L^p-Solutions of the Steady-State Navier–Stokes Equations with Rough External Forces

Abstract

In this paper we address the existence, the asymptotic behavior and stability in L p and L p,∞ , 3 2 < p ≤ ∞, for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain smallness conditions of the forcing term there exists solutions to the stationary Navier-Stokes equations in L p spaces, and we prove the stability of these solutions. Namely, we prove that such small steady state solutions attract time dependent solutions driven by the same forcing, no matter how large the initial velocity of non-stationary solutions is.We also give non-existence results of stationary solutions in L p , for 1 ≤ p ≤ 3 2 .

Asymptotics of solutions to the Navier-Stokes system in exterior domains

“…Our result is reminiscent of the result in [5] on the global-in-time existence of weak solutions to the Navier-Stokes system that holds true for initial data in L p σ (R n ) with 2 < p < n. Proof of Theorem 4.1. We borrow a method from [3]. Using [3,Lemma 4.2] we decompose u 0 = z 0 + w 0 where z 0 ∈ L 2 (Ω) and w 0 L n,∞ (Ω) ε.…”

“…We borrow a method from [3]. Using [3,Lemma 4.2] we decompose u 0 = z 0 + w 0 where z 0 ∈ L 2 (Ω) and w 0 L n,∞ (Ω) ε. Note that even though the statement of that lemma is given for n = 3, the proof goes through for n 3.…”

Large Time Behavior of the Navier-Stokes Flow

“…Wider stability results were obtained in [10] by C. Bjorland, L. Brandolese, D. Iftimie and M.E. Schonbek, where the stability for L p spaces, 3/2 < p ≤ ∞ were considered.…”

Blow-Up Criterion and Examples of Global Solutions of Forced Navier-Stokes Equations

“…the external force defined on R 3 . Since we only care about the case of U ∈ L 3 , we state the following result for ∆ −1 f ∈ L 3 without proof, which is a consequence of Theorem 2.2 in[2].Proposition 6.1. There exists an absolute constant δ > 0 with the following property.…”

“…Hence for 0 < t ≤ ∞, recalling B(u, v) := t 0 e (t−s)∆ P∇ · (u(s) ⊗ v(s))ds,Young's inequality for convolutions implies that for anyr∈ [r, ∞] ⊗ v L r ([0,T ],Ḃ s+1 p,p ) . (30)And we recall that B is a bounded operator from L ∞ ([0, T ], L 3,∞ ) × L ∞ ([0, T ], L 3,∞ ) to L ∞ ([0, T ], L 3,∞ ) for any T ∈ R + ∪ {+∞} (see[2])…”