June  2016, 36(6): 3011-3034. doi: 10.3934/dcds.2016.36.3011

Existence of solutions for a model of microwave heating

1. 

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

2. 

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

Received  June 2015 Revised  October 2015 Published  December 2015

This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RLC circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.
Citation: Pierluigi Colli, Luca Scarpa. Existence of solutions for a model of microwave heating. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3011-3034. doi: 10.3934/dcds.2016.36.3011
References:
[1]

S. Agrawal and G. A. Kriegsmann, A model for the microwave heating of a thin ceramic slab in a multimode cavity, IMA J. Appl. Math., 78 (2013), 652-664. doi: 10.1093/imamat/hxt013.

[2]

S. N. Antonsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128-1156. doi: 10.1137/S0036141092233482.

[3]

M. Badii, Existence of periodic solutions for the quasi-static thermoelastic thermistor problem, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 1-15. doi: 10.1007/s00030-008-7017-0.

[4]

F. Brezzi and G. Gilardi, Chapters 1-3, in Finite Element Handbook (eds. H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987.

[5]

G. Cimatti, Remark on the number of solutions in the thermistor problem, Matematiche (Catania), 66 (2011), 49-60.

[6]

G. Cimatti, Remarks on the existence, uniqueness and semi-explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 481-495. doi: 10.1017/S0308210509001826.

[7]

L. Consiglieri, A limit model for thermoelectric equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 57 (2011), 229-244. doi: 10.1007/s11565-011-0129-1.

[8]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089.

[9]

C. García Reimbert, M. C. Jorge, A. A. Minzoni and C. A. Vargas, Temperature modulations in a circuit model of microwave heating. Applied-mathematical perspectives on microwave processing, J. Engrg. Math., 44 (2002), 199-206. doi: 10.1023/A:1020824314529.

[10]

D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control for the thermistor problem, SIAM J. Control Optim., 48 (2009/10), 3449-3481. doi: 10.1137/080736259.

[11]

D. Hömberg and E. Rocca, A model for resistance welding including phase transitions and Joule heating, Math. Methods Appl. Sci., 34 (2011), 2077-2088. doi: 10.1002/mma.1505.

[12]

K. L. Kuttler, M. Shillor and J. R. Fernández, Existence for the thermoviscoelastic thermistor problem, Differ. Equ. Dyn. Syst., 17 (2009), 217-233. doi: 10.1007/s12591-009-0017-7.

[13]

V. S. Manoranjan, H.-M. Yin and R. Showalter, On two-phase Stefan problem arising from a microwave heating process, Discrete Contin. Dyn. Syst., 15 (2006), 1155-1168. doi: 10.3934/dcds.2006.15.1155.

[14]

L. Scarpa, A doubly nonlinear evolution problem related to a model for microwave heating, Adv. Math. Sci. Appl., 24 (2014), 251-275.

[15]

P. Shi, M. Shillor and X. Xu, Existence of a solution to the Stefan problem with Joule's heating, J. Differential Equations, 105 (1993), 239-263. doi: 10.1006/jdeq.1993.1089.

[16]

A. Sidi, R. Moulay and D. F. M. Torres, Optimal control of nonlocal thermistor equations, Internat. J. Control, 85 (2012), 1789-1801. doi: 10.1080/00207179.2012.703789.

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[18]

W. Wei, H.-M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Anal., 75 (2012), 2024-2036. doi: 10.1016/j.na.2011.10.003.

[19]

H.-M. Yin, Regularity of weak solution to Maxwell's equations and applications to microwave heating, J. Differential Equations, 200 (2004), 137-161. doi: 10.1016/j.jde.2004.01.010.

show all references

References:
[1]

S. Agrawal and G. A. Kriegsmann, A model for the microwave heating of a thin ceramic slab in a multimode cavity, IMA J. Appl. Math., 78 (2013), 652-664. doi: 10.1093/imamat/hxt013.

[2]

S. N. Antonsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128-1156. doi: 10.1137/S0036141092233482.

[3]

M. Badii, Existence of periodic solutions for the quasi-static thermoelastic thermistor problem, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 1-15. doi: 10.1007/s00030-008-7017-0.

[4]

F. Brezzi and G. Gilardi, Chapters 1-3, in Finite Element Handbook (eds. H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987.

[5]

G. Cimatti, Remark on the number of solutions in the thermistor problem, Matematiche (Catania), 66 (2011), 49-60.

[6]

G. Cimatti, Remarks on the existence, uniqueness and semi-explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 481-495. doi: 10.1017/S0308210509001826.

[7]

L. Consiglieri, A limit model for thermoelectric equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 57 (2011), 229-244. doi: 10.1007/s11565-011-0129-1.

[8]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089.

[9]

C. García Reimbert, M. C. Jorge, A. A. Minzoni and C. A. Vargas, Temperature modulations in a circuit model of microwave heating. Applied-mathematical perspectives on microwave processing, J. Engrg. Math., 44 (2002), 199-206. doi: 10.1023/A:1020824314529.

[10]

D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control for the thermistor problem, SIAM J. Control Optim., 48 (2009/10), 3449-3481. doi: 10.1137/080736259.

[11]

D. Hömberg and E. Rocca, A model for resistance welding including phase transitions and Joule heating, Math. Methods Appl. Sci., 34 (2011), 2077-2088. doi: 10.1002/mma.1505.

[12]

K. L. Kuttler, M. Shillor and J. R. Fernández, Existence for the thermoviscoelastic thermistor problem, Differ. Equ. Dyn. Syst., 17 (2009), 217-233. doi: 10.1007/s12591-009-0017-7.

[13]

V. S. Manoranjan, H.-M. Yin and R. Showalter, On two-phase Stefan problem arising from a microwave heating process, Discrete Contin. Dyn. Syst., 15 (2006), 1155-1168. doi: 10.3934/dcds.2006.15.1155.

[14]

L. Scarpa, A doubly nonlinear evolution problem related to a model for microwave heating, Adv. Math. Sci. Appl., 24 (2014), 251-275.

[15]

P. Shi, M. Shillor and X. Xu, Existence of a solution to the Stefan problem with Joule's heating, J. Differential Equations, 105 (1993), 239-263. doi: 10.1006/jdeq.1993.1089.

[16]

A. Sidi, R. Moulay and D. F. M. Torres, Optimal control of nonlocal thermistor equations, Internat. J. Control, 85 (2012), 1789-1801. doi: 10.1080/00207179.2012.703789.

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[18]

W. Wei, H.-M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Anal., 75 (2012), 2024-2036. doi: 10.1016/j.na.2011.10.003.

[19]

H.-M. Yin, Regularity of weak solution to Maxwell's equations and applications to microwave heating, J. Differential Equations, 200 (2004), 137-161. doi: 10.1016/j.jde.2004.01.010.

[1]

Yumei Liao, Wei Wei, Xianbing Luo. Existence of solution of a microwave heating model and associated optimal frequency control problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2103-2116. doi: 10.3934/jimo.2019045

[2]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[3]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[4]

V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155

[5]

Brigita Ferčec, Valery G. Romanovski, Yilei Tang, Ling Zhang. Integrability and bifurcation of a three-dimensional circuit differential system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4573-4588. doi: 10.3934/dcdsb.2021243

[6]

Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240

[7]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

[8]

Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861

[9]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227

[10]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131

[11]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[12]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[13]

Wenji Chen, Jianfeng Zhou. Global existence of weak solutions to inhomogeneous Doi-Onsager equations. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021257

[14]

Huanting Li, Yunfei Peng, Kuilin Wu. The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021289

[15]

Miroslav Bartušek. Existence of noncontinuable solutions of a system of differential equations. Conference Publications, 2009, 2009 (Special) : 54-59. doi: 10.3934/proc.2009.2009.54

[16]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

[17]

Yana Guo, Yan Jia, Bo-Qing Dong. Global stability solution of the 2D MHD equations with mixed partial dissipation. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 885-902. doi: 10.3934/dcds.2021141

[18]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

[19]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1

[20]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (244)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]