American Institute of Mathematical Sciences

Advanced
November  2003, 3(4): 481-494. doi: 10.3934/dcdsb.2003.3.481

Regularity properties of planar motions of incompressible rods

Stuart S. Antman 1,

1. 

Department of Mathematics,Institute for Physical Science and Technology, and Institute for Systems Research, University of Maryland, College Park, MD 20742, United States

Received  October 2002 Revised  March 2003 Published  August 2003

This paper treats the quasilinear evolution equations governing the planar motions of incompressible rods. Since incompressibility is here a 2-dimensional phenomenon, a thickness variable enters the governing equations in an essential and novel way. These equations have a mathematical structure strikingly different from that for compressible rods. In contrast to the case for compressible rods, the governing equations admit a priori upper and lower bounds on the stretches without the viscosity becoming singular when these stretches reach their extremes.
Keywords: Incompressible rods, a priori bounds..
Mathematics Subject Classification: 74K10, 74B20, 35B3.
Citation: Stuart S. Antman. Regularity properties of planar motions of incompressible rods. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 481-494. doi: 10.3934/dcdsb.2003.3.481
[1]

Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065
[2]

Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865
[3]

Tongren Ding, Hai Huang, Fabio Zanolin. A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 103-117. doi: 10.3934/dcds.1995.1.103
[4]

Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125
[5]

Jonatan Lenells. Traveling waves in compressible elastic rods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 151-167. doi: 10.3934/dcdsb.2006.6.151
[6]

Ferdinando Auricchio, Lourenco Beirão da Veiga, Josef Kiendl, Carlo Lovadina, Alessandro Reali. Isogeometric collocation mixed methods for rods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 33-42. doi: 10.3934/dcdss.2016.9.33
[7]

Timothy J. Healey. A rigorous derivation of hemitropy in nonlinearly elastic rods. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 265-282. doi: 10.3934/dcdsb.2011.16.265
[8]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[9]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258
[10]

Abdallah Ben Abdallah, Farhat Shel. Exponential stability of a general network of 1-d thermoelastic rods. Mathematical Control & Related Fields, 2012, 2 (1) : 1-16. doi: 10.3934/mcrf.2012.2.1
[11]

Mircea Bîrsan, Holm Altenbach. On the Cosserat model for thin rods made of thermoelastic materials with voids. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1473-1485. doi: 10.3934/dcdss.2013.6.1473
[12]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[13]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[14]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[15]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[16]

Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164
[17]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[18]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
[19]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[20]

Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences