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Global and Local Theory of Skew Mean Curvature Flows

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Abstract

In this paper, we study the skew mean curvature flow. The results are threefold. First, we prove the global regularity of solutions with initial data which are small perturbations of planes in Sobolev spaces. Second, we prove the modified scattering and the existence of wave operators for small data, which completely determines the set of asymptotic states. Third, we study the Cauchy problem for arbitrary large data.

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References

  1. Arnold, V., Khesin, B.: Topological Methods in Hydrodynamics. Appl. Math. Sci., vol. 125. Springer, New York (1998)

  2. Breuning, P.: Immersions with bounded second fundamental form. J. Geom. Anal. 25, 1344–1386 (2015)

    Article  MathSciNet  Google Scholar 

  3. Da Rios, L.: On the motion of an unbounded fluid with a vortex filament of any shape. Rend. Circ. Mat. Palermo 22, 117–135 (1906)

    Article  Google Scholar 

  4. Gomez, H.: Binormal motion of curves and surfaces in a manifold, Ph.D. thesis, University of Maryland (2004)

  5. Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    Article  MathSciNet  Google Scholar 

  6. Hasimoto, H.: A soliton on a vortex filament. J. Fluid Mech. 51, 477–485 (1972)

    Article  MathSciNet  Google Scholar 

  7. Haller, S., Vizman, C.: Non-linear Grassmannians as coadjoint orbits. Math. Ann. 329(4), 771–785 (2004)

    Article  MathSciNet  Google Scholar 

  8. Jerrard, R.: Vortex filament dynamics for Gross-Pitaevsky type equations. Ann. Sc. Norm. Super. Pisa CI. Sci. 1(4), 733–768 (2002)

  9. Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)

    Article  MathSciNet  Google Scholar 

  10. Kenig, C.E., Ponce, G., Rolvung, C., Vega, L.: The general quasilinear ultrahyperbolic Schrödinger equation. Adv. Math. 196(2), 402–433 (2005)

    Article  Google Scholar 

  11. Kenig, C.E., Ponce, G., Vega, L.: Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincar\(\acute{e}\) Anal. Non Linéaire, 10, 255–288 (1993)

  12. Kenig, C.E., Ponce, G., Vega, L.: Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math. 134, 489–545 (1998)

    Article  MathSciNet  Google Scholar 

  13. Kenig, C.E., Ponce, G., Vega, L.: The Cauchy problem for quasi-linear Schrödinger equations. Invent. Math. 158, 343–388 (2004)

    Article  MathSciNet  Google Scholar 

  14. Khesin, B.: Symplectic structures and dynamics on vortex membranes. Mosc. Math. J. 12(2), 46–462 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Khesin, B., Yang, C.: Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations. Nonlinearity 34(3) (2021)

  16. Klainerman, S., Ponce, G.: Global small amplitude solutions to nonlinear evolution equations. Commun. Pure Appl. Math. 36, 133–141 (1983)

    Article  MathSciNet  Google Scholar 

  17. Kuwert, E., Schatzle, R.: Gradient flow for the Willmore functional. Commun. Anal. Geom. 10(2), 307–340 (2002)

    Article  MathSciNet  Google Scholar 

  18. Langer, J.: A compactness theorem for surfaces with Lp-bounded second fundamental form. Math. Ann. 270, 223–234 (1985)

    Article  MathSciNet  Google Scholar 

  19. Lin, F.: Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Commun. Pure Appl. Math. 51, 385–441 (1998)

    Article  MathSciNet  Google Scholar 

  20. Li, Z.: Global transversal stability of Euclidean planes under skew mean curvature flow evolutions. Calc. Var. 60, 57 (2021)

    Article  MathSciNet  Google Scholar 

  21. Marsden, J., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Physica D 7(1–3), 305–323 (1983)

    Article  MathSciNet  Google Scholar 

  22. Marzuola, J.L., Metcalfe, J., Tataru, D.: Quasilinear Schrödinger equations I: small data and quadratic interactions. Adv. Math. 231(2), 1151–1172 (2012)

    Article  MathSciNet  Google Scholar 

  23. Marzuola, J.L., Metcalfe, J., Tataru, D.: Quasilinear Schrödinger equations II: small data and cubic nonlinearities. Kyoto J. Math. 54(3), 179–190 (2012)

    MATH  Google Scholar 

  24. Mizohata, S.: Some remarks on the Cauchy problem. J. Math. Kyoto Univ. 1, 109–127 (1961)

    MathSciNet  MATH  Google Scholar 

  25. Mizohata, S.: Sur quelques equations du type Schrödinger. Journees Equations aux derivees partielles, Saint-Jean de Monts (1981)

    Book  Google Scholar 

  26. Mizohata, S.: On the Cauchy Problem. Notes and Reports in Mathematics in Science and Engineering, vol. 3. Science Press and Academic Press (1985)

  27. Shashikanth, B.N.: Vortex dynamics in \(R^4\). J. Math. Phys. 53, 013103 (2012)

    Article  MathSciNet  Google Scholar 

  28. Song, C.: Gauss map of the skew mean curvature flow. Proc. Am. Math. Soc. 145(11), 4963–4970 (2017)

    Article  MathSciNet  Google Scholar 

  29. Song, C., Sun, J.: Skew mean curvature flow. Commun. Contemp. Math. 21(1), 1750090 (2019)

    Article  MathSciNet  Google Scholar 

  30. Song, C.: Local existence and uniqueness of skew mean curvature flow. J. Reine Angew Math. 776, 1–26 (2021)

    Article  MathSciNet  Google Scholar 

  31. Taylor, M.: Partial Differential Equations III, Nonlinear Equations, 2nd edn., Applied Mathematical Sciences, vol. 117. Springer, New York (2011)

  32. Terng, C.: Dispersive geometric curve flows. In: Surveys in Differential Geometry: Regularity and Evolution of Nonlinear Equations. Surveys in Differential Geometry, vol. 19, pp. 179–229. International Press, Somerville, MA (2014)

  33. Terng, C., Uhlenbeck, K.: Schrödinger flows on Grassmannians. In: Integrable Systems, Geometry, and Topology, AMS/IP Studies in Advanced Mathematics, vol. 36, pp. 235–256. American Mathematical Society, Providence, RI (2006). See also arxiv preprint (1999)

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Acknowledgements

The author owes sincere gratitude to the referees for the insightful comments which largely improved the presentation of this work. This work is partially supported by NSF-China Grant-1200010237 and Grant-11631007.

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  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Ze Li

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  1. Ze Li
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Correspondence to Ze Li.

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Li, Z. Global and Local Theory of Skew Mean Curvature Flows. J Geom Anal 32, 34 (2022). https://doi.org/10.1007/s12220-021-00735-9

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