Abstract
We prove the local well-posedness of a basic model for relaxational fluid vesicle dynamics by a contraction mapping argument. Our approach is based on the maximal \(L_p\)-regularity of the model’s linearization.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Amann. Linear and quasilinear parabolic problems. Vol. I, volume 89 of Monographs in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory.
R. Aris. Vectors, tensors and the basic equations of fluid mechanics. Dover Publications, 1990.
P. B. Canham. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology, 26(1):61–81, 1970.
C. H. A. Cheng, D. Coutand, and S. Shkoller. Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal., 39(3):742–800 (electronic), 2007.
R. Denk, J. Saal, and J. Seiler. Inhomogeneous Symbols, the Newton Polygon, and Maximal \({L}_p\)-Regularity. Russian J. Math. Phys., 15(2):171–192, 2008.
E. A. Evans. Bending resistance and chemically induced moments in membrane bilayers. Biophysical journal, 14(12):923–931, 1974.
L. C. Evans. Partial Differential Equations. American Mathematical Society, 1998.
G. P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Volume 1: Linearized Steady Problems. Springer, 1994.
W. Helfrich. Zeitschrift für Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie. Zeitschrift für Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie, 28(11):693–703, 1973.
N. J. Kalton and L. Weis. The \({H}^\infty \)-Calculus and Sums of Closed Operators. Math. Ann., 321:319–345, 2001.
Y. Kohsaka and T. Nagasawa. On the existence of solutions of the helfrich flow and its center manifold near spheres. Differential and Integral Equations, 19(2):121–142, 2006.
L. D. Kudrjavcev. Imbedding Theorem for a Class of Functions Defined on the Entire Space or on a Halfspace. I. Amer. Math. Soc. Transl., 74:199–225, 1968.
L. D. Kudrjavcev. Imbedding Theorem for a Class of Functions Defined on the Entire Space or on a Halfspace. II. Amer. Math. Soc. Transl., 74:227–260, 1968.
D. Lengeler. Globale Existenz für die Interaktion eines Navier-Stokes-Fluids mit einer linear elastischen Schale. PhD thesis, University of Freiburg, 2011.
D. Lengeler. Asymptotic stability of local helfrich minimizers. Preprint: arXiv:1510.01521, 2015.
D. Lengeler. On a Stokes-type system arising in fluid vesicle dynamics. Preprint: arXiv:1506.08991, 2015.
J. McCoy and G. Wheeler. Finite time singularities for the locally constrained willmore flow of surfaces. Preprint: arXiv:1201.4541, 2012.
J. McCoy and G. Wheeler. A classification theorem for Helfrich surfaces. Math. Ann., 357(4):1485–1508, 2013.
S. Meyer. Well-posedness of a moving sharp-interface problem with boussinesq-scriven surface viscosities. PhD thesis, University of Halle, 2015.
T. Nagasawa and T. Yi. Local existence and uniqueness for the n-dimensional helfrich flow as a projected gradient flow. Hokkaido Mathematical Journal, 41(2):209–226, 2012.
J. Prüss and G. Simonett. On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound., 12(3):311–345, 2010.
J. Prüss and G. Simonett. Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity. In Parabolic problems, volume 80 of Progr. Nonlinear Differential Equations Appl., pages 507–540. Birkhäuser/Springer Basel AG, Basel, 2011.
L. E. Scriven. Dynamics of a fluid interface. Chemical Engineering Science, 12(2):98–108, 1960.
U. Seifert. Configurations of fluid membranes and vesicles. Advances in physics, 46(1):13–137, 1997.
T. Tao. Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis.
H. Triebel. Theory of Function Spaces I. Birkhäuser, 1983.
H. Triebel. Theory of Function Spaces II. Birkhäuser, 1992.
W. Wang, P. Zhang, and Z. Zhang. Well-posedness of hydrodynamics on the moving elastic surface. Arch. Ration. Mech. Anal., 206(3):953–995, 2012.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author gratefully acknowledges support by DFG SPP 1506 “Transport Processes at Fluidic Interfaces”.
Rights and permissions
About this article
Cite this article
Köhne, M., Lengeler, D. Local well-posedness for relaxational fluid vesicle dynamics. J. Evol. Equ. 18, 1787–1818 (2018). https://doi.org/10.1007/s00028-018-0461-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-018-0461-3