Díaz Díaz, Jesús Ildefonso and Shmarev, Sergey
(2009)
*Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology.*
Archive for Rational Mechanics and Analysis , 194
(1).
pp. 75-103.
ISSN 0003-9527

PDF
Restricted to Repository staff only until 31 December 2020. 374kB |

Official URL: http://www.springerlink.com/content/2117372554834714/fulltext.pdf

## Abstract

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation u(t) - Delta u is an element of aH(u - mu) in Q = Omega X (0, T], where Omega subset of R(n) is a ring-shaped domain, a and mu are given positive constants, H(.) is the Heaviside maximal monotone graph: H(s) = 1 if s > 0, H(0) = [0, 1], H(s) = 0 if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets Gamma(mu) = {(x, t) : u(x, t) = mu} are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Gamma(mu) is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Gamma(mu) through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | diffusion-equations; discontinuous nonlinearities; interfaces; uniqueness |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15089 |

References: | Andreucci D., Gianni R.: Classical solutions to a multidimensional free boundary problem arising in combustion theory. Comm. Partial Differ. Equ. 19, 803–826 (1994) Bermejo, R., Carpio, J., Diaz, J.I., Tello, L.: Mathematical and numerical analysis of a nonlinear diffusive climatological energy balance model. Math. Comput. Model. (to appear) (2008) Berryman J.G.: Evolution of a stable profile for a class of nonlinear diffusion equations. III. Slow diffusion on the line. J. Math. Phys. 21, 1326–1331 (1980) Bertsch M., Hilhorst D.: The interface between regions where u < 0 and u > 0 in the porous medium equation. Appl. Anal. 41, 111–130 (1991) Budyko M.: The efects of solar radiation variations on the climate of the earth. Tellus 21, 611–619 (1969) Díaz, J.I.: Mathematical analysis of some diffusive energy balance climate models. In Mathematics, Climate and Environment. (Eds. J. Díaz and J. Lions) Masson, Paris, 28–56, 1993 Díaz J.I., Tello L.: A nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology. Collect. Math. 50, 19–51 (1999) Díaz J.I., Shmarev S.: On the behaviour of the interface in nonlinear processes with convection dominating diffusion via lagrangian coordinates. Adv. Math. Sci. Appl. 1, 19–45 (1992) Díaz J.I., Nagai T., Shmarev S.I.: On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics. Jpn. J. Indust. Appl. Math. 13, 385–415 (1996) Feireisl E., Norbury J.: Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities. Proc. R. Soc. Edinburgh Sect. A 119(1–2), 1–17 (1991) Feireisl E.: A note on uniqueness for parabolic problems with discontinuous nonlinearities. Nonlinear Anal. 16(11), 1053–1056 (1991) Gianni R., Hulshof J.: The semilinear heat equation with a Heaviside source term. Eur. J. Appl. Math. 3, 367–379 (1992) Gianni R., Ricci R.: Classical solvability of some free boundary problems through the geometry of the level lines. Adv. Math. Sci. Appl. 5, 557–567 (1995) Gurtin M.: An introduction to Fluid Mechanics. Academic Press, London (1981) Gurtin M.E., MacCamy R.C., Socolovsky E.A.: A coordinate transformation for the porous media equation that renders the free boundary stationary. Quart. Appl. Math. 42, 345–357 (1984) Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford, 2nd edn. Translated from the Russian by Howard L. Silcock, 1982 Ladyženskaja, O.A., Solonnikov, V.A., Ural′ceva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, 1967 Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Mathematics and its Applications, vol. 2, Gordon and Breach Science Publishers, New York, 1969 Ladyzhenskaya, O.A., Ural′tseva, N.N.: Linear and quasilinear elliptic equations. Leon Ehrenpreis. Academic Press, New York, 1968 Mermanov, A.M., Puhnacëv, V.V.: Lagrangian Coordinates in the Stefan Problem, Dinamika Sploshn. Sredy, (1980), pp. 90–111, 165–166 Meirmanov, A.M., Pukhnachov, V.V., Shmarev, S.I.: Evolution Equations and Lagrangian Coordinates, de Gruyter Expositions in Mathematics, 24, Walter de Gruyter, Berlin, 1997 Sellers W.: A global climatic model based on the energy balance of the earth- atmosphere system. J. Appl. Meteorol. 8, 392–400 (1969) Shmarev S.I.: Interfaces in multidimensional diffusion equations with absorption terms. Nonlinear Anal. 53, 791–828 (2003) Shmarev, S.I.: Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension. In: Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl., vol. 61. Birkhäuser, Basel, 257–273, 2005 Shmarev, S.I., Vázquez, J.L.: Lagrangian Coordinates and Regularity of Interfaces in Reaction–diffusion Equations, C. R. Acad. Sci. Paris Sér. I Math., vol. 321, pp. 993–998, 1995 Shmarev S.I., Vázquez J.L.: The regularity of solutions of reaction–diffusion equations via Lagrangian coordinates. NoDEA Nonlinear Differ. Equ. Appl. 3, 465–497 (1996) Temam R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence (2001) Xu X.: Existence and regularity theorems for a free boundary problem governing a simple climate model. Appl. Anal. 42, 33–57 (1991) |

Deposited On: | 04 May 2012 11:27 |

Last Modified: | 06 Feb 2014 10:16 |

Repository Staff Only: item control page