Double exponential growth of the vorticity gradient for the two-dimensional Euler equation. (English) Zbl 1315.35150

The author proves a double exponential growth for the gradient of the vorticity associated to a 2D Euler equation posed on the torus \(\mathbb{T} ^{2}\). He considers the Euler equation \(\dot{\theta }=\nabla \theta \cdot \nabla ^{\perp }\psi \), \(\psi =\Delta ^{-1}\theta \), with the initial condition \(\theta (x,y,0)=\theta _{0}(x,y)\) and assuming that \(\theta \) is \( 2\pi \)-periodic with respect to \(x\) and \(y\). \(\theta _{0}\) is supposed to have zero average on \(\mathbb{T}^{2}\). The author first recalls the available existence and uniqueness results for this problem, together with previously proved estimates on \(\theta \) in the \(H^{2}\)-norm and on the associated energy. The main result of the paper proves that for every large \( \lambda \) and for every \(T>0\) there exists a smooth initial data \(\theta _{0} \) with \(\left\| \theta _{0}\right\| _{\infty }<2\) such that \( \max_{t\in [ 0,T]}\left\| \nabla \theta (\cdot ,t)\right\| _{\infty }>\lambda ^{e^{T}-1}\left\| \nabla \theta _{0}\right\| _{\infty }\). Writing \(\theta (t)=\mathcal{E}_{t}\theta _{0}\), the author then proves that the above result is equivalent to a non-boundedness property of the linear operator \(\mathcal{E}_{t}\) in the Lipschitz-norm for every \(t>0\). For the proof of the main result, he mainly studies the behavior of the solution to the coupled system of ode’s \(\dot{x}=\mu _{1}(x,y)+\nu _{1}(x,y,t)\), \(\dot{y}=\mu _{2}(x,y)+\nu _{2}(x,y,t)\), with the initial data \(x(\alpha ,\beta ,0)=\alpha \) and \(y(\alpha ,\beta ,0)=\beta \). Here \(\mu =(\mu _{1},\mu _{2})=\nabla ^{\perp }\psi \) and \(\nu _{1(2)}\) are functions which satisfy \(|\nu _{1(2)}|<0.0001\upsilon r\) and \( |\nabla \nu _{1(2)}|<0.0001\upsilon \) for a small \(\upsilon \).


35Q31 Euler equations
35B45 A priori estimates in context of PDEs
76B47 Vortex flows for incompressible inviscid fluids
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