\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Solitary gravity-capillary water waves with point vortices

Abstract Related Papers Cited by
  • We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting waves are also examined: We find that they depend significantly on the position of the point vortices in the water column.
    Mathematics Subject Classification: Primary: 35Q31; Secondary: 35C07, 76B25.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude, Dokl. Akad. Nauk SSSR, 294 (1987), 1033-1037.

    [2]

    G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007.doi: 10.1002/cpa.20365.

    [3]

    A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers, preprint, arXiv:1407.0092.

    [4]

    A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.doi: 10.1088/0305-4470/34/7/313.

    [5]

    A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.doi: 10.1137/1.9781611971873.

    [6]

    A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603.doi: 10.1215/S0012-7094-07-14034-1.

    [7]

    A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.doi: 10.1002/cpa.3046.

    [8]

    A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175.doi: 10.1007/s00205-011-0412-4.

    [9]

    A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67.doi: 10.1007/s00205-010-0314-x.

    [10]

    W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.doi: 10.1006/jcph.1993.1164.

    [11]

    W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach, Nonlinearity, 5 (1992), 497-522.doi: 10.1088/0951-7715/5/2/009.

    [12]

    M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.doi: 10.1016/0022-1236(71)90015-2.

    [13]

    J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, Grenoble, 5 (1953/54), 305-370.

    [14]

    M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie.

    [15]

    M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419.doi: 10.1007/s00021-011-0068-8.

    [16]

    M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456.doi: 10.1137/100792330.

    [17]

    M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not. IMRN, (2009), 4578-4596.doi: 10.1093/imrn/rnp100.

    [18]

    M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.doi: 10.1016/j.jde.2008.01.012.

    [19]

    M. Ehrnström and E. Wahlén, Trimodal steady water waves, Arch. Ration. Mech. Anal., 216 (2015), 449-471.doi: 10.1007/s00205-014-0812-3.

    [20]

    J. Escher, A.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differential Equations, 251 (2011), 2932-2949.doi: 10.1016/j.jde.2011.03.023.

    [21]

    T. W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001.doi: 10.1007/978-0-387-21607-2.

    [22]

    F. Gerstner, Theorie der wellen, Annalen Der Physik, 32 (1809), 412-445.doi: 10.1002/andp.18090320808.

    [23]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

    [24]

    M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity, Phys. D, 237 (2008), 1530-1538.doi: 10.1016/j.physd.2008.03.015.

    [25]

    D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786.doi: 10.1017/S0308210512001990.

    [26]

    C. Hirt, S. Claessens, T. Fecher, M. Kuhn, R. Pail and M. Rexer, New ultrahigh-resolution picture of earth's gravity field, Geophysical Research Letters, 40 (2013), 4279-4283.doi: 10.1002/grl.50838.

    [27]

    V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521 (electronic).doi: 10.1137/040621168.

    [28]

    V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), 491-509.doi: 10.4310/MRL.2008.v15.n3.a9.

    [29]

    R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.doi: 10.1017/CBO9780511624056.

    [30]

    D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013.doi: 10.1090/surv/188.

    [31]

    J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978.

    [32]

    C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994.doi: 10.1007/978-1-4612-4284-0.

    [33]

    S. Mardare, On Poincaré and de Rham's theorems, Rev. Roumaine Math. Pures Appl., 53 (2008), 523-541.

    [34]

    A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.

    [35]

    A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.

    [36]

    B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity, Monatsh. Math., 174 (2014), 459-475.doi: 10.1007/s00605-013-0583-1.

    [37]

    C. C. Mei, The applied dynamics of ocean surface waves, Ocean Engineering, 11 (1984), p321.doi: 10.1016/0029-8018(84)90033-7.

    [38]

    A. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3.

    [39]

    P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416.doi: 10.1007/s00205-003-0292-3.

    [40]

    T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996.doi: 10.1515/9783110812411.

    [41]

    J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity, Nonlinearity, 26 (2013), 1529-1564.doi: 10.1088/0951-7715/26/6/1529.

    [42]

    J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698-744.doi: 10.1002/cpa.20213.

    [43]

    J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.

    [44]

    J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth, IMA J. Appl. Math., 56 (1996), 207-217.

    [45]

    K. Varholm, Water Waves with Compactly Supported Vorticity, Master's thesis, Norwegian University of Science and Technology, 2014.

    [46]

    E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.doi: 10.1016/j.jde.2008.10.005.

    [47]

    S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.doi: 10.1137/080721583.

    [48]

    S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315.doi: 10.3934/dcds.2014.34.3287.

    [49]

    V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194.doi: 10.1007/BF00913182.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(188) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return