We show that solutions u(x, t) of the non-stationnary incompressible Navier–Stokes system in R d (d ≥ 2) starting from mild decaying data a behave as |x| → ∞ as a potential field: u(x, t) = e t∆ a(x) + γ d ∇ x h,k δ h,k |x| 2 − dx h x k d|x| d+2 K h,k (t) + o 1 |x| d+1 (i) where γ d is a constant and K h,k = t 0 (u h |u k) L 2 is the energy matrix… (More)
We study the behavior at innity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in R d. We prove that if the initial magnetic eld decays suciently fast, then the plasma ow behaves as a solution of the free nonstationnary NavierStokes equations when |x| → +∞, and that the magnetic eld will govern the decay of the… (More)
We study the non-linear minimization problem on H 1 0 (Ω) ⊂ L q with q = 2n n−2 : inf u L q =1 Ω (1 + |x| β |u| k)|∇u| 2. We show that minimizers exist only in the range β < kn/q which corresponds to a dominant non-linear term. On the contrary, the linear influence for β ≥ kn/q prevents their existence.