THE EDDY CURRENT PROBLEM WITH TEMPERATURE DEPENDENT PERMEABILITY

Transcription

1 Electronic Journal of Differential Equations, Vol , No. 91, pp ISSN: URL: or ftp ejde.math.swt.edu login: ftp THE EDDY CURRENT PROBLEM WITH TEMPERATURE DEPENDENT PERMEABILITY GIOVANNI CIMATTI Abstract. We prove the uniqueness and existence of a local solution in time for a system of PDE s modelling the eddy currents and the heating of a cylindrical conductor. 1. Introduction Electric currents induced in a massive conductor by an external varying magnetic field are known as eddy currents. They in turn heat the body by Joule effect. The problem of predicting the magnetic field and the temperature in the conductor is modelled by the non linear system µh = ρ H t 1.1 u t κ u = ρ H, 1. where ρ is the electric resistivity, κ the thermal conductivity and µ the permeability. In the special situation of a very long cylindrical conductor of cross-section Ω an open, bounded and connected subset of R with a regular boundary Γ immersed in an insulating medium and with a time dependent magnetic field H given at infinity by H = h b ti 3, i 3 unit vector parallel to the axis of the cylinder. In view of the geometry, we assume H = hi 3 ; this simplifies the equations and we are led see [3and [11, to the following initial-boundary value problem, closely related to the thermistor problem: µh t = ρ h 1.3 u t κ u = ρ h, x = x 1, x 1.4 hx, t = h b t on Γ 0, T 1.5 hx, 0 = 0 in Ω 1.6 ux, t = 0 on Γ 0, T 1.7 ux, 0 = u 0 x in Ω. 1.8 The system has been the subject of a variety of investigations in the past decade. They all assume µ to be a constant and ρ a given function of the 000 Mathematics Subject Classification. 78A30, 35K50. Key words and phrases. Eddy currents, parabolic nonlinear system, temperature, permeability. c 003 Texas State University-San Marcos. Submitted June 17, 003. Published September 8,

2 GIOVANNI CIMATTI EJDE 003/91 temperature u. We refer, in particular, to the work of Hong-Ming Yin [13, [14 and also to [8 and references therein. The more special problem has been studied in [3 and [11, again with constant permeability. Now, in real conductors not only ρ and κ depend on u, but also µ, in certain cases dramatically [10; this makes the problem considerably more difficult.. Results In the stationary thermistor problem a suitable transformation linearizes the system see [4, [5 and [6 and this permits, in particular, a precise estimate of the maximum of the temperature u. Unfortunately, there does not seem to be a correspondent transformation for the parabolic system However, when ρ, κ and µ are constants the transformation useful in the thermistor problem can be applied to Let a = µ ρ, b = 1 ρ, c = κ ρ and define Hx 1, x, t = hx 1, x, at, Ux 1, x, t = ux 1, x, bt and θx 1, x, t = H x 1, x, t + c Ux 1, x, t. Under this transformation system becomes H t = H, θ t = θ. The initial and boundary conditions for θ are immediately written in terms of the same data of H and U. Thus H and θ are estimated via the parabolic maximum principle and, correspondingly, U = 1 c θ H is also estimated. In this note we present a result of uniqueness for problem assuming µ, ρ and κ to be given functions of u and a result of existence of solutions local in time. The difficulties inherent to the nonlinear problem are better understood if we rewrite equations. Equations 1.3 and 1.4 in normal form, i.e. solved with respect to h t and u t, and with the principal part in divergence form. We obtain, after easy calculations, h t = au h buh u cuh h + du u h + euh u where.1 u t = κu u + ρu h,. au = νuρu, νu = 1 µu, bu = µ uνuκu, cu = µ uνuρu, du = µ uνuκu ν uρu, eu = µ uνuκu + µ uν uκu. On the right-hand side of equation.1 the second term clearly shows the lack of uniform ellipticity in the principal part of the system. This is the main source of mathematical difficulty in these equations. We assume bu, cu, du and eu to be bounded and globally Lipschitz and au, κu to be locally bounded and to satisfy au a 0 > 0, κu κ 0 > 0..3 We denote with the norm in LΩ.

3 EJDE 003/91 THE EDDY CURRENT PROBLEM 3 Theorem.1. Under the assumptions u 0 x H 1, 0 Ω, h b t H 1, 0, T, h b 0 = 0, there exists at most one solution, h, u L 0, T ; H 1, Ω L 0, T ; H 1, 0 Ω, h t, u t L0, T ; H 1 Ω L0, T ; H 1 Ω, to problem.1,., Proof. Let h 1, u 1 and h, u be two solutions and let z = h 1 h, w = u 1 u. Subtracting equation.1 from., we obtain z t = [au 1 h 1 au h [ bu1 h 1 u 1 bu h u [ cu 1 h 1 h 1 cu h h + [ du 1 u 1 h 1 du h + [ eu 1 h 1 u 1 eu h u = {1} + {} + {3} + {4} + {5},.4 w t = [κu 1 u 1 κu u + ρu1 h 1 ρu h = {6} + {7},.5 zx, 0 = 0, wx, 0 = 0, x Ω,.6 zx, t = 0, wx, t = 0, x, t Γ 0, T..7 We multiply.4 by z and.5 by Kw, where K is a positive constant to be defined later. Integrating by parts over Ω and using the Cauchy-Schwartz inequality, we have 1 d z + Kwdx + a 0 z + Kκ 0 w dt Ω C 11 w z + C 1 w z + C z z + C 3 w z + C 31 w z + C 3 z + C 33 z z.8 + C 41 w z + C 4 z w + C 43 z z + C 51 w + C 5 z + C 53 z w + C 61 K w w + KC 71 w + KC 7 w z where the first index refers, in the various C ij, to the grouping given in.4 and.5. We apply the elementary inequality ab γa+ 1 γ b with various choices of γ and with the goal of absorbing in the left-hand side of.8 all the terms containing z and w. Clearly, in this respect, the most difficult term is C 3 w z. With ɛ > 0 we have and C 11 w z C 11 ɛ z + C 11 1 ɛ w ɛ KC 7 w z KC 7 K z + K ɛ C 7 w.

4 4 GIOVANNI CIMATTI EJDE 003/91 Treating the other groups of terms similarly, we obtain in the end 1 d z + Kwdx + ɛ a 0 Aɛ KC 7 z dt Ω K + Kκ 0 Bɛ C 3 w ɛ Lɛ, K w + Mɛ, K z where Lɛ, K and Mɛ, K are positive functions defined for ɛ > 0 and K > 0, and A, B constants easily computed. With the choice a 0 ɛ = ɛ =, K = A + C K { = max 1, 4 [ Ba 0 7 κ 0 A + C 7 + C 3A + C 7 }, a 0 we have d z + dt Kwdx + a 0 z + Kκ 0 Ω w C z + Kwdx Ω where C = max{l ɛ, K, M ɛ, K}. Hence the Gronwall lemma, gives u 1 t u t + K h 1 t h t e Ct u 1 0 u 0 + K h 1 0 h 0 and uniqueness follows. To prove a result of existence local in time, we apply a theorem by Sobolewskii in [1, which we quote below. Theorem.. For α, β, i, j = 1,, let the functions a ijαβ t, v 1, v, v 11, v 1, v 1, v, f α t, t, v 1, v, v 11, v 1, v 1, v be of class CR 7 and ṽ 1 x, ṽ x be in C Ω, with ṽ 1, ṽ = 0 on Γ. Define ã ijαβ x = a ijαβ 0, ṽ 1 x, ṽ x, ṽ 1 x, ṽ 1 x, ṽ x, ṽ x. x 1 x x 1 x Suppose that ã ijαβ xξ iα ξ jβ k ξiα, k > 0.9 ijαβ=1 iα=1 for all x Ω. Then the problem v α t a ijαβ t, v 1, v, v 1, v 1, v, v vβ x i x 1 x x 1 x x j ijαβ = f α t, v 1, v, v 1, v 1, v, v x 1 x x 1 x v α 0, x = ṽ α x x Ω v α t, x = 0, t, x Γ [0, T, α = 1, has one and only one solution defined in a suitably small interval [0, t 0, t 0 < T..10 To apply this theorem to our case we assume au, bu, cu, du, eu, κu and ρu to be CR. We define as new unknown function wt, x = ht, x h b t to obtain the homogeneous boundary condition required in.10 and suppose h b t CR and u 0 x C Ω. In view of 6 and 11 the crucial condition.9 is satisfied: indeed we have au 0 xξ 11 + ξ 1 + κu 0 xξ 1 + ξ kξ 11 + ξ 1 + ξ 1 + ξ

5 EJDE 003/91 THE EDDY CURRENT PROBLEM 5 with k = mina 0, κ 0. Hence the initial boundary value problem.1,., has a unique classical solution local in time. References [1 W. Allegretto, Existence and long term behaviour of solutions to obstacle thermistor problem, Discrete and Continuous Dynamical Systems. 8 00, [ M. Chipot, and G. Cimatti, A uniqueness result for the thermistor problem, Euro. Jnl of Applied Mathematics. 1991, [3 G. Cimatti, On two problems of electrical heating of conductors, Quart. Appl. Math , [4 G. Cimatti, A bound for the temperature in the thermistor problem, IMA J. Appl. Math , 15-. [5 H. Diesselhorst, Ueber das Probleme eines elektrisch erwaermten Leiters, Ann. Phy , [6 S. D. Howison, A note on the thermistor problem in two space dimensions, Quart. Appl. Math , [7 S. D. Howison, J. Rodrigues and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl , [8 K. Kang and S. Kim, On the Hoelder continuity of solutions of a certain system related to Maxwell equations, IMA preprint series, 1805, August. [9 l1 A. Lacey, Thermal runaway in a nonlocal problem modelling ohmic heating, Euro. Jnl of Applied Mathematics , [10 D. C. Mattis, The Theory of Magnetism, Harper-Row, New York, [11 C. Qihong, A nonlinear parabolic system arising from the eddy currents problem, Nonlinear Analysis , [1 P. E. Sobolewskii, Equations of parabolic type in Banach spaces, AMS Translations Vol. 49, 1965, pp [13 H. M. Yin, Global solutions of Maxwell s equations in an electromagnetic field with a temperature-dependent electrical conductivity, Euro. Jnl of Applied Mathematics, , [14 H. M. Yin, Optimal regularity of solutions to a degenerate elliptic system arising in electromagnetic fields. Comm. Pure Appl. Analysis Vol. 1 00, Giovanni Cimatti Department of Mathematics, University of Pisa, via Buonarroti,, Pisa, Italy address: cimatti@dm.unipi.it