EXISTENCE AND UNIQUENESS FOR THE NONSTATIONARY PROBLEM OF THE ELECTRICAL HEATING OF A CONDUCTOR DUE TO THE JOULE-THOMSON EFFECT

Existence of a weak solution is established for the initial-boundary value problem for the system ff-iut-div(O(u)7u)+r(u)a(u)TuTv=a(u)]Tvl2,div(a(u)Vv)=O. The question of uniqueness is also considered in some special cases.

Let f be a bounded domain in R N with smooth boundary 9 and T a positive number.In this paper we shall be concerned with the following problem: u div(O(u) V u) + a(u)a(u) V u V v a(u) V v 12 in QT i2 x (0, T), div(a(u) V v) 0 in u 0 on S T cgflx(O,T), v B(z,t) on ST, n(z,0) Uo(z in fix {0}. (1.1a) Here, O(u),a(u), and a(u) are known functions of their argument and B,U 0 are given data.
Problem (1.1) may be proposed as a model for the electrical heating of a conductor resulted from Thomson's effect and Joule's heating; see [1].In this situation, u is the temperature of the conductor and v the effective potential.Equation (1.1b) represents the conservation of charge, while (1.1a) says that there are two types of heat source involved in the heat conduction; the convective term in (1.1a) corresponds to Thomson's effect and the quadratic term in (1.1a) reflects Shillor, and Xu [3] asserts that the assumption that N 2 and a cl(tt) in [2] can be eliminated.
The associated stationary problem of (1.1) was first considered in [1] where tr and O are assumed to obey the Wiedemann-Franz law, i.e., (u) c for some c > 0, O(u) n and a is assumed to be linear.reformulated as Under these assumptions the stationary problem can be div(A(u,v) V u) =0O} div(A(u,v) V v) in f, u u 0,v v 0 on 0ft.
Thus a uniform bound for the temperature can be obtained, thereby establishing an existence assertion.See [1] for details.
Our main objective is to prove an existence theorem for (1.1) under rather general assumptions on the data.Indeed, if the temperature is known to be bounded, our assumptions are much weaker than those in [1].Of course, our approach is also different and is based upon an approximation scheme.We also consider the question of uniqueness, but we are only able to show that the uniqueness holds when N 2 and O(s) s.
The mathematical interest of our problem is due to the presence of quadratic gradient growth in the nonlinearity.In general, nonlinearities of this nature render the classical regularity and compactness results useless; see [4] for a detailed description in this regard.Our method makes full use of the explicit nonlinear structure of our problem, which enables us to extract enough extra information to obtain an existence assertion.We refer the reader to [4] for more related works in this direction.
Finally, let us make some comments on notation.The letter c will be used to denote the genetic constant.When distinction among different constants is needed, we add a subscript {0,1,2 to c.Other notation conventions follow those employed in [5] and [6].For example, Ilfllp,: Ilfllp=(If f Pd.) lip for y LP(9).

EXISTENCE
In this section we first establish an existence assertion for the associated stationary problem.
Then a weak solution to (1.1) is obtained via the implicit discretization in time.
Let ft be a bounded domain in R N with smooth boundary Oft.Consider the system -div(l(u,v)Vu)+K(u+v)=J(u+v)l Vvl 2 -div(J(u + v)  With respect to the data involved, we assume the following.
A weak solution to (2.1) is defined as a pair (u,v) such that u, v 6. Wl'2(fl), for all e w'2(ft) n L(fl), J(u+v) v ,dr= W'2(fl), 0 for all u u0, v v 0 on Off.
PROOF.For each k define k if Izl2>_k, Denote by V the product space wl'2(fl) xwl'2(fl) and v* its topological dual.Set E {(Ul,Vl) V:UllO u 0 and Vllo% q v0}.Clearly, E is a closed, convex subset of v.For each define an operator Ak:E--,V* by (Ak(Wl)'W2) I I(Ul'Vl) V u 7 u2dz + I {Kk(Ul + vl) J(ul + Vl)Pk( V Vl) -H(z)}u2dz where (.,.) denotes the duality pairing between V* and V.By the definition of P/,A/ is well- defined.It is not difficult to verify that for each/, A/ satisfies the following conditions: (i) A/ is bounded.
(ii) A/ is pseudomonotone. (iii) (A,(),w-wo)/lltollv.-, as Ilwllv-" for we E, where w 0 (u0,v0).Now we are in a position to invoke an existence result in [7, p. 169] to conclude that for each k there exists at least one vector-valued function w k (u k, vk)e E such that (Ak(Wk),W-Wk)>O for all we E. This is easily seen to be equivalent to the following statements: " o "o," a o, (2.6)I j(.+ ) v v a 0, (2.8)   for all e W'2(t2).Equation (2.8) allows us to use the weak maximum principle to get sup Iv(x) <c(=*,2 ).
Then the theorem follows from taking k-c in (2.7) and (2.8).
Let f, H(z), u0,v0 be given as before.Consider the following problem: (2.20) u Ofl u0' v Of v0. (2.21b) We impose the following conditions on O,a,/: (H5) O,,# are continuous and satisfy m < O(s) < M,m < a(s) < M,m < B(s) < M for some M _> m > 0 for all s E R.
A weak solution to (2.21) can be defined in the same manner as that to (2.1).THEOREM 2.2.Under the above assumptions there is a weak solution to (2.21).
Then by (HS) there exists two positive constants Cl,C 2 such that 0 < c < F'(s) _< c 2 for all s E R.
REMARK.In fact, we only need to assume that is bounded.Then we can always select a number c large enough so that O < m <c +l < M. Also, if we know that u is bounded a priori, then there is no need to assume that O,a, are bounded above.In this sense, our hypotheses are much weaker than those in [1].However, in the generality considered here it does not seem likely that u can be bounded.Now we are ready to prove an existence assertion for the following problem: (2.30a) u O,v B on ST=OI2x(O,T ), u=U Oonfx{O}. (2.30b) (2.30c) THEOREM 2.3.Let ft, O,a,/3 be given as before.Assume that B_ L2(O,T;WI'2(12))f'ILCX:)(QT), and V 0 wl'2(ft).Then there exists a weak solution to (2.30), i.e., there is a pair (u,v) such that u, v fi L2(O,T;WI'2(f)), I '(u) V vl2{dzdt + IUo(z){(z,O)dz for all e HI(o,T;W'2(fI))nL(QT)such that (z,T) 0, tr(u) V v V rldzdt 0 for all ,; E L2(O,T;W'2(I)).
To pass to the limit in (2.46), we still need to show that We infer from (2.36) that strongly in L2(OT I (un) V Vn V dzdt 0 for all L2(O,T;W,2()).
In this section, a uniqueness assertion is established for (1.1) in some special cases.
THEOREM 3.1.Let the assumptions of Theorem 2.3 hold.Assume that O(s) s and that is Lipschitz continuous.
Recall from our assumptions that V Vl, V v 2 E [L(QT)]N.I and 12 can be estimated as follows: II11_< II (#(ul)-a(u2 OFt Oft Here we used the fact that a is Lipschitz continuous.Similarly,

Oft
Clearly, f f (ui) Vvi V fdx =O for aH 6 L2(O,t;W'2(f)) O for all 0 < < T and for ( Thus u u 2, v v 2. This completes the proof.
The above theorem is not very satisfactory because it requires that xz v be bounded, which cannot be guaranteed by the existence theorem.Thus it is interesting to investigate when V v becomes bounded.We summarize our results in the following theorem.THEOREM 3.2.Let the hypothesis of Theorem 2.3 be satisfied.'Assume (i) V 0 C0"( for some 0 < A < 1; (ii) o -g-B6.L(O,T;cO'A( )) for each i; (iii) N 2; (iv) is Lipschitz continuous.
Then there is a ,X (0,1) such that L(O,T;cO'At( )) for Oz PROOF.Set =v-B.
The proof is complete.
Combining Theorems 3.1 and 3.2 yields the following: THEOREM 3.3.Let the assumptions of Theorem 3.2 hold.Assume that O(s) s.Then there exists a unique solution to (1.1).

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u u 0
on 0f, v v 0 on Ofl.

First
Round of ReviewsOctober 1, 2009 in the resulting equation to obtain view of (2.39) and (2.40), we may rewrite (2.35) to read