LOCAL WELL-POSEDNESS FOR AN ERICKSEN-LESLIE S PARABOLIC-HYPERBOLIC COMPRESSIBLE NON-ISOTHERMAL MODEL FOR LIQUID CRYSTALS

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1 Electronic Journal of Differential Equations, Vol. 017 (017), No. 3, pp ISSN: URL: or LOCAL WELL-POSEDNESS FOR AN ERICKSEN-LESLIE S PARABOLIC-HYPERBOLIC COMPRESSIBLE NON-ISOTHERMAL MODEL FOR LIQUID CRYSTALS JISHAN FAN, TOHRU OZAWA Communicated by Mitsuharo Otani Abstract. In this article we prove the local well-posedness for an Ericksen- Leslie s parabolic-hyperbolic compressible non-isothermal model for nematic liquid crystals with positive initial density. 1. Introduction We consider the following Ericksen-Leslie system modeling the hydrodynamic flow of compressible nematic liquid crystals [1,, 3, 4, 5]: t ρ + div(ρu) = 0, (1.1) t (ρu) + div(ρu u) + p(ρ, θ) µ u (λ + µ) div u = ( d d 1 ) d I 3, (1.) t (ρe) + div(ρue) + p div u θ = µ u + ut + λ(div u) + d, (1.3) d d = d( d d ), d = 1, in R 3 (0, ), (1.4) (ρ, u, θ, d, d t )(, 0) = (ρ 0, u 0, θ 0, d 0, d 1 ) in R 3, d 0 = 1, d 0 d 1 = 0. (1.5) Here ρ, u, θ is the density, velocity and temperature of the fluid, and d represents the macroscopic average of the nematic liquid crystals orientation field. e := C V θ is the internal energy and p := Rρθ is the pressure with positive constants C V and R. The viscosity coefficients µ and λ of the fluid satisfy µ > 0 and λ + 3 µ 0. The symbol d d denotes a matrix whose (i, j)th entry is i d j d, I 3 is the identity matrix of order 3, and it is easy to see that div ( d d 1 ) d I 3 = d k d k, d := d t + u d. k u t is the transpose of vector u and t u u t. System (1.1)-(1.3) is the well-known full compressible Navier-Stokes-Fourier system. When u = 0, (1.4) reduces to the wave maps system, which is one of the most beautiful and challenging nonlinear hyperbolic system. It has captured the 010 Mathematics Subject Classification. 35Q30, 35Q35, 76N10. Key words and phrases. Compressible; liquid crystals; local well-posedness. c 017 Texas State University. Submitted February 3, 017. Published September 5,

2 J. FAN, T. OZAWA EJDE-017/3 attention of mathematicians for more than thirty years now. Moreover, the wave maps system is nothing other than the Euler-Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory. When θ is a positive constant and the equation (1.4) is replaced by a harmonic heat flow d d = d d, (1.6) this problem has received many studies. Huang, Wang and Wen [6, 7] (see also [8, 9]) show the local well-posedness of strong solutions with vacuum and prove some regularity criteria. Ding, Huang, Wen and Zi [10] (also see [11, 1]) studied the low Mach number limit. Jiang, Jiang and Wang [13] (see also [14]) proved the global existence of weak solutions in R. When the fluid is incompressible, i.e., div u = 0, the similar model has been studied in [15, 16]. The aim of this article is to prove a local-well posedness result when inf ρ 0 1/C, we will prove the following result. Theorem 1.1. Let 1/C ρ 0 C, 0 θ 0, ρ 0 H, u 0, θ 0, d 0, d 0 H 3, with d 0 = 1, d 0 d 1 = 0. Then problem (1.1)-(1.5) has a unique strong solution (ρ, u, θ, d) satisfying for some T > 0. 1 ρ C, 0 θ, d = 1, C ρ L (0, T ; H ), u, θ, d, d L (0, T ; H 3 ), u, θ L (0, T ; H 4 ), u t, θ t L (0, T ; H ) Remark 1.. When n = and taking d := t ρ + div(ρu) = 0, (1.7) ( ) cos φ, System (1.1)-(1.4) reduces to sin φ t (ρu) + div(ρu u) + p(ρ, θ) µ u (λ + µ) div u = ( φ φ 1 ) φ I, t (ρe) + div(ρue) + p div u θ = µ u + ut + λ(div u) + φ, And hence the well-known wave map φ φ = 0. d tt d = d( d d t ) reduces to the wave equation φ tt φ = 0. Remark 1.3. Let d be a smooth solution to the system (1.4) with the initial data (d, d t )(, 0) = (d 0, d 1 ), if the initial data (d 0, d 1 ) obeys the conditions d 0 = 1, d 0 d 1 = 0, then we have d = 1 and d d t = 0 for all times t. Proof of Remark 1.3. Denote w := d 1, multiplying (1.4) by d, we see that ẅ w = w( d d ).

3 EJDE-017/3 ERICKSEN-LESLIE MODEL 3 Testing the above equation by ẇ, we find that (ẇ + w )dx = wẇ( d d ) dx + w(u w) dx (u )ẇ ẇ dx = wẇ( d d ) dx j u i i w j w dx + 1 w div u dx i + 1 ẇ div u dx C (w + ẇ + w ) dx. On the other hand, we observe that w dx = w(ẇ u w) dx C (w + ẇ + w ) dx. Combining the above two estimates and using the Gronwall inequality, we finish the proof. We denote { M(t) :=1 + sup 1 0 s t ρ (, s) L + ρ(, s) L + ρ(, s) H + u(, s) H 3 + θ(, s) H 3 + d(, s) H 3 + d(, s) H 3 + u L (0,t;H 4 ) + u t L (0,t;H ) + θ L (0,t;H 4 ) + θ t L (0,t;H ). } (1.8) Theorem 1.4. Let T be the maximal time of existence for problem (1.1)-(1.5) in the sense of Theorem 1.1. Then for any t [0, T ), we have that M(t) C 0 M(0) exp( tc(m(t))) (1.9) for some given nondecreasing continuous functions C 0 ( ) and C( ). It follows from (1.9) [17, 18, 19] that sup M(t) C (1.10) 0 t T for some T (0, T ). In the proofs below, we will use the following bilinear commutator and product estimates due to Kato-Ponce [0]: D s (fg) fd s g L p C( f L p 1 D s 1 g L q 1 + D s f L p g L q ), (1.11) D s (fg) L p C( f L p 1 D s g L q 1 + D s f L p g L q ) (1.1) with s > 0 and 1 p = 1 p q 1 = 1 p + 1 q and 1 < p <. The proof of the uniqueness part is standard, we omit it here. It is easy to prove Theorem 1.1 by the Galerkin method if we have (1.9) [6], thus we only need to show a priori estimates (1.9).

4 4 J. FAN, T. OZAWA EJDE-017/3. Proof of Theorem 1.4 Since the physical constants C V and R do not bring any essential difficulties in our arguments, we shall take C V = R = 1. First, it follows from (1.1) that { ρ(x, t) = ρ 0 (y(0; x, t)) exp t where y(s; x, t) is the characteristic curve defined by Then (.1) gives dy = u(y, s), y(t; x, t) = x. ds 0 } div u(y(s; x, t), s)ds, (.1) ρ, 1 ρ C 0 exp(tc(m)). (.) Applying to (1.1), testing by ρ, we see that ρ dx = div(ρu) ρ dx C(M), which yields ρ(, t) L C 0 + tc(m). (.3) Applying D 3 to (1.1), testing by D 3 ρ, using (1.11) and (1.1), we find that (D 3 ρ) dx = (D 3 (u ρ) u D 3 ρ)d 3 ρ dx u D 3 ρ D 3 ρ dx D 3 (ρ div u)d 3 ρ dx which leads to C( u L D 3 ρ L + ρ L D 3 u L ) D 3 ρ L + C( ρ L D 3 div u L + div u L D 3 ρ L ) D 3 ρ L C(M) + C(M) D 3 div u L, It is easy to show that u(, t) H = u 0 + D 3 ρ(, t) L C + tc(m). (.4) t Testing (1.4) by d and using d d = 0, we infer that ( d + d )dx = u d d dx which implies 0 u t ds H C 0 + tc(m), (.5) θ(, t) H C 0 + tc(m). (.6) (u ) d d dx C(M), d(, t) L + d(, t) L C 0 + tc(m). (.7)

5 EJDE-017/3 ERICKSEN-LESLIE MODEL 5 Taking D 3 to (1.), testing by D 3 u and using (1.1), we derive ρ D 3 u dx + µ D 3 u dx + (λ + µ) (div D 3 u) dx = D 3 p div D 3 u dx (D 3 (ρu u) ρu D 3 u)d 3 u dx (D 3 (ρu t ) ρd 3 u t )D 3 u dx (D 3 ( d d) d D 3 d)d 3 u dx (D 3 u )d D 3 d dx (.8) =: I 1 + I + I 3 + I 4 I 5. Applying D 3 to (1.4) and testing by D 3 d, we obtain ( D 3 d + D 3 d ) dx = (D 3 (u d) u D 3 d)d 3 d dx (u )D 3 d D 3 d dx + D 3 (d( d d ))D 3 d dx + D 3 d (u D 3 d)dx + D 3 d (D 3 (u d) u D 3 d (D 3 u )d) dx + I 5 (.9) =: l 1 + l + l 3 + l 4 + l 5 + I 5. Summing (.8) and (.9), we have (ρ D 3 u + D 3 d + D 3 d ) dx + µ D 3 u dx + (λ + µ) (div D 3 u) dx 4 = (I i + l i ) + l 5. i=1 (.10) Using (1.11) and (1.1), we bound I i (i = 1,, 4) and l i (i = 1,, 5) as follows. I 1 C( ρ L D 3 θ L + θ L D 3 ρ L ) div D 3 u L C(M) div D 3 u L ; I C( (ρu) L D 3 u L + u L D 3 (ρu) L ) D 3 u L C(M); I 3 C( ρ L D u t L + u t L D 3 ρ L ) D 3 u L C(M) u t H ; I 4 C d L D 4 d L D 3 u L C(M); l 1 C u L D 3 d L + C d L D 3 u L D 3 d L C(M); l = 1 D 3 d div u dx C(M); l 3 C[ d L D 3 ( d d ) L + ( d L + d L ) D3 d L ] D 3 d L C(M);

6 6 J. FAN, T. OZAWA EJDE-017/3 l 4 = i,j = i,j u i i D 3 d j D 3 d dx j u i i D 3 d j D 3 d dx + i,j 1 i u i ( j D 3 d) dx C u L D 4 d L C(M); l 5 = D 3 d(c 1 D u Dd + C Du D d)dx = i D 3 d i (C 1 D u Dd + C Du D d)dx i C D 4 d L ( D 3 u L d L + D u L 3 D 3 d L 6 + u L D 4 d L ) C(M). Inserting the above estimates into (.10), we have (ρ D 3 u + D 3 d + D 3 d )dx + µ D 3 u dx + (λ + µ) (div D 3 u) dx C(M) div D 3 u L + C(M) + C(M) u t H. Integrating the above estimates in [0, t], we arrive at t D 3 u(, t) L + D3 d(, t) L + D3 d(, t) L + D 4 u dxds C 0 exp( tc(m)). On the other hand, it follows from (1.) that u t = u u + 1 [ µ u + (λ + µ) div u p ( d d 1 ] ρ d I 3 ) which easily implies 0 (.11) (.1) u t L (0,t;H ) C 0 exp( tc(m)). (.13) Applying D 3 to (1.3), testing by D 3 θ and using (1.1), (1.11), and (1.1), we have ρ(d 3 θ) dx + D 3 θ dx = (D 3 (ρu θ) ρu D 3 θ)d 3 θ dx D 3 (p div u) D 3 θ dx (D 3 (ρθ t ) ρd 3 θ t )D 3 θ dx + D 3[ µ u + ut + λ(div u) + d D 3 θ dx C( (ρu) L D 3 θ L + θ L D 3 (ρu) L ) D 3 θ L + C( p L D 3 div u L + div u L D 3 p L ) D 3 θ L + C( ρ L D θ t L + θ t L D 3 ρ L ) D 3 θ L + C( u L D 4 u L + d L D 3 d L ) D 3 θ L

7 EJDE-017/3 ERICKSEN-LESLIE MODEL 7 which gives C(M) + C(M) D 3 div u L + C(M) θ t H + C(M) D 4 u L, t D 3 θ(, t) L + D 4 θ dx ds C 0 exp( tc(m)). (.14) 0 On the other hand, from (1.3) if follows that θ t = u θ p ρ div u + 1 [ µ ρ u + ut + λ(div u) + d ], (.15) which easily leads to This completes the proof. θ t L (0,t;H ) C 0 exp( tc(m)). (.16) Acknowledgments. J. Fan is supported by the NSFC (Grant No ). References [1] J. L. Ericksen; Conservation laws for liquid crystals, Trans. Soc. Rheo., 5 (1961) [] J. L. Ericksen; Continuum thoery of nematic liquid crystals, Res Mech., 1 (1987) [3] J. L. Ericksen; Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., 113 () (1990), [4] F. Leslie; Theory of Flow Phenomenum in Liquid Crystals, Springer, New York, NY, USA, [5] N. Jiang, Y. Luo; On well-posedness of Ericksen-Leslie s parabolic-hyperbolic liquid crystal model, arxiv: [6] T. Huang, C. Y. Wang, H. Y. Wen; Strong solutions of the compressible nematic liquid crystal flow, J. Differ. Equ. 5 (01), -65. [7] T. Huang, C. Y. Wang, H. Y. Wen; Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Ration. Mech. Anal., 04 (01), [8] X. Li, B. Guo; Well-posedness for the three-dimensional compressible liquid crystal flows, Discrete and Continuous Dynamical Systems-Series S, 9 (6) (016), [9] Y. Chu, X. Liu, X. Liu; Strong solutions to the compressible liquid crystal system, Pacific J. of Math., 57 (1) (01), [10] S. J. Ding, J. R. Huang, H. Y. Wen, R. Z. Zi; Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 64 (013), [11] G. Qi, J. Xu; The low Mach number limit for the compressible flow of liquid crystals, Appl. Math. Comput., 97 (017), [1] X. Yang; Uniform well-posedness and low Mach number limit to the compressible nematic liquid flows in a bounded domain, Nonlinear Analysis: Theory, Methods & Applications, 10 (015), [13] F. Jiang, S. Jiang, D. H. Wang; On multi-dimensional compressible flow of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 65 (1) (013), [14] F. Lin, C. Wang; Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Phil. Trans. R. Soc. A Math. Phys. Eng. Sci., 37 (09) (014). [15] E. Feireisl, M. Fremond, E. Rocca, G. Schimperna; A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 05 () (01), [16] W. Gu, J. Fan, Y. Zhou; Regularity criteria for some simplified non-isothermal models for nematic liquid crystals, Computers and Mathematics with Applications, 7 (016), [17] T. Alazard; Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (006), [18] C. Dou, S. Jiang, Y. Ou; Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Diff. Eqs., 58 (015), [19] G. Metivier, S. Schochet; The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (001),

8 8 J. FAN, T. OZAWA EJDE-017/3 [0] T. Kato, G. Ponce; Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math., 41 (1988), Jishan Fan Department of Applied Mathematics, Nanjing Forestry University, Nanjing 10037, China address: Tohru Ozawa (corresponding author) Department of Applied Physics, Waseda University, Tokyo, , Japan address: