A global theory of internal solitary waves in two-fluid systems
HTML articles powered by AMS MathViewer
- by C. J. Amick and R. E. L. Turner PDF
- Trans. Amer. Math. Soc. 298 (1986), 431-484 Request permission
Abstract:
The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity $g$. It is assumed that a fluid of constant density ${\rho _ + }$ lies above a fluid of constant density ${\rho _ - } > {\rho _ + } > 0$ and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair $(\lambda ,w)$, where the constant $\lambda = g/{c^2}$, $c$ being the wave speed, and where $w(x,\eta ) + \eta$ is the height at a horizontal position $x$ of the streamline which has height $\eta$ at $x = \pm \infty$. It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for $(\lambda ,w)$ is an unbounded connected set in ${\mathbf {R}} \times (H_0^1 \cap {C^{0,1}})$. Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.References
- Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), no. 2, 431–461. MR 732100, DOI 10.1090/S0002-9947-1984-0732100-6
- Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Jets with two fluids. I. One free boundary, Indiana Univ. Math. J. 33 (1984), no. 2, 213–247. MR 733897, DOI 10.1512/iumj.1984.33.33011
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
- Charles J. Amick, Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 3, 441–499. MR 785621
- C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9–95. MR 629699, DOI 10.1007/BF00250799
- J. Thomas Beale, The existence of solitary water waves, Comm. Pure Appl. Math. 30 (1977), no. 4, 373–389. MR 445136, DOI 10.1002/cpa.3160300402
- Hugo Beirão da Veiga, Raul Serapioni, and Alberto Valli, On the motion of nonhomogeneous fluids in the presence of diffusion, J. Math. Anal. Appl. 85 (1982), no. 1, 179–191. MR 647566, DOI 10.1016/0022-247X(82)90033-6 T. B. Benjamin, Internal waves of finite amplitude and permanent form, J. Fluid Mech. 25 (1966), 241-270.
- T. B. Benjamin, A unified theory of conjugate flows, Philos. Trans. Roy. Soc. London Ser. A 269 (1971), 587–643. MR 446075, DOI 10.1098/rsta.1971.0053
- J. L. Bona, D. K. Bose, and R. E. L. Turner, Finite-amplitude steady waves in stratified fluids, J. Math. Pures Appl. (9) 62 (1983), no. 4, 389–439 (1984). MR 735931 A. D. Frank-Kamenetskii, Diffusion and heat transfer in chemical kinetics, 2nd ed., Plenum, New York-London, 1969 [translated from Russian].
- K. O. Friedrichs and D. H. Hyers, The existence of solitary waves, Comm. Pure Appl. Math. 7 (1954), 517–550. MR 65317, DOI 10.1002/cpa.3160070305 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations, Academic Press, New York, 1968.
- A. V. Kažihov and Š. Smagulov, The correctness of boundary value problems in a certain diffusion model of an inhomogeneous fluid, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 330–332 (Russian). MR 0446059
- D. Kinderlehrer, L. Nirenberg, and J. Spruck, Regularity in elliptic free boundary problems, J. Analyse Math. 34 (1978), 86–119 (1979). MR 531272, DOI 10.1007/BF02790009 R. R. Long, Solitary waves in the one- and two-fluid systems, Tellus 8 (1956), 460-471. D. I. Meiron and P. G. Saffman, Overhanging interfacial gravity waves of large amplitude, J. Fluid Mech. 129 (1983), 213-218.
- Norman G. Meyers, An $L^{p}$e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR 159110
- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 404890, DOI 10.1137/1018076
- Charles B. Morrey Jr., On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior, Amer. J. Math. 80 (1958), 198–218. MR 106336, DOI 10.2307/2372830
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
- A. D. Peters and J. J. Stoker, Solitary waves in liquids having non-constant density, Comm. Pure Appl. Math. 13 (1960), 115–164. MR 112445, DOI 10.1002/cpa.3160130110 J. S. Russell, Report on waves, "Rep. 14th Meeting of the British Association for the Advancement of Science", p. 311, John Murray, London, 1844.
- A. M. Ter-Krikorov, Théorie exacte des ondes longues stationnaires dans un liquide hétérogène, J. Mécanique 2 (1963), 351–376 (French). MR 160400
- R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 4, 513–573. MR 656000
- R. E. L. Turner, A variational approach to surface solitary waves, J. Differential Equations 55 (1984), no. 3, 401–438. MR 766131, DOI 10.1016/0022-0396(84)90077-9
- R. E. L. Turner and J.-M. Vanden-Broeck, The limiting configuration of interfacial gravity waves, Phys. Fluids 29 (1986), no. 2, 372–375. MR 828169, DOI 10.1063/1.865721 —, On solitary waves in two-fluid systems (to appear). L. R. Walker, Interfacial solitary waves in a two-fluid medium, Phys. Fluids 16 (1973), 1796-1804.
- Gordon Thomas Whyburn, Topological analysis, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1958. MR 0099642
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 298 (1986), 431-484
- MSC: Primary 35Q99; Secondary 76C10
- DOI: https://doi.org/10.1090/S0002-9947-1986-0860375-3
- MathSciNet review: 860375