On the construction and properties of weak solutions describing dynamic cavitation Research partially supported by the EU FP7-REGPOT project ”Archimedes Center for Modeling, Analysis and Computation”, the ”Aristeia” program of the Greek Secretariat of Research, and the EU EST-project ”Differential Equations and Applications in Science and Engineering”. Part of this work was completed at the Institute of Applied and Computational Mathematics, FORTH, Greece. (In: Journal of Elasticity (2015), 118-2, 141-185, DOI: 10.1007/s10659-014-9488-z)

# On the construction and properties of weak solutions describing dynamic cavitation ††thanks: Research partially supported by the EU FP7-REGPOT project ”Archimedes Center for Modeling, Analysis and Computation”, the ”Aristeia” program of the Greek Secretariat of Research, and the EU EST-project ”Differential Equations and Applications in Science and Engineering”. Part of this work was completed at the Institute of Applied and Computational Mathematics, FORTH, Greece.(In: Journal of Elasticity (2015), 118-2, 141-185, DOI: 10.1007/s10659-014-9488-z)

Alexey Miroshnikov   and   Athanasios E. Tzavaras Department of Mathematics and Statistics, University of Massachusetts Amherst, USA. amiroshn@gmail.com Division of Computer, Electrical, Mathematical Sciences & Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. athanasios.tzavaras@kaust.edu.sa
###### Abstract

We consider the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity. Our analysis treats both stress-free cavities and cavities with contents.

Keywords: Cavitation, Shock wave, Polyconvex elasticity

Mathematics Subject Classification: 35L67, 35L70, 74B20, 74H20, 74H60

## 1 Introduction

The motion of a continuous medium with nonlinear elastic response is described by the system of partial differential equations

 ytt−div∂W∂F(∇y)=0 (1.1)

where stands for the motion, is the deformation gradient, and we have employed the constitutive theory of hyperelasticity, , that the Piola-Kirchhoff stress is given as the gradient of a stored energy function

 W:Md×d+:={F∈Rd×d:det(F)>0}⟶R.

For isotropic elastic materials the stored energy reads  , where is a symmetric function of the eigenvalues of the positive square root ; see [1, 22]. In that case (1.1) admits solutions that are radially symmetric motions,

 y(x,t)=w(|x|,t)x|x|,R=|x|,

and are generated by solving for the amplitude the scalar second-order equation

 wtt=1Rd−1∂∂R(Rd−1∂Φ∂v1(wR,wR,…,wR))−1R(d−1)∂Φ∂v2(wR,wR,…,wR). (1.2)

This equation admits the special solution corresponding to a homogeneous deformation of stretching . The question was posed [2] if discontinuous solutions of (1.2) can be constructed and it has been tied to a possible explanation of the phenomenon of cavitation in stretched rubbers [6, 7].

Ball [2] in a seminal paper proposed to use continuum mechanics for modeling cavitation and used methods of the calculus of variations and bifurcation theory to construct cavitating solutions for the equilibrium version of (1.2): There is a critical stretching such that for the homogeneous deformation is the only minimizer of the elastic stored energy; by contrast, for there exist nontrivial equilibria corresponding to a (stress-free) cavity in the material with energy less than the energy of the homogenous deformation [2]. We refer to [17, 20, 21, 13, 16, 14] and references therein for developments concerning equilibrium or quasistatic cavitating solutions.

In an important development, K.A. Pericak-Spector and S. Spector [18, 19] use the self-similar ansatz

 w(R,t)=tφ(Rt) (1.3)

to construct a weak solution for the dynamic problem (1.2) that corresponds to a spherical cavity emerging at time from a homogeneously deformed state. The cavitating solution is constructed in dimension for special classes of polyconvex energies [18, 19] and sufficiently large initial stretching. Remarkably, the cavitating solution has lower mechanical energy than the associated homogeneously deformed state from where it emerges [18], and thus provides a striking example of nonuniqueness of entropy weak solutions (for polyconvex energies). The dynamic cavitation problem is a little studied subject. Apart from [18, 19], there is an interesting almost explicit example of a dynamic solution that oscillates constructed by Chou-Wang and Horgan [4] for the dead load problem of an incompressible elastic material. Due to the incompressibility constraint the response is markedly different from the compressible case: beyond a critical load a cavity opens and then closes again, see [4]. The reader is referred to Choksi [3] for a discussion of the limit from compressible to incompressible response in radial elasticity, and to Hilgers [10] for other examples of non-uniqueness in multi-dimensional hyperbolic conservation laws due to radial point singularities.

The objective of the present work is to complement [18, 19] by establishing various further properties of weak solutions describing dynamic cavitation. First, we prove that cavity formation is always associated with a precursor shock, namely it is not possible to construct a cavitating solution that connects ”smoothly” to a uniformly deformed state. Second, we study the bifurcation diagram for dynamically cavitating solution and provide a formula that determines the critical stretch required for opening a cavity. The critical stretch turns out (for traction free cavities) to be the same as that predicted from the equilibrium cavitation analysis of Ball [2]. In a companion paper [8] we reassess the issue of nonuniqueness of weak solutions, and show that local averaging of the cavitating weak solution contributes a surface energy when opening a cavity that renders the uniform deformation the energetically preferred solution, see [8] for details and comments on the ramifications.

We now provide an outline of the technical contents of the article: Throughout we work with stored energies of the form

 Φ(v1,v2,...,vd)=d∑i=1g(vi)+h(v1v2…vd) (H0)

where satisfy

 g′′(x)>0,h′′(x)>0,limx→∞h(x)=+∞ (H1) g′′′(x)≤0,h′′′(x)<0. (H2)

Hypothesis (H1) refers to polyconvexity [2], while (H2) indicates elasticity with softening. The reader is referred to Appendix 7.2 where properties of isotropic stored energies are reviewed.

Following [18, 19], we introduce the self-similar ansatz (1.3) and the problem of cavity formation becomes to find solutions of the problem

 (s2−∂2Φ∂v21)¨φ =d−1s[(˙φ−φs)∂2Φ∂v1∂v2+∂Φ∂v1−∂Φ∂v2] (1.4) φ0 :=lims→0+φ(s)>0 (1.5)

and to check whether such solutions can be connected to a uniformly deformed state, namely

 φ(s) =λsfors>σ. (1.6)

Here, represents the speed of the cavity surface, the stretching of the (initially) uniform deformation and is the shock speed. We remark that (1.4) with (1.6) admit the special solution corresponding to a homogenous deformation ; therefore, according to this scenario, cavity formation is associated to nonuniqueness for the initial value problem of the radial elasticity equation (1.2).

To make the problem (1.4)-(1.5) determinate it is necessary to specify the value of the radial component of the Cauchy stress at the cavity surface. Two types of boundary conditions are pursued (see Section 3.1) corresponding to stress-free cavities or to a cavity with content:

Under the growth condition (see (H3) in section 3.1) and for dimension , the problem (1.4), (1.5) and (1.7) is desingularized at the origin and a solution is constructed (see Theorem 3.3). The question arises whether this cavitating solution can be connected to the uniform deformation (1.6) through a shock (or through a sonic singularity). This leads to studying the algebraic equation

 σ=  ⎷∂Φ∂v1(˙φ,φs,...,φs)−∂Φ∂v1(φs,φs,...,φs)˙φ−φs∣∣ ∣ ∣∣s=σ (1.8)

which manifests the Rankine-Hugoniot jump condition. In Theorems 3.4 and 4.1 we show there exists a unique where the connection can be effected, and that the connection either happens through a Lax shock or through a sonic singularity (i.e. a point where the coefficient in (1.4) vanishes). Then, in Theorem 4.2, we restrict to dimensions and exclude the possibility of a connection through a sonic singularity. Our analysis is inspired and extends the results of [18, 19] where cavitating solutions are constructed for sufficiently large stretchings . In particular, we show that, for dimensions , it is impossible to connect a cavitating solution smoothly to a uniformly deformed state and thus any cavitating solution is associated with a precursor shock.

The next objective is to study the bifurcation diagram of the cavitating weak solution and determine the critical stretching for dynamic cavitation. The bifurcation diagram is visualized as follows: The boundary condition (1.7) is expressed for the specific volume in the general form . Given the cavity speed , let be the cavitating solution (constructed in Section 3.1) emanating from data , . Denoting by the connection point, the associated stretching defines the map

 φ0⟼Λ(φ0,V(φ0)) where Λ(φ0,V(φ0)):=φ(σ;φ0,V(φ0))σ, (1.9)

which is precisely the dynamic bifurcation diagram (see Fig. 2 for a numerical computation of this map). The limit will determine the critical stretching. The technique of recovering the bifurcation point by computing the cavitating solution and sending the inner radius of the cavity to zero is espoused in [15], where the authors use it to devise a numerical scheme for computing in equilibrium elasticity.

To understand the limiting behavior of cavitating solutions as , we introduce the rescaling

 ψ(ξ;φ0,V(φ0)):=φ(φ0ξ;φ0,V(φ0))φ0,δ(ξ;φ0,V(φ0)):=v(φ0ξ;φ0,V(φ0)) (1.10)

which captures the inner asymptotics of the cavitating solution to (1.4)-(1.7). Rescalings have been useful in the study of cavitation for equilibrium elasticity [2] and will play an instrumental role in determining the critical stretching for dynamic cavitation.

It is proved in Proposition 5.6 that the rescaled solutions converge to a limiting profile,

 (ψ,δ)(ξ;φ0,V(φ0))→(ψ0,δ0)(ξ;V(0)),asφ0→0+,

uniformly on compact subsets of .

The limiting profile , where , is defined on and solves the initial value problem

 −∂2Φ∂v21(ψ′0,ψ0ξ,...,ψ0ξ)ψ′′0 =d−1ξ(ψ′0−ψ0ξ)[∂2Φ∂v1∂v2+∂Φ∂v1−∂Φ∂v2ψ′0−ψ0ξ](ψ′0,ψ0ξ,...,ψ0ξ), (1.11) ψ0(0) =1, δ0(0) =V(0).

The solvability of (1.11) and properties of its solutions are discussed in Proposition 5.5, where it is in particular shown that the (inner) solution is associated with a critical stretching at infinity

 Λ0(V(0)):=limξ→∞ψ0(ξ;V(0))ξ. (1.12)

Equation (1.11) is precisely the equation describing cavitating solutions in equilibrium radial elasticity, suggesting that the critical stretch for dynamic cavitation and equilibrium cavitation might conceivably coincide. The critical stretch for cavitation in equilibrium radial elasticity is studied in [2, Section 7.5] where various representation formulas for are established. In section 5.3.2 we pursue this analogy, we show that for a stress-free cavity , and establish representation formulas for the critical stretch and corresponding lower bounds.

Finally, in Theorem 5.7, we study the behavior of the cavitating solution and the associated stretch (1.9) as the cavity speed . We establish that

 limφ0→0+Λ(φ0,V(φ0))=Λ0(V(0)),

where is given by (1.12), that the speed and the strength of the precursor shock satisfy

 limφ0→0+σ(φ0,V(φ0))=√∂2Φ∂v21(Λ0(V(0)),…,Λ0(V(0))) limφ0→0+[φs−˙φ](σ(φ0,V(φ0)))=0,

and that

 φ(s;φ0,V(φ0))→Λ0(V(0))s,as φ0→0+.

Our analysis proves that the critical stretching for equilibrium and dynamic cavitation coincide.

The structure of the article is as follows: In Section 2 we introduce the equations of radial elasticity for isotropic elastic materials. In Section 3 we derive the equations for self-similar solutions of radial elasticity, describe various special solutions, and present the problem of cavitation. The analysis of Section 3 follows the ideas and extends the analysis of [18, 19] to a more general class of (polyconvex) stored energies and to boundary conditions of cavities with content. Section 4 and Section 5 contain the main new results. In Section 4 we establish various properties of weak solutions describing cavity formation from a homogeneously deformed state. In Section 5 we study the bifurcation curves associated with cavitating weak solutions and establish the properties of the critical stretching and its relation to the critical stretching predicted by the equilibrium elasticity equation. The Appendix lists some properties of radial deformations, and collects information on stored energies that is widely used in various places of the text.

## 2 The equations of radial elasticity

The stored energy of an isotropic elastic material has to satisfy the symmetry requirements

 frame indifference W(QF)=W(F),∀Q∈SO(d) isotropy W(F)=W(FQ),∀Q∈SO(d).

where is any proper rotation. These requirements are equivalent to

 W(F)=Φ(v1,v2,…,vd)

where is a symmetric function of its arguments and are the eigenvalues of called principal stretches [1, 2, 22].

For isotropic materials the system of elasticity (1.1) admits radial solutions of the form

 y(x,t)=w(R,t)xRwith R=|x|. (2.1)

The deformation gradient is computed by

 ∇y=wRx⊗xR2+wR(I−x⊗xR2) (2.2)

and has principal stretches . Using results on spectral representations of functions of matrices one computes the first Piola-Kirchhoff stress [2, p.564],

 S(∇y)=Φ1(wR,wR,...,wR)x⊗xR2+Φ2(wR,wR,...,wR)(I−x⊗xR2), (2.3)

where we used the notation , and the symmetry property (7.8). (The reader is referred to Appendix 7.2 for properties of the stored energies and details on the notation used throughout). Using the above formulas one computes that the amplitude of the radial motion (2.1) is generated by solving the second-order partial differential equation

 wtt =1Rd−1∂∂R(Rd−1∂Φ∂v1(wR,wR,...,wR))−d−1R∂Φ∂v2(wR,wR,...,wR). (2.4)

In order for solutions to be interpreted as elastic motions one needs to impose the requirement

 detF=wR(wR)d−1>0

on solutions of (2.4), which for radial motions suffices to exclude interpenetration of matter.

Equation (2.4) can also be derived by considering the action functional for radial, isotropic elastic materials, defined as the difference between kinetic and potential energy

 I[w]:=∫T0∫10Rd−1(12w2t−Φ(wR,wR,...,wR))dRdt.

Critical points of the functional are obtained by computing the first variation and setting it to zero,

 ddδ∣∣δ=0I[w+δψ]=0,

which gives the weak form of (2.4),

Finally, (2.4) can be expressed as a first order system by introducing the variables

 a=wR,b=wR,v=wt (2.5)

where is the (longitudinal) strain, is the transverse strain, and is the velocity in the radial direction. It is expressed as the equivalent first order system

 at =vR (2.6) vt =∂R(Φ1(a,b,...,b))+d−1R(Φ1(a,b,...,b)−Φ2(a,b,...,b)) bt =vR.

subject to the involution . This is a system of balance laws with geometric singularity at . Under the hypothesis the system (2.6) is hyperbolic (see [5, Def 3.1.1] for the usual definition). The characteristic speeds are genuinely nonlinear, while is linearly degenerate, see [12, 5]. The eigenvalues and the corresponding right and left eigenvectors of the flux of the system (2.6) are given by

 λ+ λ− =−√Φ11,r−=(1,−√Φ11,0)⊤,l−=(√Φ11,−1,(d−1)Φ12/√Φ11) λ0 =0,r0=((d−1)Φ12,0,−Φ11)⊤,l0=(0,0,1).

## 3 The cavitating solution of Pericak-Spector and Spector

We are interested in (2.4) subject to the initial-boundary conditions

 {w(R,0)=λRw(R,t)=λR,for|R|>¯σt. (3.1)

The symmetry of implies that and the homogeneous deformation is a special equilibrium solution of (2.4) associated to the stretching . To obtain additional solutions, it was suggested in [18] to exploit the invariance of (2.4), (3.1) under the family of the scaling transformations and to seek solutions in self-similar form

 w(R,t)=tφ(Rt). (3.2)

Introducing the ansatz (3.2) to (2.4), and using the notations and , it turns out that satisfies the singular second-order ordinary differential equation

 (3.3)

Henceforth, we will be using the short hand notations

 Φi(˙φ,φs)≡∂Φ∂vi(˙φ,φs,...,φs)Φij(˙φ,φs)≡∂2Φ∂vi∂vj(˙φ,φs,...,φs)

and so on for higher derivatives. We refer to Appendix 7.2 for details, and caution the reader that the notation together with the symmetry properties (7.8), (7.9) has implications on the differentiation of such formulas.

Moreover, we introduce the variables defined in analogy to (2.5) by

 a=˙φ,b=φs

and rewrite (3.3) in the form of the first order system

 ⎧⎪ ⎪⎨⎪ ⎪⎩(s2−Φ11(a,b))˙a=(d−1)s(a−b)P(a,b)˙b=1s(a−b) (3.4)

where

 P(a,b)={Φ12(a,b)+Φ1(a,b)−Φ2(a,b)a−b,a

is a continuous function on .

In analogy to the standard theory [12] of the Riemann problem for conservation laws, it is instructive to classify elementary solutions of (3.4). There are three classes of special solutions:

(a) Uniformly deformed states. A special class of solutions of (3.4) are the constant states , which yield a uniform deformation for the original system.

(b) Continuous solutions. The balance of the convective and the production terms in (3.4) leads to a class of solutions that are continuous (which are not present in homogeneous conservation laws and are of different origin than the rarefaction waves). These will be the main object of study here. There are two features of (3.4) that need to be addressed by the analysis: (i) the geometric singularity at , and (ii) the difficulty emerging from a potential free boundary at the sonic curve . It is well known that the resolution of the Riemann problem for multi-dimensional hyperbolic systems leads to systems that change type across sonic-curves in the self-similar variables. The analog of this phenomenon for radial solutions leads to singular ordinary differential equations across the sonic lines.

(c) Shocks. One may express the system (3.4) in the equivalent form

 dds(s2a−Φ1(a,b)) =2sa+d−1s(Φ1(a,b)−Φ2(a,b)) (3.6) dbds =1s(a−b).

Two smooth branches of solutions to (3.6) might be connected through a jump discontinuity at provided that the Rankine-Hugoniot jump conditions

 b−=b+=:b,σ2=Φ1(a+,b)−Φ2(a−,b)a+−a− (3.7)

are satisfied, where

 (a−,b−)=lims→σ−(a,b)(s),(a+,b+)=lims→σ+(a,b)(s).

According to the Lax shock admissibility criterion (see [12] where the criterion was introduced or [5, Secs 8.3, 9.4]), a shock of the characteristic family will be admissible if

 √Φ11(a+,b)<σ+=√Φ1(a+,b)−Φ2(a−,b)a+−a−<√Φ11(a−,b). (3.8)

In particular,

 if   Φ111>0,   then (3.8) is equivalent to a+a−.

Similarly, shocks of the characteristic family are admissible via the Lax criterion if

 −√Φ11(a+,b)<σ−=−√Φ1(a+,b)−Φ2(a−,b)a+−a−<−√Φ11(a−,b). (3.9)

in which case

 if   Φ111>0,   then (3.9) is equivalent to a−a+.

For radial motions shocks of the characteristic family are outgoing while shocks of the characteristic family are incoming to the origin. For the cavitation problem, it is natural to restrict to outgoing shocks and the kinematics of the cavity dictates that . Therefore, we impose the condition which corresponds to softening elastic response. Softening refers to the property that the elastic modulus decreases with an increase of the longitudinal strain and plays an important role in cavitation analysis.

### 3.1 The cavitating solution

We next consider the problem of cavitation and discuss the continuous type of solutions in this context. We employ a constitutive relation of polyconvex class

 Φ(v1,v2,...,vd)=d∑i=1g(vi)+h(v1v2…vd), (H0)

where , satisfy (H1) and (H2) and thus and . Hypothesis (H1) alludes to polyconvexity of the stored energy while Hypothesis (H2) manifests softening elastic response.

A stored energy of the form (H0) with was used in [18] to establish cavitation for . The generalization presented in (H0) is necessary in order to handle the case of , as the hypothesis of quadratic growth is too strong to allow for a cavity when . The ideas presented in this section closely follow the discussion of [18, 19], nevertheless they are presented here first for the reader’s convenience but also to set up the landscape for the forthcoming analysis in the following sections.

The differential equation (3.3) is expressed as

 Q(˙φ,φs,s)¨φ=(d−1)s(˙φ−φs)P(˙φ,φs) (3.10)

or equivalently

 ⎧⎪ ⎪⎨⎪ ⎪⎩Q(a,b,s)˙a=(d−1)s(a−b)P(a,b)˙b=1s(a−b) (3.11)

where

 Q(a,b,s) (3.12)

Desingularization at the origin. We next transform (3.10) into a system for the quantities

 φ(s),v(s)=˙φ(φs)d−1with dataφ(0)=φ0>0,v(0)=v0>0, (3.13)

henceforth restricting to stored energies of class (H0). A lengthy but straightforward calculation shows that satisfies the initial-value problem

 (3.14)

In view of (H1) and the assumption , the only term on the right-hand side of (3.14) that might be singular at is the term . This motivates to impose the growth condition

 limx→∞(g′(x)xd−2)=γ≥0. (H3)

Doing that the emerging system is not singular and one may apply the standard existence theory for ordinary differential equations to obtain

###### Lemma 3.1.

Suppose and hypotheses (H0), (H1), and (H3) hold. Then there exists a unique solution of the system (3.14) defined on a maximal interval of existence.

###### Proof.

Under hypotheses (H0)-(H1) we have , . Moreover, by (H3), the limit exists and is finite and the right hand side of (3.14) is continuous for (up to the boundary ). A careful review of the various terms indicates that the initial value problem (3.14) is expressed as

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩˙φ=A(s,v,φ)˙v=B(s,v,φ)+C(s,v,φ)H(s,φ)φ(0)=φ0>0v(0)=v0>0. (3.15)

where are functions on with a neighbourhood of . The term

 (3.16)

carries the singular behaviour in and by (H3) it is continuous on .

Moreover, the assumptions , imply

 0≤g′′(x)x≤∫10g′(sx)xds=g′(x)−g′(0),x>0

and, using once again (H3), it automatically implies

 (3.17)

Now observe that

 ∣∣∂H∂φ(s,φ)∣∣=1φ∣∣g′′(φs)(sφ)d−3−(d−2)g′(φs)(sφ)d−2∣∣≤C

for and a suitable neighbourhood of . The standard existence and uniqueness theory for systems of ordinary differential equations then provides the result. ∎

###### Remark 3.2.

The sign requirements in hypotheses (H1), (H2) place the restriction that . Also, if then (H3) enforces that . This is consistent with (H1)-(H2).

Boundary data at the cavity surface. A natural assumption motivated from mechanical considerations is to impose that the radial Cauchy stress vanishes at the cavity surface. Using the standard formula relating the Cauchy stress tensor to the Piola-Kirchhoff stress (e.g. [2, 1])

 T(F)=1detFS(F)F⊤

and (2.3) it follows that for given by (2.2) we have

The radial component of the Cauchy stress is given by

For the solution of (3.14) it is easy to see that

and therefore

This motivates to impose the growth condition

 h′(x)→−∞asx→0+,h′(x) →+∞asx→+∞. (H4)

Under (H1), (H4) the inverse is a well-defined function on and the boundary condition becomes

One may consider more general boundary conditions that are referred in [18] as cavities with content and require that , where is some prescribed function. Such conditions could model at a phenomenological level the effect of remnant plasticity inside the cavity, and are postulated in analogy to the form of kinetic relations in the motion of phase boundaries. For physical reasons the remnant plasticity at the cavity should correspond to tensile forces, which dictates that . One checks that

It is not entirely clear if such an assumption is mechanically justified, nevertheless it can be analyzed by the mathematical theory at no additional effort. Note that both (3.19) and (3.20) decrease the freedom of the data by one degree. For the bifurcation analysis in section 5 we assume that is continuous at . This implies that is also continuous at .

A class of self-similar solutions. We now construct a class of self-similar solutions to (3.10). Proceeding along the lines of [18, Thm 5.1] we have:

###### Theorem 3.3.

Assume that , satisfies (H0)-(H3) and let , . Then, there exists a unique solution of (3.10) satisfying the initial data (3.13) and defined on a maximal interval of existence , with . The solution has the following properties:

• solves (3.14) and there holds

 φ(s)s∼φ0sand˙φ(s)∼v0(sφ0)d−1ass→0+. (3.21)
• , , are strictly monotonic and satisfy

 (3.22) 0<˙φ(s)<˙φ(t)<φ(t)t<φ(s)sfor0

and

 Q(˙φ,φs,s)<0,˙φ(s)−φ(s)s<0for0
• The following limits exist

 lims→T−Q(˙φ,φs,s)=0,lims→T−(˙φ−φs)=c0≤0 (3.25)

(but it is not known if ).

###### Proof.

By Lemma 3.1 there exists a unique local solution of (3.14) with , which of course satisfies (i). Next, we observe that solves the system (3.11) where, in view of (3.12) and (3.21), we have . This together with (3.21) implies that and for for sufficiently small . We now check that satisfying (H0)-(H3) has the properties

 Φ11(a,b)>0,Φ111(a,b)<0,P(a,b)>0 (3.26)

which together with (3.11) imply , and

 dds(a−b)=1sQ(a−b)[(d−1)P−Q]>0.

Thus, (3.22), (3.23) and (3.24) must hold for . The solution can be continued in that manner for so long as ,   , and on a maximal interval of existence , with . It is also clear that the solution cannot hit the diagonal unless . On the interval we clearly have

 a(ε)

that is and stay away from zero and in a bounded range as increases. Then (3.27) and the fact that , imply that   and

 limsups→T−Q(a(s),b(s),s)=0. (3.28)

Also, in view of (3.22) and (3.23) we have as .

A computation shows that

 12ddsQ2=2sQ−d−1s(a−b)[P(a,b)Φ111(a,b)+Φ112(a,b)Q]

Using (3.12) and the bounds (3.27) we see that for

 ∣∣ddsQ2∣∣≤Cs∈(ε,T). (3.29)

Now if and

then (3.29) would be violated. We thus deduce that as . ∎

### 3.2 Connection to a uniformly deformed state.

Next, the smooth cavitating solution of (3.11) constructed in Theorem 3.3 will be connected to a uniformly deformed state through an outgoing shock of speed . At the connection, the Rankine-Hugoniot relations (3.7) must be satisfied and they imply that

 φ−(σ)=φ+(σ),σ2(˙φ+(σ)−˙φ−(σ))=Φ1(˙φ+,φ+σ)−Φ1(˙φ−,φ−σ)

where