Uniqueness criteria for the Oseen vortex in the 3d Navier-Stokes equations

References

• Gallagher, I., Gallay, T., Lions, P.-L. (2005). On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Math. Nachr. 278(14):1665–1672. DOI: 10.1002/mana.200410331.
   Web of Science ®Google Scholar
• Gallagher, I., Gallay, T. (2005). Uniqueness for the two-dimensional navier-stokes equation with a measure as initial vorticity. Math. Ann. 332(2):287–327. DOI: 10.1007/s00208-004-0627-x.
   Web of Science ®Google Scholar
• Devenport, W. J., Rife, M. C., Liapis, S. I., Follin, G. J. (1996). The structure and development of a wing-tip vortex. J. Fluid Mech. 312:67–106. DOI: 10.1017/S0022112096001929.
   Web of Science ®Google Scholar
• Moffatt, H. K., Kida, S., Ohkitani, K. (1994). Stretched vortices–the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259:241–264. DOI: 10.1017/S002211209400011X.
   Web of Science ®Google Scholar
• Helmholtz, H. (1858). Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen. J. Reine Angew. 55:25–55.
   Google Scholar
• Thomson, W. (1868). VI. On vortex motion. Trans. R Soc. Edinb. 25(1):217–260. DOI: 10.1017/S0080456800028179.
   Google Scholar
• Thomson, W. (1880). XXIV. Vibrations of a columnar vortex. The London, Edinburgh, and Dublin Philosophical Magazine, J. Sci. 10(61):155–168. DOI: 10.1080/14786448008626912.
   Google Scholar
• Da Rios, L. (1906). Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque. Rend. Circ. Matem. Palermo. 22(1):117–135. DOI: 10.1007/BF03018608.
   Google Scholar
• Jerrard, R. L., Seis, C. (2017). On the vortex filament conjecture for Euler flows. Arch. Rational Mech. Anal. 224(1):135–172. DOI: 10.1007/s00205-016-1070-3.
   Web of Science ®Google Scholar
• Majda, A. J., Bertozzi, A. L. (2002). Vorticity and Incompressible Flow, Vol. 27. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Ricca, R. L. (1996). The contributions of da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dyn. Res. 18(5):245–268. DOI: 10.1016/0169-5983(96)82495-6.
   Web of Science ®Google Scholar
• Widnall, S. E., Bliss, D. B., Tsai, C.-Y. (1974). The instability of short waves on a vortex ring. J. Fluid Mech. 66(1):35–47. DOI: 10.1017/S0022112074000048.
   Web of Science ®Google Scholar
• Banica, V., Vega, L. (2009). On the stability of a singular vortex dynamics. Commun. Math. Phys. 286(2):593–627. DOI: 10.1007/s00220-008-0682-3.
   Web of Science ®Google Scholar
• Banica, V., Vega, L. (2012). Scattering for 1D cubic NLS and singular vortex dynamics. J. Eur. Math. Soc. 14(1):209–253.
   Web of Science ®Google Scholar
• Banica, V., Vega, L. (2012). Selfsimilar solutions of the binormal flow and their stability. In Singularities in Mechanics: Formation, Propagation and Microscopic Description, volume 38 of Panor. Synthèses, p. 1–35. Paris, France: Soc. Math. France.
   Google Scholar
• Banica, V., Vega, L. (2013). Stability of the self-similar dynamics of a vortex filament. Arch. Rational Mech. Anal. 210(3):673–712. DOI: 10.1007/s00205-013-0660-6.
   Web of Science ®Google Scholar
• Banica, V., Vega, L. (2015). The initial value problem for the binormal flow with rough data. Ann. Sci. École Norm. Sup. 48(6):1423–1455. DOI: 10.24033/asens.2273.
   Web of Science ®Google Scholar
• Banica, V., Vega, L. (2018). Evolution of polygonal lines by the binormal flow. arXiv preprint, arXiv:1807.06948.
   Google Scholar
• Hasimoto, H. (1972). A soliton on a vortex filament. J. Fluid Mech. 51(3):477–485. DOI: 10.1017/S0022112072002307.
   Web of Science ®Google Scholar
• Jerrard, R. L., Smets, D. (2012). On Schrödinger maps from T1 to S2. Ann. Sci. École Norm. Sup. 45(4):637–680. DOI: 10.24033/asens.2175.
   Web of Science ®Google Scholar
• Jerrard, R. L., Smets, D. (2015). On the motion of a curve by its binormal curvature. J. Eur. Math. Soc. 17(6):1487–1515. DOI: 10.4171/JEMS/536.
   Web of Science ®Google Scholar
• Giga, Y., Miyakawa, T. (1989). Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces. Comm. Partial Differential Equations. 14:577–618.
   Web of Science ®Google Scholar
• Feng, H., Šverák, V. (2015). On the Cauchy problem for axi-symmetric vortex rings. Arch. Rational Mech. Anal. 215(1):89–123. DOI: 10.1007/s00205-014-0775-4.
   Web of Science ®Google Scholar
• Gallay, T., Sverák, V. (2016). Uniqueness of axisymmetric viscous flows originating from circular vortex filaments. arXiv preprint arXiv:1609.02030.
   Google Scholar
• Bedrossian, J., Germain, P., Harrop-Griffiths, B. (2018). Vortex filament solutions of the Navier-Stokes equations. arXiv preprint, arXiv:1809.04109.
   Google Scholar
• Fujita, H., Kato, T. (1964). On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal. 16(4):269–315. DOI: 10.1007/BF00276188.
   Web of Science ®Google Scholar
• Kato, T. (1984). Strong Lp-solutions of the Navier-Stokes equations in Rm, with applications to weak solutions. Math. Z. 187:471–480.
   Web of Science ®Google Scholar
• Koch, H., Tataru, D. (2001). Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1):22–35. DOI: 10.1006/aima.2000.1937.
   Web of Science ®Google Scholar
• Cannone, M. (1995). Ondelettes, paraproduits, et Navier-Stokes. Diderot Editeur.
   Google Scholar
• Cannone, M. (1997). A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat. Iberoamericana. 15:515–541.
   Google Scholar
• Cannone, M., Planchon, F. (1996). Self-similar solutions for Navier-Stokes equations in R3. Comm. Partial Differential Equations. 21:179–193.
   Web of Science ®Google Scholar
• Chemin, J.-Y. (1999). Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. Anal. Math. 77(1):27–50. DOI: 10.1007/BF02791256.
   Web of Science ®Google Scholar
• Kozono, H., Yamazaki, M. (1994). Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Part. Diff. Eq. 19(5–6):959–1014. DOI: 10.1080/03605309408821042.
   Web of Science ®Google Scholar
• Planchon, F. (1996). Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in R3. Ann. Inst. H. Poincaré Anal. Non Linéaire. 13:319–336.
   Web of Science ®Google Scholar
• Taylor, M. E. (1992). Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Part. Differ. Eq. 17(9–10):1407–1456. DOI: 10.1080/03605309208820892.
   Web of Science ®Google Scholar
• Jia, H., Sverák, V. (2014). Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math. 196(1):233–265. DOI: 10.1007/s00222-013-0468-x.
   Web of Science ®Google Scholar
• Bourgain, J., Pavlović, N. (2008). Ill posedness of the Navier-Stokes equations in a critical space in 3d. J. Func. Anal. 255(9):2233–2247. DOI: 10.1016/j.jfa.2008.07.008.
   Web of Science ®Google Scholar
• Cottet, G.-H. (1986). équations de Navier-Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sci. Paris Sér. I Math. 303:105–108.
   Google Scholar
• Giga, Y., Miyakawa, T., Osada, H. (1988). Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Rational Mech. Anal. 104(3):223–250. DOI: 10.1007/BF00281355.
   Web of Science ®Google Scholar
• Gallagher, I., Planchon, F. (2002). On global infinite energy solutions to the Navier-Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 161(4):307–337. DOI: 10.1007/s002050100175.
   Web of Science ®Google Scholar
• Gallay, T. (2012). Stability and interaction of vortices in two-dimensional viscous flows. Discrete Contin. Dyn. Syst. Ser. S. 5(6):1091–1131. DOI: 10.3934/dcdss.2012.5.1091.
   Web of Science ®Google Scholar
• Germain, P. (2006). Équations de Navier-Stokes dans R2: existence et comportement asymptotique de solutions d’énergie infinie. Bull. Sci. Math. 130(2):123–151.
   Web of Science ®Google Scholar
• Guillod, J., Šverák, V. (2017). Numerical investigations of non-uniqueness for the navier-stokes initial value problem in borderline spaces. arXiv preprint arXiv:1704.00560.
   Google Scholar
• Jia, H., Sverak, V. (2015). Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space? J. Funct. Anal. 268(12):3734–3766. DOI: 10.1016/j.jfa.2015.04.006.
   Web of Science ®Google Scholar
• Gallay, T., Wayne, C. (2002). Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2. Arch. Rational Mech. Anal. 163:209–258.
   Web of Science ®Google Scholar
• Gallay, T., Wayne, C. (2005). Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Commun. Math. Phys. 255(1):97–129. DOI: 10.1007/s00220-004-1254-9.
   Web of Science ®Google Scholar
• Gallay, T., Smets, D. (2018). Spectral stability of inviscid columnar vortices. arXiv preprint, arXiv:1805.05064.
   Google Scholar
• Gallay, T., Smets, D. (2019). On the linear stability of vortex columns in the energy space. J. Math. Fluid Mech. 21(4): Paper No. 48. DOI: 10.1007/s00021-019-0453-2.
   Web of Science ®Google Scholar
• Simon, J. (1986). Compact sets in the space Lp(0,T;B). Ann. Mat. Pura. Appl. 146:65–96.
   Web of Science ®Google Scholar