# CONFLUENTES MATHEMATICI

Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations
Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 67-95.

Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity ${\omega }_{\theta }$ is integrable with respect to the two-dimensional measure $\mathrm{d}r\phantom{\rule{0.166667em}{0ex}}\mathrm{d}z$, where $\left(r,\theta ,z\right)$ denote the cylindrical coordinates in ${ℝ}^{3}$, we show the existence of a unique global solution, which converges to zero in ${L}^{1}$ norm as $t\to \infty$. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between both situations. The solutions we construct have infinite energy in general, so that energy dissipation cannot be invoked to control the long-time behavior. We also treat the more general case where the initial vorticity is a finite measure whose atomic part is small enough compared to viscosity. Such data include point masses, which correspond to vortex filaments in the three-dimensional picture.

Reçu le : 2014-12-31
Révisé le : 2014-12-31
Accepté le : 2014-12-31
Publié le : 2016-02-15
DOI : https://doi.org/10.5802/cml.25
Classification : 35Q30,  76D05,  35B07,  35B40,  35B45
@article{CML_2015__7_2_67_0,
author = {Thierry Gallay and Vladim\'\i r \v Sver\'ak},
title = {Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {7},
number = {2},
year = {2015},
pages = {67-95},
doi = {10.5802/cml.25},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2015__7_2_67_0/}
}
Thierry Gallay; Vladimír Šverák. Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations. Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 67-95. doi : 10.5802/cml.25. https://cml.centre-mersenne.org/item/CML_2015__7_2_67_0/

[1] H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. Math. 132 (2008), 592–624 (in French).

[2] H. Abidi, T. Hmidi, S. Keraani, On the global well-posedness for the axisymmetric Euler equations, Math. Ann. 347 (2010), 15–41.

[3] M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128 (1994), 329–358.

[4] H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: “Global solutions of two-dimensional Navier-Stokes and Euler equations”, Arch. Rational Mech. Anal. 128 (1994), 359–360.

[5] H. Feng and V. Šverák, On the Cauchy problem for axi-symmetric vortex rings, Arch. Rational Mech. Anal. 215 (2015), 89–123.

[6] I. Gallagher and Th. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann. 332 (2005), 287–327.

[7] Th. Gallay, Stability and interaction of vortices in two-dimensional viscous flows, Discr. Cont. Dyn. Systems Ser. S 5 (2012), 1091-1131.

[8] Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on ${ℝ}^{2}$, Arch. Ration. Mech. Anal. 163 (2002), 209–258.

[9] Th. Gallay and C. E. Wayne, Global Stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys. 255 (2005), 97–129.

[10] Th. Gallay and C. E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on ${ℝ}^{3}$, Phil. Trans. R. Soc. Lond. A 360 (2002), 2155–2188.

[11] Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal. 104 (1988), 223–250.

[12] Y. Giga and T. Miyakawa, Navier-Stokes flow in ${ℝ}^{3}$ with measures as initial vorticity and Morrey spaces, Comm. Partial Diff. Equations 14 (1989), 577–618.

[13] T. Kato, Strong ${L}^{p}$-solutions of the Navier-Stokes equation in ${ℝ}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), 471–480.

[14] T. Kato, The Navier-Stokes equation for an incompressible fluid in ${ℝ}^{2}$ with a measure as the initial vorticity, Differential Integral Equations 7 (1994), 949–966.

[15] G. Koch, N. Nadirashvili, G. Seregin, and V. Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math. 203 (2009), 83–105.

[16] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), 22–35.

[17] O. Ladyzhenskaya, Unique solvability in the large of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 7 (1968), 155–177 (in Russian).

[18] S. Leonardi, J. Málek, J. Nečas, M. Pokorný, On axially symmetric flows in ${ℝ}^{3}$, Z. Anal. Anwendungen 18 (1999), 639–649.

[19] Jian-Guo Liu and Wei-Cheng Wang, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation, SIAM J. Math. Anal. 41 (2009), 1825–1850.

[20] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954.

[21] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York, 1978.

[22] V. Šverák, Selected Topics in Fluid Mechanics, lectures notes of an introductory graduate course taught in 2011/2012, available at the following URL :

[23] M. Ukhovskii and V. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech. 32 (1968), 52–61.