Skip to main content
Log in

Regularized Euler-\(\alpha \) motion of an infinite array of vortex sheets

  • Published:
Bollettino dell'Unione Matematica Italiana Aims and scope Submit manuscript

Abstract

We consider the Euler-\(\alpha \) regularization of the Birkhoff–Rott equation and compare its solutions with the dynamics of the non regularized vortex-sheet. For a flow induced by an infinite array of planar vortex-sheets we analyze the complex singularities of the solutions.Through the singularity tracking method we show that the regularized solution has several complex singularities that approach the real axis. We relate their presence to the formation of two high-curvature points in the vortex sheet during the roll-up phenomenon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Notes

  1. In both cases the imaginary part of the complex characterizations due to the asymptotic behavior in (3.17) is negligible.

  2. For \(\alpha =0.01\) the singularity \(\tilde{p}_1^{\alpha }\) is clearly detected only from \(t=1.3\).

References

  1. Bailey, D.H., Borwein, J.M., Crandall, R.E., Zucker, I.J.: Lattice sums arising from the Poisson equation. J. Phys. A Math. Theor. 46(11), 115201 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, G.R., Beale, J.T.: Vortex blob methods applied to interfacial motion. J. Comp. Phys. 196(1), 233–258 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, G.R., Caflisch, R.E., Siegel, M.: Singularity formation during Rayleigh–Taylor instability. J. Fluid Mech. 252, 51–75 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baker, G.R., Meiron, D.I., Orszag, S.A.: Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baker, G.R., Pham, L.D.: A comparison of blob methods for vortex sheet roll-up. J. Fluid Mech. 547, 297–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baker, G.R., Shelley, M.J.: On the connection between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161–194 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baker, G.A. Jr., Graves-Morris, P.: Padé approximants, 2nd edn. In: Encyclopedia of Mathematics and its Applications, vol. 59, Cambridge University Press, Cambridge (1996)

  8. Bardos, C., Linshiz, J.S., Titi, E.S.: Global regularity for a Birkhoff–Rott-\(\alpha \) approximation of the dynamics of vortex sheets of the 2d Euler equations. Phys. D Nonlinear Phenom. 237(14–17), 1905–1911 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bardos, C., Linshiz, J.S., Titi, E.S.: Global regularity and convergence of a Birkhoff–Rott-\(\alpha \) approximation of the dynamics of vortex sheets of the two-dimensional Euler equations. Commun. Pure Appl. Math. 63(6), 697–746 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Benedetto, D., Caglioti, E., Marchioro, C.: On the motion of a vortex ring with a sharply concentrated vorticity. Math. Methods Appl. Sci. 23(2), 147–168 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benedetto, D., Marchioro, C., Pulvirenti, M.: The \(2\)-D incompressible Euler flow for singular initial conditions, Nonlinear variational problems and partial differential equations (Isola d’Elba, 1990), Pitman Res. Notes Math. Ser., vol. 320, Longman Sci. Tech. Harlow, pp. 57–74 (1995)

  12. Benedetto, D., Pulvirenti, M.: From vortex layers to vortex sheets. SIAM J. Appl. Math. 52(4), 1041–1056 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Birkhoff, Garrett, Fisher, Joseph: Do vortex sheets roll up? Rend. Circ. Mat. Palermo 8(1), 77–90 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caflisch, R.E.: Singularity formation for complex solutions of the 3D incompressible Euler equations. Phys. D 67(1–3), 1–18 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Caflisch, R.E., Gargano, F., Sammartino, M., Sciacca, V.: Complex singularities and PDEs. Riv. Mat. Univ. Parma 6(1), 69–133 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Caflisch, R.E., Gargano, F., Sammartino, M., Sciacca, V.: Complex singularity analysis of vortex layer flow (2016) (in preparation)

  17. Caflisch, R.E., Lombardo, M.C., Sammartino, M.: Vortex layers of small thickness (2016) (in preparation)

  18. Caflisch, R.E., Lowengrub, J.S.: Convergence of the vortex method for vortex sheets. SIAM J. Numer. Anal. 26(5), 1060–1080 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Caflisch, R.E., Orellana, O.F.: Long time existence for a slightly perturbed vortex sheet. Comm. Pure Appl. Math 39, 807–838 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Caflisch, R.E., Orellana, O.F.: Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20(2), 293–307 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cannone, M., Lombardo, M.C., Sammartino, M.: Existence and uniqueness for the Prandtl equations. Comptes Rendus de l’Acadmie des Sciences-Series I-Mathematics 332(3), 277–282 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Cannone, M., Lombardo, M.C., Sammartino, M.: Well-posedness of Prandtl equations with non-compatible data. Nonlinearity 26(3), 3077–3100 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable: Theory and Technique. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  24. Chen, M.J., Forbes, L.K.: Accurate methods for computing inviscid and viscous Kelvin–Helmholtz instability. J. Comput. Phys. 230(4), 1499–1515 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cichowlas, C., Brachet, M.-E.: Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows. Fluid Dyn. Res. 36(4–6), 239–248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Coclite, G.M., Gargano, F., Sciacca, V.: Analytic solutions and singularity formation for the peakon b-family equations. Acta Appl. Math. 122, 419–434 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Cowley, S.J.: Computer extension and analytic continuation of Blasius’ expansion for impulsively flow past a circular cylinder. J. Fluid Mech. 135, 389–405 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cowley, S.J., Baker, G.R., Tanveer, S.: On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233–267 (1999)

    Article  MathSciNet  Google Scholar 

  29. DellaRocca, G., Lombardo, M.C., Sammartino, M., Sciacca, V.: Singularity tracking for Camassa–Holm and Prandtl’s equations. Appl. Numer. Math. 56(8), 1108–1122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dhanak, M.R.: Equation of motion of a diffusing vortex sheet. J. Fluid Mech. 269, 265–281 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. DiPerna, R.J., Majda, A.J.: Concentrations in regularizations for \(2\)-D incompressible flow. Commun. Pure Appl. Math. 40(3), 301–345 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Duchon, J., Robert, R.: Global vortex sheet solutions of euler equations in the plane. J. Differ. Equ. 73(2), 215–224 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes-\(\alpha \) model of fluid turbulence. Phys. D. 152–153, 505–519 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Frisch, U., Matsumoto, T., Bec, J.: Singularities of Euler flow? Not out of the blue!. J. Stat. Phys. 113(5), 761–781 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gargano, F., Ponetti, G., Sammartino, M., Sciacca, V.: Complex singularities in KdV solutions. Ricerche Mat. (2016). doi:10.1007/s11587-016-0269-9

  37. Gargano, F., Sammartino, M., Sciacca, V.: Singularity formation for Prandtl’s equations. Phys. D Nonlinear Phenom. 238(19), 1975–1991 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gargano, F., Sammartino, M., Sciacca, V.: High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex. Comput. Fluids 52, 73–91 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gargano, F., Sammartino, M., Sciacca, V., Cassel, K.W.: Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions. J. Fluid Mech. 747, 381–421 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gargano, F., Sammartino, M., Sciacca, V., Cassel, K.W.: Viscous-inviscid interactions in a boundary-layer flow induced by a vortex array. Acta Appl. Math. 132, 295–305 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gerard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23, 591–609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gerard-Varet, D., Nguyen, T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77, 71–88 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Grard-Varet, D., Masmoudi, N.: Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. ENS 48(6), 1273–1325 (2015)

    MathSciNet  MATH  Google Scholar 

  44. Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137(1), 1–81 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. Holm, D.D., Marsden, J.E., Ratiu, T.S.: Euler–Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80(19), 4173–4176 (1998)

    Article  Google Scholar 

  46. Holm, D.D., Nitsche, M., Putkaradze, V.: Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion. J. Fluid Mech. 555, 149–176 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ignatova, M., Vicol, V.: Almost global existence for the prandtl boundary layer equations. Arch. Ration. Mech. Anal. 220(2), 809–848 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Klein, C., Roidot, K.: Numerical study of shock formation in the dispersionless Kadomtsev–Petviashvili equation and dispersive regularizations. Phys. D 265, 1–25 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Klein, C., Roidot, K.: Numerical study of the semiclassical limit of the Davey–Stewartson II equations. Nonlinearity 27(9), 2177–2214 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Klein, C., Roidot, K.: Numerical study of the long wavelength limit of the toda lattice. Nonlinearity 28(8), 2993–3025 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  51. Krasny, R.: Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65(2), 292–313 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  52. Krasny, R.: A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  53. Kukavica, I., Lombardo, M.C., Sammartino, M.: Zero viscosity limit for analytic solutions of the primitive equations. Arch. Ration. Mech. Anal. 222(1), 15–45 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kukavica, I., Vicol, V.: On the local existence of analytic solutions to the Prandtl boundary layer equations. Commun. Math. Sci. 11, 269–292 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  55. Lebeau, G.: Régularité du problème de Kelvin–Helmholtz pour l’équation d’Euler 2d, ESAIM Control Optim. Calc. Var. 8, : 801–825 (electronic). A tribute to J. L, Lions (2002)

  56. Lombardo, M.C., Cannone, M., Sammartino, M.: Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 35(4), 987–1004 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  57. Lopes Filho, M.C., Lowengrub, J., Nussenzveig Lopes, H.J., Zheng, Y.: Numerical evidence of nonuniqueness in the evolution of vortex sheets. ESAIM Math. Mod. Num. Anal. 40(2), 225–237 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  58. Maekawa, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure Appl. Math. 67(7), 1045–1128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. Marchioro, C.: Euler evolution for singular initial data and vortex theory: a global solution. Commun. Math. Phys. 116(1), 45–55 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  60. Marchioro, C.: On the inviscid limit for a fluid with a concentrated vorticity. Commun. Math. Phys. 196(1), 53–65 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  61. Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, vol. 96. Springer, New York (1994)

    Book  MATH  Google Scholar 

  62. Matsumoto, T., Bec, J., Frisch, U.: The analytic structure of 2D Euler flow at short times. Fluid Dyn. Res. 36(4–6), 221–237 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  63. Moore, D.W.: The equation of motion of a vortex layer of small thickness. Stud. Appl. Math. 58(2), 119–140 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  64. Moore, D.W.: The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. Ser. A 365(1720), 105–119 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  65. Moore, D.W.: Numerical and analytical aspects of Helmholtz instability. Lyngby, Denmark. In: Proceedings of the Sixteenth International Congress of Theoretical and Applied Mechanics, pp. 263–274 (1985)

  66. Nitsche, M.: Singularity formation in a cylindrical and a spherical vortex sheet. J. Comput. Phys. 173(1), 208–230 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  67. Oliver, M., Shkoller, S.: The vortex blob method as a second-grade non-Newtonian fluid. Commun. Partial Differ. Equ. 26(1–2), 295–314 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  68. Pauls, W., Frisch, U.: A Borel transform method for locating singularities of Taylor and Fourier series. J. Stat. Phys. 127(6), 1095–1119 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  69. Pauls, W., Matsumoto, T., Frisch, U., Bec, J.: Nature of complex singularities for the 2D Euler equation. Phys. D 219(1), 40–59 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  70. Pólya, G.: Untersuchungen über Lücken und Singularitäten von Potenzreihen. Math. Z. 29, 549–640 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  71. Roidot, K., Mauser, N.: Numerical study of the transverse stability of NLS soliton solution in several classes of NLS-type equations. arxiv:1401.5349 (2015)

  72. Saffman, P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1993)

  73. Sammartino, M., Caflisch, R.E.: Zero viscosity limit for analytic solutions, of the Navier–Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192(2), 433–461 (1998)

  74. Shelley, M.J.: A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244, 493–526 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  75. Sohn, S.-I.: Singularity formation and nonlinear evolution of a viscous vortex sheet model. Phys. Fluids 25(1), 014106 (2013)

    Article  MATH  Google Scholar 

  76. Sulem, C., Sulem, P.-L., Bardos, C., Frisch, U.: Finite time analyticity for the two- and three-dimensional Kelvin–Helmholtz instability. Commun. Math. Phys. 80(4), 485–516 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  77. Sulem, C., Sulem, P.-L., Frisch, H.: Tracing complex singularities with spectral methods. J. Comput. Phys. 50(1), 138–161 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  78. Tryggvason, G., Dahm, W.J.A., Sbeih, K.: Fine structure of vortex sheet rollup by viscous and inviscid simulation. J. Fluids Eng. 113(1), 31–36 (1991)

    Article  Google Scholar 

  79. Weideman, J.A.C.: Computing the dynamics of complex singularities of nonlinear PDEs. SIAM J. Appl. Dyn. Syst. 2(2), 171–186 (2003). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  80. Sijue, Wu: Mathematical analysis of vortex sheets. Commun. Pure Appl. Math. 59(8), 1065–1206 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  81. Zhong, X.: Additive semi-implicit Runge–Kutta methods for computing high-speed nonequilibrium reactive flows. J. Comp. Phys. 128(1), 19–31 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The work of FG and VS has been partially supported by the Progetti Giovani GNFM 2015 grant.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Sammartino.

Appendix: Singularity tracking methods

Appendix: Singularity tracking methods

The singularity tracking method based on the asymptotic ansatz (3.17) has the advantage to be of easy implementation, but it has the relevant drawback that it gives information only on the complex singularity closest to the real axis. To retrieve information on the possible other singularities outside the width of the analyticity strip, one can use two other methods which, as we shall explain in this Appendix, should be used together to obtain more robust results.

The first method used is the so called BPH (Borel–Polya–Van der Hoeven) method, originally proposed in [68] to characterize all the complex singularities for the Burgers equation. It is based on an asymptotic ansatz given by Polya ([70]) for a Borel series obtained from a Taylor series. In particular, given the inverse Taylor series \(f(z) = \sum _{k=0}^N f_k/z^{k+1}\) that has n complex singularities \(c_j= |c_j|\)e\(^{-i\rho _j}\) for \(j=1,2,\ldots , n\), its Borel transform is given by \(U_B(\zeta )=\sum _{k=0}^N f_k\zeta ^{k}/k!\). Evaluating the modulus of the Borel series \(G(r)=|U_B(r\text {e}^{i\phi })|\) along the rays \(r\text {e}^{i\phi }\), one obtains, through a steepest descent argument, the following asymptotic behaviour

$$\begin{aligned} G(r)\approx C(\phi )r^{-(\mu (\phi )+1)}\text {e}^{h(\phi )r} \quad \text {for}\quad r \rightarrow \infty . \end{aligned}$$
(A.37)

The indicatrix function \(h(\phi )\) is the piecewise cosine function

$$\begin{aligned} h(\phi )=|c_j|\cos (\phi -\rho _j) \quad \text {for} \quad \phi _{j-1}<\phi <\phi _j, \end{aligned}$$
(A.38)

where the angular intervals \((\phi _{j-1},\phi _{j})\) depend on the complex positions of the singularities (we refer to [68] for a deeper explanation on how the set \(\phi _j\), \(j=1,2,\ldots ,n\) is determined). Therefore, through numerical interpolation we can determine the parameters \(|c_j|\) and \(\rho _j\) that give the locations of the complex singularities \(c_j\). In practice, for each direction \(\phi \) we need to determine the exponential rate of (A.37) that allows for construction of the indicatrix function h. Moreover, an estimate of \(\mu (\rho _j)\) in (A.37) returns the characterization of the singularity \(c_j\). The BPH method can be easily applied to Fourier series \(u(z)=\sum _{k=-K/2}^{k=K/2}u_k\text {e}^{ikz}\) by writing u as a Taylor series. This is accomplished by introducing the complex variables \(Z_+=e^{iz},Z_-=e^{-iz}\) so that

$$\begin{aligned} u(z)=\sum _{k=0}^{K/2}u_ke^{ikz}+\sum _{k=1}^{K/2}u_ke^{-ikz}=\sum _{k=0}^{K/2} u_k/Z_-^k+\sum _{k=1}^{K/2}u_k/Z_+^k \end{aligned}$$
(A.39)

The advantage of this methodology with respect to the singularity tracking method lies in the fact that it is possible to capture information on all the singularities located in the convex hull outside the radius of convergence of a Taylor series (or the strip of analyticity of a Fourier series). However some drawbacks are present. In fact singularities close to each other could be difficult to distinguish, and the various cosine function in (A.38) can be numerically determined along different range of r.

As already done in [39] to detect the complex singularities of the wall shear of Navier-Stokes equation, it is strongly suggested to obtain preliminary information on the position of the complex singularities, so that one can search for the specific cosine functions forming the indicatrix function. This is accomplished with the second method we have used, the Padé approximation . We recall that the Padé approximant

$$\begin{aligned} P_{L/M}=\frac{\sum _{i=0}^{L}a_i z^i}{1+\sum _{j=1}^{M}b_j z^j} \end{aligned}$$
(A.40)

is a rational function which approximates a complex function f(z) as

$$\begin{aligned} f(z) - P_{L/M} (z) = O (z^{L+M+1} ). \end{aligned}$$
(A.41)

If \(f(z)=\sum _{k=0}^{\infty }f_kz^k\), the M unknown denominator coefficients \(b_j, j=1\ldots ,M\) and the \(L+1\) unknown numerator coefficients \(a_i, i=0,\ldots ,L\) are determined uniquely by (A.41). This means that the following set of linear equations must be solved

$$\begin{aligned} f_{\alpha }+ \sum _{i=1}^{\min (\alpha ,M)}b_{i} f_{\alpha -i}=a_{\alpha } \quad \alpha =0,\dots ,L; \qquad f_{L+\beta }+ \sum _{i=1}^{M}b_{i}f_{L+\beta -i}=0, \quad \beta =1,\dots ,M. \end{aligned}$$

Padé approximants are easy compute for Fourier series: given \(u(x)= \sum _{k=-N}^{N} \hat{u}_k e^{i k x}\), then its Padé approximants can be derived considering the Fourier series as the sum of two power series in the complex variables \(z_{+} = e^{ix}\) and \(z_{-} = e^{-ix}\), i.e.

$$\begin{aligned} u(z) \approx P_{L/M}(z_{+})+Q_{L/M}(z_{-}) - \hat{u}_0 , \end{aligned}$$
(A.42)

with \(L+M+1\le N\), and \( P_{L/M}(z_{+})\) and \(Q_{L/M}(z_{-})\) are respectively the Padé approximants of \(\sum _{k=0}^{N} \hat{u}_k z_{+}^k\) and \(\sum _{k=0}^{N} \hat{u}_k z_{-}^k\), see [7, 79]. The main drawbacks of the Padé approximation is that it gives no information about the characterizations of the singularities. Moreover it is unable to distinguish between poles and branch singularities, as branch cut are approximated by a sequence of poles where the cut should be. However, for our purpose, we need only to detect the position of the singularities, and this can be done by simply extrapolate the zeros of the denominator in (A.40).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caflisch, R.E., Gargano, F., Sammartino, M. et al. Regularized Euler-\(\alpha \) motion of an infinite array of vortex sheets. Boll Unione Mat Ital 10, 113–141 (2017). https://doi.org/10.1007/s40574-016-0097-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40574-016-0097-6

Keywords

Navigation