论文标题
liouville型定理用于固定的navier-stokes方程
Liouville type theorems for stationary Navier-Stokes equations
论文作者
论文摘要
我们表明，3D不可压缩的Navier在整个空间，半空间或周期性平板都必须消失的条件下，任何平稳的固定解决方案都必须消失，该条件在某些$ 0 \leδ\ le 1 <l $和$ q = 6（3-δ）/（3-δ）/（6-δ）/（6-δ）$，$ $ \ liminf_ \ liminf_ { \ | u \ |^{3-δ} _ {l^{q}（r <| x | <lr）} =0。$$我们还证明了足够的条件，允许缩小半径比率$ l = 1+r^{ - α} $。通过假设更强的衰减速率，相似的结果在零边界条件的平板上保持。我们不假定速度的全局界限。关键是要估计Annuli中当地的压力，而RADII比$ L $任意接近1。
We show that any smooth stationary solution of the 3D incompressible Navier-Stokes equations in the whole space, the half space, or a periodic slab must vanish under the condition that for some $0 \le δ\le 1<L$ and $q=6(3-δ)/(6-δ)$, $$\liminf_{R \to \infty} \frac 1R \|u\|^{3-δ}_{L^{q}(R<|x|<LR)}=0.$$ We also prove sufficient conditions allowing shrinking radii ratio $L= 1+R^{-α}$. Similar results hold on a slab with zero boundary condition by assuming stronger decay rates. We do not assume global bound of the velocity. The key is to estimate the pressure locally in the annuli with radii ratio $L$ arbitrarily close to 1.
