Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor

Cimatti, Giovanni; Prodi, G. A.

doi:10.1007/bf01766151

Cited by 68 publications

(38 citation statements)

References 2 publications

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“…Moreover if we proceed as in Lemma 3.1 of [2], where a similar situation occurs, we can show that un > 0 in Q. where again C is a constant not depending on n .…”

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confidence: 66%

On two problems of electrical heating of conductors

Cimatti¹

1991

Quart. Appl. Math.

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Abstract. The thermal effects of the currents induced in a massive conductor by an external slowly varying magnetic field are studied with regard to existence and uniqueness of solutions. In the first part a theorem of existence of solution is also given for the thermistor problem with a current limiting device.1. Introduction. In this paper we study various problems related to the Joule heating of a conductor. In Sec. 2 a theorem of existence is given for a version of the "thermistor problem" first proposed in [3] and then studied for special boundary conditions in [1], The boundary conditions here considered are quite general and the same methods apply even to more general situations. In Sec. 3 a model is proposed for describing the Joule heating due to the eddy currents in a very long cylinder when the external variable magnetic field is parallel to the axis of the cylinder. A theorem of existence is given for this problem.

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“…Moreover if we proceed as in Lemma 3.1 of [2], where a similar situation occurs, we can show that un > 0 in Q. where again C is a constant not depending on n .…”

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confidence: 66%

On two problems of electrical heating of conductors

Cimatti¹

1991

Quart. Appl. Math.

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“…(1.3) For the physical background of the thermistor problem and some explicit solutions we refer to [1], [9], [10], [11], and the references therein. There has been recent mathematical interest in the problem in case o(u) is uniformly positive; see [2], [3], [4], [7], [8]. Cimatti and Prodi in [2] and Cimatti in [3] considered the Dirichlet boundary conditions for both (p and u and proved existence of a solution.…”

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confidence: 99%

“…There has been recent mathematical interest in the problem in case o(u) is uniformly positive; see [2], [3], [4], [7], [8]. Cimatti and Prodi in [2] and Cimatti in [3] considered the Dirichlet boundary conditions for both (p and u and proved existence of a solution. In [4] Cimatti extended the existence result to the case where cp = <p°, u -u° onrD, r^cdQ, d1> n du n r -in\r -= 0, ^-=o onryv = dQ\ro.…”

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The thermistor problem for conductivity which vanishes at large temperature

Chen¹,

Friedman²

1993

Quart. Appl. Math.

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Abstract. The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient a(u) vanishes (u = temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set {a(u) = 0} and also prove uniqueness.

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“…To our knowledge, there is no general result on the numerical analysis of problem (1.1). For mathematical and numerical analyses of simpler problems consisting of nonlinear coupled systems of two scalar elliptic equations, we refer to [6][7][8][9][10][11][12].…”

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confidence: 99%

Finite element analysis of a coupled thermally dependent viscosity flow problem

Zhu¹,

Loula²,

F.³

et al. 2007

Mat. apl. comput.

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Abstract. In this work, a stationary Stokes flow with thermal effects is studied both mathematically and numerically. First, existence, uniqueness and regularity of the weak solution of the problem are established. Next, fi nite element approximation to the problem, based on a fi xed point algorithm, is proposed. Then, an error estimate between continuous solution and discrete one is obtained. Finally, some numerical tests are presented to confi rm the theoretical results.Mathematical subject classification: 35J, 35Q, 65N30, 76.

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Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor

Cited by 68 publications

References 2 publications

On two problems of electrical heating of conductors

On two problems of electrical heating of conductors

The thermistor problem for conductivity which vanishes at large temperature

Finite element analysis of a coupled thermally dependent viscosity flow problem

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