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Nonparabolic dissipative systems modeling the flow of liquid crystals. (English) Zbl 0842.35084
Summary: We study a simplified system which retains most of the interesting mathematical properties of the original Ericksen-Leslie equations for the flow of liquid crystals. This is a coupled nonparabolic dissipative dynamic system. We derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions. We also discuss uniqueness and some stability properties of the system.

35Q35 PDEs in connection with fluid mechanics
76A15 Liquid crystals
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B35 Stability in context of PDEs
Full Text: DOI
[1] Chen, Comm. Anal. Geom. 1 pp 327– (1993) · Zbl 0845.35049 · doi:10.4310/CAG.1993.v1.n3.a1
[2] Chen, Math. Z. 201 pp 83– (1989)
[3] Ericksen, Res. Mechanica 21 pp 381– (1987)
[4] Ericksen, Trans. Soc. Rheol. 5 pp 22– (1961)
[5] Equilibrium theory of liquid crystals, pp. 233–398 in: Advances in Liquid Crystals, Vol. 2, ed., Academic Press, New York, 1975.
[6] Ericksen, Arch. Rational Mech. Anal. 113 pp 97– (1991)
[7] and , eds., Theory and Applications of Liquid Crystals, IMA Vol. 5, Springer-Verlag, New York, 1986.
[8] and , Elliptic Partial Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin, 1983. · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[9] Hardt, Comm. Math. Phys. 105 pp 547– (1986)
[10] Keller, SIAM J. Appl. Math. 49 pp 116– (1989)
[11] The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
[12] , and , Linear and Quasilinear Equations Of Parabolic Type, Transl. Math. Monographs, Vol. 23, American Mathematical Society, 1968.
[13] Theory of flow phenomena in liquid crystals, pp. 1–81 in: Advances in Liquid Crystals, Vol. 4, ed., Academic Press, New York, 1979.
[14] Leslie, Arch. Rational Mech. Anal. 28 pp 265– (1968)
[15] Lin, Comm Pure. Appl. Math. 42 pp 789– (1989)
[16] Mathematics theory of liquid crystals, in: Applied Mathematics at the Turn of the Century, to appear.
[17] Serrin, Arch. Rational Mech. Anal. 9 pp 187– (1962)
[18] The initial value problem for Navier-Stokes equations, pp. 69–98 in: Nonlinear Problems, ed., University of Wisconsin Press, 1963.
[19] Navier-Stokes Equations, rev. ed., Studies in Mathematics and its Applications 2, North-Holland, Amsterdam, 1977.
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