September  2014, 19(7): 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

Singular limit of an integrodifferential system related to the entropy balance

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia

Received  January 2013 Revised  May 2013 Published  August 2014

A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Convergence results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.
Citation: Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
References:
[1]

Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.  Google Scholar

[2]

Noordhoff International Publishing, Leyden, 1976.  Google Scholar

[3]

Math. Models Methods Appl. Sci., 13 (2003), 1565-1588. doi: 10.1142/S0218202503003033.  Google Scholar

[4]

Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026. doi: 10.3934/dcdsb.2006.6.1001.  Google Scholar

[5]

Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035.  Google Scholar

[6]

J. Differential Equations, 246 (2009), 3260-3295. doi: 10.1016/j.jde.2009.02.007.  Google Scholar

[7]

J. Math. Pures Appl., 88 (2007), 455-481. doi: 10.1016/j.matpur.2007.09.005.  Google Scholar

[8]

North-Holland Math. Stud., 5, North-Holland, Amsterdam, 1973.  Google Scholar

[9]

Commun. Pure Appl. Anal., 11 (2012), 1959-1982. doi: 10.3934/cpaa.2012.11.1959.  Google Scholar

[10]

Asymptot. Anal., 82 (2013), 139-162. doi: 10.3233/ASY-2012-1142.  Google Scholar

[11]

Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[12]

Math. Methods Appl. Sci., 29 (2006), 2149-2179. doi: 10.1002/mma.765.  Google Scholar

[13]

Proc. Roy. Soc. Lond. A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.  Google Scholar

[14]

Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[15]

Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.  Google Scholar

[16]

Contin. Mech. Thermodyn., 20 (2009), 479-487. doi: 10.1007/s00161-009-0093-5.  Google Scholar

[17]

Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.  Google Scholar

[2]

Noordhoff International Publishing, Leyden, 1976.  Google Scholar

[3]

Math. Models Methods Appl. Sci., 13 (2003), 1565-1588. doi: 10.1142/S0218202503003033.  Google Scholar

[4]

Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026. doi: 10.3934/dcdsb.2006.6.1001.  Google Scholar

[5]

Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035.  Google Scholar

[6]

J. Differential Equations, 246 (2009), 3260-3295. doi: 10.1016/j.jde.2009.02.007.  Google Scholar

[7]

J. Math. Pures Appl., 88 (2007), 455-481. doi: 10.1016/j.matpur.2007.09.005.  Google Scholar

[8]

North-Holland Math. Stud., 5, North-Holland, Amsterdam, 1973.  Google Scholar

[9]

Commun. Pure Appl. Anal., 11 (2012), 1959-1982. doi: 10.3934/cpaa.2012.11.1959.  Google Scholar

[10]

Asymptot. Anal., 82 (2013), 139-162. doi: 10.3233/ASY-2012-1142.  Google Scholar

[11]

Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[12]

Math. Methods Appl. Sci., 29 (2006), 2149-2179. doi: 10.1002/mma.765.  Google Scholar

[13]

Proc. Roy. Soc. Lond. A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.  Google Scholar

[14]

Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[15]

Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.  Google Scholar

[16]

Contin. Mech. Thermodyn., 20 (2009), 479-487. doi: 10.1007/s00161-009-0093-5.  Google Scholar

[17]

Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[1]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[2]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031

[3]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057

[4]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[5]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051

[6]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[7]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[8]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[9]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020

[10]

Yangrong Li, Fengling Wang, Shuang Yang. Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3643-3665. doi: 10.3934/dcdsb.2020250

[11]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

[12]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[13]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[14]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019

[15]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[16]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[17]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012

[18]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021025

[19]

Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034

[20]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]