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In this paper we study the Liouville type problem for stationary magnetohydrodynamic equations in R n . We show that under certain anisotropic integrability conditions on the components of the velocity and the magnetic field the solution is trivial. [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 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 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]

We study the backward discretely self-similar solutions to the generalized Euler equations, where usual Biot-Savart kernel representing the velocity in terms of the vorticity is replaced by various power of the laplacian. Under milder sufficient conditions than the previous results on the Euler equations we show that Liouville type theorems hold for the time periodic solutions to the profile equations, which means that there exists no backward discretely self-similar solutions to the generalized Euler system having such profile. [ABSTRACT FROM AUTHOR]

In this brief note we study the Liouville type problem for the stationary Navier–Stokes equations in R 3. We show that under certain anisotropic integrability conditions on the components of the velocity implies that the solution is trivial. [ABSTRACT FROM AUTHOR]

We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $${\mathbb{R}^2}$$ . We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations. [ABSTRACT FROM AUTHOR]

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ru̇žička-Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $${L^{p, \infty} (\mathbb{R}^3)}$$ with $${p > \frac{3}{2}}$$ . [ABSTRACT FROM AUTHOR]

In this paper we prove three different Liouville type theorems for the steady Navier–Stokes equations in R 3 . In the first theorem we improve logarithmically the well-known L 9 2 ( R 3 ) result. In the second theorem we present a sufficient condition for the trivially of the solution ( v = 0 ) in terms of the head pressure, Q = 1 2 | v | 2 + p . The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee v = 0 . [ABSTRACT FROM AUTHOR]

We present a simple, short, and elementary proof that if v is a Beltrami flow with a finite energyin R3,then v=0. In the case of the Beltrami flows satisfying v∈Lloc∞(R3)∩Lq(R3) with q ∈[2,3),or [ABSTRACT FROM AUTHOR]

Abstract: We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions. [Copyright &y& Elsevier]

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $${\mathbb {R}^N}$$ . If we assume 'single signedness condition' on the force, then we can show that a $${C^1 (\mathbb {R}^N)}$$ solution ( v, p) with $${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$$ is trivial, v = 0. For the solution of the steady Navier-Stokes equations, satisfying $${v(x) \to 0}$$ as $${|x| \to \infty}$$ , the condition $${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$$ , which is stronger than the important D-condition, $${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203-215, ), using the self-similar Euler equations directly. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove the Liouville type theorem for the stationary Navier–Stokes equations on . More specifically, if a solution to the stationary Navier–Stokes system satisfies additional conditions characterized by the decays near infinity and by the oscillation, then we show that . [Copyright &y& Elsevier]

In this paper we study the Euler-Poincaré equations in $${\mathbb {R}^N}$$ . We prove local existence of weak solutions in $${W^{2,p}(\mathbb {R}^N), p > N}$$ , and local existence of unique classical solutions in $${H^k (\mathbb {R}^N)}$$ , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation ( α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to $${C([0, T);H^k(\mathbb {R}^N))}$$ with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution $${\mathbf{u} \in H^1(\mathbb{R}^N)}$$ is u=0; for α= 0 any weak solution $${\mathbf{u} \in L^2(\mathbb{R}^N)}$$ is u=0. [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]