论文标题
2D Boussinesq方程的小尺度形成
Small scale formation for the 2D Boussinesq equation
论文作者
论文摘要
我们研究了没有热扩散的2D不可压缩的Boussinesq方程，并旨在随着时间的流逝而构建严格的小尺度形成实例。在粘性情况下，我们构造了满足$ \ sup_ {τ\ in [0，t]} \ | \nablaρ（τ）\ | _ {l^2} \ gtrsim t^α$的示例。对于条带中的无关方程式，我们构造了满足$ \ |ω（t）\ | _ {l^\ infty} \ gtrsim t^3 $和$ \ sup_ {τ\ in [0，t]} \ | \ nablaρ（f.nablaρ（f）平滑解决方案。这些增长结果适用于广泛的初始数据，我们只需要某些对称性和标志条件。作为一种应用，我们还构建了3D轴对称欧拉方程的解决方案，该方程的速度具有无限的时间生长。
We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying $\sup_{τ\in[0,t]} \|\nabla ρ(τ)\|_{L^2}\gtrsim t^α$ for some $α>0$. For the inviscid equation in the strip, we construct examples satisfying $\|ω(t)\|_{L^\infty}\gtrsim t^3$ and $\sup_{τ\in[0,t]} \|\nabla ρ(τ)\|_{L^\infty} \gtrsim t^2$ during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth.
