WEAK-STRONG UNIQUENESS FOR THREE DIMENSIONAL INCOMPRESSIBLE ACTIVE LIQUID CRYSTALS

doi:10.1007/s10473-024-0413-7

摘要/Abstract

摘要： 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.

关键词: analysis of parabolic and elliptic types, weak-strong uniqueness, active liquid crystals, weak solution, energy equality

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.

Key words: analysis of parabolic and elliptic types, weak-strong uniqueness, active liquid crystals, weak solution, energy equality

中图分类号:

35A02

Fan YANG, Congming LI. WEAK-STRONG UNIQUENESS FOR THREE DIMENSIONAL INCOMPRESSIBLE ACTIVE LIQUID CRYSTALS[J]. 数学物理学报(英文版), 2024, 44(4): 1415-1440.

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.

参考文献

[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