On certain Neumann problems for Quasilinear
Parabolic Systems Modeling Multiphase Flows
Hermano Frid∗
Juan Gonzalez Marin†
Vladimir Shelukhin‡
We study the well-posedness of certain Neumann problems for N × N quasilinear parabolic systems
uit + div (fi (u)v) =
N
X
div (Bij (u)∇uj ),
i = 1, . . . , N,
j=1
which naturally appear as models in applications such as polydisperse suspensions and multiphase flows in porous media, among others.
We obtain existence and uniqueness of smooth solutions assuming values on
a physical invariant domain with no smallness assumption. The boundary value
problems considered
fi (u)v(x) −
2
X
Bij (u)∇uj · ~n(x) = k0 u v(x) · ~n(x),
i = 1, . . . , N,
j=1
include the zero-flux Neumann problem for a model of polydisperse suspensions.
The strategy is the use of the Leray-Schauder fixed point theorem, based on a
priori estimates implying regularity in Hölder Spaces. As a first step towards
the still open general multidimensional case, we consider the case of domains
with radial symmetry.
References
[1] Berres, S., Bürger, R. and Frid, H. Neumann problems for quasi-linear
parabolic systems modeling polydisperse suspensions. SIAM J. Math. Anal.
38 (2006), 557–573.
[2] Chueh, K., Conley, C., Smoller, J., Positively invariant regions for systems
of nonlinear diffusion equations, Ind. Univ. Math. J. 26 (1977), 373-392.
[3] Frid, H. and Shelukhin, V., A quasi-linear parabolic system for three-phase
capillary flow in porous media. SIAM J. Math. Anal. 35 (2003), no. 4,
1029–1041.
∗ Instituto de Matemática Pura e Aplicada - IMPA. Estrada Dona Castorina, 110. Rio de
Janeiro, RJ 22460-320, Brazil, [email protected].
† Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros
s/n, Salvador, BA 40170-110 , Brazil, [email protected].
‡ Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia, [email protected].
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[4] Frid, H. and Shelukhin, V., Initial boundary value problems for a quasilinear parabolic system in three-phase capillary flow in porous media, SIAM
J. Math. Anal., 36 (2005), pp. 1407-1425.
[5] Frid, H. Gonzalez, J. and Shelukhin, V., On certain Neumann problems for
Quasilinear Parabolic Systems Modeling Multiphase Flows, to appear.
[6] Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N., “Linear and
Quasi-linear Equations of Parabolic Type”, Providence, R.I.: Amer. Math.
Soc. 1968.
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