On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations

• Published in: Journal of nonlinear science Vol. 30; no. 4; pp. 1503 - 1517
• Main Authors: Chae, Dongho, Wolf, Jörg
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2020; Springer Nature B.V
• Subjects: Analysis; Classical Mechanics; Economic Theory/Quantitative Economics/Mathematical Methods; Liouville theorem; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Newtonian fluids; Non Newtonian fluids; Theoretical; Non-Newtonian fluid equations; 76D03; Liouville type theorem; 76D05; 35Q30
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-020-09615-y

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p ( u ) = ∇ · ( | D ( u ) | p - 2 D ( u ) ) with D ( u ) = 1 2 ( ∇ u + ( ∇ u ) ⊤ ) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution u ∈ W 1 , p ( R 3 ) satisfying lim inf R → ∞ | u B ( R ) | = 0 is trivial, i.e., u ≡ 0 . On the other hand, for 9 / 5 < p < 3 we prove the following Liouville type theorem: if there exists a matrix valued function V = { V ij } such that ∂ j V ij = u i (summation convention), whose L 3 p 2 p - 3 mean oscillation has the following growth condition at infinity, ∫ - B ( r ) | V - V B ( r ) | 3 p 2 p - 3 d x ≤ C r 9 - 4 p 2 p - 3 ∀ 1 < r < + ∞ , then u ≡ 0 .