Journal of Mathematical Analysis and Applications 247, 517᎐556 Ž2000. doi:10.1006rjmaa.2000.6872, available online at http:rrwww.idealibrary.com on
On Strongly Degenerate Convection᎐Diffusion Problems Modeling Sedimentation᎐Consolidation Processes R. Burger ¨ Institute of Mathematics A, Uni¨ ersity of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany E-mail:
[email protected]
S. Evje RF-Rogaland Research, Thormøhensgt. 55, N-5008 Bergen, Norway E-mail:
[email protected]
and K. Hvistendahl Karlsen RF-Rogaland Research, Thormøhensgt. 55, N-5008 Bergen, Norway; and Department of Mathematics, Uni¨ ersity of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway E-mail:
[email protected] Submitted by W. L. Wendland Received March 20, 2000
We investigate initial-boundary value problems for a quasilinear strongly degenerate convection᎐diffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentation᎐ consolidation processes. The existence of entropy solutions belonging to BV is shown by the vanishing viscosity method. The existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient. New uniqueness proofs for entropy solutions are also presented. These proofs rely on a recent extension to second-order equations of Kruzkov’s method of ‘‘doubling the ˇ variables.’’ The application to a sedimentation᎐consolidation model is illustrated by two numerical examples. 䊚 2000 Academic Press Key Words: degenerate convection᎐diffusion equation; entropy solutions; discontinuous diffusion coefficient; sedimentation᎐consolidation processes; BV solutions. 517 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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BURGER , EVJE, AND KARLSEN ¨
1. INTRODUCTION In this paper we consider quasilinear strongly degenerate parabolic equations of the type
⭸ t u q ⭸x Ž q Ž t . u q f Ž u . . s ⭸x2 A Ž u . , u
Ž x, t . g QT , A Ž u . [ H aŽ s . ds, aŽ u . G 0,
Ž 1.1.
0
where QT [ ⍀ = T , ⍀ [ Ž0, 1., and T [ Ž0, T .. In general, we allow that the diffusion coefficient aŽ u. vanishes on intervals of solution values u, where Ž1.1. is then of hyperbolic type; therefore this equation is also called hyperbolic᎐parabolic. Although equations of this type occur in a variety of applications, we focus here on the application to sedimentation᎐consolidation processes w3, 8, 9x, which leads to an initial-boundary value problem ŽIBVP. with mixed Dirichlet-flux boundary conditions Ž‘‘Problem A’’. or alternatively to an IBVP with two flux conditions Ž‘‘Problem B’’.. It is well known that solutions of Ž1.1. develop discontinuities due to the nonlinearity of the flux density function f Ž u. and the degeneracy of the diffusion coefficient. Therefore one has to consider entropy solutions in order to have a well-posed problem. Moreover, in regions where Ž1.1. is hyperbolic, solution values propagate along straight-line characteristics which might intersect the lateral boundaries of QT from the interior and require the treatment of Dirichlet boundary conditions as entropy boundary conditions w2, 7x. A review of properties and known existence and uniqueness results related to the concept of entropy solutions for Eq. Ž1.1., as well as an overview of numerical methods for strongly degenerate parabolic equations, is provided in w11x. Our particular application justifies various assumptions on the coefficients of Ž1.1. and on the initial and boundary data. Most notably, many constitutive equations proposed for these processes imply that aŽ u. s 0 for u F u c and that aŽ u. jumps at u c to a positive value, where u c is a given constant, the so-called critical concentration. We therefore insist on using a discontinuous diffusion coefficient aŽ u.. This case had not been covered by the previous existence and uniqueness analysis of Problem A by Burger and Wendland w6x, which relies on relatively strong assumptions on ¨ the regularity of the coefficients of Eq. Ž1.1. and on the initial and boundary data; in particular, aŽ u. is assumed to be continuously differentiable. We point out that the previous analysis w6x was limited to Problem A and that Problem B has not been treated so far. The first objective of this paper is to show existence of entropy solutions belonging to BV Ž QT . for these problems when the diffusion coefficient is
CONVECTION ᎐ DIFFUSION PROBLEMS
519
discontinuous. We show that smoothing out the jumps of aŽ u. and of the initial and boundary data by a standard mollifier technique will not cause new singularities when the smoothing parameter tends to zero in the vanishing viscosity method. As a part of the existence proof of Problem B, we show that the integrated diffusion coefficient AŽ u. belongs to the Holder space C 1, 1r2 ŽQT .. This is a significantly better regularity property ¨ compared to the result ⭸x AŽ u. g L2 Ž QT . valid for Problem A. The second objective of this paper is to present new uniqueness proofs for both problems based on the technique known as ‘‘doubling of the variables.’’ This technique was introduced in Kruzkov’s pioneering work ˇ w13x as a tool for proving the L1 contraction principle for entropy solutions of scalar conservation laws and very recently was extended elegantly by Carrillo w10x to a class of degenerate parabolic equations. It is the extension in w10x that we adopt here to our initial-boundary value problems. We emphasize that these uniqueness proofs merely require that the functions f Ž u. and AŽ u. are locally Lipschitz continuous Ž aŽ u. may be discontinuous. and that they are not based on deriving jump conditions as in w24x. In fact, continuity of aŽ u. has been assumed in previous papers w6, 24, 25x in order to derive such jump conditions. Furthermore, the jump conditions ᎏand thus the corresponding uniqueness proofᎏderived by Wu and Yin w25x Žsee also w7x. have at present no multidimensional analogue, whereas the uniqueness approach presented here also works in multidimensions w4x. Having said this, some new results dedicated to the solution of this problem are available; see Vol’pert w20x. We mention that to produce an entropy solution belonging to BV Ž QT ., it is necessary to require that the initial function u 0 belongs to the class B of functions for which TVŽ ⭸x AŽ u.. is uniformly bounded with respect to regularization. This condition is rather restrictive but is satisfied by most initial data occurring in the context of the sedimentation᎐consolidation problems. Our problems are also solvable for u 0 f B Žsay u 0 g BV Ž ⍀ .., but then it is only possible to show the existence of an entropy solution in the larger class C 1r2 Ž T ; L1 Ž ⍀ .. l L⬁Ž T ; BV Ž ⍀ .., also referred to as BV1, 1r2 Ž QT . w22x. In this larger class, one cannot assume a priori that the traces of the entropy solution at the boundaries of QT exist. To resolve this problem one needs a reformulation of the concept of the solution that avoids these traces. Such a solution concept has been employed by Wu w22x, but will not be considered here since it is not obvious how to prove the uniqueness of such solutions. This paper is organized as follows. In Section 2, we recall some properties of mollifiers and related functions, state the initial boundary value problems with the respective pertaining assumptions on the data, and formulate definitions of entropy solutions. In Section 3, existence of entropy solutions is shown by the vanishing viscosity method, and the
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improved regularity result valid for entropy solutions of Problem B is derived. Uniqueness of entropy solutions is shown in Section 4. In Section 5 we present two numerical solutions of the IBVP modeling sedimentation with compression, in which the assumptions for the existence of BV solutions are satisfied.
2. MATHEMATICAL PRELIMINARIES AND DEFINITION OF ENTROPY SOLUTIONS 2.1. Mollifiers and Related Functions Let g C0⬁Ž⺢. be a function satisfying G 0, supp ; Žy1, 1., and 5 5 L1 Ž⺢ . s 1, and define a standard mollifier w16x with support in Žyh, h. by hŽ x . s Ž xrh.rh. A C⬁ regularization of a bounded function bŽ u. is then given by the convolution h
Ž b) h . Ž u . [ H b Ž u y ¨ . h Ž ¨ . d¨ . yh
Moreover, we define for sufficiently small h ) 0 the functions
D hŽ x . [
x
Hy⬁ Ž . d , h
h Ž x . [ 1 y D h Ž x y 2 h . ,
Ž 2.1.
hŽ x . [ D hŽ x y Ž 1 y 2 h. . , which have the property stated in the following lemma given in w24x. LEMMA 1. Let ¨ g L1 Ž T ; L⬁Ž ⍀ ... If the traces ␥ 0¨ [ Ž␥ ¨ .Ž0, t . and ␥ 1¨ [ Ž␥ ¨ .Ž1, t . exist a.e. in T , then we ha¨ e for g C⬁Ž QT ., lim hx0
HHQ s
⭸x Ž Ž x, t . Ž h Ž x . q h Ž x . . . ¨ Ž x, t . dt dx T
T
H0
Ž Ž 1, t . ␥ 1¨ y Ž 0, t . ␥ 0¨ . dt.
2.2. Statement of Problem A PROBLEM A. We consider the IBVP
⭸ t u q ⭸x Ž q Ž t . u q f Ž u . . s ⭸x2 A Ž u . ,
Ž x, t . g QT ,
Ž A1.
u Ž x, 0 . s u 0 Ž x . ,
x g ⍀,
Ž A2.
u Ž 1, t . s 1 Ž t . ,
t g Ž 0, T x ,
Ž A3.
t g Ž 0, T x ,
Ž A4.
f Ž u Ž 0, t . . y ⭸x A Ž u Ž 0, t . . s 0,
CONVECTION ᎐ DIFFUSION PROBLEMS
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where we assume that f is continuous and piecewise differentiable , f F 0, supp f ; w 0, u max x , 5 f X 5 ⬁ F ⬁, a Ž u . G 0, supp a : supp f ,
Ž 2.2.
a Ž u . s 0 for u F u c , 0 - u c - u max ,
Ž 2.3. qŽ t. F 0 ᭙t g T,
TVT Ž qX . - ⬁.
TVT Ž q . - ⬁,
Ž 2.4.
Since A is monotonically non-increasing, sgnŽ k 1 y k 2 .Ž AŽ k 1 . y AŽ k 2 .. s < AŽ k 1 . y AŽ k 2 .
