Abstract
Continuum physics, which includes the disciplines of continuum mechanics, continuum thermodynamics, continuum electromagnetism, and certain fields of chemistry, furnishes refined mathematical models for the behavior of material bodies that are not invisibly small. The governing equations for theories of continuum physics typically involve partial differential equations and generalizations thereof.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. S. Antman, Nonlinear Problems of Elasticity, Springer, 1995.
V. I. Arnol’d and B. A. Khesin, Topological Methods in Hydrodynamics, Springer, 1998.
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, 1992.
J. M. Ball and R. D. James, From Microscales to Macroscales in Materials, in preparation.
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford, 1998.
C.-M. Brauner and C. Schmidt-Lainé, eds., Mathematical Modeling in Combustion and Related Topics, Nijhoff, 1988.
A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford, 2000.
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, 1996.
H. D. Bui, Inverse Problems in the Mechanics of Materials: An Introduction, CRC, 1994.
A. J. Chorin, Vorticity and Turbulence, Springer, 1994.
P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity; Vol. II: Theory of Plates; Vol. III: Theory of Shells; 1998, 1997, 2000, North-Holland.
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, 1989.
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, 2000.
T. Dubois, F. Jauberteau, and R. Temam, Dynamic Multilevel Methods and the Numerical Simulation of Turbulence, Cambridge, 1999.
A. C. Eringen and G. A. Maugin, Electrodynamics of Continua, Springer, 1990.
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1991.
L. B. Freund, Dynamic Fracture Mechanics, Cambridge, 1989.
E. Fried, Fifty years of research on evolving phase interfaces, Evolving Phase Interfaces in Solids, edited by J. M. Ball, D. Kinderlehrer, P. Podio-Guidugli, and M. Slemrod, Springer, 1999.
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, 1994.
J. Glimm and A. J. Majda, eds., Multidimensional Hyperbolic Problems and Computations, Springer, 1991.
M. E. Gurtin, Configurational Forces as Basic Concepts of Continuum Physics, Springer, 2000.
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., 1989.
K. Havner, Finite Plastic Deformation of Crystalline Solids, Cambridge, 1992.
D. D. Holm, J. E. Marsden and T. Ratiu, The Euler-Poisson equations and semidirect products with applications to continuum theories, Advances in Mathematics 137 (1998) 1–81.
K. Hutter and A. van de Ven, Field Matter Interactions in Thermoelastic Solids, Springer Lecture Notes in Physics 88, 1978.
V. Isakov, Inverse Problems for Partial Differential Equations, Springer, 1998.
D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer, 1990.
J. P. Keener and J. Sneyd, Mathematical Physiology, Springer, 1998.
N. Kikuchi and J. T. Oden, Contact Problems in Elasticity, SIAM, 1988.
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge, 1991.
R. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992.
P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford, 1996.
I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer, 1991.
G. W. Milton, The Theory of Composites, Cambridge, 2000.
I. Müller and T. Ruggeri, Rational Extended Thermodynamics, 2nd ed., Springer, 1998.
P. Pedregal, Optimization, relaxation and Young measures, Bulletin of the American Mathematical Society 36 (1999) 27–58.
B. Perthame, ed., Advances in Kinetic Theory and Computing, World Scientific, 1994.
R. Racke, Lectures on Nonlinear Evolution Equations, Vieweg, 1992.
K. R. Rajagopal and C. A. Truesdell, An Introduction to the Mechanics of Fluids, Birkhäuser, 2000.
M. Renardy, W. Hrusa, and J. Nohel, Mathematical Problems in Viscoelasticity, Longman, 1987.
D. Ruelle, Turbulence, Strange Attractors, and Chaos, World Scientific, 1995.
D. Serre, Systems of Conservation Laws, Cambridge, 1999, 2000.
J. B. Serrin, ed., New Perspectives in Thermodynamics, Springer, 1986.
M. Šilhavý, The Mechanics and Thermomechanics of Continuous Media, Springer, 1997.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd. edn., Springer, 1997.
C. Truesdell, Rational Thermodynamics, 2nd ed., Springer, 1984.
C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Handbuch der Physik, Vol. III/3, Springer, 1965.
C. Truesdell and R. Toupin, The Classical Field Theories of Mechanics, Handbuch der Physik, Vol. III/1, Springer, 1960.
A. Visintin, Differential Models of Hysteresis, Springer, 1994.
M. Wiegner, The Navier-Stokes equations-a neverending challenge? Jahresbericht der Deutschen Mathematiker-Vereinigung 101 (1999) 1–25.
Zhouping Xin, Some current topics in nonlinear conservation laws, in Current Topics in Nonlinear Conservation Laws, L. Hsiao and Z. Xin, editors, International Press, 2000.
Songmu Zheng, Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems, Longman, 1995.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Antman, S.S. (2001). Nonlinear Continuum Physics. In: Engquist, B., Schmid, W. (eds) Mathematics Unlimited — 2001 and Beyond. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56478-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-56478-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63114-6
Online ISBN: 978-3-642-56478-9
eBook Packages: Springer Book Archive