Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations

Gallay, Thierry ; Šverák, Vladimír

Confluentes Mathematici, Tome 7 (2015), p. 67-92 / Harvested from Numdam

Résumé

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 ${\rm d} r\,{\rm d} z$, where $(r,\theta ,z)$ denote the cylindrical coordinates in $\mathbb{R}^3$, we show the existence of a unique global solution, which converges to zero in $L^1$ norm as $t \rightarrow \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.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/cml.25
Classification:  35Q30,  76D05,  35B07,  35B40,  35B45

Bibliographie

[1] H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. Math. 132 (2008), 592–624 (in French). | MR 2455899 | Zbl 1159.35052

[2] H. Abidi, T. Hmidi, S. Keraani, On the global well-posedness for the axisymmetric Euler equations, Math. Ann. 347 (2010), 15–41. | MR 2593281

[3] M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128 (1994), 329–358. | MR 1308857 | Zbl 0837.35110

[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. | MR 1308858 | Zbl 0837.35112

[5] H. Feng and V. Šverák, On the Cauchy problem for axi-symmetric vortex rings, Arch. Rational Mech. Anal. 215 (2015), 89–123. | MR 3296145

[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. | MR 2178064 | Zbl 1096.35102

[7] Th. Gallay, Stability and interaction of vortices in two-dimensional viscous flows, Discr. Cont. Dyn. Systems Ser. S 5 (2012), 1091-1131. | MR 2968938 | Zbl 1269.35027

[8] Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbb{R}^2$, Arch. Ration. Mech. Anal. 163 (2002), 209–258. | MR 1912106 | Zbl 1042.37058

[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. | MR 2123378 | Zbl 1139.35084

[10] Th. Gallay and C. E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbb{R}^3$, Phil. Trans. R. Soc. Lond. A 360 (2002), 2155–2188. | MR 1949968 | Zbl 1048.35055

[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. | MR 1017289 | Zbl 0666.76052

[12] Y. Giga and T. Miyakawa, Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Diff. Equations 14 (1989), 577–618. | MR 993821 | Zbl 0681.35072

[13] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z. 187 (1984), 471–480. | MR 760047 | Zbl 0545.35073

[14] T. Kato, The Navier-Stokes equation for an incompressible fluid in $\mathbb{R}^2$ with a measure as the initial vorticity, Differential Integral Equations 7 (1994), 949–966. | MR 1270113 | Zbl 0826.35094

[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. | MR 2545826 | Zbl 1208.35104

[16] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), 22–35. | MR 1808843 | Zbl 0972.35084

[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). | MR 241833 | Zbl 0202.37302

[18] S. Leonardi, J. Málek, J. Nečas, M. Pokorný, On axially symmetric flows in $\mathbb{R}^3$, Z. Anal. Anwendungen 18 (1999), 639–649. | MR 1718156 | Zbl 0943.35066

[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. | MR 2564196 | Zbl 1197.35195

[20] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. | MR 100158 | Zbl 0096.06902

[21] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York, 1978. | MR 493421 | Zbl 0242.46001

[22] V. Šverák, Selected Topics in Fluid Mechanics, lectures notes of an introductory graduate course taught in 2011/2012, available at the following URL : http://www.math.umn.edu/~sverak/course-notes2011.pdf

[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. | MR 239293 | Zbl 0172.53405