An Inverse Problem for a Nonlinear Diffusion-Convection Equation
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An Inverse Problem for a Nonlinear Diffusion-Convection Equation Silvana De Lillo · Diletta Burini
Received: 24 November 2011 / Accepted: 26 January 2012 / Published online: 7 June 2012 © Springer Science+Business Media B.V. 2012
Abstract A method for constructing the Dirichlet-to-Neumann map for a nonlinear diffusion-convection equation is presented. The problem is reduced to the solution of a nonlinear integral equation in one independent variable. Existence and uniqueness of the solution may be proven for small times via a contraction mapping technique. Keywords Inverse problem · Nonlinear diffusion-convection · Dirichlet-to-Neuman map
1 Introduction The nonlinear diffusion-convection equation θt = θ 2 (θxx − θx ),
θ ≡ θ (x, t)
(1)
is a well known model describing the flow of two immiscible fluids through a porous medium [1, 2]. In [3, 4] one and two-phase free boundary problems for (1) were considered and shown to admit a unique solution for small times; moreover, exact travelling wave solutions were also obtained. In this contribution we consider an Inverse Problem for (1). Namely, in the following we construct the Dirichlet-to-Neumann map [5, 6] for (1), defined over the domain −∞ < x < s(t) (s(0) = b > 0), characterized by the initial datum θ (x, 0) = θ0 (x) > 0,
−∞ < x < b
θ0 (b) = β2 > 0 S. De Lillo · D. Burini () Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, via Vanvitelli 1, 06123 Perugia, Italy e-mail: [email protected] S. De Lillo Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Perugia, Italy e-mail: [email protected]
(2a) (2b)
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S. De Lillo, D. Burini
and by the following set of boundary conditions: θ s(t), t = f (t), t > 0, θ (−∞, t) = β1 > 0,
f (0) = β2
(β1 > β2 ).
(3a) (3b)
In the above relations we assume that θ0 (x) is a regular, bounded function of its argument, s(t) is a continuously, differentiable function of time, s˙ (t) is bounded (|˙s (t)| ≤ α). In (3a) the Dirichlet datum θ (s(t), t) is given. We denote by (3c) θx s(t), t = g(t) the Neumann datum, which is unknown. In the following, g(t) will be determined in terms of f (t), in order to be compatible with the motion of the boundary s(t).
2 Linearization We start our analysis by introducing the so-called hodograph transform θ (x, t) = ψ(z, t),
z ≡ z(x, t)
(4a)
and 1 , zt = θ − θx . θ Under this change of variables, we obtain the Burgers equation for ψ(z, t) zx =
ψt = ψzz − 2ψz ψ
(4b)
(5)
over the domain −∞ < z < z¯ (t). The moving boundary z¯ (t) is a-priori unknown. Indeed, due to (4b), it has the form t s˙ (τ ) z¯ (t) ≡ z s(t), t = θ s(τ ), τ − θx s(τ ), τ + dτ (6) θ (s(τ ), τ ) 0 which, when (3a) and (3c) are used, becomes t z¯ (t) = f (τ ) − g(τ ) + 0
s˙ (τ ) dτ. θ (s(τ ), τ )
(7)
Moreover, the Burgers equation (5) is characterized by the initial datum ψ(z, 0) = ψ(z0 ) = θ0 (x) where
z0 ≡ z0 (x) = 0
x
1 dx , θ0 (x )
and by the boundary conditions ψ z¯ (t), t = θ s(t), t = f (t)
(8a)
(8b)
(9)
An I
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