On the splash singularity for the free-surface of a Navier–Stokes fluid
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 475-503.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for d-dimensional flows, d=2 or 3, the free-surface of a viscous water wave, modeled by the incompressible Navier–Stokes equations with moving free-boundary, has a finite-time splash singularity for a large class of specially prepared initial data. In particular, we prove that given a sufficiently smooth initial boundary (which is close to self-intersection) and a divergence-free velocity field designed to push the boundary towards self-intersection, the interface will indeed self-intersect in finite time.

DOI : 10.1016/j.anihpc.2018.06.004
Mots-clés : Navier–Stokes, Free-boundary problem, Finite-time singularity, Splash singularity, Interface singularity 
@article{AIHPC_2019__36_2_475_0,
     author = {Coutand, Daniel and Shkoller, Steve},
     title = {On the splash singularity for the free-surface of a {Navier{\textendash}Stokes} fluid},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {475--503},
     publisher = {Elsevier},
     volume = {36},
     number = {2},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.06.004},
     mrnumber = {3913195},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.004/}
}
TY  - JOUR
AU  - Coutand, Daniel
AU  - Shkoller, Steve
TI  - On the splash singularity for the free-surface of a Navier–Stokes fluid
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 475
EP  - 503
VL  - 36
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.004/
DO  - 10.1016/j.anihpc.2018.06.004
LA  - en
ID  - AIHPC_2019__36_2_475_0
ER  - 
%0 Journal Article
%A Coutand, Daniel
%A Shkoller, Steve
%T On the splash singularity for the free-surface of a Navier–Stokes fluid
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 475-503
%V 36
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.004/
%R 10.1016/j.anihpc.2018.06.004
%G en
%F AIHPC_2019__36_2_475_0
Coutand, Daniel; Shkoller, Steve. On the splash singularity for the free-surface of a Navier–Stokes fluid. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 475-503. doi : 10.1016/j.anihpc.2018.06.004. http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.004/

[1] Abels, H. The initial-value problem for the Navier–Stokes equations with a free surface in Lq-Sobolev spaces, Adv. Differ. Equ., Volume 10 (2005), pp. 45–64 | MR | Zbl

[2] Amrouche, C.; Girault, V. The existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Jpn. Acad., Volume 67 (1991) no. Ser. A, pp. 171–175 | MR | Zbl

[3] Amrouche, C.; Seloula, N.E.H. On the Stokes equations with the Navier-type boundary conditions, Differ. Equ. Appl., Volume 3 (2011), pp. 581–607 | MR | Zbl

[4] Bae, H. Solvability of the free boundary value problem of the Navier–Stokes equations, Discrete Contin. Dyn. Syst., Volume 29 (2011), pp. 769–801 | MR | Zbl

[5] Beale, J. The initial value problem for the Navier–Stokes equations with a free surface, Commun. Pure Appl. Math., Volume 34 (1981) no. 3, pp. 359–392 | DOI | MR | Zbl

[6] Beale, J.T. Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., Volume 84 (1983/84), pp. 307–352 | MR | Zbl

[7] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; Gómez-Serrano, M. Finite time singularities for water waves with surface tension, J. Math. Phys., Volume 53 (2012) | DOI | MR | Zbl

[8] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; Gómez-Serrano, M. Finite time singularities for the free boundary incompressible Euler equations, Ann. Math., Volume 178 (2013), pp. 1061–1134 | DOI | MR | Zbl

[9] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; Gómez-Serrano, M. Splash singularities for the free boundary Navier–Stokes equations, 2015 | arXiv

[10] Córdoba, D.; Enciso, A.; Grubic, N. Splash and almost-splash stationary solutions to the Euler equations, 2014 (Arxiv preprint) | arXiv

[11] Cheng, C.H.A.; Shkoller, S. The interaction of the 3D Navier–Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal., Volume 42 (2010), pp. 1094–1155 | MR | Zbl

[12] Coutand, D.; Shkoller, S. Unique solvability of the free-boundary Navier–Stokes equations with surface tension, 2002 | arXiv

[13] Coutand, D.; Shkoller, S. On the motion of an elastic solid inside of an incompressible viscous fluid, Arch. Ration. Mech. Anal., Volume 176 (2005), pp. 25–102 | DOI | MR | Zbl

[14] Coutand, D.; Shkoller, S. The interaction between quasilinear elastodynamics and the Navier–Stokes equations, Arch. Ration. Mech. Anal., Volume 179 (2006), pp. 303–352 | DOI | MR | Zbl

[15] Coutand, D.; Shkoller, S. Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Am. Math. Soc., Volume 20 (2007), pp. 829–930 | DOI | MR | Zbl

[16] Coutand, D.; Shkoller, S. On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., Volume 325 (2014), pp. 143–183 | DOI | MR | Zbl

[17] Coutand, D.; Shkoller, S. On the impossibility of finite-time splash singularities for vortex sheets, Arch. Ration. Mech. Anal., Volume 221 (2016), pp. 987–1033 | DOI | MR

[18] Dacorogna, B.; Moser, J. On a partial differential equation involving the Jacobian determinant, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 7 (1990), pp. 1–26 | DOI | Numdam | MR | Zbl

[19] Elgindi, T.; Lee, D. Uniform regularity for free-boundary Navier–Stokes equations with surface tension, 2014 | arXiv

[20] Fefferman, C.; Ionescu, A.D.; Lie, V. On the absence of “splash” singularities in the case of two-fluid interfaces, Duke Math. J., Volume 165 (2016), pp. 417–462 | DOI | MR

[21] Hataya, Y. Decaying solution of a Navier–Stokes flow without surface tension, J. Math. Kyoto Univ., Volume 49 (2009), pp. 691–717 | MR | Zbl

[22] Guo, Y.; Tice, I. Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., Volume 207 (2013), pp. 459–531 | MR | Zbl

[23] Guo, Y.; Tice, I. Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, Volume 6 (2013), pp. 1429–1533 | MR | Zbl

[24] Guo, Y.; Tice, I. Local well-posedness of the viscous surface wave problem without surface tension, Anal. PDE, Volume 6 (2013), pp. 287–369 | MR | Zbl

[25] Masmoudi, N.; Rousset, F. Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal., Volume 223 (2017), pp. 301–417 | DOI | MR

[26] Nishida, T.; Teramoto, Y.; Yoshihara, H. Global in time behavior of viscous surface waves: horizontally periodic motion, J. Math. Kyoto Univ., Volume 44 (2004) no. 2, pp. 271–323 | MR | Zbl

[27] Padula, M.; Solonnikov, V.A. Navier–Stokes Equations and Related Nonlinear Problems, Ann. Univ. Ferrara Sez. VII (N.S.), Volume 46 (2000), pp. 307–336 (Ferrara, 1999) | MR | Zbl

[28] Solonnikov, V.A. Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 41 (1977), pp. 1388–1424 | MR | Zbl

[29] Solonnikov, V.A. On an initial boundary value problem for the Stokes systems arising in the study of a problem with a free boundary, Proc. Steklov Inst. Math., Volume 3 (1991), pp. 191–239 | MR | Zbl

[30] Solonnikov, V.A. Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, St. Petersburg Math. J., Volume 3 (1992), pp. 189–220 | MR | Zbl

[31] Solonnikov, V.A.; Scadilov, V.E. On a boundary value problem for a stationary system of Navier–Stokes equations, Proc. Steklov Inst. Math., Volume 125 (1973), pp. 186–199 | MR | Zbl

[32] Tani, A.; Tanaka, N. Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Ration. Mech. Anal., Volume 130 (1995) no. 4, pp. 303–314 | DOI | MR | Zbl

[33] Wang, Y.; Xin, Z. Vanishing viscosity and surface tension limits of incompressible viscous surface waves, 2015 | arXiv

Cité par Sources :