Letuuunnnbe the solution of theN-membranes problem (1.5), under the assumptions (1.3), with given data fffnnn and gggnnn satisfying (1.4). Assuming that fffnnn converges to fff in [Lp(Ω)]N and that gggnnn converges toggg in [Lq(Γ)]N, we shall extend now the following stability result in Ls(Ω) (1 ≤s < ∞) of [1] for the corresponding coincidence sets (defined in (1.11)),
χ{unk=ããã=unl} −−−−→
n
χ{uk=ããã=ul}, for 1≤k < l≤N.
Recalling the inequalities (2.14), Auuu = FFF a.e. in Ω, for some function FFF∈ [Lp(Ω)]N, as in Lemma 2 of [8], we have
Auk=Auk+1 a.e. in{x∈Ω :uk(x) =uk+1(x)}
and so we can characterize a.e. in Ω eachFk in terms offland the characteristic functionsχ
{ur=ããã=us}, 1≤l≤N, 1≤r < s≤N.
In what follows, we use, as before, the convention,u0= +∞anduN+1=−∞. We define the following sets
Θk,l={x∈Ω :uk−1(x)> uk(x) =ã ã ã=ul(x)> ul+1(x)}, (3.1) the sets of contact of exactly the membranesuk, . . . , ul.
Proposition 3.1. If k, l∈Nare such that1≤k≤l≤N , we have 1. Aur=
fk,l a.e. in Θk,l if r∈ {k, . . . , l}, fr a.e. in Θk,l if r∈ {k, . . . , l}.
2. Ifk < l then for allr∈ {k, . . . , l} fr+1,l≥ fk,r a.e. in Θk,l.
Proof. Because of the regularity resultAuuu∈ [Lp(Ω)]N, the proof of this propo- sition is the same as for the case with boundary Dirichlet condition, done in [1],
since it was done locally at a.e. pointx∈Ω.
Remark 3.2. It is well known that a necessary condition for existing contact in the case of two membranesu1andu2, subject to external forcesf1andf2respectively, is thatf2≥f1. Depending on the boundary conditions, this condition may be (or not) sufficient for contact.
TheN-membranes Problem with Neumann Type Boundary Condition 63 We would like to emphasize that condition 2. of the preceding proposition is a necessary condition for the firstr−kmembranes (k < r≤l) to be in contact with the otherl−r+ 1 membranes. We can interpret physically the condition 2.
by regarding the firstr−kmembranes as one membrane where a force with the intensity of the average of the forcesfk, . . . , fris applied and all the otherl−r+ 1 as another one where it was applied a force with the intensity equal to the average of the remaining forcesfr+1, . . . , fl.
As for the boundary Dirichlet condition case, we may characterize the varia- tional inequality (1.5) as a system ofN equations, coupled through the characteris- tic functions of the coincidence setsIk,l. In (1.13) we presented the system forN= 3, containing as a special caseN= 2. The next theorem presents the general case.
Theorem 3.3. Under the assumptions (1.3), let uuu be the solution of the problem (1.5)with datafff andggg satisfying (1.4). Then
Aur=fr+
1≤k<l≤N, k≤r≤l
bk,lr χ
k,l a.e. inΩ, (3.2) where
bk,lr =bk,lr [f] =
⎧⎪
⎨
⎪⎩
fk,l− fk,l−1 if r=l
fk,l− fk+1,l if r=k
2 (l−k)(l−k+1)
fk+1,l−1−12(fk+fl)
if k < r < l.
Also exactly as in [1], using the variational convergenceuuunnn →uuuin
H1(Ω) N, we may prove the continuous dependence of the coincidence sets with respect to the external data.
Theorem 3.4. Assuming (1.3) and given n ∈ N, let uuunnn denote the solution of problem(1.5) with given datafffnnn ∈[Lp(Ω)]N,gggnnn∈[Lq(Γ)]N, withp, qas in (1.4).
Suppose that fn fn
fn −−−−→
n fff in [Lp(Ω)]N, gggnnn −−−−→
n ggg in [Lq(Γ)]N. Then
un
uunn −−−−→
n uuu in
H1(Ω) N. (3.3)
If, in addition, the limit forces satisfy
fk,r=fr+1,l for allk, r, l∈ {1, . . . , N} withk≤r < l, (3.4) then, for any1≤s <∞,∀k, l∈ {1, . . . , N}, k < l,
χ{unk=ããã=unl} −−−−→
n
χ{uk=ããã=ul} in Ls(Ω). (3.5) Remark 3.5. The condition (3.4) for the stability of the coincidence sets forN = 2 is simplyf2=f1and forN = 3, the condition (1.12) (see [2] for a direct proof).
Remark 3.6. It would be interesting to prove a condition analogous to the system (3.2) for the boundary operatorB (under additional regularity of the solutionuuu),
i.e., to find sufficient conditions for some coefficientsγrj,k involving the averages gk,l such that, if ˆIk,l={x∈Γ :uk(x) =ã ã ã=ul(x)},then
Bur=gr+
1≤k<l≤N, k≤r≤l
γrk,l χ
Iˆk,l a.e. on Γ.
References
[1] Azevedo, A. & Rodrigues, J.F. & Santos, L. The N-membranes problem for quasi- linear degenerate systems Interfaces and Free Boundaries Volume 7Issue 3 (2005) 319–337.
[2] Azevedo, A. & Rodrigues, J.F. & Santos, L.Remarks on the two and three membranes problem (Taiwan 2004) Elliptic and parabolic problems: recent advances, Chen, C.
C. & Chipot, M. & Lin, C. S. (Eds.), World Scientific Singapore (2005) 19–33.
[3] Chipot, M. & Vergara-Cafarelli, G.The N-membranes problem Appl. Math. Optim.
13no3 (1985) 231–249.
[4] Gilbard, D. & Trudinger, N.S. Elliptic partial differential equations of second order 2nd edition, Springer-Verlag, Berlin, 1983.
[5] Hanouzet, B. & Joly, J.L. M´ethodes d’ordre dans l’interpr´etation de certains in´equations variationnelles et applicationsJ. Functional Analysis34(1979) 217–249 (see also C. R. Ac. Sci. Paris281(1975) 373–376).
[6] Lewy, H. & Stampacchia, G. On the smoothness of superharmonics which solve a minimum problem J. Analyse Math.23(1970) 227–236.
[7] Rodrigues, J.F.Obstacle problems in mathematical physicsNorth Holland, Amster- dam, 1987.
[8] Rodrigues, J.F. Stability remarks to the obstacle problem for the p-Laplacian type equationsCalc. Var. Partial Differential Equations23no. 1 (2005) 51–65
[9] Troianiello, G.M.Elliptic differential equations and obstacle problemsPlenum Press, New York, 1987.
[10] Vergara-Caffarelli, G.Regolarit`a di un problema di disequazioni variazionali relativo a due membrane Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 50 (1971) 659–662.
A. Azevedo and L. Santos University of Minho Campus de Gualtar 4700–030 Braga, Portugal e-mail:assis@math.uminho.pt e-mail:lisa@math.uminho.pt J.F. Rodrigues
CMUC/University of Coimbra and University of Lisbon Av. Prof. Gama Pinto, 2
1649–003 Lisboa, Portugal e-mail:rodrigue@fc.ul.pt
International Series of Numerical Mathematics, Vol. 154, 65–74 c 2006 Birkh¨auser Verlag Basel/Switzerland
Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport
Markus Bause and Willi Merz
Abstract. In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, chemical and biological processes of contam- inant transport and degradation in the subsurface. Our latest results for the existence, uniqueness and regularity of solutions to the model equations are summarized; cf. [4, 9]. The basic idea of the proof of regularity is sketched briefly. Moreover, our numerical discretization scheme that has proved its ca- pability of approximating reliably and efficiently solutions of the mathematical model is described shortly, and an error estimate is given; cf. [2, 3]. Finally, to illustrate our approach of modelling and simulating bioreactive transport in the subsurface, the movement and expansion of am-xylene plume is studied numerically under realistic field-scale assumptions.
1. Introduction
In particular in industrialized countries, groundwater and soil pollution has be- come a major environmental threat. In many cases groundwater and soil contain a mixture of organic and anorganic substances. Usually, the contamination itself is hardly accessible in the subsurface. But, fortunately, biodegradation tends to attenuate at least some contaminants during groundwater transport. However, its potential is difficult to predict. Mathematical models and numerical simulations can be used to predict the long-term evaluation of contaminant plumes and help to design remediation techniques for field scale problems.
In the sequel, a mathematical model incorporating relevant processes of con- taminant transport and (bio-)degradation in the subsurface is presented, the math- ematical properties of its solutions are given and a reliable and efficient approx- imation scheme for numerical simulations is proposed. Finally, a realistic aquifer contamination scenario is investigated numerically.