The occurrence of interfaces in nonlinear diffusion-advection processes. (English) Zbl 0672.76094

The nonlinear diffusion-advection processes referred to in the title of this paper are those described by the equation (*) \(u_ t=(a(u))_{xx}+(b(u))_ x\), in which subscripts denote partial differentiation. The functions a and b belong to \(C([0,\infty))\cap C^ 2(0,\infty)\), and are such that \(a''\) and \(b''\) are locally Hölder continuous on (0,\(\infty)\), and \(a'(s)>0\) for \(s>0\). Furthermore, without any loss of generality, it is supposed that \(a(0)=0\) and \(b(0)=0\). Because of its resemblance to the celebrated equation arising in statistical mechanics, (*) is often referred to as the nonlinear Fokker- Planck equation. Equation (*) models a number of different physical phenomena. For instance, when u denotes unsaturated soil-moisture content, the equation describes the infiltration of water in a homogeneous porous medium. It also appears with \(a(s)=s^ 4\) and \(b(s)=- s^ 3\) in the theory of the flow of a thin viscous film over an inclined bed.
Equation (*) is parabolic when \(u>0\), but may degenerate for \(u=0\). Hence the equation need not admit classical solutions. Under appropriate conditions though, the equation is known to possess a unique generalized solution which is nonnegative and continuous. This solution is a classical solution of (*) in a neighbourhood of any point where it is positive. It is trivially also a classical solution in the interior of the set of points where it is zero. However, the derivatives of the solution may be undefined or discontinuous at points separating a region where the solution is positive from one where it is zero. The characteristic that equation (*) admit solutions possessing interfaces separating a region where the solution is positive from one where it is zero, is, itself, a peculiarity associated with the degeneracy of the equation. For instance, solutions of the linear heat equation do not display such behaviour. Given an initial-boundary value problem for the linear heat equation with nontrivial nonnegative initial data, the solution is positive everywhere in the problem domain. In this context, the linear heat equation is often said to propagate perturbations with infinite speed. An equation which does admit solutions possessing interfaces of the type described is said to have finite speed of propagation of perturbations.
In this paper we shall establish necessary and sufficient conditions for equation (*) to admit solutions possessing interfaces separating a region where the solution is positive from one where it is zero. We also study some properties characterizing such an interface. For convenience, we restrict the discussion to the Cauchy problem for equation (*) and to interfaces which provide an upper bound for the support of a solution. Nonetheless, our results may be extended to the Cauchy-Dirichlet problem and the first boundary-value problem for equation (*), and to general interfaces.


76R50 Diffusion
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI


[1] S. Anoenent, Analyticity of the interface of the porous media equation after the waiting time, Mathematical Institute University of Leiden Report 30 (1985), 12 pp.
[2] S. N. Antoncev, On the localization of solutions of nonlinear degenerate elliptic and parabolic equations, Soviet Math. Dokl. 24 (1981), 420–424. Translation of: Dokl. Akad. Nauk SSSR 260 (1981), 1289–1293.
[3] D. G. Aronson, L. A. Caffarelli & J. L. Vazquez, Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure Appl. Math. 38 (1985), 375–404. · Zbl 0544.35058
[4] D.G. Aronson & J.
[5] J. Bear, Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, New York London Amsterdam, 1972. · Zbl 1191.76001
[6] P. Benilan & J. L. Vazquez, Concavity of solutions of the porous medium equation, Trans. Amer. Math. Soc. 299 (1987), 81–93. · Zbl 0628.76092
[7] J. Buckmaster, Viscous sheets advancing over dry beds, J. Fluid Mech. 81 (1977), 735–756. · Zbl 0379.76029
[8] L. A. Caffarelli& A. Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101 (1979), 1193–1218. · Zbl 0439.76084
[9] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys. 15 (1943), 1–89. · Zbl 0061.46403
[10] J. I. Diaz & R. Kersner, Non existence d’une des frontières libres dans une équation dégénérée en théorie de la filtration, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 505–508. · Zbl 0543.35102
[11] J. I. Diaz & L. Véron, Compacité du support des solutions d’équations quasi linéaires elliptiques ou paraboliques, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), 149–152.
[12] J. I. Diaz & L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc. 290 (1985), 787–814. · Zbl 0579.35003
[13] L. C. Evans & B. F. Knerr, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979), 153–166. · Zbl 0403.35052
[14] B. H. Gilding, Properties of solutions of an equation in the theory of infiltration, Arch. Rational Mech. Anal. 65 (1977), 203–225. · Zbl 0366.76074
[15] B. H. Gilding, A nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 393–432. · Zbl 0364.35027
[16] B. H. Gilding, Improved theory for a nonlinear degenerate parabolic equation, submitted for publication. Also appearing as: Twente University of Technology Department of Applied Mathematics Memorandum 587 (1986), 42 pp.
[17] K. Höll · Zbl 0597.35107
[18] A. S. Kalashnikov, The occurrence of singularities in solutions of the non-steady seepage equation, U.S.S.R. Comput. Math. and Math. Phys. 7 (1967), 269–275. Translation of: Zh. Vychisl. Mat. i Mat. Fiz. 7 (1967), 440–444.
[19] A. S. Kalashnikov, The nature of the propagation of perturbations in processes that can be described by quasilinear degenerate parabolic equations (in Russian), Trudy Sem. Petrovsk. 1 (1975), 135–144. · Zbl 0318.35047
[20] R. Kersner, Filtration with absorption: necessary and sufficient condition for the propagation of perturbations to have finite velocity, J. Math. Anal. Appl. 90 (1982), 463–479. · Zbl 0512.35048
[21] B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), 381–415. · Zbl 0365.35030
[22] O. A. Oleinik, A. S. Kalashnikov & Chzhou Y.-L., The Cauchy problem and boundary problems for equations of the type of non-stationary filtration (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667–704.
[23] L. A. Peletier, A necessary and sufficient condition for the existence of an interface in flows through porous media, Arch. Rational Mech. Anal. 56 (1974), 183–190. · Zbl 0294.35040
[24] L. A. Peletier, On the existence of an interface in nonlinear diffusion processes, Ordinary and Partial Differential Equations (edited by B. D. Sleeman & I. M. Michael), Lecture Notes in Mathematics 415, Springer-Verlag, Berlin Heidelberg New York, 1974, pp. 412–416.
[25] M. H. Protter & H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967. · Zbl 0153.13602
[26] J. L. Vázquez, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285 (1984), 717–737. · Zbl 0524.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.