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Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

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Abstract

For all \(\epsilon >0\), we prove the existence of finite-energy strong solutions to the axi-symmetric 3D Euler equations on the domains \( \{(x,y,z)\in {\mathbb {R}}^3: (1+\epsilon |z|)^2\le x^2+y^2\}\) which become singular in finite time. The solutions we construct have bounded vorticity before a certain time when the vorticity becomes unbounded. We further show that solutions with 0 swirl are always globally regular in the setting we consider. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a 1D system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system Elgindi and Jeong (Finite-time Singularity Formation for Strong Solutions to the Boussinesq System, 2017) though a number of modifications must be made due to anisotropy and since our domains are not scale-invariant. This seems to be the first construction of singularity formation for finite-energy strong solutions to the actual 3D Euler system.

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Notes

  1. See, for example, http://www.claymath.org/sites/default/files/navierstokes.pdf.

  2. It is important to remark that, to avoid ill-posedness issues (as in [23] and [3]), it is necessary to ask that \(\nabla u\) is bounded on a time-interval and not just at the initial time. In fact, one could simply ask whether there is a Banach space \(X\subset L^2\cap W^{1,\infty }({\mathbb {R}}^3)\) where the 3D Euler equations are locally well-posed but not globally well-posed.

  3. While Yudovich [69] solutions are usually called weak solutions, we feel that classifying them as such is slightly misleading in the present context. Besides, the Yudovich theory does not extend to 3D even locally in time.

  4. We are aware that the incompressible Euler equations satisfies a two-parameter family of scaling invariances. However, using the time scaling invariance introduces a number of difficulties which are still not fully understood.

  5. Unfortunately this error appears in a few books and papers in mathematical fluid mechanics. We thank Dongyi Wei for pointing this out to us.

  6. Note that \(\arctan {{\frac{1}{\epsilon }}}<{\frac{\pi }{2}}\) for every \(\epsilon >0\).

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Acknowledgements

We thank the anonymous referee for helpful comments. T.M. Elgindi acknowledges funding from the NSF grants DMS-1817134 and DMS-1402357. I.-J. Jeong has been supported by the POSCO Science Fellowship of POSCO TJ Park Foundation and the National Research Foundation of Korea(NRF) Grant (No. 2019R1F1A1058486).

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  1. Department of Mathematics, UC San Diego, San Diego, USA

    Tarek M. Elgindi

  2. Department of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea

    In-Jee Jeong

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Elgindi, T.M., Jeong, IJ. Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations. Ann. PDE 5, 16 (2019). https://doi.org/10.1007/s40818-019-0071-6

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