本文選題：Navier-Stokes方程 + 古代解　； 參考：《中國科學(xué):數(shù)學(xué)》2017年10期

【摘要】：本文研究不可壓縮Navier-Stokes方程的古代解所具有的Liouville性質(zhì).在二維情形以及三維軸對稱具平凡角向速度(v_θ=0)情形下,本文證明了光滑的溫和古代解的"最優(yōu)"Liouville定理,即當(dāng)渦度滿足一定條件且速度場v關(guān)于空間變量次線性增長時,v恒為常向量,并且在速度場線性增長條件下給出了非平凡古代解的反例.其中,在二維情形下,渦度w需要滿足的條件為,對所有的t∈(-∞,0)一致成立lim_(|x|→+∞)|w(x,t)|=0;在三維軸對稱具平凡角向速度情形下,渦度w需要滿足的條件為,對所有的t∈(-∞,0)一致成立lim_(r→+∞)(|w(x,t)|)/r=0.在三維軸對稱具非平凡角向速度(v_θ≠0)的情形下,本文證明了,若Γ=rv_θ∈L_t~∞L_x~p(R~3×(-∞,0)),其中1≤p∞,則有界的溫和古代解必為常向量.
[Abstract]:In this paper, we study the Liouville properties of the ancient solutions of incompressible Navier-Stokes equations. In the case of two-dimensional and three-dimensional axisymmetric angular velocities v _ 胃 _ 0, we prove the "optimal" Liouville theorem for smooth and mild ancient solutions. That is, when the vorticity satisfies certain conditions and the velocity field v increases sublinearly with respect to spatial variables, v is constant vector, and a counterexample of the nontrivial ancient solution is given under the condition of linear growth of velocity field. Where, in two-dimensional case, the condition that vorticity w needs to be satisfied is that for all t 鈭，

本文編號：2033284