Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow

Boussandel, Sahbi ; Chill, Ralph ; Fašangová, Eva

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 335-346

Abstract (VO)

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and $L^2$-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented in the special situation only.

Détail

DOI : 10.1007/s10587-012-0033-6

Classification : 35B30, 35B65, 35K90, 35K93, 46T20
Keywords: curve shortening flow; maximal regularity; local inverse function theorem

Bibliographie

[1] Almgren, F., Taylor, J. E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optimization 31 (1993), 387-438. | DOI | MR | Zbl

[2] Amann, H.: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4 (2004), 417-430. | DOI | MR | Zbl

[3] Amann, H.: Maximal regularity and quasilinear parabolic boundary value problems. Recent advances in elliptic and parabolic problems. Hackensack, NJ: World Scientific (2005), 1-17. | MR | Zbl

[4] Amann, H.: Quasilinear parabolic problems via maximal regularity. Adv. Differ. Equ. 10 (2005), 1081-1110. | MR | Zbl

[5] Amann, H.: Existence and regularity for semilinear parabolic evolution equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11 (1984), 593-676. | MR | Zbl

[6] Amann, H.: Quasilinear evolution equations and parabolic systems. Trans. Am. Math. Soc. 293 (1986), 191-227. | DOI | MR | Zbl

[7] Angenent, S. B.: Nonlinear analytic semiflows. Proc. R. Soc. Edinb., Sect. A 115 (1990), 91-107. | DOI | MR | Zbl

[8] Angenent, S. B.: Parabolic equations for curves on surfaces I. Curves with {$p$}-integrable curvature. Ann. Math. (2) 132 (1990), 451-483. | MR | Zbl

[9] Arendt, W., Chill, R., Fornaro, S., Poupaud, C.: {$L^p$}-maximal regularity for nonautonomous evolution equations. J. Differ. Equations 237 (2007), 1-26. | DOI | MR

[10] Bothe, D., Prüss, J.: {$L_P$}-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39 (2007), 379-421. | DOI | MR | Zbl

[11] Brakke, K. A.: The Motion of a Surface by its Mean Curvature. Princeton, New Jersey: Princeton University Press. Tokyo: University of Tokyo Press (1978). | MR | Zbl

[12] Chou, K.-S., Zhu, X.-P.: The Curve Shortening Problem. Boca Raton, FL: Chapman & Hall/CRC. ix (2001). | MR | Zbl

[13] Clément, P., Li, S.: Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3 (1994), 17-32. | MR | Zbl

[14] Prato, G. Da, Grisvard, P.: Equations d'évolution abstraites non linéaires de type parabolique. Ann. Mat. Pura Appl., IV. Ser. 120 (1979), 329-396 French. | DOI | MR | Zbl

[15] Simon, L. De: Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rend. Sem. Mat. Univ. Padova 34 (1964), 205-223 Italian. | MR | Zbl

[16] Deckelnick, K.: Weak solutions of the curve shortening flow. Calc. Var. Partial Differ. Equ. 5 (1997), 489-510. | DOI | MR | Zbl

[17] Deckelnick, K., Dziuk, G., Elliott, C. M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numerica 14 (2005), 139-232. | DOI | MR | Zbl

[18] DeTurck, D. M.: Deforming metrics in the direction of their Ricci tensors. J. Differ. Geom. 18 (1983), 157-162. | DOI | MR | Zbl

[19] Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations and Their Applications 57. Boston, MA: Birkhäuser (2004). | MR | Zbl

[20] Escher, J., Prüss, J., Simonett, G.: A new approach to the regularity of solutions for parabolic equations. Evolution Equations. Proceedings in honor of the 60th birthdays of P. Bénilan, J. A. Goldstein and R. Nagel. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 234 (2003), 167-190. | MR | Zbl

[21] Giga, Y.: Surface Evolution Equations. A level set approach. Monographs in Mathematics 99. Basel: Birkhäuser (2006). | MR | Zbl

[22] Guidetti, D.: A maximal regularity result with applications to parabolic problems with nonhomogeneous boundary conditions. Rend. Semin. Mat. Univ. Padova 84 (1990), 1-37 (1991). | MR

[23] Hieber, M., Rehberg, J.: Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains. SIAM J. Math. Anal. 40 (2008), 292-305. | DOI | MR | Zbl

[24] Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. Calculus of variations and geometric evolution problems. (Cetraro, 1996), Berlin: Springer. Lect. Notes Math. 1713 (1999), 45-84. | MR | Zbl

[25] Ladyženskaja, O. A., Solonnikov, V. A., Ural'ceva, N. N.: Linear and Quasilinear Equations of Parabolic Type. Moskva: Izdat. `Nauka' (1967), Russian. | MR

[26] Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differ. Equ. 3 (1995), 253-271. | DOI | MR | Zbl

[27] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications. 16. Basel: Birkhäuser (1995). | MR | Zbl

[28] Lunardi, A.: Interpolation Theory. 2nd ed. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) 9. Pisa: Edizioni della Normale (2009). | MR | Zbl

[29] Mikula, K., Ševčovič, D.: Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Vis. Sci. 6 (2004), 211-225. | DOI | MR

[30] Prüss, J., Schnaubelt, R.: Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time. J. Math. Anal. Appl. 256 (2001), 405-430. | DOI | MR | Zbl

[31] Prüss, J.: Maximal regularity for evolution equations in {$L_p$}-spaces. Conf. Semin. Mat. Univ. Bari (2002), 1-39 (2003). | MR

[32] Saal, J.: Strong solutions for the Navier-Stokes equations on bounded and unbounded domains with a moving boundary. Electron. J. Differ. Equ., Conf. 15 (2007), 365-375. | MR

[33] Simonett, G.: The Willmore flow near spheres. Differ. Integral Equ. 14 (2001), 1005-1014. | MR | Zbl

[34] Zeidler, E.: Nonlinear Functional Analysis and its Applications. Volume I: Fixed-point theorems. Translated from the German by Peter R. Wadsack. New York: Springer-Verlag (1993). | MR | Zbl

[35] Zhu, X.-P.: Lectures on Mean Curvature Flows. AMS/IP Studies in Advanced Mathematics 32. Providence, RI: American Mathematical Society (AMS), Somerville: International Press (2002). | MR | Zbl