American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A proximal-like algorithm for vibro-impact problems with a non-smooth set of constraints
  • PROC Home
  • This Issue
  • Next Article
    Second order oscillation of mixed nonlinear dynamic equations with several positive and negative coefficients
2011, 2011(Special): 1176-1185. doi: 10.3934/proc.2011.2011.1176

Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion

J. F. Padial 1,

1. 

Dpto. Matemática Aplicada, Fac. de Matemáticas, U. Politécnica de Madrid, 28040 Madrid, Spain

Received  August 2010 Revised  July 2011 Published  October 2011

We consider an inverse 2D free-boundary elliptic-parabolic problem, modeling the transient regime of a magnetically con ned plasma in a non ideal Stellarator device. The inverse nature of the problem comes from the fact that the associated Grad-Shafranov equation involves some unknown nonlinear terms which must be determined by the current-carrying Stellarator condition. One of the main diculties for the mathematical approach is due to the fact that these nonlinear terms are neither a monotone function and are nor a Lipschitz function either. Nevertheless, we can introduce an auxiliary problem for which the solution of our original problem is a supersolution. On the other hand, we can to obtain a subsolution for this new problem. Finally, by applying the comparison principle for quasi-linear problems, we obtain some appropriate estimates on the location and size of the plasma region.
Keywords: Current-carrying Stellarators, free boundary problem, relative rearrangement, Inverse nonlinear elliptic-parabolic problem, location of free boundary.
Mathematics Subject Classification: Primary: 35J85; Secondary: 35R30.
Citation: J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176
[1]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920
[2]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122
[3]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[4]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667
[5]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431
[6]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[7]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks & Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655
[8]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337
[9]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44
[10]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
[11]

Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020103
[12]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[13]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
[14]

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 & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[15]

Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261
[16]

Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238
[17]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
[18]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[19]

Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789
[20]

Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 997-1008. doi: 10.3934/dcdsb.2016.21.997

 Impact Factor: 

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences