Abstract
We present the necessary and sufficient conditions of the well-posedness of the initial value problem for certain fourth-order linear dispersive systems on the one-dimensional torus. This system is related with a dispersive flow for closed curves into compact Riemann surfaces. For this reason, we study not only the general case but also the corresponding special case in detail. We apply our results on the linear systems to the fourth-order dispersive flows. We see that if the sectional curvature of the target Riemann surface is constant, then the equation of the dispersive flow satisfies our conditions of the well-posedness.
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Chang, N.-H., Shatah, J., Uhlenbeck, K.: Schrödinger maps. Comm. Pure Appl. Math. 53, 590–602 (2000)
Chihara, H.: The initial value problem for Schrödinger equations on the torus. Int. Math. Res. Not. 2002(15), 789–820 (2002)
Chihara, H.: The initial value problem for a third order dispersive equation on the two dimensional torus. Proc. Am. Math. Soc. 133, 2083–2090 (2005)
Chihara, H.: Schrödinger flow into almost Hermitian manifolds. Bull. Lond. Math. Soc. 45, 37–51 (2013)
Chihara, H., Onodera, E.: A third order dispersive flow for closed curves into almost Hermitian manifolds. J. Funct. Anal. 257, 388–404 (2009)
Chihara, H., Onodera, E.: A fourth-order dispersive flow into Kähler manifolds. Z. Anal. Anwend (2015, to appear)
Doi, S.-I.: Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. Duke Math. J. 82, 679–706 (1996)
Koiso, N.: The vortex filament equation and a semilinear Schrödinger equation in a Hermitian symmetric space. Osaka J. Math. 34, 199–214 (1997)
Koiso, N.: Long time existence for vortex filament equation in a Riemannian manifold. Osaka J. Math. 45, 265–271 (2008)
Koiso, N.: Vortex filament equation in a Riemannian manifold. Tohoku Math. J. 55, 311–320 (2003)
Kumano-go, H.: Pseudo-Differential Operators. The MIT Press, Cambridge (1981)
Mizohata, S.: On the Cauchy Problem. Academic Press, New York (1985)
Mizuhara, R.: The initial value problem for third and fourth order dispersive equations in one space dimension. Funkcial. Ekvac. 49, 1–38 (2006)
Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces. Birkhäuser, Basel (2010)
Onodera, E.: A third-order dispersive flow for closed curves into Kähler manifolds. J. Geom. Anal. 18, 889–918 (2008)
Onodera, E.: Generalized Hasimoto transform of one-dimensional dispersive flows into compact Riemann surfaces. In: SIGMA Symmetry Integrability Geom. Methods Appl. 4, article No. 044 (2008)
Onodera, E.: A remark on the global existence of a third order dispersive flow into locally Hermitian symmetric spaces. Commun. Partial Differ. Equ. 35, 1130–1144 (2010)
Onodera, E.: A curve flow on an almost Hermitian manifold evolved by a third order dispersive equation. Funkcial. Ekvac. 55, 137–156 (2012)
Singer, I.M., Thorpe, J.A.: Lecture Notes on Elementary Topology and Geometry. Springer, Berlin (1967)
Takeuchi, J.: A necessary condition for the well-posedness of the Cauchy problem for a certain class of evolution equations. Proc. Japan Acad. 50, 133–137 (1974)
Tarama, S.: Remarks on L2-wellposed Cauchy problem for some dispersive equations. J. Math. Kyoto Univ. 37, 757–765 (1997)
Tarama, S.: \(L^2\)-well-posed Cauchy problem for fourth order dispersive equations on the line. Electron. J. Differ. Equ. 2011, 1–11 (2011)
Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)
Acknowledgments
The author would like to thank Eiji Onodera for invaluable comments and helpful information on the holomomy. The author is also grateful to the referees for careful reading the first version of the manuscript of this paper. The author was supported by JSPS Grant-in-Aid for Scientific Research #23340033.
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Chihara, H. Fourth-order dispersive systems on the one-dimensional torus. J. Pseudo-Differ. Oper. Appl. 6, 237–263 (2015). https://doi.org/10.1007/s11868-015-0112-1
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DOI: https://doi.org/10.1007/s11868-015-0112-1