On the Solution of a Hyperbolic One-Dimensional Free Boundary Problem for a Maxwell Fluid

Abstract

We study a hyperbolic (telegrapher's equation) free boundary problem describing the pressure-driven channel flow of a Bingham-type fluid whose constitutive model was derived in the work of Fusi and Farina (2011). The free boundary is the surface that separates the inner core (where the velocity is uniform) from the external layer where the fluid behaves as an upper convected Maxwell fluid. We present a procedure to obtain an explicit representation formula for the solution. We then exploit such a representation to write the free boundary equation in terms of the initial and boundary data only. We also perform an asymptotic expansion in terms of a parameter tied to the rheological properties of the Maxwell fluid. Explicit formulas of the solutions for the various order of approximation are provided.

1. Introduction

In this paper we study the well posedness of a hyperbolic free boundary problem arisen from a one-dimensional model for the channel flow of a rate-type fluid with stress threshold presented in [1]. The model describes the one-dimensional flow of a fluid which behaves as a nonlinear viscoelastic fluid if the stress is above a certain threshold 𝜏_𝑜 and like a rate type fluid if the stress is below that threshold. The problem investigated here belongs to a series of extensions of the classical Bingham model we have proposed in recent years (see [2–5]).

In particular, in [1] we describe the one-dimensional flow of such a fluid in an infinite channel, assuming that in the outer part of the channel the material behaves as a viscoelastic upper convected Maxwell fluid, while in the inner core as a rate-type Oldroyd-B fluid. The general mathematical model is derived within the framework of the theory of natural configurations developed by Rajagopal and Srinivasa (see [6]). The constitutive equations are obtained imposing how the system stores and dissipates energy and exploiting the criterion of the maximization of the dissipation rate.

The main practical motivation behind this study comes from the analysis of materials like asphalt or bitumen which exhibit a stress threshold beyond which they change its rheological properties. Indeed from the papers [7–9], it is clear that such materials have a viscoelastic behaviour (for instance, upper convected Maxwell fluid) which is observed if the applied stress is greater than a certain threshold (see, in particular, [7]).

The mathematical formulation for the channel flow driven by a constant pressure gradient consists in a free boundary problem involving a hyperbolic telegrapher's equation (Maxwell fluid) and a third-order equation (Oldroyd-B fluid). The free boundary is the surface dividing the two domains: the inner channel core and the external layer. Due to the high complexity of the general problem, here we have considered a simplified version which arises when the order of magnitude of some physical parameters involved in the general model ranges around particular values. In such a case we have that the velocity of the inner core is constant in space and time, while the outer part behaves as a viscoelastic upper convected Maxwell fluid (see [1] for more details). The mathematical formulation turns out to be a hyperbolic free boundary problem which, in the authors knowledge, is new since it involves a telegrapher's equation coupled with an ODE describing the evolution of the interface.

The paper is structured as follows. In Section 2 we formulate the problem, namely, problem (2.1), and specify the basic assumptions. In Section 3 we give an equivalent formulation of the problem which leads to a nonlinear integrodifferential equation for the free boundary. We prove local existence and uniqueness for such an equation (see Theorem 3.3), under specific assumptions on the data.

The interesting aspect of the mathematical analysis lies on the technique we employ to reduce the complete problem to a single integrodifferential equation from which some mathematical properties can be derived (the free boundary equation can be solved autonomously from the governing equation of the velocity field). Such a methodology is a generalization of a technique already introduced in [2].

In Section 4 we perform an asymptotic expansion in terms of a coefficient 𝜔 (representing the ration between the elastic characteristic time and the relaxation time of the viscoelastic material), which typically is of the order 𝑂(10⁻¹). This procedure allows to obtain approximations of the actual solution up to any order through an iterative procedure. We do not prove the convergence of the asymptotic approximations to the actual solution (whose existence is proved in Theorem 3.3), limiting ourselves to develop only the formal procedure. Indeed, the main advantage of this procedure is that, for each order of approximation, the governing equation for the velocity field is the “standard” wave equation, which is by far easier to handle than the telegrapher's equation. We end the paper with few conclusive remarks.

2. Mathematical Formulation

In this section we state the mathematical problem. We refer the reader to [1] for all the details describing how this simplified model was derived from the general one. In the general case, in the region [0,𝑠], the fluid behaves as an Oldroyd-B type fluid. The problem we are studying here is a particular case of such a model, which stems from some specific assumptions on the physical parameters (fulfilled by materials like asphalt and bitumen). Under such assumptions, the inner core [0,𝑠] moves with uniform constant velocity 𝑉_𝑜.

We consider an orthogonal coordinate system 𝑥𝑜𝑦 and assume that the fluid is confined between two parallel plates placed at distance 2𝐿. We assume that the motion takes place along the 𝑥-direction and that the velocity field has the form 𝑣(⃗𝑦,t)=𝑣(𝑦,𝑡)𝑖⃗. We rescale the problem (in a nondimensional form) and, because of symmetry, we study the upper part of the layer 𝑦∈[0,1] (the space variable is rescaled by 𝐿). The geometry of the system we investigate is depicted in Figure 1.

Figure 1 

Upper part of the channel.

The mathematical model is written for the velocity field 𝑣(𝑦,𝑡) in the viscoelastic region which is separated from the region with zero strain rate (uniform velocity) by the moving interface 𝑦=𝑠(𝑡).

The nondimensional formulation is the following:𝑣𝑣(_𝑡𝑡𝑦+,02)𝜔=𝑣_𝑡𝑣=_𝑜𝑣_𝑦𝑦+𝛽²𝑦∈(𝑠,1),𝑡>0,𝑣(𝑦)𝑦∈𝑠_𝑜,1_𝑡,𝑣𝑣(((1𝑠𝑦,,,𝑡𝑡)0))===𝑉00_𝑜𝑡𝑦>∈0,𝑠_𝑜,1,𝑣𝑡>0,_𝑦𝑠(0(𝑠),=𝑡)𝑠+_𝑜𝑠𝑣̇_𝑡(𝑠,𝑡)=−𝛽²𝙱𝚗𝑡>0,,𝑠_𝑜∈(0,1).(2.1) where (i)𝜌 is the material density, (ii)𝜂 is the viscosity of the fluid, (iii)𝜇 is the elastic modulus, (iv)𝛽² is a positive parameter depending on the viscosity 𝜂 (see [1]), (v)𝙱𝚗 is the Bingham number, (vi)𝑉_𝑜 is the velocity of the inner core, (vii)2𝜔=𝑡_𝑒/𝑡_𝑟, (viii)𝑡_𝑒=𝐿√𝜌/𝜇 is the characteristic elastic time, (ix)𝑡_𝑟=𝜂/2𝜇 is the relaxation time.

In the case of asphalt typical values are (see [8, 9])𝜇=1MPa,𝜌=1.5×10³K𝑔/m³,𝜂=10²MPa⋅s.(2.2)

Taking 𝐿=500m we get𝑡_𝑒=15s,𝑡_𝑟=50s,⟹𝜔=0.15.(2.3)

Remark 2.1. In [1] we have proved that problem (2.1) admits a stationary solution provided 𝑉_𝑜⩽𝛽²12+𝙱𝚗(2.4) and that the stationary solution is given by 𝑣_∞(𝑦)=−𝛽²2(𝑠−𝙱𝚗−𝑦)²+𝛽²𝙱𝚗2+𝑉^𝑜𝑠_∞,=1+𝙱𝚗−𝙱𝚗²+2𝑉^𝑜𝛽².(2.5)

3. An Equivalent Formulation

Before proceeding in proving analytical results of problem (2.1) we introduce the new coordinate system𝑥=1−𝑦,⟺𝑦=1−𝑥,(3.1) and the new variable𝑈(𝑥,𝑡)=exp(𝜔𝑡)𝑣(1−𝑥,𝑡),⟺𝑣(𝑦,𝑡)=𝑈(1−𝑦)exp(−𝜔𝑡).(3.2) With transformations (3.1)-(3.2), problem (2.1) becomes𝑈_𝑥𝑥−𝑈_𝑡𝑡+𝑈𝜔(𝑥²𝑈,0=)=−𝑈𝛽²_𝑜exp(𝜔𝑡),𝑥∈(0,𝜉),𝑡>0,𝑈_𝑡(𝑥),𝑥∈0,𝜉_𝑜,𝑈(𝑥(𝜉,0,𝑈𝑡))=(=0,𝑈e𝑡x)₁p(=𝑥()0𝜔,,𝑡)𝑉𝑥𝑡_𝑜>∈0,0,𝜉_𝑜,𝑈,𝑡>0,_𝑥(𝜉,𝑡)+𝜉̇𝑈_𝑡(𝜉𝜉,𝑡()0−)=𝜉̇𝜔𝜉𝑈_𝑜(𝜉,𝑡)=exp(𝜔𝑡)𝛽²𝙱𝚗,𝑡>0,,𝜉_𝑜∈(0,1),(3.3)

where𝜉(𝑡)=1−𝑠(𝑡),𝜉_𝑜=1−𝑠_𝑜,𝑈_𝑜(𝑥)=𝑣_𝑜(1−𝑥),𝑈₁(𝑥)=𝜔𝑈_𝑜(𝑥).(3.4) Notice that, by means of (3.2), the evolution equation for the new variable 𝑈(𝑥,𝑡) has become a nonhomogeneous Klein-Gordon equation [10].

The domain of problem (3.3) is depicted in Figure 2. We begin by considering the domain 𝐷_𝐼 (see Figure 3). Here the solution has the representation formula (see [11])𝑈(𝑥,𝑡)=12𝑈^𝑜(𝑥−𝑡)+𝑈_𝑜(𝑥+𝑡)+12^{𝑥+𝑡𝑥−𝑡}𝑅(𝑥,𝑡;𝜁,0)𝑈₁+𝛽(𝜁)−𝑅_𝜃(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁₂2^𝑡0exp(𝜔𝜃)𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅(𝑥,𝑡;𝜁,𝜃)𝑑𝜁,(3.5) where 𝑅(𝑥,𝑡;𝜁,𝜃) is the Riemann's function that solves the problem (see again Figure 3)Ω(𝑥,𝑡)=𝑅{𝑅𝑅(_𝜁𝜁𝑥−((,𝑥𝑥𝑡)𝑅,,𝑡𝑡∶;;_𝜃𝜃𝑥𝑥𝑥++−−𝜔𝑡𝑡²𝑡+−𝑅+𝜃𝜃=𝜃,,𝜃𝜃⩽0))==𝜁(⩽11𝜁,𝑥𝜃𝜃𝜃)+∈∈∈𝑡−Ω[[00,,𝜃(𝑥𝑡𝑡,]],,,0𝑡)⩽,𝜃⩽𝑡}.(3.6)

Figure 2 

Sketch of the domain.

Figure 3 

The domain 𝐷𝐼.

To determine the solution of problem (3.6) we set𝑧=(𝑡−𝜃)²−(𝜁−𝑥)²,(3.7) where𝑧𝑧^2𝜃−𝑧^2𝜁=1,_𝜃𝜃−𝑧_𝜁𝜁=1𝑧.(3.8)

By means of (3.7) problem (3.6) becomes𝑅^(𝑧)+𝑅^(𝑧)𝑧−𝜔²𝑅(0)=1,𝑅(𝑧)=0(3.9) where (3.9)₍₁₎ is the modified Bessel equation of zero order. The solution of (3.9) is given by𝑅(𝑧)=𝑅(𝑥,𝑡;𝜁,𝜃)=𝐼_𝑜𝜔(𝑡−𝜃)²−(𝜁−𝑥)²,(3.10) where 𝐼_𝑜 is the modified Bessel function of zero order. It is easy to prove that the function defined by (3.10) satisfies problem (3.6). Moreover, since [12]𝐼^𝑜(𝑥)𝑥=21𝐼^𝑜(𝑥)−𝐼₂(𝑥),(3.11)

one can prove that𝑅_𝜃(𝑥,𝑡;𝜁,𝜃)=𝜔²(𝜃−𝑡)2𝐼^𝑜𝜔(𝑡−𝜃)²−(𝜁−𝑥)²−𝐼²𝜔(𝑡−𝜃)²−(𝜁−𝑥)²,(3.12) where 𝐼₂ is the modified Bessel function of second order. Recalling that 𝑈(𝑥,0)=𝑈_𝑜(𝑥) and, by (3.3)₍₃₎, (3.4), that𝑈_𝑡(𝑥,0)=𝜔𝑈_𝑜(𝑥),(3.13)we see that𝑅(𝑥=,𝑡;𝐼𝜁,0)𝜔−𝑅_𝜃(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)_𝑜𝜔𝑡²−(𝜁−𝑥)²𝜔+𝜔²𝑡2−𝜔²𝑡2𝐼²𝜔𝑡²−(𝜁−𝑥)²𝑈^𝑜(𝜁),(3.14)and representation formula (3.5) can be rewritten as 𝑈(𝑥,𝑡)=12𝑈^𝑜+1(𝑥−𝑡)+𝑈_𝑜(𝑥+𝑡)2^{𝑥+𝑡𝑥−𝑡}𝐼^𝑜𝜔𝑡²−(𝜁−𝑥)²𝜔+𝜔²𝑡2−𝜔²𝑡2𝐼²𝜔𝑡²−(𝜁−𝑥)²𝑈^𝑜+𝛽(𝜁)𝑑𝜁₂2^𝑡0exp𝑘𝜃2𝑑𝜃⋅^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝐼_𝑜𝜔(𝑡−𝜃)²−(𝜁−𝑥)²𝑑𝜁.(3.15)

Let us now write a representation formula for 𝑈(𝑥,𝑡) in the domain 𝐷_𝐼𝐼 (see Figure 4). We once again make use of (3.6), where now 𝑈_𝑜 has to be extended to the domain [−𝜉_𝑜,0]. Following [2], we extend 𝑈_𝑜 imposing condition (3.3)₍₄₎, that is, 𝑈(0,𝜃)=0. From the representation formula we get0=12𝑈^𝑜−𝑡^∗+𝑈_𝑜𝑡^∗+12^𝑡∗_−𝑡∗𝑅0,𝑡^∗+𝛽;𝜁,0𝜔−𝑅_𝜃0,𝑡^∗;𝜁,0𝑈_𝑜(𝜁)𝑑𝜁₂2^𝑡∗₀exp(𝜔𝜃)𝑑𝜃^{𝑡∗−𝜃−𝑡}_∗+𝜃𝑅0,𝑡^∗;𝜁,𝜃𝑑𝜁,(3.16) where 𝑡^∗=𝑡−𝑥(3.17)is the coordinate of the intersection of the characteristic 𝜁=𝑥−𝑡+𝜃 with 𝜁=0. Relation (3.16) can be rewritten as0=12𝑈^𝑜(𝑥−𝑡)+𝑈_𝑜(𝑡−𝑥)+12^{𝑡−𝑥𝑥−𝑡}𝑅(0,𝑡−𝑥;𝜁,0)𝜔−𝑅_𝜃+𝛽(0,𝑡−𝑥;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁₂2^0𝑡−𝑥exp(𝜔𝜃)𝑑𝜃^{𝑡−𝑥−𝜃𝑥−𝑡+𝜃}𝑅(0,𝑡−𝑥;𝜁,𝜃)𝑑𝜁.(3.18) From (3.18), the extended function 𝑈^𝑠x_𝑜(𝑥), defined in [−𝜉_𝑜,0], fulfills the following Volterra integral equation of second type: 𝑈=^𝑜𝑠𝑥(−𝑥𝑈−_𝑜𝑡)−^0𝑥−𝑡𝑅(0,𝑡−𝑥;𝜁,0)𝜔−𝑅_𝜃(0,𝑡−𝑥;𝜁,0)𝑈^𝑜𝑠𝑥(𝜁)𝑑𝜁−𝛽(𝑡−𝑥)+^0𝑡−𝑥𝑅(0,𝑡−𝑥;𝜁,0)𝜔−𝑅_𝜃(0,𝑡−𝑥;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁₂2^0𝑡−𝑥exp(𝜔𝜃)𝑑𝜃^{𝑡−𝑥−𝜃𝑥−𝑡+𝜃}𝑅(0,𝑡−𝑥;𝜁,𝜃)𝑑𝜁.(3.19) Equation (3.19) can be put in the more compact form𝑈^𝑜𝑠𝑥(𝜒)−^𝜒0𝐾^𝑠𝑥(𝜒,𝜁)𝑈^𝑜𝑠𝑥(𝜁)𝑑𝜁=𝐹^𝑠𝑥(𝜒),(3.20)where 𝜒=𝑥−𝑡∈[−𝜉_𝑜,0] and 𝐾^𝑠𝑥(𝜒,𝜁)=𝐼_𝑜𝜔√𝜒²−𝜁²𝜔−𝜔²𝜒2+𝜔²𝜒2𝐼²𝜔√𝐹_𝑠𝑥𝜒²−𝜁²,(𝜒)=−−𝑈𝛽_𝑜(−𝜒)+^0−𝜒𝑅(0,−𝜒;𝜁,0)𝜔−𝑅_𝜃(0,−𝜒;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁₂2^0−𝑥exp(𝜔𝜃)𝑑𝜃^{−𝑥−𝜃𝑥+𝜃}𝑅(0,−𝑥;𝜁,𝜃)𝑑𝜁.(3.21)

Figure 4 

The domain 𝐷𝐼𝐼.

Due to the regularity of the kernel 𝐾^𝑠𝑥(𝜒,𝜁) the function 𝑈^𝑜𝑠𝑥(𝜒) (which can be determined using the iterated kernels method, [13]) is a smooth function. Thus we extend 𝑈_𝑜(𝑥) as 𝑈_𝑜(𝑥)=𝑈^𝑜𝑠𝑥𝑈(𝑥),𝑥∈−𝜉_𝑜,0_𝑜,(𝑥),𝑥∈0,𝜉_𝑜,(3.22)and the solution 𝑈(𝑥,𝑡) in the domain 𝐷_𝐼𝐼 is given by 𝑈(𝑥,𝑡)=12𝑈^𝑜(𝑥+𝑡)−𝑈_𝑜(𝑡−𝑥)+12^{𝑥−𝑡𝑡−𝑥}𝑅(0,𝑡−𝑥;𝜁,0)𝜔−𝑅_𝜃+1(0,𝑡−𝑥;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁2^{𝑥+𝑡𝑥−𝑡}𝑅(𝑥,𝑡;𝜁,0)𝜔−𝑅_𝜃(𝑡,𝑥;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^0𝑡−𝑥+𝛽𝑒^𝜔𝜃𝑑𝜃^{𝑥−𝑡+𝜃𝑡−𝑥−𝜃}𝑅(0,𝑡−𝑥;𝜁,𝜃)𝑑𝜁₂2^𝑡0𝑒^𝜔𝜃𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅(𝑥,𝑡;𝜁,𝜃)𝑑𝜁.(3.23)

Remark 3.1. We notice that, considering the representation formulae (3.5) and (3.23), lim_𝑥→𝑡+𝑈(𝑥,𝑡)=lim_{𝑥→𝑡−}𝑈(𝑥,𝑡).(3.24) Moreover, taking the first derivatives (with respect to time 𝑡 and space 𝑥) of 𝑈(𝑥,𝑡) for the domains 𝐷_𝐼 and 𝐷_𝐼𝐼 it is easy to prove that, assuming the compatibility condition 𝑈_𝑜(0)=0, lim_𝑥→𝑡+𝑈_𝑥(𝑥,𝑡)=lim_{𝑥→𝑡−}𝑈_𝑥(𝑥,𝑡),𝑡∈0,𝜉^𝑜lim_𝑥→𝑡+2,𝑈_𝑡(𝑥,𝑡)=lim_{𝑥→𝑡−}𝑈_𝑡(𝑥,𝑡),𝑡∈0,𝜉^𝑜2,(3.25) where the derivatives in limits on the l.h.s. of (3.25) are evaluated using (3.5), while the ones on the r.h.s. using (3.23). This implies that the solution is 𝐶¹ across the characteristic 𝑥=𝑡, that is, the line that separates the domains 𝐷_𝐼 and 𝐷_𝐼𝐼.

We now write the representation formula for 𝑈(𝑥,𝑡) in the domain 𝐷_𝐼𝐼𝐼. We proceed as in [2] assuming that the velocity of the free boundary 𝑥=𝜉(𝑡) is less than the velocity of the characteristics (i.e., |𝜉̇|<1) and extending 𝑈_𝑜 to the domain [𝜉_𝑜,𝜉(𝜉_𝑜)+𝜉_𝑜] (see Figure 2) in a way such that 𝑈(𝜉,𝑡)=exp(𝜔𝑡)𝑉_𝑜 (i.e., imposing the free boundary condition (3.3)₍₅₎).

Given a point (𝑥,𝑡) in the domain 𝐷_𝐼𝐼𝐼 we define the point (𝜉^∗,𝑡^∗) as the intersection of the characteristic (with negative slope) passing from (𝑥,𝑡) and the free boundary 𝑥=𝜉(𝑡) (see Figure 5). It is easy to check that 𝜕𝜉𝑡^∗_∗+𝑡^∗=𝑥+𝑡,⟹𝑡^∗=𝑡^∗(𝑥,𝑡),(3.26)𝜕𝑡=𝜉̇(𝑡^∗1)+1,𝜕𝜕𝑡𝑥^∗=𝜉̇(𝑡^∗1)+1.(3.27)We consider once again the representation formula (3.5) and impose condition (3.3)₍₅₎, getting2𝑒^𝜔𝑡∗𝑉_𝑜=+𝑈𝛽_𝑜₂𝜉^∗−𝑡^∗+𝑈_𝑜𝜉^∗+𝑡^∗+^{𝜉∗+𝑡∗}_{𝜉∗−𝑡∗}𝑅𝜉^∗,𝑡^∗;𝜁,0𝜔−𝑅_𝜃𝜉^∗,𝑡^∗;𝜁,0𝑈_𝑜(𝜁)𝑑𝜁^𝑡∗₀𝑒^𝜔𝜃𝑑𝜃^{𝜉∗+𝑡∗𝜉−𝜃}_{∗−𝑡∗+𝜃}𝑅𝜉^∗,𝑡^∗;𝜁,𝜃𝑑𝜁.(3.28)From (3.28) we see that the extension 𝑈^𝑜𝑑𝑥 to the domain [𝜉_𝑜,𝜉(𝜉_𝑜)+𝜉_𝑜] is the solution of the following Volterra integral equation of second kind: 𝑈=^𝑜𝑑𝑥2𝜉𝑒^∗+_𝜔𝑡∗𝑡^∗+^{𝜉∗+𝑡∗}_𝜉𝑜𝑅𝜉^∗,𝑡^∗;𝜁,0𝜔−𝑅_𝜃𝜉^∗,𝑡^∗;𝜁,0𝑈^𝑜𝑑𝑥(𝜁)𝑑𝜁−𝛽₂𝑉_𝑜−𝑈_𝑜𝜉^∗−𝑡^∗−^𝜉𝑜_{𝜉∗−𝑡∗}𝑅𝜉^∗,𝑡^∗;𝜁,0𝜔−𝑅_𝜃𝜉^∗,𝑡^∗;𝜁,0𝑈_𝑜(𝜁)𝑑𝜁^𝑡∗₀𝑒^𝜔𝜃𝑑𝜃^{𝜉∗+𝑡∗𝜉−𝜃}_{∗−𝑡∗+𝜃}𝑅𝜉^∗,𝑡^∗;𝜁,𝜃𝑑𝜁.(3.29)Recalling (3.26) and proceeding as for the domain 𝐷_𝐼𝐼, the above can be rewritten as𝑈^𝑜𝑑𝑥(𝜒)+^𝜒𝜉_𝑜𝐾^𝑑𝑥(𝜒,𝜁)𝑈^𝑜𝑑𝑥(𝜁)𝑑𝜁=𝐹^𝑑𝑥(𝜒),(3.30)where 𝜒=𝑥+𝑡 and𝐾^𝑑𝑥(𝜒,𝜁)=𝐼_𝑜𝜔(𝑡^∗(𝜒))²−𝜁−𝜉^∗(𝜒)²𝜔+𝜔²𝑡^∗(𝜒)−𝜔₂2𝑡^∗(𝜒)2𝐼²𝜔(𝑡^∗(𝜒))²−𝜉^∗(𝜒)−𝜁²𝐹_𝑑𝑥,(𝜒)=−2𝑒𝛽^𝜔𝑡₂𝑉_𝑜−𝑈_𝑜(𝑥−𝑡)−^𝜉𝑜_𝑥−𝑡𝑅(𝑥,𝑡;𝜁,0)𝜔−𝑅_𝜃(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁^𝑡0𝑒^𝜔𝜃𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅(𝑥,𝑡;𝜁,𝜃)𝑑𝜁||||_{(𝑥=𝜉∗(𝜒),𝑡=𝑡∗(𝜒))}.(3.31) Once again the regularity of the kernel 𝐾^𝑑𝑥(𝜒,𝜁) ensures the regularity of the solution 𝑈^𝑜𝑑𝑥(𝜒). The function 𝑈_𝑜(𝑥) can thus be defined in the interval [−𝜉_𝑜,𝜉(𝜉_𝑜)+𝜉_𝑜] as𝑈_𝑜(𝑥)=⎧⎪⎪⎨⎪⎪⎩𝑈^𝑜𝑠𝑥𝑈(𝑥),𝑥∈−𝜉_𝑜,0_𝑜,𝑈_𝑜𝑑𝑥(𝑥),𝑥∈0,𝜉_𝑜,(𝑥),𝑥∈𝜉_𝑜,𝜉𝜉_𝑜+𝜉_𝑜.(3.32) The solution in the domain 𝐷_𝐼𝐼𝐼 is thus given by𝑈(𝑥,𝑡)=𝑒^𝜔𝑡∗𝑉_𝑜+12𝑈^𝑜(𝑥−𝑡)−𝑈_𝑜𝜉^∗−𝑡^∗+12^{𝑥+𝑡𝑥−𝑡}−1𝑃(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁2^{𝜉𝑥+𝑡∗}_−𝑡∗𝑃𝜉^∗,𝑡^∗;𝜁,0𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0−𝛽𝑒^𝜔𝜃𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅(𝑥,𝑡;𝜁,𝜃)𝑑𝜁₂2^𝑡∗₀𝑒^𝜔𝜃𝑑𝜃^{𝜉𝑥+𝑡−𝜃}_{∗−𝑡∗+𝜃}𝑅𝜉^∗,𝑡^∗;𝜁,𝜃𝑑𝜁,(3.33) where for simplicity of notation we have introduced𝑃(𝑥,𝑡;𝜁,𝜃)=𝑅(𝑥,𝑡;𝜁,𝜃)𝜔−𝑅_𝜃(𝑥,𝑡;𝜁,𝜃),(3.34)and where 𝑈_𝑜(𝑥) is given by (3.32). Therefore for any fixed 𝐶¹ function 𝜉(𝑡) with |𝜉̇|<1 we have that the solution to problem (3.3)_(1–5) is given by (3.5), (3.23), (3.33) with 𝑈_𝑜 defined by (3.32). At this point we make use of (3.3)₍₆₎ to determine the evolution equation of the free boundary 𝑥=𝜉(𝑡). We begin writing the derivatives 𝑈_𝑡(𝑥,𝑡) and 𝑈_𝑥(𝑥,𝑡). To this aim we exploit formula (3.33) since 𝑈_𝑡 and 𝑈_𝑥 have to be evaluated on 𝑥=𝜉(𝑡), which belongs to domain 𝐷_𝐼𝐼𝐼. Differentiating (3.33) with respect to 𝑥 we get𝑈_𝑥(𝑥,𝑡)=12𝑈^𝑜(𝑥−𝑡)−𝑈^𝑜𝜉^∗−𝑡^∗𝜉̇𝑡^∗−1𝜉̇(𝑡^∗)+1+𝜔𝑉^𝑜𝑒^𝜔𝑡∗+1𝜉̇^∗+12𝑃(𝑥,𝑡;𝑥+𝑡,0)𝑈^𝑜−1(𝑥+𝑡)−𝑃(𝑥,𝑡;𝑥−𝑡,0)𝑈_𝑜(𝑥−𝑡)2𝑃𝜉^∗,𝑡^∗;𝑥+𝑡,0𝑈_𝑜(𝑥+𝑡)−𝑃𝜉^∗,𝑡^∗;𝜉^∗−𝑡^∗,0𝑈_𝑜𝜉^∗−𝑡^∗𝜉̇𝑡^∗−1−1𝜉̇(𝑡^∗)+12^{𝜉𝑥+𝑡∗}_−𝑡∗𝑃_𝑥𝜉^∗,𝑡^∗;𝜁,0𝜉̇𝑡^∗+𝑃_𝑡𝜉^∗,𝑡^∗;𝜁,0𝑈^𝑜(𝜁)+1𝜉̇(𝑡^∗)+1𝑑𝜁2^{𝑥+𝑡𝑥−𝑡}𝑃_𝑥(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0−𝛽𝑒^𝜔𝜃𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅_𝑥(𝑥,𝑡;𝜁𝜃)𝑑𝜁₂2^𝑡∗₀𝑒^𝜔𝜃𝑅𝜉^∗,𝑡^∗;𝑥+𝑡−𝜃,𝜃−𝑅𝜉^∗,𝑡^∗;𝜉^∗−𝑡^∗+𝜃,𝜃𝜉̇𝑡^∗−1−𝛽𝜉̇(𝑡^∗)+1𝑑𝜃₂2^𝑡∗₀𝑒^𝜔𝜃^{𝜉𝑥+𝑡−𝜃}_{∗−𝑡∗+𝜃}𝑅_𝑥𝜉^∗,𝑡^∗;𝜁,𝜃𝜉̇𝑡^∗−𝑅_𝑡𝜉^∗,𝑡^∗;𝜁,𝜃𝑑𝜁𝜉̇(𝑡^∗)+1,(3.35)while, differentiating (3.33) with respect to 𝑡, we obtain 𝑈_𝑡(𝑥,𝑡)=12−𝑈^𝑜(𝑥−𝑡)−𝑈^𝑜𝜉^∗−𝑡^∗𝜉̇𝑡^∗−1𝜉̇(𝑡^∗)+1+𝜔𝑉^𝑜𝑒^𝜔𝑡∗+1𝜉̇^∗+12𝑃(𝑥,𝑡;𝑥+𝑡,0)𝑈^𝑜−1(𝑥+𝑡)+𝑃(𝑥,𝑡;𝑥−𝑡,0)𝑈_𝑜(𝑥−𝑡)2𝑃𝜉^∗,𝑡^∗;𝑥+𝑡,0𝑈_𝑜(𝑥+𝑡)−𝑃𝜉^∗,𝑡^∗;𝜉^∗−𝑡^∗,0𝑈_𝑜𝜉^∗−𝑡^∗𝜉̇𝑡^∗−1−1𝜉̇(𝑡^∗)+12^{𝜉𝑥+𝑡∗}_−𝑡∗𝑃_𝑥𝜉^∗,𝑡^∗;𝜁,0𝜉̇𝑡^∗+𝑃_𝑡𝜉^∗,𝑡^∗;𝜁,0𝑈^𝑜(𝜁)𝜉̇(𝑡^∗)+1𝑑𝜁+𝛽²^𝑡0+1𝑒^𝜔𝜃𝑑𝜃2^{𝑥+𝑡𝑥−𝑡}𝑃_𝑡(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0−𝛽𝑒^𝜔𝜃𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅_𝑡(𝑥,𝑡;𝜁𝜃)𝑑𝜁₂2^𝑡∗₀𝑒^𝜔𝜃𝑅𝜉^∗,𝑡^∗;𝑥+𝑡−𝜃,𝜃−𝑅𝜉^∗,𝑡^∗;𝜉^∗−𝑡^∗+𝜃,𝜃𝜉̇𝑡^∗−1−𝛽𝜉̇(𝑡^∗)+1𝑑𝜃₂2^𝑡∗₀𝑒^𝜔𝜃^{𝜉𝑥+𝑡−𝜃}_{∗−𝑡∗+𝜃}𝑅_𝑥𝜉^∗,𝑡^∗;𝜁,𝜃𝜉̇𝑡^∗−𝑅_𝑡𝜉^∗,𝑡^∗;𝜁,𝜃𝑑𝜁𝜉̇(𝑡^∗)+1.(3.36) Notice that 𝑃(𝑥,𝑡;𝑥+𝑡,0)=𝑃(𝑥,𝑡;𝑥−𝑡,0)=𝜔+𝜔²𝑡𝑅(𝑥,𝑡;𝑥+𝑡−𝜃,𝜃)−𝑅(𝑥,𝑡;𝑥−𝑡+𝜃,𝜃)=20,.(3.37) Now we evaluate (3.35) and (3.36) on the free boundary 𝑥=𝜉(𝑡), that is,𝑈_𝑥(𝜉,𝑡)=𝑈^𝑜(𝜉−𝑡)𝜉̇+1−𝜔+𝜔²𝑡2𝑈^𝑜(𝜉−𝑡)1𝜉̇+1+𝜔𝑉^𝑜𝑒^𝜔𝑡+1𝜉̇+12^{𝜉+𝑡𝜉−𝑡}𝑃_𝑥(𝜉,𝑡;𝜁,0)−𝑃_𝑡(𝜉,𝑡;𝜁,0)𝑈^𝑜(𝜁)𝑑𝜁𝜉̇+1−𝛽²^𝑡0𝑒^𝜔𝜃+𝛽𝜉̇+1𝑑𝜃₂2^𝑡0𝑒^𝜔𝜃^{𝜉+𝑡−𝜃𝜉−𝑡+𝜃}𝑅_𝑥(𝜉,𝑡;𝜁,𝜃)−𝑅_𝑡(𝜉,𝑡;𝜁,𝜃)𝑑𝜁𝑈𝜉̇+1_𝑡,(𝜉,𝑡)=−𝑈^𝑜(𝜉−𝑡)𝜉̇𝜉̇+1+𝜔+𝜔²𝑡2𝑈^𝑜(𝜉−𝑡)𝜉̇𝜉̇+1+𝜔𝑉^𝑜𝑒^𝜔𝑡+1𝜉̇+12^{𝜉+𝑡𝜉−𝑡}𝑃_𝑡(𝜉,𝑡;𝜁,0)−𝑃_𝑥(𝜉,𝑡;𝜁,0)𝜉̇𝑈^𝑜(𝜁)𝑑𝜁𝜉̇+1+𝛽²^𝑡0𝑒^𝜔𝜃𝜉̇+𝛽𝜉̇+1𝑑𝜃₂2^𝑡0𝑒^𝜔𝜃^{𝜉+𝑡−𝜃𝜉−𝑡+𝜃}𝑅_𝑡(𝜉,𝑡;𝜁,𝜃)−𝑅_𝑥(𝜉,𝑡;𝜁,𝜃)𝜉̇𝑑𝜁𝜉̇+1.(3.38) At this point we insert (3.38), (3.3)₍₅₎ in (3.3)₍₆₎, obtaining 𝜉̇−1𝜔+𝜔²𝑡2𝑈^𝑜(𝜉−𝑡)−𝑈^𝑜(𝜉−𝑡)+𝛽²𝜔𝑒^𝜔𝑡−1+12^{𝜉+𝑡𝜉−𝑡}𝑃_𝑡+𝛽(𝜉,𝑡;𝜁,0)−𝑃_𝑥(𝜉,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁₂2^𝑡0𝑒^𝜔𝜃𝑑𝜃^{𝜉+𝑡−𝜃𝜉−𝑡+𝜃}𝑅_𝑡(𝜉,𝑡;𝜁,𝜃)−𝑅_𝑥(𝜉,𝑡;𝜁,𝜃)𝑑𝜁=𝑒^𝜔𝑡𝛽²𝙱𝚗,(3.39) which is a nonlinear integrodifferential equation of the first order and where 𝑈_𝑜 is defined by (3.32). Equation (3.39) is the free boundary equation which, as we mentioned in the introduction, does no longer depend on the velocity field 𝑈(𝑥,𝑡).

Figure 5 

The domain 𝐷𝐼𝐼𝐼.

Next we remark that (3.39) can be further simplified. Indeed, recalling (3.10) and (3.12), 𝑅_𝑥(𝑥,𝑡;𝜁,𝜃)=−𝑅_𝜁(𝑥,𝑡;𝜁,𝜃),𝑅_𝜃𝑥(𝑥,𝑡;𝜁,𝜃)=−𝑅_𝜃𝜁(𝑥,𝑡;𝜁,𝜃),(3.40)so that, on (𝜉(𝑡),𝑡;𝜁,𝜃), we have ^{𝜉+𝑡−𝜃𝜉−𝑡+𝜃}𝑅_𝑥𝑑𝜁=−^{𝜉+𝑡−𝜃𝜉−𝑡+𝜃}𝑅_𝜁𝑑𝜁=𝑅(𝜉,𝑡;𝜉+𝑡−𝜃,𝜃)−𝑅(𝜉,𝑡;𝜉−𝑡+𝜃,𝜃)=0,(3.41)while, on (𝜉(𝑡),𝑡;𝜁,𝜃),^{𝜉+𝑡𝜉−𝑡}𝑃_𝑥𝑑=𝜁=𝜔[−𝑅(𝜉,^{𝜉+𝑡𝜉−𝑡}𝑡𝑅;𝜉_𝜁𝜔−𝑑𝑡𝜁,0+)−𝑅^{𝜉+𝑡𝜉−𝑡}(𝑅𝜉,_𝜃𝜁𝑡;𝑑𝜉𝜁+𝑡,0)]+𝑅_𝜃(𝜉,𝑡;𝜉+𝑡,0)−𝑅_𝜃(𝜉,𝑡;𝜉−𝑡,0)=0.(3.42)Hence (3.39) reduces to 𝜉̇−1𝜔+𝜔²𝑡2𝑈^𝑜(𝜉−𝑡)−𝑈^𝑜(𝜉−𝑡)+𝛽²𝜔𝑒^𝜔𝑡+1−12^{𝜉+𝑡𝜉−𝑡}𝑃_𝑡(𝜉,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0𝑒^𝜔𝜃𝑑𝜃^{𝜉+𝑡−𝜃𝜉−𝑡+𝜃}𝑅_𝑡(𝜉,𝑡;𝜁,𝜃)𝑑𝜁=𝑒^𝜔𝑡𝛽²𝙱𝚗.(3.43)

Remark 3.2. The function 𝑈(𝑥,𝑡) is continuous across the characteristic 𝑥+𝑡=𝜉_𝑜. Indeed lim_{𝑥+𝑡→𝜉+𝑜}𝑈(𝑥,𝑡)=lim_{𝑥+𝑡→𝜉−𝑜}𝑈(𝑥,𝑡),(3.44)where the limit lim_{𝑥+𝑡→𝜉}^+𝑜 is evaluated using (3.33) and the limit lim_{𝑥+𝑡→𝜉}^−𝑜 using (3.5) or (3.23). If we evaluate the derivatives 𝑈_𝑥 and 𝑈_𝑡 on the characteristic 𝑥+𝑡=𝜉_𝑜 we get two different results depending on whether we are evaluating such derivatives in 𝐷_𝐼 or 𝐷_𝐼𝐼𝐼. We can prove that lim_{𝑥+𝑡→𝜉−𝑜}𝑈_𝑥(𝑥,𝑡)=12𝑈^𝑜+1(𝑥−𝑡)+𝑈^𝑜𝜉_𝑜+𝑃𝑥,𝑡;𝜉_𝑜,0𝑈_𝑜𝜉_𝑜−𝑃(𝑥,𝑡;𝑥−𝑡,0)𝑈_𝑜(𝑥−𝑡)2^𝜉𝑜_𝑥−𝑡𝑃_𝑥(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0lim_{𝑥+𝑡→𝜉+𝑜}𝑒^𝜔𝜃𝑑𝜃^{𝜉𝑜−𝜃𝑥−𝑡+𝜃}𝑅_𝑥(𝑥,𝑡;𝜁,𝜃)𝑑𝜁,(3.45)𝑈_𝑥(𝑥,𝑡)=12𝑈^𝑜(𝑥−𝑡)−𝑈^𝑜𝜉_𝑜𝜉̇^𝑜−1𝜉̇_𝑜+1+𝑃𝑥,𝑡;𝜉^𝑜+𝜔𝑉_𝑜,0𝑈_𝑜𝜉_𝑜−𝑃(𝑥,𝑡;𝑥−𝑡,0)𝑈_𝑜(𝑥−𝑡)𝜉̇_𝑜+1+12^𝜉𝑜_𝑥−𝑡𝑃_𝑥(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0lim_{𝑥+𝑡→𝜉−𝑜}𝑒^𝜔𝜃𝑑𝜃^{𝜉𝑜−𝜃𝑥−𝑡+𝜃}𝑅_𝑥(𝑥,𝑡;𝜁,𝜃)𝑑𝜁,(3.46)𝑈_𝑡(𝑥,𝑡)=12𝑈^𝑜𝜉_𝑜−𝑈^𝑜+𝛽₂(𝑥−𝑡)+𝑃𝑥,𝑡;𝜉_𝑜,0𝑈_𝑜𝜉_𝑜+𝑃(𝑥,𝑡;𝑥−𝑡,0)𝑈_𝑜(𝑥−𝑡)^𝑡0𝑒^𝜔𝜃𝑑𝜃+12^𝜉𝑜_𝑥−𝑡𝑃_𝑡(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²2^𝑡0lim_{𝑥+𝑡→𝜉+𝑜}𝑒^𝜔𝜃𝑑𝜃^{𝜉𝑜−𝜃𝑥−𝑡+𝜃}𝑅_𝑡(𝑥,𝑡;𝜁,𝜃)𝑑𝜁,(3.47)𝑈_𝑡(𝑥,𝑡)=12−𝑈^𝑜(𝑥−𝑡)−𝑈^𝑜𝜉_𝑜𝜉̇^𝑜−1𝜉̇_𝑜+1+𝑃𝑥,𝑡;𝜉^𝑜+𝜔𝑉_𝑜,0𝑈_𝑜𝜉_𝑜−𝑃(𝑥,𝑡;𝑥−𝑡,0)𝑈_𝑜(𝑥−𝑡)𝜉̇_𝑜+1+12^𝜉𝑜_𝑥−𝑡𝑃_𝑡(𝑥,𝑡;𝜁,0)𝑈_𝑜(𝜁)𝑑𝜁+𝛽²^𝑡0+𝛽𝑒^𝜔𝜃𝑑𝜃₂2^𝑡0𝑒^𝜔𝜃𝑑𝜃^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑅_𝑥(𝑥,𝑡;𝜁,𝜃)𝑑𝜁,(3.48) where 𝜉̇_𝑜=𝜉̇(0). It is easy to check that, imposing that (3.45) equals (3.46) and that (3.47) equals (3.48), we get the following condition: 𝜉̇_𝑜𝑈^𝑜𝜉_𝑜=𝜔𝑈_𝑜𝜉_𝑜=𝜔𝑉_𝑜,(3.49) which is the condition that must be fulfilled if we want the first derivatives of 𝑈(𝑥,𝑡) to be continuous across the characteristic 𝑥+𝑡=𝜉_𝑜.

If we assume that the free boundary equation (3.3)₍₆₎ holds up to 𝑡=0 we get𝑈^𝑜𝜉_𝑜=𝛽²𝙱𝚗.(3.50) Moreover, from (3.43) we have that, when 𝑡=0,𝜉̇_𝑜−1𝜔𝑈_𝑜𝜉_𝑜−𝑈^𝑜𝜉_𝑜=𝛽²𝙱𝚗.(3.51) We can therefore prove the following.

Theorem 3.3. If one assumes that compatibility condition 𝑈_𝑜(𝜉_𝑜)=𝑉_𝑜 and hypotheses (3.49), (3.50), (3.51) hold, then necessarily either 𝑉_𝑜=0 or 𝜔=0 and problem (3.3) admits a unique local 𝐶¹ solution (𝑈,𝜉), such that 𝜉̇(0)=0. If one does not assume hypothesis (3.49) (meaning that the first derivatives of 𝑈 are not continuous along the characteristic 𝑥+𝑡=𝜉_𝑜), then problem (3.3) admits a unique local solution (𝑈,𝜉), such that −1<𝜉̇(𝑡)<0 if and only if 𝑉_𝑜<𝛽²𝙱𝚗2𝜔.(3.52)

Proof. If we suppose that (3.49), (3.50), (3.51) hold then we have 𝜉̇_𝑜−1𝑈^𝑜𝜉_𝑜𝜉̇_𝑜−𝑈^𝑜𝜉_𝑜=𝛽²𝙱𝚗,⟹𝜉̇_𝑜=0or𝜉̇_𝑜=2.(3.53) The initial velocity 𝜉̇_𝑜=2 is not physically acceptable since existence of a solution requires that |𝜉̇|<1. Therefore 𝜉̇_𝑜=0 and, recalling (3.49), we have either 𝑉_𝑜=0 or 𝜔=0, since 𝑈^𝑜(𝜉_𝑜)=𝛽²𝙱𝚗≠±∞. If, on the other hand, we suppose that condition (3.49) does not hold, but we assume (3.52), then it is easy to show that −1<𝜔𝑉^𝑜𝜔𝑉_𝑜−𝛽²𝙱𝚗=1+𝛽²𝙱𝚗𝜔𝑉_𝑜−𝛽²𝙱𝚗=𝜉̇^𝑜<0,(3.54) so that for a sufficiently small time 𝑡>0 there exists a unique solution with −1<𝜉̇<0. The existence of such a solution can be proved using classical tools like iterated kernels method (see [13]).

Remark 3.4. Let us consider the limit case in which 𝜔=0 and 𝛽²=0. In this particular situation the Riemann's function 𝑅(𝑥,𝑡;𝜁,𝜃)≡1 and the solution 𝑈(𝑥,𝑡) is given by 𝑈(𝑥,𝑡)=⎧⎪⎪⎨⎪⎪⎩12𝑈^𝑜1(𝑥+𝑡)+𝑈_𝑜(𝑥−𝑡),in𝐷_𝐼,2𝑈^𝑜1(𝑥+𝑡)−𝑈_𝑜(𝑡−𝑥),in𝐷_𝐼𝐼,2𝑈^𝑜(𝑥−𝑡)−𝑈_𝑜(𝜉−𝑡)+𝑉_𝑜,in𝐷_𝐼𝐼𝐼,(3.55) and the free boundary equation is the characteristic with positive slope passing through (𝜉_𝑜,0), that is, 𝜉̇−1𝑈^𝑜(𝜉−𝑡)=0⟹𝑈_𝑜(𝜉−𝑡)=𝑈_𝑜𝜉_𝑜,(3.56) namely, 𝜉(𝑡)=𝜉_𝑜+𝑡⟹𝜉̇(𝑡)=1.(3.57) So, setting 𝑡_𝑜=1−𝜉_𝑜, for 𝑡≥𝑡_𝑜, the region with uniform velocity (the inner core) has disappeared. For 𝑡≥𝑡_𝑜, the solution 𝑈(𝑥,𝑡) is thus found solving 𝑈_𝑥𝑥=𝑈_𝑡𝑡,0⩽𝑥⩽1,𝑡⩾1−𝜉_𝑜𝑈𝑈𝑥,1−𝜉_𝑜=𝑈^∗𝑜(𝑥),0⩽𝑥⩽1,_𝑡𝑥,1𝑈−(0𝜉,_𝑜𝑡)==𝑈0^∗1𝑡(𝑥⩾),1−0𝜉⩽_𝑜𝑥⩽1,𝑈(1,𝑡)=𝑉_𝑜,𝑡⩾1−𝜉_𝑜,(3.58) where 𝑈^∗𝑜(𝑥) and 𝑈^∗1(𝑥) are determined evaluating (3.55) at time 𝑡=𝑡_𝑜. To solve problem (3.58) we introduce the new variable 𝑊(𝑥,𝑡)=𝑈(𝑥,𝑡)−𝑥𝑉_𝑜(3.59) and rescale time with 𝜃=𝑡−𝑡_𝑜.(3.60) Problem (3.58) becomes 𝑊(𝑊𝑥,0_𝑥𝑥)==𝑊𝑈_{𝜃𝜃∗𝑜},0⩽𝑥⩽1,𝜃⩾0,𝑊(𝑥)−𝑉_𝑜(𝑥),0⩽𝑥⩽1,_𝜃(𝑥,0)=𝑊𝑊𝑊((01₁,,(𝜃𝜃𝑥)))===00𝑈,,^∗1(𝜃𝜃𝑥)⩾⩾,00,,0⩽𝑥⩽1,(3.61) whose solution is [11] 𝑊(𝑥,𝜃)=^∞_𝑖=1𝐴_𝑛cos(𝜋𝑛𝜃)+𝐵_𝑛sin(𝜋𝑛𝜃)sin(𝜋𝑛𝑥),(3.62) where 𝐴_𝑛=2^𝜃0𝑊_𝑜(𝑧)sin(𝜋𝑛𝑧)𝑑𝑧,𝐵_𝑛=2𝜋𝑛^𝜃0𝑈(𝑥,𝑡)=𝑊𝑥,𝑡−1+𝜉_𝑜𝑊₁(𝑧)sin(𝜋𝑛𝑧)𝑑𝑧,+𝑥𝑉_𝑜.(3.63)

4. Asymptotic Expansion

In this section we look for a solution to problem (3.3)₍₁₎ in the following form: 𝑈(𝑥,𝑡)=^∞_𝑖=0𝜔^𝑖𝑈^(𝑖)(𝑥,𝑡).(4.1) This allows to obtain a sequence of problems for each 𝑖=0,1,2… with the free boundary being given by 𝜉^(𝑖)(𝑡). (We remark that the sequence {𝜉^(𝑖)(𝑡)} is not, in general, an asymptotic sequence.) Such an analysis is motivated by the fact that, in practical cases (asphalt and bitumen), 𝜔=𝑂(10⁻¹) (see (2.3)). Hence, it makes sense to look for a “perturbative” approach for the system (3.3).

We do not discuss the issue of the convergence of series (4.1) and of the sequence {𝜉^(𝑖)(𝑡)}, which is beyond the scope of the present paper. We limit ourselves to a formal derivation of the free boundary problems that can be obtained plugging (4.1)₍₁₎ into (3.3): ^∞_𝑖=0𝜔^𝑖𝑈^{(𝑖)𝑥𝑥}(𝑥,𝑡)−𝜔^𝑖𝑈^{(𝑖)𝑡𝑡}(𝑥,𝑡)+𝜔^𝑖+2𝑈^(𝑖)(𝑥,𝑡)=−𝛽^2∞_𝑖=0(𝜔𝑡)^𝑖𝑖!.(4.2) Hence, for each 𝑖=0,1,2,…, we have 𝑖𝑖==10,,𝑈𝑈^{(𝑜)𝑥𝑥}_(1)𝑥𝑥(𝑥,𝑡)−𝑈^{(𝑜)𝑡𝑡}(𝑥,𝑡)=−𝛽²,𝑖=2,𝑈_(2)𝑥𝑥(𝑥,𝑡)−𝑈^(1)𝑡𝑡(𝑥,𝑡)=−𝛽²𝑡,(𝑥,𝑡)−𝑈^(2)𝑡𝑡(𝑥,𝑡)=−𝛽²𝑡²2!−𝑈^(𝑜)𝑖>2,⋮𝑈_{(𝑖)𝑥𝑥}(𝑥,𝑡),(𝑥,𝑡)−𝑈^{(𝑖)𝑡𝑡}(𝑥,𝑡)=−𝛽²𝑡^𝑖𝑖!−𝑈^(𝑖−2)(𝑥,𝑡)(4.3)

and the following free boundary problems 𝑖=0,⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩𝑈^{(𝑜)𝑥𝑥}(𝑥,𝑡)−𝑈^{(𝑜)𝑡𝑡}(𝑥,𝑡)=−𝛽²𝑈𝑈^(𝑜)(𝑥,0)=𝑈_𝑜(𝑥),_𝑡(𝑜)𝑈(𝑥,0)=0,_(𝑜)𝑈(0,𝑡)=0,_(𝑜)𝜉^(𝑜),𝑡=𝑉_𝑜𝑈𝜉^𝑥(𝑜)𝜉^(𝑜),𝑡+𝜉̇^(𝑜)𝑈^𝑡(𝑜)𝜉^(𝑜),𝑡=𝛽²𝙱𝚗,_(𝑜)𝑖=1,⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩𝑈_(1)𝑥𝑥(0)=𝜉_𝑜,(4.4)(𝑥,𝑡)−𝑈^(1)𝑡𝑡(𝑥,𝑡)=−𝑡𝛽²𝑈𝑈⁽¹⁾(𝑥,0)=0,_𝑡(1)𝑈(𝑥,0)=𝑈_𝑜(𝑥),₍₁₎𝑈(0,𝑡)=0,₍₁₎𝑈_𝑥(1)𝜉⁽¹⁾,𝑡=𝑉_𝑜𝑡𝜉𝜉⁽¹⁾,𝑡+𝜉̇⁽¹⁾𝑈^𝑡(1)𝜉⁽¹⁾,𝑡−𝜉̇⁽¹⁾𝑉_𝑜=𝑡𝛽²𝙱𝚗,₍₁₎𝑖≥2,⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩𝑈_{(𝑖)𝑥𝑥}(0)=𝜉_𝑜,(4.5)(𝑥,𝑡)−𝑈^{(𝑖)𝑡𝑡}(𝑥,𝑡)=−𝑡^𝑖𝑖!𝛽²𝑈−𝑈^(𝑖−2)(𝑥,𝑡)_(𝑖)𝑈(𝑥,0)=0,_𝑡(𝑖)𝑈(𝑥,0)=0,_(𝑖)𝑈(0,𝑡)=0,_(𝑖)𝜉^(𝑖),𝑡=𝑖𝑡!^𝑖𝑉^𝑜𝑈^𝑥(𝑖)𝜉^(𝑖),𝑡+𝜉̇^(𝑖)𝑈^𝑡(𝑖)𝜉^(𝑖),𝑡−𝜉̇^(𝑖)𝑉_𝑜𝑡^𝑖−1(𝑖−1)!=𝑖𝑡!^𝑖𝛽²𝜉𝙱𝚗,_(𝑖)(0)=𝜉_𝑜.(4.6) We immediately remark that, in each problem, the governing equation is no longer a telegrapher's equation, but a nonhomogeneous wave equation. Hence, using classical d'Alembert formula, we can write the representation formula for each domain 𝐷_𝐼, 𝐷_𝐼𝐼, 𝐷_𝐼𝐼𝐼 and for each order of approximation 𝑖=0,1,.... In particular, in 𝐷_𝐼 we have 𝐷_𝐼⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑈^(𝑜)(𝑥,𝑡)=12𝑈^𝑜(𝑥+𝑡)+𝑈_𝑜(𝑥−𝑡)+𝛽²𝑡²𝑈2!₍₁₎,(𝑥,𝑡)=12^{𝑥+𝑡𝑥−𝑡}𝑈_𝑜(𝜁)𝑑𝜁+𝛽²𝑡³⋮𝑈3!_(𝑖),(𝑥,𝑡)=𝛽²2^𝑡0^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑈^(𝑖−2)(𝜁,𝜃)𝑑𝜁𝑑𝜃+𝛽²𝑡^𝑖+2(𝑖+2)!,𝑖⩾2,(4.7) while in 𝐷_𝐼𝐼𝐷_𝐼𝐼⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩𝑈^(𝑜)(𝑥,𝑡)=12𝑈^𝑜(𝑥+𝑡)−𝑈_𝑜(𝑡−𝑥)+𝛽²𝑡²2!−𝛽²(𝑥−𝑡)²𝑈2!₍₁₎,(𝑥,𝑡)=12^{𝑥+𝑡𝑡−𝑥}𝑈_𝑜(𝜁)𝑑𝜁+𝛽²𝑡³3!−𝛽²(𝑡−𝑥)³⋮𝑈3!_(𝑖),(𝑥,𝑡)=𝛽²2^𝑡0^{𝑥−𝑡+𝜃𝑡−𝑥+𝜃}𝑈^(𝑖−2)(𝜁,𝜃)𝑑𝜁𝑑𝜃+𝛽²2^𝑡0^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}+𝛽𝑈^(𝑖−2)(𝜁,𝜃)𝑑𝜁𝑑𝜃₂𝑡^𝑖+2(𝑖+2)!−𝛽²(𝑡−𝑥)^𝑖+2(𝑖+2)!,𝑖⩾2,(4.8) and in 𝐷_𝐼𝐼𝐼𝐷_𝐼𝐼𝐼⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩𝑈^(𝑜)(𝑥,𝑡)=𝑉_𝑜+12𝑈^𝑜(𝑥−𝑡)−𝑈_𝑜𝜉^(𝑜)∗−𝑡^∗+𝛽²𝑡²2!−𝛽²𝑡^∗2𝑈2!₍₁₎,(𝑥,𝑡)=𝑉_𝑜𝑡^∗+12^{𝜉(1)∗−𝑡∗}_𝑥−𝑡𝑈_𝑜(𝜁)𝑑𝜁+𝛽²𝑡³3!−𝛽²𝑡^∗3⋮𝑈3!_(𝑖),(𝑥,𝑡)=𝛽²2^𝑡0^{𝑥+𝑡−𝜃𝑥−𝑡+𝜃}𝑈^(𝑗−2)(𝜁,𝜃)𝑑𝜁𝑑𝜃−𝛽²2^𝑡∗₀^{𝜉𝑥+𝑡−𝜃}_{(𝑖)∗−𝑡∗+𝜃}+𝑉_𝑜𝑈^(𝑖−2)(𝜁,𝜃)𝑑𝜁𝑑𝜃𝑡^∗𝑖𝑖!+𝛽²𝑡^𝑖+2(𝑖+2)!−𝛽²𝑡^∗^𝑖+2(𝑖+2)!.𝑖≥2.(4.9) Proceeding as in Section 3 we can show that the evolution equations of the free boundary 𝑥=𝜉^(𝑖)(𝑡) at each step are given by𝑖=1𝑖,=0,𝜉̇₍₁₎𝜉̇⁽⁰⁾−1𝛽²𝑡−𝑈^𝑜𝜉^(𝑜)−𝑡=𝛽²𝙱𝚗,(4.10)−1𝑈_𝑜𝜉⁽¹⁾−𝑡−𝑉_𝑜+𝛽²𝑡²2!=𝑡𝛽²𝑖⩾2,𝜉̇_(𝑖)𝙱𝚗,(4.11)−1𝛽²(𝑖+2)𝑡^𝑖+1(𝑖+1)!−𝑉^𝑜𝑡^𝑖−1(𝑖−1)!+𝛽²^𝑡0𝑈^(𝑖−2)𝜉^(𝑖)−𝑡+𝜃,𝜃𝑑𝜃=𝑡^𝑖𝑖!𝛽²𝙱𝚗.(4.12) At the zero order, assuming the compatibility condition 𝑈^(𝜉_𝑜)=𝛽²Bn (see problem (4.4)), we have 1−𝜉̇^𝑜(𝑜)𝑈^𝑜𝜉_𝑜=𝛽²𝙱𝚗,⟹𝜉̇^𝑜(𝑜)=0.(4.13)

At the first order (see problem (4.5)), we assume that the compatibility condition of second order holds in the corner (𝜉_𝑜,0). This means that we can differentiate the free boundary equation (4.5)₍₆₎ and take the limit for 𝑡→0. We have 𝑈^(1)𝑥𝑥𝜉⁽¹⁾,𝑡𝜉̇⁽¹⁾+𝑈^(1)𝑥𝑡𝜉⁽¹⁾,𝑡+𝜉̈⁽¹⁾𝑈^𝑡(1)𝜉⁽¹⁾,𝑡+𝜉̇⁽¹⁾²𝑈^(1)𝑥𝑡𝜉⁽¹⁾,𝑡−𝜉̈⁽¹⁾𝑉_𝑜=𝛽²𝙱𝚗,(4.14)

which, when 𝑡→0, reduces to𝑈^′_𝑜𝜉_𝑜1+𝜉̇⁽¹⁾²_𝑜=𝛽²𝙱𝚗,⟹𝜉̇^𝑜(1)=0.(4.15)

For the generic 𝑖th order (see problem (4.6)), we assume that the compatibility conditions in the corner (𝜉_𝑜,0) hold up to order (𝑖−1). Therefore we can take the (𝑖−1)th derivative of (4.6)₍₆₎, obtaining 𝑑^𝑖−1𝑑𝑡^𝑖−1𝑈^𝑥(𝑖)𝜉^(𝑖),𝑡+𝑑^𝑖−1𝑑𝑡^𝑖−1𝜉̇^(𝑖)𝑈^𝑡(𝑖)𝜉^(𝑖),𝑡+𝜉̇^(𝑖)𝑑^𝑖−1𝑑𝑡^𝑖−1𝑈^𝑡(𝑖)−𝜉̇𝜉^(𝑖),𝑡_(𝑖)𝑉_𝑜−𝑉_𝑜𝑡^𝑖−1(𝑖−1)!𝑑𝑑𝑡^{𝑖−1𝑖−1}𝜉̇^(𝑖)=𝑡𝛽²𝙱𝚗,(4.16)

which, in the limit 𝑡→0, reduces to−𝜉̇^𝑜(𝑖)𝑉_𝑜=0⟹𝜉̇^𝑜(𝑖)=0.(4.17) We therefore conclude that, assuming enough regularity for each problem 𝑖=0,1,2,…, (4.10)–(4.12) posses a unique local solution with 𝜉^(𝑖)(0)=𝜉_𝑜 and 𝜉̇^(𝑖)(0)=0.

Before proceeding further we suppose that 𝑈(𝑥) has the following properties:

(H1)  𝑈_𝑜(𝑥)∈𝐶^∞([0,𝜉_𝑜]),

(H2) 0<𝑈_𝑜(𝑥)<𝑉_𝑜forall𝑥∈(0,𝜉_𝑜), 𝑈_𝑜(0)=0,𝑈_𝑜(𝜉_𝑜)=𝑉_𝑜,

(H3) 𝑈^𝑜(𝑥)>0forall𝑥∈[0,𝜉_𝑜] and 𝑈^(𝜉_𝑜)=𝛽²𝙱𝚗,

(H4) 𝑈_𝑜(𝑥) satisfies all the compatibility conditions up to any order in the corner (𝜉_𝑜,0).

4.1. Zero-Order Approximation

We introduce the new variable 𝜙^(𝑜)=𝜉^(𝑜)−𝑡, so that (4.10) can be rewritten as𝜙̇^(𝑜)𝛽²𝑡−𝑈^𝑜𝜙^(𝑜)=𝛽²𝙱𝚗,(4.18)

with 𝜙^(𝑜)(0)=𝜉_𝑜. Then we look for the solution 𝑡=𝑡(𝜙^(𝑜)) which fulfills the following Cauchy problem:𝛽²𝙱𝚗𝑑𝑡𝑑𝜙^(𝑜)𝑡𝜉=𝛽²𝑡−𝑈^𝑜𝜙^(𝑜)_𝑜,=0,(4.19) that is,𝑡𝜙^(𝑜)=−1𝛽²𝙱𝚗^𝜙(𝑜)_𝜉𝑜𝑈^𝑜(𝑧)exp𝜙^(𝑜)−𝑧𝙱𝚗𝑑𝑧.(4.20) Recalling that |𝜉̇^(𝑜)|<1, that is,|||𝜙̇^(𝑜)+1|||<1,⟹−2<𝜙̇^(𝑜)<0,⟹𝑑𝑡𝑑𝜙^(𝑜)<−12,(4.21) from (4.19)₍₁₎ we realize that (4.21) is fulfilled if𝜉_𝑜+𝙱𝚗2<𝛽1²inf_{𝑧∈0,𝜉𝑜}𝑈^𝑜(𝑧).(4.22) Therefore, under hypothesis (4.22), local existence of a classical solution is guaranteed. Such a solution is given by 𝜉^(𝑜)(𝑡)=𝜙^(𝑜)(𝑡)+𝑡, where 𝜙^(𝑜)(𝑡) is determined inverting (4.20).

4.2. First-Order Approximation

We now have to solve the problem𝜉̇⁽¹⁾−1𝑈_𝑜𝜉⁽¹⁾−𝑡−𝑉_𝑜+𝛽²𝑡²2!=𝑡𝛽²𝙱𝚗,(4.23) with 𝜉⁽¹⁾(0)=𝜉_𝑜. Proceeding as in Section 4.1 we introduce the new variable 𝜙⁽¹⁾=𝜉⁽¹⁾−𝑡, so that (4.23) becomes𝜙̇⁽¹⁾𝑈^𝑜𝜙⁽¹⁾−𝑉_𝑜+𝛽²𝑡²2!=𝑡𝛽²𝙱𝚗,(4.24)

and we have to solve the following Cauchy problem:𝑑𝑡𝑑𝜙⁽¹⁾=1𝑡𝛽²𝙱𝚗𝑈^𝑜𝜙⁽¹⁾−𝑉_𝑜+𝛽²𝑡²𝑡𝜉2!_𝑜,=0.(4.25) We notice that (4.25)₍₁₎ is a Bernoulli equation. Therefore, setting 𝑤=𝑡², problem (4.25) becomes𝑑𝑤𝑑𝜙⁽¹⁾=𝑤𝙱𝚗+2𝑈^𝑜𝜙⁽¹⁾−𝑉_𝑜𝑤𝜉𝛽²𝙱𝚗_𝑜,=0.(4.26) whose solution is given by𝑡²𝜙⁽¹⁾=^𝜙(1)_𝜉𝑜2exp𝜙⁽¹⁾−𝑠𝙱𝚗𝑈^𝑜(𝑠)−𝑉_𝑜𝛽²𝙱𝚗𝑑𝑠,(4.27) which make sense only if 𝜙⁽¹⁾⩽𝜉_𝑜. Integrating (4.27) by parts we get𝑡²𝜙⁽¹⁾=2^𝜙(1)_𝜉𝑜exp𝜙⁽¹⁾−𝑠𝙱𝚗𝑈^𝑜(𝑠)𝛽²𝑑𝑠+𝛽2²𝑉_𝑜−𝑈_𝑜𝜙⁽¹⁾.(4.28) We recall from the previous section that the condition |𝜉̇⁽¹⁾(𝑡)|<1 is guaranteed if𝑑𝑡𝑑𝜙⁽¹⁾<−12,(4.29)

which, by virtue of(4.25)₍₁₎, is equivalent to require that𝑡²+𝙱𝚗𝑡+2𝛽²𝑈𝜙⁽¹⁾−𝑉_𝑜<0.(4.30) Hence, under assumption (𝐻2), the discriminant Δ=𝙱𝚗²+8𝛽⁻²(𝑉_𝑜−𝑈_𝑜(𝜙⁽¹⁾))>𝙱𝚗²>0, and (4.30) is fulfilled when−𝙱𝚗−√Δ2<0⩽𝑡<−𝙱𝚗+√Δ2.(4.31) Therefore, in order to have a unique local solution, we must require that𝑡²<𝙱𝚗²+Δ−2√Δ𝙱𝚗4<𝙱2𝚗²+𝛽2²𝑉_𝑜−𝑈_𝑜𝜙⁽¹⁾,(4.32) which, exploiting (4.28), becomes2^𝜙(1)_𝜉𝑜exp𝜙⁽¹⁾−𝑠𝙱𝚗𝑈^𝑜(𝑠)𝛽²𝑑𝑠<𝙱2𝚗².(4.33) The latter is automatically satisfied, under assumption (𝐻3), recalling that 𝜙⁽¹⁾⩽𝜉_𝑜. So, also for the first order we have local uniqueness and existence of the solution 𝜉⁽¹⁾(𝑡)=𝜙⁽¹⁾(𝑡)+𝑡, where 𝜙⁽¹⁾(𝑡) is obtained inverting (4.28).

4.3. 𝑖th-Order Approximation

We now consider here the 𝑖th-order approximation. The evolution equation of the free boundary is given by (4.12). Proceeding as in the previous sections we set 𝜙^(𝑖)=𝜉^(𝑖)−𝑡, so that (4.12) can be rewritten as𝜙̇^(𝑖)𝛽²(𝑖+2)𝑡^𝑖+1(𝑖+1)!−𝑉^𝑜𝑡^𝑖−1(𝑖−1)!+𝛽²^𝑡0𝑈^(𝑖−2)𝜙^(𝑖)−𝑡+𝜃,𝜃𝑑𝜃=𝑡^𝑖𝑖!𝛽²𝙱𝚗,(4.34) with 𝜙^(𝑖)(0)=𝜉_𝑜. Once again we look for 𝑡=𝑡(𝜙^(𝑖)), solving this Cauchy problem𝑑𝑡𝑑𝜙^(𝑖)=(𝑖+2)(𝑖+1)𝙱𝑡𝚗−𝑉^𝑜𝑖𝛽²𝙱𝚗𝑡+𝑡^𝑖𝑖𝙱!𝚗^𝑡0𝑡𝜉𝑈^(𝑖−2)𝜙^(𝑖)−𝑡+𝜃,𝜃𝑑𝜃𝑡𝜉_𝑜=0,_𝑜=0.(4.35) Now, hypothesis (𝐻4) and (4.17) entaillim_𝑡→0+𝑑𝜙^(𝑖)𝑑𝑡=lim_𝑡→0⁺𝑑𝑡𝑑𝜙^(𝑖)=−1.(4.36)Therefore𝑉_𝑜𝑖𝛽²=lim_𝑡→0+𝑖!𝑡^𝑖−1^𝑡0𝑈^(𝑖−2)𝜙^(𝑖)−𝑡+𝜃,𝜃𝑑𝜃.(4.37) So for 𝑡 sufficiently small, we can approximate the integral on the r.h.s. of (4.37) in the following way:^𝑡0𝑈^(𝑖−2)𝜙^(𝑖)−𝑡+𝜃,𝜃𝑑𝜃=𝐶_𝑖𝜙^(𝑖)𝑡^𝑖−1,(4.38) where 𝐶_𝑖 is a smooth function of 𝜙^(𝑖), determined exploiting (4.7), (4.8), and (4.9). In particular,𝐶_𝑖