active hydrodynamics of liquid crystals. Journal of Differential Equations, 2020, 268(8): 4194-4221
[31] Lions P L, Masmoudi N. Uniqueness of mild solutions of the Navier-Stokes system in $L^N$. Communications in Partial Differential Equations, 2001, 26(11/12): 2211-2226
[32] Lin F, Wang C. Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2014, 372(2029): 20130361
[33] Lemarié-Rieusset P G. The Navier-Stokes Problem in the 21st Century. Boca, Raton: CRC Press, 2018
[34] Luo X. Stationary solutions and nonuniqueness of weak solutions for the Navier-Stokes equations in high dimensions. Archive for Rational Mechanics and Analysis, 2019, 233(2): 701-747
[35] Prodi G. Un teorema di unicià per le equazioni di Navier-Stokes. Annali di Matematica Pura ed Applicata, 1959, 48(1): 173-182
[36] Paicu M, Zarnescu A. Global existence and regularity for the full coupled Navier-Stokes and $Q$-tensor system. SIAM Journal on Mathematical Analysis, 2011, 43(5): 2009-2049
[37] Paicu M, Zarnescu A. Energy dissipation and regularity for a coupled Navier-Stokes and $Q$-tensor system. Archive for Rational Mechanics and Analysis, 2012, 203: 45-67
[38] Serrin J. On the interior regularity of weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 1962, 9: 187-191
[39] Serrin J.The initial value problem for the Navier-Stokes equations//Langer R E. Nonlinear Problems. Wisconsin: Univ Wisconsin Press, 1963: 69-98
[40] Shinbrot M. The energy equation for the Navier-Stokes system. SIAM Journal on Mathematical Analysis, 1974, 5(6): 948-954
[41] Struwe M. On partial regularity results for the Navier-Stokes equations. Communications on Pure and Applied Mathematics, 1988, 41(4): 437-458
[42] Wilkinson M. Strictly physical global weak solutions of a Navier-Stokes $Q$-tensor system with singular potential. Archive for Rational Mechanics and Analysis, 2015, 218(1): 487-526
[43] Wang D, Xu X, Yu C. Global weak solution for a coupled compressible Navier-Stokes and $Q$-tensor system. Communications in Mathematical Sciences, 2015, 13(1): 49-82
[44] Xiao Y. Global strong solution to the three-dimensional liquid crystal flows of $Q$-tensor model. Journal of Differential Equations, 2017, 262(3): 1291-1316
Options
Outlines

/