On the Long Time Dynamics of the Landau-De Gennes Gradient Flow

• Published in: Journal of statistical physics 2023, Vol.190 (1), Article 26
• Main Authors: Stanislavova, Milena, Stefanov, Atanas G.
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Gradient flow; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-022-03046-7

We study the gradient flow, generated by the Landau-De Gennes energy functional, in the physically relevant spatial dimensions d = 2 , 3 . We establish global well-posedness and global exponential time decay bounds for large H 1 data in the 2D case, and uniform bounds for large data in 3D. This is indeed the best possible outcome for unrestricted coefficients in 3D, given that steady states do exist, at least for some coefficient configurations. We also establish leading order terms and in particular sharp asymptotics for the said dynamics in 2D. In 3D, we similarly isolate the leading order term, under the necessary assumption that a given, possibly large, solution converges to zero as t → ∞ . As a corollary, we prove an asymptotic formula for the correlation functional, c ( y , t ) = ∫ R d tr ( Q ( x + y , t ) Q ( x , t ) ) d x ∫ R d tr ( Q 2 ( x , t ) ) d x = e - | y | 2 8 t + O L y ∞ ( t - 1 2 ) , d = 2 , 3 for (potentially large) solutions Q ( t ) obeying a natural asymptotic condition. Such a formula was established in Kirr (J Stat Phys 155:625–657 (2014) for d = 3 and small initial data Q 0 , subject to the non-degeneracy condition ∫ R 3 Q 0 ( x ) d x ≠ 0 .