# Finite-Time Singularity Formation for Incompressible Euler Moving Interfaces in the Plane

- 65 Downloads

## Abstract

This paper provides a new general method for establishing a finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. This methodology is applied to two different problems. The first problem considered is the two-phase vortex sheet problem with surface tension, for which, under suitable assumptions of smallness of the initial height of the heaviest phase and velocity fields, is proved the finite-time singularity of the natural norm of the problem. This is in striking contrast with the case of finite-time splash and splat singularity formation for the one-phase Euler equations of [4] and [8], for which the natural norm (in the one-phase fluid) stays finite all the way until contact. The second problem considered involves the presence of a heavier rigid body moving in the inviscid fluid. For a very general set of geometries (essentially the contact zone being a graph) we first establish that the rigid body will hit the bottom of the fluid domain in finite time. Compared to the previous paper [20] for the rigid body case, the present paper allows for small square integrable vorticity and provides a characterization of acceleration at contact. A surface energy is shown to blow up and acceleration at contact is shown to oppose the motion: it is either strictly positive and finite if the contact zone is of non zero length, or infinite otherwise.

## Notes

### Acknowledgements

The authorwants to thank the anonymous referee for the careful reading of this manuscript, for identifying imprecisions and confusing points and for making useful suggestions that have improved the clarity of this work. The author also wants to thank the editor for useful comments that have further improved the clarity of this work.

### Compliance with ethical standards

### Ethical Responsibilities and Ethical Standard

The author satisfies all required ethical responsibilities and standards required by Archive for Rational Mechanics and Analysis.

### Conflict of interest

The author declares that he has no conflict of interest.

## References

- 1.Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. Invent. Math.
**198**, 71–163 (2014)ADSMathSciNetCrossRefGoogle Scholar - 2.Ambrose, David M.: Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal.
**35**, 211–244 (2003)MathSciNetCrossRefGoogle Scholar - 3.Ambrose, D.M., Masmoudi, N.: Well-posedness of 3D vortex sheets with surface tension. Commun. Math. Sci.
**5**, 391–430 (2007)MathSciNetCrossRefGoogle Scholar - 4.Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, M.: Finite time singularities for the free boundary incompressible Euler equations. Ann. Math.
**178**, 1061–1134 (2013)MathSciNetCrossRefGoogle Scholar - 5.Cheng, C.H.A., Coutand, D., Shkoller, S.: On the motion of vortex sheets with surface tension in the 3D Euler equations with vorticity. Commun. Pure Appl. Math.
**61**, 1715–1752 (2008)CrossRefGoogle Scholar - 6.Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Commun. Pure Appl. Math.
**53**, 1536–1602 (2000)MathSciNetCrossRefGoogle Scholar - 7.Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc.
**20**, 829–930 (2007)MathSciNetCrossRefGoogle Scholar - 8.Coutand, D., Shkoller, S.: On the finite-time splash and splat singularities for the 3-D free-surface euler equations. Commun. Math. Phys.
**325**, 143–183 (2014)ADSMathSciNetCrossRefGoogle Scholar - 9.Coutand, D., Shkoller, S.: On the impossibility of finite-time splash singularities for vortex sheets. Arch. Ration. Mech. Anal.
**221**, 987–1033 (2016)MathSciNetCrossRefGoogle Scholar - 10.Fefferman, C., Ionescu, A.D., Lie, V.: On the absence of "splash" singularities in the case of two-fluid interfaces. Duke Math. J.
**165**, 417–462 (2016)MathSciNetCrossRefGoogle Scholar - 11.Gérard-Varet, D., Hillairet, M.: Regularity issues in the problem of fluid-structure interaction. Arch. Ration. Mech. Anal.
**195**, 375–407 (2010)MathSciNetCrossRefGoogle Scholar - 12.Gérard-Varet, D., Hillairet, M., Wang, C.: The influence of boundary conditions on the contact problem in a \(3 D\) Navier–Stokes flow.
*J. Math. Pures Appl.***103**, 1–38 (2015)Google Scholar - 13.Hillairet, M.: Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Partial Differ. Equ.
**32**, 1345–1371 (2007)MathSciNetCrossRefGoogle Scholar - 14.Hillairet, M., Takahashi, T.: Collisions in three-dimensional fluid structure interaction problems. SIAM J. Math. Anal.
**40**, 2451–2477 (2009)MathSciNetCrossRefGoogle Scholar - 15.Hillairet, M., Takahashi, T.: Blow up and grazing collision in viscous fluid solid interaction systems.
*Ann. Inst. H. Poincaré Anal. Non Linéaire***27**, 291–313 (2010)ADSMathSciNetCrossRefGoogle Scholar - 16.Houot, J., Munnier, A.: On the motion and collisions of rigid bodies in an ideal fluid. Asymptot. Anal.
**56**, 125–158 (2008)MathSciNetzbMATHGoogle Scholar - 17.Glass, O., Sueur, F.: Uniqueness results for weak solutions of two-dimensional fluid-solid systems. Arch. Ration. Mech. Anal.
**218**, 907–944 (2015)MathSciNetCrossRefGoogle Scholar - 18.Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc.
**18**, 605–654 (2005)MathSciNetCrossRefGoogle Scholar - 19.Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math.
**162**, 109–194 (2005)MathSciNetCrossRefGoogle Scholar - 20.Munnier, A., Ramdani, K.: Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid.
*SIAM J. Math. Anal.***47**, 4360–4403 (2015)MathSciNetCrossRefGoogle Scholar - 21.Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler equation. Commun. Pure Appl. Math.
**61**, 698–744 (2008)MathSciNetCrossRefGoogle Scholar - 22.Shatah, J., Zeng, C.: A priori estimates for fluid interface problems. Commun. Pure Appl. Math.
**61**(6), 848–876 (2008)MathSciNetCrossRefGoogle Scholar - 23.Shatah, J., Zeng, C.: Local well-posedness for fluid interface problems. Arch. Ration. Mech. Anal.
**199**, 653–705 (2011)MathSciNetCrossRefGoogle Scholar - 24.Sideris, T.C.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys.
**101**, 475–485 (1985)ADSMathSciNetCrossRefGoogle Scholar - 25.Starovoitov, V.N.: Behavior of a rigid body in an incompressible viscous fluid near a boundary.
*Free boundary problems (Trento, 2002), Internat. Ser. Numer. Math. Birkhuser Basel*,**147**, 313–327 (2004)Google Scholar - 26.Xin, Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math.
**51**, 229–240 (1998)MathSciNetCrossRefGoogle Scholar - 27.Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math.
**130**, 39–72 (1997)ADSMathSciNetCrossRefGoogle Scholar - 28.Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc.
**12**, 445–495 (1999)MathSciNetCrossRefGoogle Scholar - 29.Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math.
**61**, 877–940 (2008)MathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.