## 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 $$\upsilon$$.

### MSC:

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

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