The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

 Access to full text

 Full (PDF)

1. Alouges, F., DeSimone, A. and Lefebvre, A. (2008). Optimal Strokes for low Reynolds number swimmers: An example, Journal of Nonlinear Science 18: 27-302. Zbl1146.76062
2. Becker, L.E, Koehler, S.A. and Stone, H.A. (2003). On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer, Journal of Fluid Mechanics 490: 15-35. Zbl1063.76690
3. Belter, D. and Skrzypczyński, P (2010). A biologically inspired approach to feasible gait learning for a hexapod robot, International Journal of Applied Mathematics and Computer Science 20(1): 69-84, DOI: 10.2478/v10006-010-0005-7. Zbl1300.93116
4. Childress, S. (1981). Mechanics of Swimming and Flying, Cambridge University Press, Cambridge. Zbl0499.76118
5. Dal Maso, Gi., DeSimone, A. and Morandotti, M. (2011). An existence and uniqueness result for the motion of self-propelled micro-swimmers, SIAM Journal of Mathematical Analysis 43: 1345-1368. Zbl05969681
6. Fukuda, T., Kawamoto, A., Arai, F. and Matsuura, H. (1995). Steering mechanism and swimming experiment of micro mobile robot in water, Proceedings of Micro Electro Mechanical Systems (MEMS'95), Amsterdam, The Netherlands, pp. 300-305. 
7. Fauci, L. and Peskin, C.S. (1988). A computational model of aquatic animal locomotion, Journal of Computational Physics 77: 85-108. Zbl0641.76140
8. Fauci, L. (1993). Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics 141: 91-102. Zbl0786.76105
9. Galdi, G.P. (1999). On the steady self-propelled motion of a body in a viscous incompressible fluid, Archive for Rational Mechanics and Analysis 1448: 53-88. Zbl0957.76012
10. Galdi, G.P. (2002). On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in S. Friedlander and D. Serre (Eds.), Handbook of Mathematical Fluid Mechanics, Elsevier Science, Amsterdam, pp. 653-791. Zbl1230.76016
11. Gray, J. (1932). Study in animal locomotion IV: The propulsive power of the dolphin, Journal of Experimental Biology 10: 192-199. 
12. Gray, J. and Hancock, G.J. (1955). The propulsion of sea-urchin spermatozoa, Journal of Experimental Biology 32: 802. 
13. Guo, S. (2002). Afin type of micro-robot in pipe, Proceedings of the 2002 International Symposium on Micromechatronics and Human Science (MHS 2002), Ngoya, Japan, pp. 93-98. 
14. Gurtin, M.E. (1981). An Introduction to Continuum Mechanics, Academic Press, New York, NY. Zbl0559.73001
15. Hawthorne, M.F., Zink, J.I., Skelton, J.M., Bayer, M.J., Liu, Ch., Livshits, E., Baer, R. and Neuhauser, D. (2004). Electrical or photocontrol of rotary motion of a metallacarborane, Science 303: 1849. 
16. Happel, V. and Brenner, H. (1965). Low Reynolds Number Hydrodynamics, Prentice Hall, Upper Saddle River, NJ. Zbl0612.76032
17. Hirose, S. (1993). Biologically Inspired Robots: Snake-like Locomotors and Manipulators, Oxford University Press, Oxford. 
18. Kanso. E., Marsden, J.E., Rowley, C.W. and Melli-Huber, J. (2005). Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science 15: 255-289. Zbl1181.76032
19. Khapalov, A.Y. (1999). Approximate controllability properties of the semilinear heat equation with lumped controls, International Journal of Applied Mathematics and Computer Science 9(4): 751-765. Zbl0951.93007
20. Khapalov, A.Y. (2005). The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid, Technical Report 2005-5, Washington State University, Department of Mathematics, http://www.math.wsu.edu/TRS/2005-5.pdf. 
21. Khapalov, A.Y. and Eubanks, S. (2009). The wellposedness of a 2-D swimming model governed in the nonstationary Stokes fluid by multiplicative controls, Applicable Analysis 88: 1763-1783, DOI:10.1080/00036810903401222. 
22. Khapalov, A.Y. (2010). Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Mathematics Series, Vol. 1995, Springer-Verlag, Berlin/Heidelberg. Zbl1210.93005
23. Khapalov, A.Y. and Trinh, G. (2013). Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid, Discrete and Continuous Dynamical Systems Series A 33: 1513-1544. Zbl1277.35278
24. Khapalov, A.Y. (2013). Micro motions of a swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, arXiv: 1205.1088 [math.AP], (preprint). Zbl1287.35057
25. Koiller, J., Ehlers, F. and Montgomery, R. (1996). Problems and progress in microswimming, Journal of Nonlinear Science 6: 507-541. Zbl0867.76099
26. Ladyzhenskaya, O.A. (1963). The Mathematical Theory of Viscous Incompressible Flow, Cordon and Breach, New York, NY. 
27. Lighthill, M.J. (1975). Mathematics of Biofluid Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA. 
28. Mason, R. and Burdick, J.W. (2000). Experiments in carangiform robotic fish locomotion, Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, CA, USA, pp. 428-435. 
29. McIsaac, K.A. and Ostrowski, J.P. (2000). Motion planning for dynamic eel-like robots, Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, CA, USA, pp. 1695-1700. 
30. Martinez, S. and Cortes, J. (2001). Geometric control of robotic locomotion systems, Proceedings of the 10th Fall Workshop on Geometry and Physics, Miraflores de la Sierra, Spain, Vol. 4, pp. 183-198. Zbl1148.93009
31. Morgansen, K.A., Duindam, V., Mason, R.J., Burdick, J.W. and Murray, R.M. (2001). Nonlinear control methods for planar carangiform robot fish locomotion, Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, Korea, pp. 427-434. 
32. Peskin, C.S. (1977). Numerical analysis of blood flow in the heart, Journal of Computational Physics 25: 220-252. Zbl0403.76100
33. Peskin, C.S. and McQueen, D.M. (1994). A general method for the computer simulation of biological systems interacting with fluids, SEB Symposium on Biological Fluid Dynamics, Leeds, UK. 
34. San Martin, J., Takashi, T. and Tucsnak, M. (2007). A control theoretic approach to the swimming of microscopic organisms, Quarterly Applied Mathematics 65: 405-424. Zbl1135.76058
35. San Martin, J., Scheid, J.-F., Takashi, T. and Tucsnak, M. (2008). An initial and boundary value problem modeling of fish-like swimming, Archive of Rational Mechanics and Analysis 188: 429-455. Zbl1138.76072
36. Shapere, A. and Wilczeck, F. (1989). Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics 198: 557-585. Zbl0674.76114
37. Sigalotti, M. and Vivalda, J.-C. (2009). Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM: Control, Optimisation and Calculus of Variations 16: 1053-1076. Zbl1210.37013
38. Taylor, G.I. (1951). Analysis of the swimming of microscopic organisms, Proceeding of the Royal Society of London A 209: 447-461. Zbl0043.40302
39. Taylor, G.I. (1952). Analysis of the swimming of long and narrow animals, Proceedings of the Royal Society of London A 214(1117): 158-183. Zbl0047.43901
40. Temam, R. (1984). Navier-Stokes Equations, North-Holland, Amsterdam. 
41. Trintafyllou, M.S., Trintafyllou, G.S. and Yue, D.K.P. (2000). Hydrodynamics of fishlike swimming, Annual Review of Fluid Mechanics 32: 33-53. Zbl0988.76102
42. Tytell, E.D., Hsu, C.-Y, Williams, T.L., Cohen, A.H. and Fauci, L.J. (2010). Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming, Proceedings of the National Academy Sciences, USA 107: 19832-19837. 
43. Wu, T.Y. (1971). Hydrodynamics of swimming fish and cetaceans, Advances in Applied Mathematics 11: 1-63. 

1. Alexander Khapalov, Addendum to "The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation"