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In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics (MHD) system in R 3. Let (v , B , p) be a smooth solution to the stationary MHD equations in R 3. We show that if there exist smooth matrix valued potential functions Φ , Ψ such that ∇ ⋅ Φ = v and ∇ ⋅ Ψ = B , whose L 6 mean oscillations have certain growth condition near infinity, namely ⨍ B (r) | Φ − Φ B (r) | 6 d x + ⨍ B (r) | Ψ − Ψ B (r) | 6 d x ≤ C r ∀ 1 < r < + ∞ , then v = B = 0 and p =constant. With additional assumption of r − 8 ∫ B (r) | B − B B (r) | 6 d x → 0 as r → + ∞ , similar result holds also for the Hall-MHD system. [ABSTRACT FROM AUTHOR]

In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 (1) 1 (R n) , which corresponds to a critical case of the generalized Campanato spaces L q (N) s (R n). The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 (R n) ↪ L 1 (1) 1 (R n) ↪ C 0 , 1 (R n) , where B ∞ , 1 1 (R n) and C 0 , 1 (R n) are the Besov space and the Lipschitz space respectively. In particular L 1 (1) 1 (R n) contains non- C 1 (R n) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 (1) 1 (R n) , for which the solution to the Euler equations blows up in finite time. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in R 3 with the viscous part of the stress tensor A p (u) = div (| D (u) | p − 2 D (u)) , where D (u) = 1 2 (∇ u + (∇ u) ⊤) and 9 5 < p < 3. We consider a weak solution u ∈ W l o c 1 , p (R 3) and its potential function V = (V i j) ∈ W l o c 2 , p (R 3) , i.e. ∇ ⋅ V = u. We show that there exists a constant s 0 = s 0 (p) such that if the L s mean oscillation of V for s > s 0 satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results [5,6] as particular cases. [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary Navier–Stokes equations in R 3 . Let V = (V ij) be a potential function of a smooth solution u, which means u = ∇ · V . We show that if there exists 3 < s < + ∞ such that the L s mean oscillation of the potential function has certain growth condition near infinity, namely 1 | B (r) | ∫ B (r) | V - V B (r) | s d x ≤ C r min { s - 3 3 , s 6 } ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]