Abstract
We consider the classical Cauchy problem for the three dimensional Navier–Stokes equation with the initial vorticity ω 0 concentrated on a circle, or more generally, a linear combination of such data for circles with common axis of symmetry. We show that natural approximations of the problem obtained by smoothing the initial data satisfy good uniform estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution. We impose no restriction on the size of the initial data.
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References
Ben-Artzi M.: Global solutions of two-dimensional Navier–Stokes and Euler equations. Arch. Ration. Mech. Anal. 128, 329–358 (1994)
Cottet G.-H.: Equations de Navier–Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sci. Paris Ser. I Math. 303, 105–108 (1986)
Fabes E.B., Stroock D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96(4), 327–338 (1986)
Gallagher I., Gallay Th., Lions P.-L.: 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 (2005)
Gallay Th., Wayne C.E.: Global stability of vortex solutions of the two-dimensional Navier–Stokes equation. Commun. Math. Phys. 255, 97–129 (2005)
Gallagher I., Gallay T.: Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity. Math. Ann. 332, 287–327 (2005)
Giga Y., Miyakawa T., Osada H.: Two-dimensional Navier–Stokes flow with measures as initial vorticity. Arch. Ration. Mech. Anal. 104, 223–250 (1988)
Giga Y., Miyakawa T.: Navier–Stokes flow in R3 with measures as initial vorticity and Morrey spaces. Commun. Partial Differ. Equ. 14, 577–618 (1989)
Kato T.: The Navier–Stokes equation for an incompressible fluid in R2 with a measure as the initial vorticity. Differ. Integral Equ. 7, 949–966 (1994)
Koch H., Tataru D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35 (2001)
Ladyzhenskaya, O.: Unique global solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry. Zap. Naucn. Sem. LOMI 7, 155–177 (1968). (in Russian)
Leonardi S., Malek J., Necas J., Pokorny M.: On axially symmetric flows in R3. Z. Anal. Anwendungen 18, 639–649 (1999)
Liu J.G., Wang W.C.: Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier–Stokes equation. SIAM J. Math. Anal. 41(5), 182–1850 (2009)
Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Stein E.: Harmonic Analysis. Princeton University Press, Princeton (1993)
Šverák, V.: Lecture Notes on “Topics in Mathematical Physics”. http://math.umn.edu/sverak/course-notes2011
Taylor M.E.: Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations. Commun. Partial Differ. Equ. 17(9–10), 1407–1456 (1992)
Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2. Amsterdam New York Oxford, North–Holland, 1977
Torchinsky, A.: Real-variable methods in harmonic analysis. In: Pure and Applied Mathematics, vol. 123. Academic Press, Inc., Orlando, 1986
Ukhovskii M., Yudovich V.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 52–61 (1968)
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Feng, H., Šverák, V. On the Cauchy Problem for Axi-Symmetric Vortex Rings. Arch Rational Mech Anal 215, 89–123 (2015). https://doi.org/10.1007/s00205-014-0775-4
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DOI: https://doi.org/10.1007/s00205-014-0775-4