-
Previous Article
Effect of intracellular diffusion on current--voltage curves in potassium channels
- DCDS-B Home
- This Issue
-
Next Article
Foreword
Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour
1. | Dipartimento di Matematica, Università di Pisa, Pisa, Italy, Italy |
2. | Dipartimento di Scienze di Base e Applicate, per l'Ingegneria - Sezione Matematica, Sapienza Università di Roma, Rome, Italy |
3. | School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland |
References:
[1] |
G. Amendola, The minimum free energy for incompressible viscoelastic fluids, Math. Methods Appl. Sci., 29 (2006), 2201-2223.
doi: 10.1002/mma.769. |
[2] |
G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. of Mech. and Appl. Math., 57 (2004), 429-446.
doi: 10.1093/qjmam/57.3.429. |
[3] |
G. Amendola and M. Fabrizio, Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model, Diff. Int. Eq., 20 (2007), 445-466. |
[4] |
G. Amendola, M. Fabrizio, J. M. Golden and B. Lazzari, Free energies and asymptotic behaviour for incompressible viscoelastic fluids, Appl. Anal., 88 (2009), 789-805.
doi: 10.1080/00036810903042117. |
[5] |
G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[6] |
S. Breuer and E. T. Onat, On the determination of free energy in viscoelastic solids, Z. angew. Math. Phys., 15 (1964), 184-191.
doi: 10.1007/BF01602660. |
[7] |
S. Breuer and E. T. Onat, On recoverable work in viscoelasticity, Z. Angew. Math. Phys., 15 (1981), 13-21. |
[8] |
S. Carillo, Existence, Uniqueness and Exponential Decay: An Evolution Problem in Heat Conduction with Memory, Quarterly of Appl. Math., 69 (2011), 635-649. S 0033-569X(2011)01223-1, Article Electronically published on July 7, (2011).
doi: 10.1090/S0033-569X-2011-01223-1. |
[9] |
S. Carillo, An evolution problem in materials with fading memory: solution's existence and uniqueness, Complex Variables and Elliptic Equations An International Journal, 56 (2011), 481-492.
doi: 10.1080/17476931003786667. |
[10] |
S. Carillo, Materials with Memory: Free energies & solutions' exponential decay, Communications on Pure And Applied Analysis, 9 (2010), 1235-1248.
doi: 10.3934/cpaa.2010.9.1235. |
[11] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[12] |
W. A. Day, Some results on the least work needed to produce a given strain in a given time in a viscoelastic material and a uniqueness theorem for dynamic viscoelasticity, Quart. J. Mech. Appl. Math., 23 (1970), 469-479.
doi: 10.1093/qjmam/23.4.469. |
[13] |
G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity, 43 (1996), 247-278.
doi: 10.1007/BF00042503. |
[14] |
L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity, 54 (1999), 141-185.
doi: 10.1023/A:1007646017347. |
[15] |
L. Deseri, M. Fabrizio and J. M. Golden, The concept of minimal state in viscoelasticity: New free energies and applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[16] |
M. Fabrizio, G. Gentili and J. M. Golden, The minimum free energy for a class of compressible viscoelastic fluids, Advances Diff. Eq., 7 (2002), 319-342. |
[17] |
M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373.
doi: 10.1007/BF00375062. |
[18] |
M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., 60 (2002), 341-381. |
[19] |
M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Int. Eq., 6 (1993), 491-505. |
[20] |
M. Fabrizio and A. Morro, Reversible processes in thermodynamics of continuous media, J. Nonequil. Thermodyn., 16 (1991), 1-12.
doi: 10.1515/jnet.1991.16.1.1. |
[21] |
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992.
doi: 10.1137/1.9781611970807. |
[22] |
G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math., 60 (2002), 153-182. |
[23] |
J. M. Golden, Free energy in the frequency domain: The scalar case, Quart. Appl. Math., 58 (2000), 127-150. |
[24] |
D. Graffi and M. Fabrizio, On the notion of state for viscoelastic materials of "rate'' type, Atti. Accad. Naz. Lincei, 83 (1990), 201-208. |
[25] |
D. Graffi and M. Fabrizio, Nonuniqueness of free energy for viscoelastic materials, Atti Accad. Naz. Lincei, 83 (1990), 209-214. |
[26] |
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50. |
[27] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[28] |
M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal., 62 (1976), 303-321. |
show all references
References:
[1] |
G. Amendola, The minimum free energy for incompressible viscoelastic fluids, Math. Methods Appl. Sci., 29 (2006), 2201-2223.
doi: 10.1002/mma.769. |
[2] |
G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. of Mech. and Appl. Math., 57 (2004), 429-446.
doi: 10.1093/qjmam/57.3.429. |
[3] |
G. Amendola and M. Fabrizio, Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model, Diff. Int. Eq., 20 (2007), 445-466. |
[4] |
G. Amendola, M. Fabrizio, J. M. Golden and B. Lazzari, Free energies and asymptotic behaviour for incompressible viscoelastic fluids, Appl. Anal., 88 (2009), 789-805.
doi: 10.1080/00036810903042117. |
[5] |
G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[6] |
S. Breuer and E. T. Onat, On the determination of free energy in viscoelastic solids, Z. angew. Math. Phys., 15 (1964), 184-191.
doi: 10.1007/BF01602660. |
[7] |
S. Breuer and E. T. Onat, On recoverable work in viscoelasticity, Z. Angew. Math. Phys., 15 (1981), 13-21. |
[8] |
S. Carillo, Existence, Uniqueness and Exponential Decay: An Evolution Problem in Heat Conduction with Memory, Quarterly of Appl. Math., 69 (2011), 635-649. S 0033-569X(2011)01223-1, Article Electronically published on July 7, (2011).
doi: 10.1090/S0033-569X-2011-01223-1. |
[9] |
S. Carillo, An evolution problem in materials with fading memory: solution's existence and uniqueness, Complex Variables and Elliptic Equations An International Journal, 56 (2011), 481-492.
doi: 10.1080/17476931003786667. |
[10] |
S. Carillo, Materials with Memory: Free energies & solutions' exponential decay, Communications on Pure And Applied Analysis, 9 (2010), 1235-1248.
doi: 10.3934/cpaa.2010.9.1235. |
[11] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[12] |
W. A. Day, Some results on the least work needed to produce a given strain in a given time in a viscoelastic material and a uniqueness theorem for dynamic viscoelasticity, Quart. J. Mech. Appl. Math., 23 (1970), 469-479.
doi: 10.1093/qjmam/23.4.469. |
[13] |
G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity, 43 (1996), 247-278.
doi: 10.1007/BF00042503. |
[14] |
L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity, 54 (1999), 141-185.
doi: 10.1023/A:1007646017347. |
[15] |
L. Deseri, M. Fabrizio and J. M. Golden, The concept of minimal state in viscoelasticity: New free energies and applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[16] |
M. Fabrizio, G. Gentili and J. M. Golden, The minimum free energy for a class of compressible viscoelastic fluids, Advances Diff. Eq., 7 (2002), 319-342. |
[17] |
M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373.
doi: 10.1007/BF00375062. |
[18] |
M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., 60 (2002), 341-381. |
[19] |
M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Int. Eq., 6 (1993), 491-505. |
[20] |
M. Fabrizio and A. Morro, Reversible processes in thermodynamics of continuous media, J. Nonequil. Thermodyn., 16 (1991), 1-12.
doi: 10.1515/jnet.1991.16.1.1. |
[21] |
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992.
doi: 10.1137/1.9781611970807. |
[22] |
G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math., 60 (2002), 153-182. |
[23] |
J. M. Golden, Free energy in the frequency domain: The scalar case, Quart. Appl. Math., 58 (2000), 127-150. |
[24] |
D. Graffi and M. Fabrizio, On the notion of state for viscoelastic materials of "rate'' type, Atti. Accad. Naz. Lincei, 83 (1990), 201-208. |
[25] |
D. Graffi and M. Fabrizio, Nonuniqueness of free energy for viscoelastic materials, Atti Accad. Naz. Lincei, 83 (1990), 209-214. |
[26] |
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50. |
[27] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[28] |
M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal., 62 (1976), 303-321. |
[1] |
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793 |
[2] |
John Murrough Golden. Constructing free energies for materials with memory. Evolution Equations and Control Theory, 2014, 3 (3) : 447-483. doi: 10.3934/eect.2014.3.447 |
[3] |
Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086 |
[4] |
Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235 |
[5] |
Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253 |
[6] |
Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks and Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941 |
[7] |
Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361 |
[8] |
Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505 |
[9] |
Xia Chen, Tuoc Phan. Free energy in a mean field of Brownian particles. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 747-769. doi: 10.3934/dcds.2019031 |
[10] |
Kongzhi Li, Xiaoping Xue. The Łojasiewicz inequality for free energy functionals on a graph. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022066 |
[11] |
Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 |
[12] |
Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 |
[13] |
Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170 |
[14] |
Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799 |
[15] |
Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure and Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027 |
[16] |
Ugo Boscain, Thomas Chambrion, Grégoire Charlot. Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 957-990. doi: 10.3934/dcdsb.2005.5.957 |
[17] |
Maria Cameron. Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree. Networks and Heterogeneous Media, 2014, 9 (3) : 383-416. doi: 10.3934/nhm.2014.9.383 |
[18] |
Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355 |
[19] |
Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687 |
[20] |
Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]