Global existence for a nonlocal model for adhesive contact

TL;DR

This paper proves the global-in-time existence of solutions for a complex nonlocal PDE model describing adhesive contact with damage and elongation effects, advancing the mathematical understanding of such contact problems.

Contribution

It introduces a novel analytical approach to establish global existence for a nonlinear nonlocal PDE system modeling adhesive contact with damage.

Findings

Global existence of solutions established

Handles complex nonlinearities and nonlocal damage effects

Extends local solutions to global solutions using a nonstandard method

Abstract

In this paper we address the analytical investigation of a model for adhesive contact, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the unilateral contact conditions, the physical constraints on the internal variables, as well as the integral contributions related to the nonlocal forces. For the associated initial-boundary value problem we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument.

Equations

$$(ϵ(\mathbf{u}),\text{raise}1.94444pt\chi ,\nabla \text{raise}1.94444pt\chi ,{\mathbf{u}}_{{|}_{{\Gamma }_{\mathrm{C}}}},\mathbf{g})$$

$$\mathbf{g}(x,y)=2(x-y){\mathbf{u}}_{{|}_{{\Gamma }_{\mathrm{C}}}}(x),$$

$${\Psi }_{\Omega }(ϵ(\mathbf{u}))=\frac{1}{2}ϵ(\mathbf{u})\mathbb{K}ϵ(\mathbf{u}),$$

$${\Psi }_{{\Gamma }_{\mathrm{C}}}(\text{raise}1.94444pt\chi ,\nabla \text{raise}1.94444pt\chi ,{\mathbf{u}}_{{|}_{{\Gamma }_{\mathrm{C}}}})=I_{[0,1]}(\text{raise}1.94444pt\chi )+\gamma (\text{raise}1.94444pt\chi )+\frac{1}{2}|\nabla \text{raise}1.94444pt\chi {|}^2+\frac{1}{2}\text{raise}1.94444pt\chi |{\mathbf{u}}_{{|}_{{\Gamma }_{\mathrm{C}}}}{|}^2+I_{(-\infty ,0]}({\mathbf{u}}_{{|}_{{\Gamma }_{\mathrm{C}}}}\cdot \mathbf{n}),$$

$${\Psi }_{\mathrm{nl}}(\text{raise}1.94444pt\chi (x),\text{raise}1.94444pt\chi (y),\mathbf{g}(x,y))=\frac{1}{2}{\mathbf{g}}^2(x,y)\text{raise}1.94444pt\chi (x)\text{raise}1.94444pt\chi (y){\mathrm{e}}^{\frac{-|x-y{|}^2}{d^2}},$$

$${\Phi }_{\Omega }(ϵ({\mathbf{u}}_t))=\frac{1}{2}ϵ({\mathbf{u}}_t){\mathbb{K}}_vϵ({\mathbf{u}}_t),$$

$${\Phi }_{{\Gamma }_{\mathrm{C}}}({\text{raise}1.94444pt\chi }_t)=\frac{1}{2}|{\text{raise}1.94444pt\chi }_t{|}^2+I_{(-\infty ,0]}({\text{raise}1.94444pt\chi }_t).$$

$${\mathcal{P}}_{int}$$

$$+{\int }_{{\Gamma }_{\mathrm{C}}}{\int }_{{\Gamma }_{\mathrm{C}}}2M(x,y)(x-y)\mathbf{v}(x)\mathrm{d}x\mathrm{d}y+{\int }_{{\Gamma }_{\mathrm{C}}}{\int }_{{\Gamma }_{\mathrm{C}}}(B_{\mathrm{nl}}^1(x,y)w(x)+B_{\mathrm{nl}}^2(x,y)w(y))\mathrm{d}x\mathrm{d}y.$$

$${\mathcal{P}}_{ext}={\int }_{\Omega }\mathbf{f}\mathbf{v}\mathrm{d}\Omega +{\int }_{{\Gamma }_{\mathrm{N}}}\mathbf{h}\mathbf{v}\mathrm{d}\Gamma ,$$

$$-\mathrm{div}\Sigma =\mathbf{f}\text{in }\Omega ,$$

$$\Sigma \mathbf{n}(x)=\mathbf{R}(x)+{\int }_{{\Gamma }_{\mathrm{C}}}2(x-y)M(x,y)\mathrm{d}y\text{in }{\Gamma }_{\mathrm{C}},$$

$$\mathbf{u}=0\text{in }{\Gamma }_{\mathrm{D}},\Sigma \mathbf{n}=\mathbf{h}\text{in }{\Gamma }_{\mathrm{N}}.$$

$$B(x)-\mathrm{div}\mathbf{H}(x)={\int }_{{\Gamma }_{\mathrm{C}}}\left(B_{\mathrm{nl}}^1(x,y)+B_{\mathrm{nl}}^2(y,x)\right)\mathrm{d}y\text{ in }{\Gamma }_{\mathrm{C}},\mathbf{H}\cdot {\mathbf{n}}_s=0\text{ on }\partial {\Gamma }_{\mathrm{C}}.$$

$$\Sigma =\frac{\partial {\Psi }_{\Omega }}{\partial ϵ(\mathbf{u})}+\frac{\partial {\Phi }_{\Omega }}{\partial ϵ({\mathbf{u}}_t)}=\mathbb{K}ϵ(\mathbf{u})+{\mathbb{K}}_vϵ({\mathbf{u}}_t),$$

$$\mathbf{R}=-\frac{\partial {\Psi }_{{\Gamma }_{\mathrm{C}}}}{\partial \mathbf{u}}\in -\text{raise}1.94444pt\chi \mathbf{u}-\partial I_{(-\infty ,0]}(\mathbf{u}\cdot \mathbf{n})\mathbf{n}.$$

$$M(x,y)=-\frac{\partial {\Psi }_{\mathrm{nl}}}{\partial \mathbf{g}(x,y)}=-\mathbf{g}(x,y)\text{raise}1.94444pt\chi (x)\text{raise}1.94444pt\chi (y){\mathrm{e}}^{-\frac{|x-y{|}^2}{d^2}},$$

$$B_{\mathrm{nl}}^1(x,y)=-\frac{\partial {\Psi }_{\mathrm{nl}}}{\partial \text{raise}1.94444pt\chi (x)}=-\frac{{\mathbf{g}}^2(x,y)}{2}{\mathrm{e}}^{-\frac{|x-y{|}^2}{d^2}}\text{raise}1.94444pt\chi (y),$$

$$B_{\mathrm{nl}}^2(x,y)=-\frac{\partial {\Psi }_{\mathrm{nl}}}{\partial \text{raise}1.94444pt\chi (y)}=-\frac{{\mathbf{g}}^2(x,y)}{2}{\mathrm{e}}^{-\frac{|x-y{|}^2}{d^2}}\text{raise}1.94444pt\chi (x).$$

$$B=\frac{\partial {\Psi }_{{\Gamma }_{\mathrm{C}}}}{\partial \text{raise}1.94444pt\chi }+\frac{\partial {\Phi }_{{\Gamma }_{\mathrm{C}}}}{\partial {\text{raise}1.94444pt\chi }_t}\in \partial I_{[0,1]}(\text{raise}1.94444pt\chi )+{\gamma }^{\prime }(\text{raise}1.94444pt\chi )+\frac{1}{2}|\mathbf{u}{|}^2+{\text{raise}1.94444pt\chi }_t+\partial I_{(-\infty ,0]}({\text{raise}1.94444pt\chi }_t),$$

$$\mathbf{H}=\frac{\partial {\Psi }_{{\Gamma }_{\mathrm{C}}}}{\partial \nabla \text{raise}1.94444pt\chi }=\nabla \text{raise}1.94444pt\chi .$$

$$-div(\mathbb{K}ϵ(\mathbf{u}(x,t))+{\mathbb{K}}_vϵ({\mathbf{u}}_t(x,t)))=\mathbf{f}(x,t),$$

$$\mathbf{u}(x,t)=0,$$

$$(\mathbb{K}ϵ(\mathbf{u}(x,t))+{\mathbb{K}}_vϵ({\mathbf{u}}_t(x,t)))\mathbf{n}=\mathbf{h}(x,t),$$

$$(\mathbb{K}ϵ(\mathbf{u}(x,t))+{\mathbb{K}}_vϵ({\mathbf{u}}_t(x,t)))\mathbf{n}+\text{raise}1.94444pt\chi (x,t)\mathbf{u}(x,t)+\partial I_{(-\infty ,0]}(\mathbf{u}(x,t)\cdot \mathbf{n})\mathbf{n}$$

$$+{\int }_{{\Gamma }_{\mathrm{C}}}2(x-y)\mathbf{g}(x,y)\text{raise}1.94444pt\chi (x,t)\text{raise}1.94444pt\chi (y,t){\mathrm{e}}^{-\frac{|x-y{|}^2}{d^2}}\mathrm{d}y\ni 0,$$

$${\text{raise}1.94444pt\chi }_t(x,t)+\partial I_{(-\infty ,0]}({\text{raise}1.94444pt\chi }_t(x,t))-\Delta \text{raise}1.94444pt\chi (x,t)+\partial I_{[0,1]}(\text{raise}1.94444pt\chi (x,t))+{\gamma }^{\prime }(\text{raise}1.94444pt\chi (x,t))$$

$$\ni -\frac{1}{2}|\mathbf{u}(x,t){|}^2-\frac{1}{2}{\int }_{{\Gamma }_{\mathrm{C}}}\left({\mathbf{g}}^2(x,y)+{\mathbf{g}}^2(y,x)\right)\text{raise}1.94444pt\chi (y,t){\mathrm{e}}^{-\frac{|x-y{|}^2}{d^2}}\mathrm{d}y,$$

$${\partial }_{{\mathbf{n}}_s}\text{raise}1.94444pt\chi (x,t)=0$$

$$\{\begin{array}{l}\text{raise}1.94444pt\chi (x,t)\mathbf{u}(x,t){\int }_{{\Gamma }_{\mathrm{C}}}\eta (x-y)\text{raise}1.94444pt\chi (y,t)\mathrm{d}y,\\ -\frac{1}{2}|\mathbf{u}{|}^2(x,t){\int }_{{\Gamma }_{\mathrm{C}}}\eta (x-y)\text{raise}1.94444pt\chi (y,t)\mathrm{d}y-\frac{1}{2}{\int }_{{\Gamma }_{\mathrm{C}}}\eta (x-y)\text{raise}1.94444pt\chi (y,t)|\mathbf{u}{|}^2(y,t)\mathrm{d}y\end{array}\text{with }\eta (\zeta )=4{\zeta }^2{\mathrm{e}}^{-{\zeta }^2/{\mathrm{d}}^2}.$$

$$\mathcal{K}[w](x):={\int }_{{\Gamma }_{\mathrm{C}}}k(x,y)w(y)\mathrm{d}y,w\in L^1({\Gamma }_{\mathrm{C}}).$$

$$\begin{array}{c}\text{ Ω is a bounded Lipschitz domain in R3, with }\\ \partial \Omega ={\overline{\Gamma }}_{\mathrm{D}}\cup {\overline{\Gamma }}_{\mathrm{N}}\cup {\overline{\Gamma }}_{\mathrm{C}},\text{ ΓD, ΓN, ΓC, open disjoint subsets in the relative topology of ∂Ω, such that }\\ {\mathcal{H}}^2({\Gamma }_{\mathrm{D}}),{\mathcal{H}}^2({\Gamma }_{\mathrm{C}})>0,\text{and ΓC⊂R2 a sufficiently smooth flat surface.}\end{array}$$

$$\begin{array}{c}\mathbf{H}:=L^2(\Omega ;{\mathbb{R}}^3),\mathbf{V}:=\{\mathbf{v}\in H^1(\Omega ;{\mathbb{R}}^3):\mathbf{v}=0a.e.\; on{\Gamma }_{\mathrm{D}}\},\mathbf{Y}:=H_{00,{\Gamma }_{\mathrm{D}}}^{1/2}({\Gamma }_{\mathrm{C}};{\mathbb{R}}^3),\\ H:=L^2({\Gamma }_{\mathrm{C}}),V:=H^1({\Gamma }_{\mathrm{C}}),W:=\{\text{raise}1.94444pt\chi \in H^2({\Gamma }_{\mathrm{C}}):{\partial }_{{\mathbf{n}}_s}\text{raise}1.94444pt\chi =0\},\end{array}$$

$$H^{1/2}_{00,\Gamma_{\mathrm{D}}}(\Gamma_{\mathrm{C}};\mathbb{R}^{3})=\Big{\{}\mathbf{w}\in H^{1/2}(\Gamma_{\mathrm{C}};\mathbb{R}^{3})\,:\ \exists\,\tilde{\mathbf{w}}\in H^{1/2}(\Gamma;\mathbb{R}^{3})\text{ with }\tilde{\mathbf{w}}=\mathbf{w}\text{ in $\Gamma_{\mathrm{C}}$,}\ \tilde{\mathbf{w}}=\mathbf{0}\text{ in $\Gamma_{\mathrm{D}}$}\Big{\}}\,.$$

$$\mathbf{V}\subset L^4({\Gamma }_{\mathrm{C}})\text{ continuously, and }\mathbf{V}⋐L^{4-s}({\Gamma }_{\mathrm{C}})\text{ compactly for all }s\in (0,3],$$

Full text

Key words: Contact, adhesion, nonlocal damage, existence results. AMS (MOS) Subject Classification: 35K55, 35Q72, 74A15, 74M15.

1. Introduction

The mathematical field of contact mechanics has flourished over the last decades, as illustrated for instance in the monographs [9, 21, 22]. In this paper, we focus on a PDE system pertaining to a subclass of contact models, i.e.  models for adhesive contact. This phenomenon plays an important role in the analysis of the stability of structures. Indeed, it is well known that interfacial regions between materials are fundamental to ensure the strength and stability of structural elements. We describe adhesive contact by using a “surface damage theory”, as proposed by M. Frémond (see e.g. [12]), for the action of the adhesive substance (one may think of glue), located on the contact surface. This “damage approach” provides an efficient and predictive theory for the mechanical behavior of the structure.

The main idea underlying this modeling perspective is that, while the basic unilateral contact theory does not allow for any resistance to tension, in adhesive contact resistance to tension is related to the action of microscopic bonds between the surfaces of the adhering solids. The resulting description of the state of the bonds between the two bodies is in the spirit of damage models. In fact, it is given in terms of a phase parameter $\chi$ akin to a damage parameter, characterizing the state of the cohesive bonds. The PDE system is recovered from the balance laws of continuum mechanics, including (in a generalized principle of virtual power) the effects of micro-forces and micro-motions which are responsible for the breaking (or possibly repairing) of the microscopic bonds on the contact surface.

The mathematical analysis of models for adhesive contact and delamination à la Frémond, pioneered by [17], has attracted remarkable attention over the last 15 years, both in the case of rate-independent (cf., e.g., [13, 15, 19]) and rate-dependent (see, among others, [18], [10]) evolution.

The type of model investigated in the present paper was first rigorously derived and analyzed in [1] (cf. also [2]). The associated PDE system couples an equation for the macroscopic deformations of the body and a “boundary” equation on the contact surface, describing the evolution of the state of the glue in terms of a surface damage parameter. The system is highly nonlinear, mainly due to the presence of nonlinear boundary conditions and nonsmooth constraints on the internal variables, providing the unilaterality of the contact, the physical consistency of the damage parameter and, possibly, the unidirectional character of the degradation process). The model from [1] has been generalized to the non-isothermal case, with the temperature evolution described by an entropy balance equation (cf. e.g. [3]) or by the more classical energy balance equation (see [6]). Furthermore, in [4], [5], and [6] friction effects have also been included, even in the temperature-dependent case. There, we deal with the coupling between the Signorini condition for adhesive contact and a nonlocal regularization of the Coulomb law where the friction coefficient may depend on the temperature.

In this paper we again focus the isothermal and frictionless case. In fact, we address the analysis of a model for adhesive contact that generalizes that from [1] by assuming that also nonlocal forces act on the contact surface. This leads to  the presence of novel nonlocal (nonlinear) contributions in the resulting PDE system. More precisely, on the contact surface we consider interactions between damage at a point and damage in its neighborhood (i.e. we use a gradient surface damage theory) and moreover we admit an interaction between the adhesive substance and the two bodies. An example of this kind of behavior is given by experiments showing that elongation, i.e. a variation of the distance of two distinct points on the contact surface, may have damaging effects. Thus, a nonlocal quantity corresponding to the elongation is considered in the energy functionals.

This model was introduced in [12, 11]. While referring to the latter paper for details on the modeling and comments on the applications and computational results, in the following lines we will outline the derivation of the model for the sake of completeness. We will confine the discussion to the reduced case when only one body is considered in adhesive contact with a rigid support on a part of its boundary. We observe that this choice has the advantage of simplifying the exposition in comparison to the two-body case, while affecting neither the relevance of the model nor its analytical investigation.

1.1. The model and the PDE system

Let us consider a body which is located in a sufficiently smooth and bounded domain $\Omega\subset{\mathbb{R}}^{3}$ and lying on a rigid support during a finite time interval $(0,T)$. We denote its boundary by $\partial\Omega=\overline{\Gamma}_{\mathrm{D}}\cup\overline{\Gamma}_{{\mathrm{N}}}\cup\overline{\Gamma}_{\mathrm{C}}$. Here $\Gamma_{\mathrm{D}},\,\Gamma_{\mathrm{N}},\,\Gamma_{\mathrm{C}}$ are open subsets in the relative topology of $\partial\Omega$, each of them with a smooth boundary and disjoint one from each other. In particular, $\Gamma_{\mathrm{C}}$ is the contact surface, which is considered as a flat surface and identified with a subset of ${\mathbb{R}}^{2}$, cf. (2.1) ahead. Hereafter we shall suppose that both $\Gamma_{\mathrm{C}}$ and $\Gamma_{\mathrm{D}}$ have positive measure. We let ${\bf u}={\bf 0}$ on $\Gamma_{\mathrm{D}}$, while a known traction is applied on $\Gamma_{\mathrm{N}}$.

As already mentioned, in the model we neither encompass thermal evolution nor frictional effects. Thus, the state variables of the model are

[TABLE]

where we denote by ${\bf u}$ the vector of small displacements, $\epsilon({\bf u})$ the symmetric linearized strain tensor, $\chi$ a damage parameter defined on the contact surface, \nabla{\hbox{\raise 1.94444pt\hbox{\chi}}} its gradient, and ${\bf u}_{|_{\Gamma_{\mathrm{C}}}}$ the trace of the displacement ${\bf u}$ on $\Gamma_{\mathrm{C}}$. The parameter $\chi$ is assumed to take values in $[0,1]$, with {\hbox{\raise 1.94444pt\hbox{\chi}}}=0 for completely damaged bonds, {\hbox{\raise 1.94444pt\hbox{\chi}}}=1 for undamaged bonds, and {\hbox{\raise 1.94444pt\hbox{\chi}}}\in(0,1) for partially damaged bonds. The nonlocal term $\mathbf{g}$, defined for $(x,y)\in\Gamma_{\mathrm{C}}\times\Gamma_{\mathrm{C}}$ by

[TABLE]

describes the damaging effects due to the elongation. The free energy of the system is given by the sum of a bulk, a local surface, and a nonlocal surface contributions. More precisely, the free energy (density) in $\Omega$ is the classical one in elasticity theory

[TABLE]

where $\mathbb{K}=(a_{ijkh})$ stands for the elasticity tensor. Moreover, the local surface part of the free energy (density) is given by

[TABLE]

where the indicator function $I_{[0,1]}$ of the interval $[0,1]$ accounts for a physical constraint on $\chi$, being I_{[0,1]}({\hbox{\raise 1.94444pt\hbox{\chi}}})=0 if {\hbox{\raise 1.94444pt\hbox{\chi}}}\in[0,1] and I_{[0,1]}({\hbox{\raise 1.94444pt\hbox{\chi}}})=+\infty otherwise. Analogously, denoting by $I_{(-\infty,0]}$ the indicator function of the interval $(-\infty,0]$, the term $I_{(-\infty,0]}({\bf u}_{|_{\Gamma_{\mathrm{C}}}}\cdot{\bf n})$ renders the impenetrability condition on the contact surface, as it enforces that ${\bf u}_{|_{\Gamma_{\mathrm{C}}}}\cdot{\bf n}\leq 0$ (${\bf n}$ is the outward unit normal vector to $\Gamma_{\mathrm{C}}$). Finally, the function $\gamma$, sufficiently smooth and possibly nonconvex, is related to non-monotone dynamics for $\chi$ (from a physical point of view, it includes some cohesion in the material). Finally, let us consider the nonlocal surface free energy (density), which is given by

[TABLE]

where the exponential function with distance $d$ describes the attenuation of nonlocal actions with distance $|x-y|$ between points $x$ and $y$ on the contact surface.

As far as dissipation, we assume that there is no dissipation due to changes of the nonlocal variable, while we consider the dissipative variables $\epsilon({\bf u}_{t})$ in $\Omega$ and {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t} in $\Gamma_{\mathrm{C}}$. We follow the approach proposed by J.J. Moreau to prescribe the dissipated energy by means of the so-called pseudo-potential of dissipation which is a convex, nonnegative functional, attaining its minimal value [math] when the dissipation (described by the dissipative variables) is zero. More precisely, we define the volume part $\Phi_{\Omega}$ of the pseudo-potential of dissipation by

[TABLE]

where $\mathbb{K}_{v}=(b_{ijkh})$ denotes the viscosity tensor. The surface part $\Phi_{\Gamma_{\mathrm{C}}}$ of the pseudo-potential of dissipation depends on {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t} via

[TABLE]

The quadratic term \frac{1}{2}|{\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}|^{2} encodes rate-dependent evolution of $\chi$, where the indicator term I_{(-\infty,0]}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}) forces {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t} to take nonpositive values and renders the unidirectional character of the damage process.

Hereafter, we shall omit for simplicity the index $v_{|_{\Gamma_{\mathrm{C}}}}$ to denote the trace on $\Gamma_{\mathrm{C}}$ of a function $v$, defined in $\Omega$. The equations are recovered by a generalization of the principle of virtual powers, in which microscopic forces (also nonlocal ones) responsible for the damage process in the adhesive substance are included. More precisely, for any virtual bulk velocity ${\bf v}$ with ${\bf v}=\bf{0}$ on $\Gamma_{\mathrm{D}}$ and for any virtual microscopic velocity $w$ on the contact surface, we define the power of internal forces in $\Omega$ and $\Gamma_{\mathrm{C}}$ as follows

[TABLE]

Here, $\Sigma$ is the Cauchy stress tensor, ${\bf R}$ the classical macroscopic reaction on the contact surface, $B$ and ${\bf H}$ are local interior forces, responsible for the degradation of the adhesive bonds between the body and the support. The terms $M(x,y)$ and $B^{i}_{\mathrm{nl}}(x,y),i=1,2,$ are new scalar nonlocal contributions: they stand for internal microscopic nonlocal forces on the contact surface and describe the effects of the elongation as a source of damage. The power of the external forces is given by

[TABLE]

where ${\bf f}$ is a bulk known external force, while ${\bf h}$ is a given traction on $\Gamma_{\mathrm{N}}$. Note that, here we have disregarded external forces acting on the microscopic level and confined ourselves to the case of null accelerations power.

The principle of virtual powers, holding for every virtual microscopic and macroscopic velocities and every subdomain in $\Omega$, leads to the quasistatic momentum balance

[TABLE]

which will be also posed in a time interval $(0,T)$, and supplemented by the following boundary conditions

[TABLE]

Observe that in (1.10) the boundary condition for the stress tensor on the contact surface combines a local contribution involving the (pointwise) reaction ${\bf R}(x)$ and a nonlocal force (defined in terms of the new variable $M(x,y)$), related to the elongation.

Again, the principle of virtual powers leads to a micro-force balance on the contact surface, also posed in the time interval $(0,T)$, given by

[TABLE]

Here, ${\bf n}_{s}$ denotes the outward unit normal vector to $\partial\Gamma_{\mathrm{C}}$.

Constitutive relations for $\Sigma,{\bf R},B,{\bf H},M$, and $B^{i}_{\mathrm{nl}},i=1,2,$ are given in terms of the free energies and the pseudo-potentials of dissipation. More precisely, the constitutive relation for the stress tensor $\Sigma$ accounts for the dissipative (viscous) dynamics of the deformations

[TABLE]

while the local reaction ${\bf R}$ is given by

[TABLE]

As for as the nonlocal force on $\Gamma_{\mathrm{C}}$, we prescribe

[TABLE]

while the terms $B_{\mathrm{nl}}^{1}$ and $B_{\mathrm{nl}}^{2}$ are (formally) defined as derivatives of $\Psi_{\mathrm{nl}}$ with respect to the values of the surface damage parameter in $x$ and $y\in\Gamma_{\mathrm{C}}$, respectively, as it follows

[TABLE]

We further prescribe $B$ by

[TABLE]

($\partial I_{(-\infty,0]}$ and $\partial I_{[0,1]}$ denoting the convex analysis subdifferentials of $I_{(-\infty,0]}$ and $I_{[0,1]}$, resp.), and let ${\bf H}$ be

[TABLE]

Combining the previous constitutive relations with the balance laws, we obtain the following boundary value problem

[TABLE]

where all integrals on $\Gamma_{\mathrm{C}}$ involve the Lebesgue measure, which coincides with the Hausdorff measure on $\Gamma_{\mathrm{C}}$ by the flatness requirement, cf. (2.1) ahead. Let us stress once again, with respect to the ‘standard’ Frémond’s system for adhesive contact, (1.20) encompasses nonlocal terms both in the normal reaction on the contact surface (cf. with (1.20d)) and in the flow rule (1.20e) for $\chi$. In particular, we note that the right-hand side of (1.20e) (see also the second of (1.21)) may be different from zero even if ${\bf u}(x,t)={\bf 0}$, due to the integral contributions which render the damaging effects of the elongation.

Taking into account the explicit expression (1.1) of $\mathbf{g}$, observe the integral terms on the left-hand side of (1.20d) and on the right-hand side of (1.20e) can be rewritten as

[TABLE]

In view of (1.21), it is natural to address the analysis of a generalization of system (1.20), cf. (2.16) ahead, where the kernel $\eta(x-y)$, featuring the bounded, even, and positive function $\eta\in\mathrm{C}^{0}(\mathbb{R})$, is replaced by a bounded, symmetric, and positive kernel $k:\Gamma_{\mathrm{C}}\times\Gamma_{\mathrm{C}}\to[0,\infty)$, inducing the nonlocal operator

[TABLE]

1.2. Analytical results and plan of the paper

Our main result, Theorem 2.1****, states the existence of global-in-time solutions for the Cauchy problem associated with a generalized version of system (1.20), featuring the nonlocal operator $\mathcal{K}$, and where the various, concrete, subdifferential operators are replaced by maximal monotone nonlinearities. It is stated in Section 2, where we collect all the assumptions on the problem data and introduce a suitable variational formulation of our PDE system. In Section 3 we set up a suitable approximation, for which we prove a local-in-time well-posedness result by means of Schauder fixed point tecnique. In Section 4, by a priori estimates combined with compactness and monotonicity tools, we develop a passage to the limit argument and we obtain a local-in-time solution for the original problem. Finally, in Section 5 we complete the proof of Theorem 2.1, extending the local solution to a global one by a carefully devised prolongation procedure.

2. Setup and main result

Throughout the paper we shall assume that

[TABLE]

More precisely, by flat we mean that $\Gamma_{\mathrm{C}}$ is a subset of a hyperplane of $\mathbb{R}^{3}$ and on $\Gamma_{\mathrm{C}}$ the Lebesgue and Hausdorff measures $\mathcal{L}^{2}$ and $\mathcal{H}^{2}$ coincide. As for smoothness, we require that $\Gamma_{\mathrm{C}}$ has a $\mathrm{C}^{1,1}$-boundary.

Hereafter, will use the following

Notation 2.1**.**

Given a Banach space $X$, we denote by $\sideset{}{{}_{X}}{\mathop{\langle\cdot,\cdot\rangle}}$ the duality pairing between its dual space $X^{\prime}$ and $X$ itself and by $\|\cdot\|_{X}$ both the norm in $X$ and in any power of it. In particular, we shall use short-hand notation for some function spaces

[TABLE]

where we recall that

[TABLE]

The space $\mathbf{V}$ is endowed with the natural norm induced by $H^{1}(\Omega;\mathbb{R}^{3})$. Throughout the paper, we will also use that

[TABLE]

where the above embeddings have to be understood in the sense of traces.

Finally, we will use the symbols $c,c^{\prime},C,C^{\prime},\ldots$, with meaning possibly varying in the same line, to denote several positive constants only depending on known quantities. Analogously, with the symbols $I_{1},I_{2},\ldots$ we will denote several integral terms appearing in the estimates.

The bilinear forms in the momentum balance.

We recall the definition of the standard bilinear forms of linear viscoelasticity, which are involved in the variational formulation of equation (1.20a). Dealing with an anisotropic and inhomogeneous material, we assume that the fourth-order tensors $\mathbb{K}=(a_{ijkh})$ and $\mathbb{K}_{v}=(b_{ijkh})$, denoting the elasticity and the viscosity tensor, respectively, satisfy the classical symmetry and ellipticity conditions

[TABLE]

By the previous assumptions on the elasticity and viscosity coefficients, the following bilinear forms $a,b:H^{1}(\Omega;\mathbb{R}^{3})\times H^{1}(\Omega;\mathbb{R}^{3})\to\mathbb{R}$, defined by

[TABLE]

turn out to be continuous and symmetric. In particular, we have

[TABLE]

Moreover, since $\Gamma_{\mathrm{D}}$ has positive measure, by Korn’s inequality we deduce that $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$ are $\mathbf{V}$-elliptic, i.e., there exist $C_{a},C_{b}>0$ such that

[TABLE]

Assumptions on the nonlinearities of the system.

We will consider an extended version of system (1.20), where the subdifferentials $\partial I_{(-\infty,0]}$ in the boundary condition (1.20d) and in the flow rule (1.20e) for $\chi$, and $\partial I_{[0,1]}$ in (1.20e), are replaced by more general subdifferential operators.

1. (1)

We consider a function

[TABLE]

Note that as soon as $0\in\mathrm{dom}(\widehat{\alpha})$, we can always reduce to the case $\widehat{\alpha}(0)=0$ by a translation. Then, we introduce the proper, convex and lower semicontinuous functional

[TABLE]

We set {\mbox{\boldmath\alpha}}:=\partial{\mbox{\boldmath\widehat{\alpha}}}:\mathbf{Y}\rightrightarrows\mathbf{Y}^{\prime}. It follows from (2.7) that \mathbf{0}\in{\mbox{\boldmath\alpha}}(\mathbf{0}). 2. (2)

We consider

[TABLE]

and set $\rho:=\partial\widehat{\rho}:\mathbb{R}\rightrightarrows\mathbb{R}$. 3. (3)

We let

[TABLE]

and set $\beta:=\partial\widehat{\beta}:\mathbb{R}\rightrightarrows\mathbb{R}$.

The operator {\mbox{\boldmath\alpha}}:\mathbf{Y}\rightrightarrows\mathbf{Y}^{\prime} will replace $\partial I_{(-\infty,0]}$ in the boundary condition (1.20d). Thus, as soon as $\mathrm{dom}(\widehat{\alpha})\subset(-\infty,0]$, with $\alpha$ we render the impenetrability (unilateral) constraint ${\bf u}\cdot\mathbf{n}\leq 0$ a.e. on $\Gamma_{\mathrm{C}}\times(0,T)$. The operators $\rho$ and $\beta$ will generalize the subdifferentials $\partial I_{(-\infty,0]}$ and $\partial I_{[0,1]}$ in (1.20e). On the one hand, the requirement $\mathrm{dom}(\widehat{\beta})\subset[0,+\infty)$ guarantees {\hbox{\raise 1.94444pt\hbox{\chi}}}\geq 0 a.e. in $\Gamma_{\mathrm{C}}\times(0,T)$. On the other hand, starting from an initial datum {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0} fulfilling 0\leq{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}\leq 1 a.e. on $\Gamma_{\mathrm{C}}$ (cf. (2.12b) below) and taking into account that {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}\leq 0 a.e. in $\Gamma_{\mathrm{C}}\times(0,T)$ since $\mathrm{dom}(\widehat{\rho})\subset(-\infty,0]$, we will ultimately deduce that {\hbox{\raise 1.94444pt\hbox{\chi}}}\in[0,1] a.e. on $\Gamma_{\mathrm{C}}\times(0,T)$.

1. (4)

As for the kernel $k$ defining the operator $\mathcal{K}$ from (1.22), we will require that

[TABLE] 2. (5)

Finally, we will suppose that

[TABLE]

Assumptions on the problem data.

We suppose that

[TABLE]

where \beta^{0}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) denotes the minimal section of \beta({\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}). Then, from

[TABLE]

(which follows from the positivity of $\widehat{\beta}$ and from the fact that $\widehat{\beta}(0)=0$), we immediately deduce that

[TABLE]

As far as the body force $\mathbf{f}$ and the surface traction $\mathbf{h}$ are concerned, we prescribe that

[TABLE]

and we introduce $\mathbf{F}:(0,T)\to\mathbf{V}^{\prime}$ by

[TABLE]

Of course, thanks to (2.13), $\mathbf{F}\in L^{2}(0,T;\mathbf{V}^{\prime})$.

We are now in a position to give the variational formulation of the initial-boundary value problem for the (generalized) version of system (1.20) tackled in this paper.

Problem 2.2**.**

Starting from initial data ({\bf u}_{0},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) fulfilling (2.12), find a quintuple ({\bf u},{\mbox{\boldmath\zeta}},{\hbox{\raise 1.94444pt\hbox{\chi}}},\omega,\xi) with

[TABLE]

such that ${\bf u}(0)={\bf u}_{0}$, {\hbox{\raise 1.94444pt\hbox{\chi}}}(0)={\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}, and fulfilling a.e. in (0,T)

[TABLE]

Now we state the main result of this paper, which ensures the existence of solutions to Problem 2.2. Let us mention in advance that, for the selections \omega\in\rho({\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}) and \xi\in\beta({\hbox{\raise 1.94444pt\hbox{\chi}}}) we will obtain stronger integrability properties than those required with (2.15d), cf. (2.17) below. They are obtained by means of a special a priori estimate, first introduced in ****[7]****, which is tailored to the doubly nonlinear structure of (2.16c) and has to be performed in order to separately estimate the selections in \rho({\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}) and \beta({\hbox{\raise 1.94444pt\hbox{\chi}}}), cf. also Remark 3.2 ahead. It is in view of this estimate that the conditions {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}\in W (in accordance with the property {\hbox{\raise 1.94444pt\hbox{\chi}}}\in L^{\infty}(0,T;W)\cap W^{1,\infty}(0,T;H)\subset\mathrm{C}_{\mathrm{weak}}^{0}([0,T];W)) and \beta^{0}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{0})\in H from (2.12b) are needed.

Theorem 2.1**.**

Assume (2.1), (2.4), (2.7)–(2.11), and (2.13). Then, for any pair ({\bf u}_{0},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) of initial data fulfilling (2.12) there exists a quintuple ({\bf u},{\mbox{\boldmath\zeta}},{\hbox{\raise 1.94444pt\hbox{\chi}}},\omega,\xi) solving Problem 2.2 and

1. (1)

enjoying the enhanced integrability properties

[TABLE]

and such that

[TABLE] 2. (2)

fulfilling the energy-dissipation inequality

[TABLE]

for all $0\leq s\leq t\leq T$, with $\mathcal{R}$ and $\mathcal{E}$ given by (2.21) and (2.22) below.

In fact, (2.19) holds as a balance if $\widehat{\rho}$ is positively homogeneous of degree $1$ (i.e., $\widehat{\rho}(\lambda v)=\lambda\widehat{\rho}(v)$ for all $\lambda\geq 0$), e.g. in the particular case $\widehat{\rho}=I_{(-\infty,0]}$.

Remark 2.3**.**

The energy-dissipation inequality (2.19) in fact holds along any solution to Problem 2.2. It reflects the fact that system (2.16) has a (generalized) gradient system structure. In fact, it can be rewritten as the (abstract) doubly nonlinear differential inclusion

[TABLE]

involving the dissipation potential $\mathcal{R}:\mathbf{V}\times H\to[0,+\infty]$

[TABLE]

and the energy functional

[TABLE]

In (2.20), $\partial\mathcal{R}:\mathbf{V}\times H\rightrightarrows\mathbf{V}^{\prime}\times H$ is the subdifferential of $\mathcal{R}$ in the sense of convex analysis, while $\partial^{-}\mathcal{E}:\mathbf{V}\times H\rightrightarrows\mathbf{V}^{\prime}\times H$ is the Fréchet subdifferential (cf., e.g., [16]) of $\mathcal{E}$. We will not specify its definition, here, but only mention that, in the case of the specific energy functional driving system (2.16), $\partial^{-}\mathcal{E}$ is given by the sum of the convex analysis subdifferentials of the convex contributions to $\mathcal{E}$, with the Gâteau-derivatives of the nonconvex, but smooth contributions. Thus, (2.20) yields (2.16).

However, for technical reasons, we will not directly exploit the structure (2.20) in the proof of our existence result for system (2.16). Nonetheless, (2.20) underlies (2.19).

Remark 2.4**.**

No uniqueness result seems to be available for Problem 2.2, due to the doubly nonlinear character of the flow rule for $\chi$. Nonetheless, arguing in the very same way as in the proof of [1, Prop. 2.3] (cf. also the proof of Prop. 4.1 ahead), it should be possible to prove a local-in-time uniqueness result, in the particular (physical) case of $\beta=\partial I_{[0,1]}$. Namely, that any two quintuples of solutions ({\bf u}_{1},{\mbox{\boldmath\zeta}}_{1},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{1},\omega_{1},\xi_{1}) and ({\bf u}_{2},{\mbox{\boldmath\zeta}}_{2},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{2},\omega_{2},\xi_{2}) such that there exists $T_{0}\in(0,T]$ with

[TABLE]

do coincide on $[0,T_{0}]$. In turn, the above separation property is guaranteed as soon as the initial datum {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0} fulfills {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}\in[\delta,1) on $\Gamma_{\mathrm{C}}$, for some $\delta\in(0,1)$ (note that (2.12b) allows for {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0} taking the values [math] and $1$, instead).

The proof of Theorem 2.1 will be developed throughout Sections 3–5 by setting up an approximate system and proving the existence (and uniqueness) of local-in-time solutions for it (Sec. 3), passing to the limit with the approximation parameter and concluding the existence of local solutions for Problem 2.2 (Sec. 4), extending the local solution to a global one via a non-standard prolongation argument (Sec. 5).

3. The approximate system

After introducing the approximate problem in Sec. 3.1, in Sec. 3.2 we prove its local-in-time well-posedness by means of a fixed point argument combined with continuous dependence estimates.

3.1. Setup of the approximate problem

In system (3.2) below, we will replace the operators $\rho,\,\beta:\mathbb{R}\rightrightarrows\mathbb{R}$ featuring in the flow rule for $\chi$ by their Yosida regularizations $\rho_{\varepsilon}$ and $\beta_{\varepsilon}$, cf. e.g. ****[8]****. We will exploit that $\rho_{\varepsilon},\,\beta_{\varepsilon}:\mathbb{R}\to\mathbb{R}$ are Lipschitz continuous functions on $\mathbb{R}$, and denote by $\widehat{\rho}_{\varepsilon}$ and $\widehat{\beta}_{\varepsilon}$ their primitives fulfilling $\widehat{\rho}_{\varepsilon}(0)=0$ and $\widehat{\beta}_{\varepsilon}(0)=0$. The convex, $\mathrm{C}^{2}$-functions $\widehat{\rho}_{\varepsilon}$ and $\widehat{\beta}_{\varepsilon}$ are in fact the Yosida approximations of the functions $\widehat{\rho}$ and $\widehat{\beta}$, defined by

[TABLE]

We are now in a position to give the variational formulation of the approximate problem.

Problem 3.1**.**

Starting from initial data ({\bf u}_{0},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) fulfilling (2.12), find a triple ({\bf u},{\mbox{\boldmath\zeta}},{\hbox{\raise 1.94444pt\hbox{\chi}}}) as in (2.15a)–(2.15c) such that ${\bf u}(0)={\bf u}_{0}$, {\hbox{\raise 1.94444pt\hbox{\chi}}}(0)={\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}, and fulfilling a.e. in (0,T)

[TABLE]

Remark 3.2**.**

On the one hand, with these regularizations, we will able to render all a priori estimates rigorously within the frame of the approximate system. In particular, we refer to a regularity estimate that consists in testing the flow rule (3.2c) by \partial_{t}(-\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}}+\beta_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}})) (cf. the proof of Proposition 4.1). This estimate allows us to bound both -\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}} and (a selection) \xi\in\beta({\hbox{\raise 1.94444pt\hbox{\chi}}}) in $L^{\infty}(0,T;H)$, thus giving rise to (2.17) (since the estimates for {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t} and \omega\in\rho({\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}) then follow by comparison in (2.16c)). On the other hand, since $\mathrm{dom}(\beta_{\varepsilon})=\mathbb{R}$, in the approximate system (3.2) the constraint {\hbox{\raise 1.94444pt\hbox{\chi}}}\geq 0 will be no longer enforced. Because of this, the first a priori estimate performed on the approximate system (cf. again the proof of Proposition 4.1) will not have a global-in-time character, and the procedure for extending a local solution to a global one, developed in Sec. 5, will be more complex than the standard one.

3.2. Local-in-time existence for the approximate problem

This section is devoted to the proof of the following result.

Proposition 3.3**.**

Assume (2.1), (2.4), (2.7)–(2.11), and (2.13). Let ({\bf u}_{0},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) be a pair of initial data fulfilling (2.12). Then, there exists a final time $\widehat{T}>0$ such that, for every $\varepsilon>0$, Problem 3.1 admits a unique solution (u,{\mbox{\boldmath\zeta}},{\hbox{\raise 1.94444pt\hbox{\chi}}}) on $(0,\widehat{T})$, with the enhanced regularity {\hbox{\raise 1.94444pt\hbox{\chi}}}\in L^{\infty}(0,\widehat{T};W)\cap H^{1}(0,\widehat{T};V)\cap W^{1,\infty}(0,\widehat{T};H).

We will prove Proposition 3.3 by constructing an operator, defined between suitable function spaces, whose fixed points yield solutions to system (3.2), and by showing that it does admit fixed points thanks to the Schauder theorem. This procedure will yield a local-in-time solution to Problem 3.1 defined on the interval $(0,\widehat{T})$, with $\widehat{T}$ in fact independent of $\varepsilon$, cf. Remark 3.8 ahead. In view of this, in Sec. 4 we will pass to the limit in system (3.2) as $\varepsilon\downarrow 0$ and obtain the existence of local-in-time solutions to the original Problem 2.2.

Our construction of the Schauder operator will be based on two results, Lemma 3.5 and Lemma 3.6 ahead, tackling the separate solvability, on a given interval $(0,\mathsf{T})$, of the momentum balance (3.2a), with $\chi$ replaced by a given function \overline{{\hbox{\raise 1.94444pt\hbox{\chi}}}}\in L^{4}(0,\mathsf{T};H), and of the flow rule (3.2c) for the adhesive parameter, with (the trace of) ${\bf u}$ replaced by a given $\overline{{\bf u}}\in H^{1}(0,\mathsf{T};L^{4}(\Gamma_{\mathrm{C}}))$. Preliminarily, we fix the properties of the nonlocal operator $\mathcal{K}$ from (1.22) in the following result, whose proof is left to the reader.

Lemma 3.4**.**

Assume (2.10). Then, $\mathcal{K}$ is well defined, linear and continuous from $L^{1}(\Gamma_{\mathrm{C}})$ to $L^{\infty}(\Gamma_{\mathrm{C}})$, with

[TABLE]

Furthermore, for every $1\leq p<\infty$ the operator $\mathcal{K}$ is continuous from $L^{1}(\Gamma_{\mathrm{C}})$, equipped with the weak topology, to $L^{p}(\Gamma_{\mathrm{C}})$ with the strong topology (i.e.  if $w_{n}\,\rightharpoonup\,w$ in $L^{1}(\Gamma_{\mathrm{C}})$ then $\mathcal{K}[w_{n}]\rightarrow\mathcal{K}[w]$ in $L^{p}(\Gamma_{\mathrm{C}})$). Finally, there holds

[TABLE]

We start by tackling the momentum balance (3.2a). In what follows we will denote by $Q_{i}$ and $\widetilde{Q}_{i}$, $i=1,\ldots$, computable, non-negative and continuous functions, monotone increasing w.r.t. each of their variables, that will enter in the a priori estimates holding for the solutions to the momentum balance/adhesive flow rule.

Lemma 3.5**.**

Assume (2.1), (2.4), (2.7), (2.10), and (2.13). Let ${\bf u}_{0}$ fulfill (2.12a). Then, for every \overline{{\hbox{\raise 1.94444pt\hbox{\chi}}}}\in L^{4}(0,\mathsf{T};H) there exists a unique pair ({\bf u},{\mbox{\boldmath\zeta}})\in H^{1}(0,\mathsf{T};\mathbf{V})\times L^{2}(0,\mathsf{T};\mathbf{Y}^{\prime}) fulfilling ${\bf u}(0)={\bf u}_{0}$ and

[TABLE]

for a.a. $t\in(0,\mathsf{T}),$ with

[TABLE]

Furthermore, there exists a positive function $\widetilde{Q}_{1}$ such that for every \overline{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{1}\,,\overline{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{2}\in L^{4}(0,\mathsf{T};H), with ${\bf u}_{1},\,{\bf u}_{2}$ the associated solutions of (3.5) starting from ${\bf u}_{0}$, there holds

[TABLE]

We denote by

[TABLE]

Proof.

A standard modification of the proof of [1, Prop. 4.2], which is in turn based on a time-discretization procedure, yields the existence statement. Estimate (3.6) follows from testing (3.5) by ${\bf u}_{t}$ and integrating on a time interval $(0,t)$, which gives

[TABLE]

thanks to the chain-rule formula

[TABLE]

In view of (2.6) and of the positivity of \widehat{{\mbox{\boldmath\alpha}}}, we have that

[TABLE]

Moreover, applying the Young inequality, we get

[TABLE]

We then perform the following estimates for the last two terms on the right-hand side of (3.9)

[TABLE]

where we have used (2.3), as well as estimate (3.3), yielding

[TABLE]

All in all, we obtain

[TABLE]

whence (3.6) by the Gronwall Lemma.

In order to show the continuous dependence estimate (3.7), we subtract (3.5) for given \overline{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{2} from (3.5) for \overline{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{1} and test the resulting equation by ${\bf u}_{1}-{\bf u}_{2}$. With calculations similar to those above, we obtain, by virtue of (3.3) and the Young inequality, that

[TABLE]

whence (3.7). Finally, a comparison in (3.5) yields the uniqueness of $\zeta$, too. ∎

We now address the flow rule (3.2c) for the adhesion parameter.

Lemma 3.6**.**

Assume (2.1), (2.8)–(2.11). Let {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0} fulfill (2.12b). Then, for every $\overline{u}\in H^{1}(0,\mathsf{T};L^{4}(\Gamma_{\mathrm{C}};\mathbb{R}^{3}))$ there exists a unique {\hbox{\raise 1.94444pt\hbox{\chi}}}\in L^{\infty}(0,\mathsf{T};W)\cap H^{1}(0,\mathsf{T};V)\cap W^{1,\infty}(0,\mathsf{T};H) fulfilling {\hbox{\raise 1.94444pt\hbox{\chi}}}(0)={\hbox{\raise 1.94444pt\hbox{\chi}}}_{0} and

[TABLE]

with

[TABLE]

Furthermore, there exists a function $\widetilde{Q}_{2}$ such that for every $\overline{{\bf u}}_{1},\,\overline{{\bf u}}_{2}\in L^{4}(0,\mathsf{T};L^{4}(\Gamma_{\mathrm{C}};\mathbb{R}^{3}))$, with {\hbox{\raise 1.94444pt\hbox{\chi}}}_{1},\,{\hbox{\raise 1.94444pt\hbox{\chi}}}_{2} the associated solutions to (3.14) emanating from the same initial datum {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}, there holds

[TABLE]

We denote by

[TABLE]

Proof.

Observe that (3.14) can be recast as the abstract doubly nonlinear equation

[TABLE]

with the dissipation potential \mathcal{R}_{\mathsf{{\hbox{\raise 1.3611pt\hbox{\chi}}}}}^{\varepsilon}(\dot{\hbox{\raise 1.94444pt\hbox{\chi}}}):=\tfrac{1}{2}\|\dot{\hbox{\raise 1.94444pt\hbox{\chi}}}\|_{H}^{2}+\int_{\Gamma_{\mathrm{C}}}\widehat{\rho}_{\varepsilon}(\dot{\hbox{\raise 1.94444pt\hbox{\chi}}})\,\mathrm{d}x (cf. (2.21)), and the driving energy $\mathcal{E}_{2}$ from (2.22). In fact, here both the convex analysis subdifferential \partial\mathcal{R}_{\mathsf{{\hbox{\raise 1.3611pt\hbox{\chi}}}}}^{\varepsilon} and the Fréchet subdifferential $\partial^{-}\mathcal{E}_{2}$ reduce to singletons. Under the condition that $\overline{\bf u}\in H^{1}(0,\mathsf{T};L^{4}(\Gamma_{\mathrm{C}};\mathbb{R}^{3}))$, we may apply [14, Theorem 2.2] to conclude the existence of a solution {\hbox{\raise 1.94444pt\hbox{\chi}}}\in L^{2}(0,\mathsf{T};W)\cap L^{\infty}(0,\mathsf{T};V)\cap H^{1}(0,\mathsf{T};H) for (the Cauchy problem for) (3.18), via a time discretization procedure. The enhanced regularity $L^{\infty}(0,\mathsf{T};W)\cap H^{1}(0,\mathsf{T};V)\cap W^{1,\infty}(0,\mathsf{T};H)$ may be inferred by performing, on the time discrete level, the regularity estimate that consists in testing the flow rule by \partial_{t}(-\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}}+\beta_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}})) (cf. the proof of Prop. 4.1).

Estimate (3.15) follows from testing (3.14) by {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t} and integrating in time. Taking into account that

[TABLE]

since $\rho_{\varepsilon}(0)=0$ and $\rho_{\varepsilon}$ is increasing, and using that \int_{0}^{t}\int_{\Gamma_{\mathrm{C}}}\beta_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}}){\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}=\int_{\Gamma_{\mathrm{C}}}\widehat{\beta}_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}}(t))\,\mathrm{d}x-\int_{\Gamma_{\mathrm{C}}}\widehat{\beta}_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{0})\,\mathrm{d}x by the chain rule, we get

[TABLE]

where we have also used that \int_{\Gamma_{\mathrm{C}}}\widehat{\beta}_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{0})\,\mathrm{d}x\leq\int_{\Gamma_{\mathrm{C}}}\widehat{\beta}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{0})\,\mathrm{d}x. With Young’s inequality, we estimate

[TABLE]

with (1) due to the Lipschitz continuity of $\gamma^{\prime}$, (2) to estimate (3.12), and (3) again following from (3.3), via \|\mathcal{K}[{\hbox{\raise 1.94444pt\hbox{\chi}}}|\overline{\bf u}|^{2}]\|_{L^{\infty}(\Gamma_{\mathrm{C}})}\leq C\|{\hbox{\raise 1.94444pt\hbox{\chi}}}|\overline{\bf u}|^{2}\|_{L^{1}(\Gamma_{\mathrm{C}})}\leq C\|\overline{\bf u}\|_{L^{4}(\Gamma_{\mathrm{C}})}^{2}\|{\hbox{\raise 1.94444pt\hbox{\chi}}}\|_{H}. Moreover, a comparison in (3.14) also provides a bound for \|\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}}\|_{L^{2}(0,\mathsf{T};H)}, whence the estimate for $\chi$ in $L^{2}(0,\mathsf{T};W)$ by standard elliptic regularity results, relying on the assumption of $\mathrm{C}^{1,1}$-boundary for $\Gamma_{\mathrm{C}}$.

Finally, in order to prove (3.16), let us preliminarily introduce, for fixed $\overline{\bf u}\in L^{4}(0,\mathsf{T};L^{4}(\Gamma_{\mathrm{C}};\mathbb{R}^{3}))$, the Lipschitz continuous mapping

[TABLE]

Then, we subtract equation (3.14), written for a given $\overline{{\bf u}}_{2}$ in $L^{4}(0,\mathsf{T};L^{4}(\Gamma_{\mathrm{C}};\mathbb{R}^{3}))$, from (3.14) for $\overline{{\bf u}}_{1}$, test the resulting relation by \partial_{t}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{1}-{\hbox{\raise 1.94444pt\hbox{\chi}}}_{2}). We use that

[TABLE]

a.e. in $(0,\mathsf{T})$ by the monotonicity of $\rho_{\varepsilon}$, and, on the right-hand side, we estimate

[TABLE]

All in all, we obtain

[TABLE]

a.e. in $(0,\mathsf{T})$. For (1), we have used the Lipschitz continuity of $\gamma^{\prime}$ and of $\beta_{\varepsilon}$ (with Lipschitz constant $\tfrac{1}{\varepsilon}$), and we have for example estimated

[TABLE]

where (2) follows from (3.3). Therefore, from (3.19) we deduce, via the Gronwall Lemma, that

[TABLE]

for almost all $t\in(0,\mathsf{T})$. Upon integrating in time we then conclude estimate (3.16) for \|{\hbox{\raise 1.94444pt\hbox{\chi}}}_{1}-{\hbox{\raise 1.94444pt\hbox{\chi}}}_{2}\|_{H^{1}(0,\mathsf{T};H)}.

We then also recover the bound for \|{\hbox{\raise 1.94444pt\hbox{\chi}}}_{1}-{\hbox{\raise 1.94444pt\hbox{\chi}}}_{2}\|_{L^{\infty}(0,\mathsf{T};V)}. ∎

Solution operator for Problem 3.1 and application of the Schauder theorem.

**For fixed $M>0$ we consider the closed ball **

[TABLE]

In view of (3.8) and (3.17), the operator

[TABLE]

Our next result shows that, at least for small times, the map $\mathcal{T}$ complies with the conditions of the Schauder theorem.

Lemma 3.7**.**

Assume (2.1), (2.4), (2.7)–(2.11), and (2.13). Let the initial data ({\bf u}_{0},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) fulfill (2.12). Then, there exists $\widehat{T}>0$ such that the operator $\mathcal{T}$

1. (1)

maps $\mathcal{S}_{M}(\widehat{T})$ into itself; 2. (2)

is continuous with respect to the strong topology of $L^{4}(0,\widehat{T};H)$; 3. (3)

maps $\mathcal{S}_{M}(\widehat{T})$ into a compact subset of $L^{4}(0,\widehat{T};H)$.

Proof.

Ad (1): Combining estimates (3.6) and (3.15) we find for every $\mathsf{T}\in(0,T]$

[TABLE]

Therefore,

[TABLE]

so that property (1) follows by choosing

[TABLE]

Ad (2): Let ({\hbox{\raise 1.94444pt\hbox{\chi}}}_{n})_{n},\,{\hbox{\raise 1.94444pt\hbox{\chi}}}\subset\mathcal{S}_{M}(\widehat{T}) fulfill {\hbox{\raise 1.94444pt\hbox{\chi}}}_{n}\to{\hbox{\raise 1.94444pt\hbox{\chi}}} in $L^{4}(0,\widehat{T};H)$. Combining (3.6) with (3.7) we find that \mathcal{T}_{1}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{n})\to\mathcal{T}_{1}({\hbox{\raise 1.94444pt\hbox{\chi}}}) in $L^{\infty}(0,\widehat{T};\mathbf{V})$. We then use (3.15) and (3.16) to conclude that \mathcal{T}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{n})=\mathcal{T}_{2}(\mathcal{T}_{1}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{n}))\to\mathcal{T}({\hbox{\raise 1.94444pt\hbox{\chi}}})=\mathcal{T}_{2}(\mathcal{T}_{1}({\hbox{\raise 1.94444pt\hbox{\chi}}})) in $L^{\infty}(0,\widehat{T};V)\cap H^{1}(0,\widehat{T};H)$, and thus in $L^{4}(0,\widehat{T};H)$.

Ad (3): The compactness property easily follows from combining estimates (3.6) and (3.15). ∎

**Eventually, we are in a position to conclude the proof of Proposition 3.3. The existence statement follows from Lemma 3.7 via the Schauder theorem. Uniqueness of solutions to Problem 3.1 (the same argument would also yield continuous dependence on the initial and problem data) ensues by adding up estimates (3.13) and (3.19) and applying the Gronwall Lemma. **** **

Remark 3.8**.**

Estimate (3.23) shows that the local-existence time $\widehat{T}$ does not depend on $\varepsilon$.

A closer examination of the proof of Lemma 3.7 (based on Lemmas 3.5 and 3.6) in fact reveals that $\widehat{T}$ does not depend on the specific initial conditions ${\bf u}_{0}$, {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0} either, but only on quantities related to them. In other words, for any “ball” of initial data

[TABLE]

there exists a final time $\widehat{T}_{r}>0$, only depending on $r$ and on known constants, such that, for any ({{\bf u}}_{0},{{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}})\in\mathcal{I}(r) and for any $\varepsilon>0$, the approximate Problem 3.1, supplemented with the initial data ({{\bf u}}_{0},{{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}}) admits a solution on the interval $(0,\widehat{T}_{r})$.

4. Local existence for Problem 2.2

Throughout this section we will highlight the dependence of the local solution for Problem 3.1 (found in Proposition 3.3) on the approximation parameter $\varepsilon$, by denoting it with ({\bf u}_{\varepsilon},{\mbox{\boldmath\zeta}}_{\varepsilon},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}). In this section, we perform an asymptotic analysis as $\varepsilon\downarrow 0$ of the sequence ({\bf u}_{\varepsilon},{\mbox{\boldmath\zeta}}_{\varepsilon},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon})_{\varepsilon} on the existence interval $(0,\widehat{T})$, which does not depend on $\varepsilon$, cf. Remark 3.8.

We thus obtain the following existence result of local-in-time solutions for Problem 2.2.

Proposition 4.1**.**

Assume (2.1), (2.4), (2.7)–(2.11), and (2.13). Let ({\bf u}_{0},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}) fulfill (2.12). Then, for every vanishing sequence $\varepsilon_{k}\downarrow 0$ as $k\to\infty$ there exists a quintuple (\widehat{{\bf u}},\widehat{{\mbox{\boldmath\zeta}}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}},\widehat{\omega},\widehat{\xi}), with

[TABLE]

such that the following convergences hold as $k\to\infty$

[TABLE]

Besides, (\widehat{{\bf u}},\widehat{{\mbox{\boldmath\zeta}}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}},\widehat{\omega},\widehat{\xi}) is a solution of Problem 2.2, fulfilling the energy-dissipation inequality (2.19) (holding as a balance if $\widehat{\rho}$ is $1$-positively homogeneous), on $(0,\widehat{T})$.

Proof.

Step $1$: a priori estimates. We derive some a priori estimates for (suitable norms of) the family ({\bf u}_{\varepsilon},{\mbox{\boldmath\zeta}}_{\varepsilon},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon})_{\varepsilon} on the interval $(0,\widehat{T})$. First of all, recall that the solution components {\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon} are found as fixed points of the Schauder operator $\mathcal{T}$ from (3.21) in the ball $\mathcal{S}_{M}(\widehat{T})$. Therefore, in view of estimate (3.6) there exists a positive constant $C$, independent of $\varepsilon$, such that

[TABLE]

Then, (3.15) (cf. also (3.22)) yields

[TABLE]

Secondly, we perform a comparison argument in (3.2a). Observe that, by the second of (3.3) and (4.3), we have

[TABLE]

Therefore, on account of (4.2)–(4.4) we obtain

[TABLE]

Next, we establish a further regularity estimate on {\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}. We multiply (3.2c) by \partial_{t}(-\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}+\beta_{\varepsilon}({\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon})) and we integrate over $\Gamma_{\mathrm{C}}\times(0,t)$. Observe that all the calculations below can be made rigorous, for the very same system (3.2), by arguing with difference quotients. We have

[TABLE]

where $I_{i},i=1,2,3$, will be defined and estimated in what follows. We note that the integral terms on the left-hand side of (4.6) are non-negative, thanks to the monotonicity of $\rho_{\varepsilon}$ and of $\beta_{\varepsilon}$ and to the fact that $\rho_{\varepsilon}(0)=0$. Moreover, integrating by parts and taking into account (2.11), (2.12b), (4.2), and (4.3) we have

[TABLE]

Arguing in a similar way and using (2.12a), (2.12b), (3.3), (4.2), and (4.4), we infer

[TABLE]

Prior to estimating $I_{3}$, we observe that, again by the second of (3.3) combined with (4.2)–(4.3), we have

[TABLE]

Then, exploiting (2.12a), (2.12b), (3.3), (4.2)–(4.3), and (4.8), we get

[TABLE]

We plug the above estimates for $I_{i},i=1,2,3$ into (4.6), take into account the previously obtained (4.2), (4.3), and apply the Gronwall Lemma. In this way, we conclude that

[TABLE]

On the other hand, by the monotonicity of $\beta_{\varepsilon}$, we have for a.e. $t\in(0,\widehat{T})$

[TABLE]

Hence, by (4.3) and the aforementioned elliptic regularity results valid on $\Gamma_{\mathrm{C}}$, we deduce that

[TABLE]

and, besides,

[TABLE]

for some positive constant $c$ independent of $\varepsilon$. Finally, a comparison in (3.2c), on account of (4.2), (4.11), and (4.12), entails

[TABLE]

Step $2$: compactness argument.

Thanks to the above estimates, by well-known weak and weak* compactness results, we deduce that, for every vanishing sequence $(\varepsilon_{k})_{k}$ there exists a (not relabeled) subsequence of \{({\bf u}_{\varepsilon_{k}},{\mbox{\boldmath\zeta}}_{\varepsilon_{k}},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon_{k}})\}_{k} such that convergences (4.1a), (4.1d), and (4.1f)–(4.1h) hold as $\varepsilon_{k}\downarrow 0$. Moreover, using Aubin-Lions compactness arguments and the generalized Ascoli theorem (cf., e.g., [20]), we also obtain (4.1b), (4.1c) and (4.1e), entailing that the pair (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}) (see Prop. 4.1) fulfills the initial conditions ${\bf u}(0)={\bf u}_{0}$ and {\hbox{\raise 1.94444pt\hbox{\chi}}}(0)={\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}.

Step $3$: limit passage in the approximate system (3.2).

We exploit convergences (4.1). To simplify notation we omit the subindex $k$ and denote the sequences of approximate solutions directly by the index $\varepsilon$. We first discuss the convergence of the non-local terms \mathcal{K}[{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}] and \mathcal{K}[{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}|{\bf u}_{\varepsilon}|^{2}]. We mainly exploit the continuity properties of the operator $\mathcal{K}$ stated by Lemma 3.4. We first observe that (4.1e) implies that {\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}\rightarrow\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}} in $\mathrm{C}^{0}([0,\widehat{T}];L^{1}(\Gamma_{\mathrm{C}}))$, so that we can deduce

[TABLE]

We analogously proceed to deal with the term \mathcal{K}[{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}|{\bf u}_{\varepsilon}|^{2}]. Indeed, (4.1c) and (4.1e) lead to the strong convergence {\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon}|{\bf u}_{\varepsilon}|^{2}\rightarrow\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}|\widehat{{\bf u}}|^{2} in $\mathrm{C}^{0}([0,\widehat{T}];L^{1}(\Gamma_{\mathrm{C}}))$, since by Sobolev embeddings {\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon} strongly converges in $\mathrm{C}^{0}([0,\widehat{T}];L^{p}(\Gamma_{\mathrm{C}}))$ for any $1\leq p<\infty$, while $|{\bf u}_{\varepsilon}|^{2}\rightarrow|\widehat{{\bf u}}|^{2}$ in $\mathrm{C}^{0}([0,\widehat{T}];L^{q}(\Gamma_{\mathrm{C}}))$ for all $1\leq q<2$. Thus, we can deduce

[TABLE]

Finally, (4.14) and (4.1c) with (4.4), yield

[TABLE]

Analogously, we have that (4.14) and (4.1c), (4.1e) imply (at least) that

[TABLE]

Now, it is a standard matter to pass to the limit (weakly) in the momentum balance (3.2a) and in the flow rule (3.2c) and conclude that (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}},\widehat{{\mbox{\boldmath\zeta}}},\widehat{\xi},\widehat{w}) fulfill (2.16a) and (2.16c). Actually, to complete our proof it remains to identify (\widehat{{\mbox{\boldmath\zeta}}},\widehat{\xi},\widehat{w}). We point out that (4.1e) and (4.1g) lead to

[TABLE]

for all $t\in(0,\widehat{T}]$, whence the identification \widehat{\xi}\in\beta(\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}) a.e. in $\Gamma_{\mathrm{C}}\times(0,\widehat{T})$. Then, we aim to prove that

[TABLE]

We first prove that

[TABLE]

whence (4.18). Let us test (3.2a) by ${\bf u}_{\varepsilon}$ and integrate over $(0,t)$. This gives

[TABLE]

Taking the $\limsup$ as $\varepsilon\searrow 0$, exploiting the lower semicontinuity of the bilinear forms $a$ and $b$, as well as (4.1a)–(4.1e), (4.15), and (3.4), we can infer that

[TABLE]

as desired. For the above inequality, we have in particular used

[TABLE]

due to (3.4), (4.1c), (4.1e), and (4.15). Analogously we proceed by testing (3.2c) by \partial_{t}{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon} and integrate over $(0,t)$ to prove that

[TABLE]

whence (4.19). We have

[TABLE]

First, we recall that by Mosco convergence of $\widehat{\beta}_{\varepsilon}$ to $\widehat{\beta}$, we have

[TABLE]

Thus, taking the limsup as $\varepsilon\searrow 0$ of both sides of (4.24) and exploiting (2.11), (4.1a)–(4.1e), (4.15), (4.16) and the lower semicontinuity properties of the Lebesgue norms, we get

[TABLE]

which gives (4.23).

The energy-dissipation inequality (2.19) follows from testing the momentum balance (2.16a) by ${\bf u}_{t}$, the flow rule (2.16c) by {\hbox{\raise 1.94444pt\hbox{\chi}}}_{t}, integrating along the time interval $(0,\widehat{T})$, and adding the above relations. The key ingredients to deduce (2.19) are the chain rules $\int_{s}^{t}a({\bf u},{\bf u}_{t})\,\mathrm{d}r=\tfrac{1}{2}a({\bf u}(t),{\bf u}(t))-\tfrac{1}{2}a({\bf u}(s),{\bf u}(s))$, (3.10), and

[TABLE]

where for (1) we have also used the symmetry property (3.4). In particular, observe the first and second chain-rule identities allow us to combine the contributions from the momentum balance with those from the adhesive flow rule. We also use that

[TABLE]

since $\widehat{\rho}(0)=0$. Then, (2.19) follows with straightforward calculations.

Observe that, if $\widehat{\rho}$ is $1$-positively homogeneous, (4.27) in fact holds with an equality sign. Therefore, (2.19) is valid as a balance. ∎****

5. From a local to a global solution: conclusion of the proof of Theorem 2.1

The standard procedure for extending the local solution of Problem 2.2 found with Prop. 4.1 to a global one would involve deriving suitable a priori estimates on (local) solutions of Prob. 2.2 on a generic interval $(0,\mathsf{T})$, $\mathsf{T}\in(0,T]$, and showing that such estimates are in fact independent of the final time $\mathsf{T}$. This would allow us to consider some maximal extension of the local solution from Prop. 4.1 and conclude, by a classical contradiction argument, that it must be defined on the whole interval $(0,T)$.

In the present case, in order to carry out the maximal extension argument, it would be necessary to rely on a global-in-time estimate (1) in $\mathrm{C}^{0}([0,\mathsf{T}];\mathbf{V})$ for ${\bf u}$; (2) in $L^{\infty}(0,\mathsf{T};W)\cap H^{1}(0,\mathsf{T};V)\subset\mathrm{C}_{\mathrm{weak}}^{0}([0,\mathsf{T}];W)$ for $\chi$, in accordance with the requirements ${\bf u}_{0}\in\mathbf{V}$ and {\hbox{\raise 1.94444pt\hbox{\chi}}}_{0}\in W on the initial data. However, while for ${\bf u}$ a global estimate in $H^{1}(0,\mathsf{T};\mathbf{V})$ directly follows from the energy-dissipation inequality (2.19) (cf. Lemma 5.2 ahead), from (2.19) we can derive global estimates for $\chi$ only in the spaces $L^{\infty}(0,\mathsf{T};V)\cap H^{1}(0,\mathsf{T};H)$ (and, as a by product, cf. Lemma 3.6, in $L^{2}(0,\mathsf{T};W)$, as well). In turn, the enhanced estimate $L^{\infty}(0,\mathsf{T};W)\cap H^{1}(0,\mathsf{T};V)$ seems to be obtainable only upon performing the (formally written, here) test by \partial_{t}(-\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}}+\xi), with \xi\in\beta({\hbox{\raise 1.94444pt\hbox{\chi}}}). As shown by the proof of Prop. 4.1, such a test can be made rigorous only in the frame of the approximate system (3.2). Unfortunately, for (3.2) the basic “energy” estimates for ${\bf u}$ in $H^{1}(0,\mathsf{T};\mathbf{V})$ and for $\chi$ in $L^{\infty}(0,\mathsf{T};V)\cap H^{1}(0,\mathsf{T};H)$ do not possess a global-in-time character, cf.  Remark 3.2 and again the proof of Prop. 4.1, essentially because the lack of the positivity constraint on $\chi$ does not allow one to control from below the energy \mathcal{E}({\bf u}(t),{\hbox{\raise 1.94444pt\hbox{\chi}}}(t)) on the left-hand side of (2.19).

These technical difficulties related to the extension procedure were already manifest in the framework of the adhesive contact system first investigated in ****[1]****, where nonlocal effects were disregarded. In Sec. 5 therein, to overcome them we have developed a careful prolongation argument, where the key idea is to extend the (${\bf u}$- and $\chi$-components of a) local solution (\widehat{{\bf u}},\widehat{{\mbox{\boldmath\zeta}}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}},\widehat{\omega},\widehat{\xi}), along with its ‘approximability properties’. The latter play a key role in recovering the enhanced regularity estimate for the $\chi$-component in $L^{\infty}(0,\mathsf{T};W)\cap H^{1}(0,\mathsf{T};V)$.

In what follows, we will adapt the argument from ****[1]**** to our own situation. To avoid overburdening the paper, we will explain the steps of the procedure but omit several details and often refer to the calculations in ****[1, Sec. 5]****, which could indeed be repeated with minor changes in the present framework.

Scheme of the proof of the extension procedure.

For the intents and purposes of our extension argument, we slightly modify the terminology introduced in Sec. 2 and call solution of Problem 2.2 on an interval $(0,\mathsf{T})$

[TABLE]

we will adopt the very same convention for the solutions of the approximate Problem 3.1.

Recall that, in Prop. 4.1 we have shown that, along a sequence $\varepsilon_{k}\downarrow 0$, the (unique) solutions ({\bf u}_{\varepsilon_{k}},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon_{k}}) of Prob. 3.1 (with regularization parameter $\varepsilon_{k}$) converge to a solution (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}) of Prob. 2.2 on the time interval $(0,\widehat{T})$. In the following lines, for a fixed $\mathsf{T}\in(0,T]$ we will denote by ({\bf u}_{\varepsilon_{k}}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon_{k}}^{\mathsf{T}}) the unique solution of Prob.  $\ref{prob:approx}_{\varepsilon_{k}}$ on $(0,\mathsf{T})$: clearly, for $\mathsf{T}\geq\widehat{T}$, ({\bf u}_{\varepsilon_{k}}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon_{k}}^{\mathsf{T}}) is the (unique) extension of ({\bf u}_{\varepsilon_{k}},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon_{k}}).

We are now in a position to define the solution concept for Problem 2.2 that we are going to extend on the whole interval $[0,T]$.

Definition 5.1**.**

Let $\mathsf{T}\in(0,T]$. We call a solution (in the sense of (5.1)) ({\bf u}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{\mathsf{T}}) of Problem 2.2 on $(0,\mathsf{T})$ an approximable solution if there exists a (not-relabeled) subsequence of $(\varepsilon_{k})_{k}$ such that the related sequence ({\bf u}_{\varepsilon_{k}}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}_{\varepsilon_{k}}^{\mathsf{T}})_{k} of solutions of Problem $\ref{prob:approx}_{\varepsilon_{k}}$ on $(0,\mathsf{T})$ converge to ({\bf u}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{\mathsf{T}}) in the sense

[TABLE]

It is immediate to check that, for all $\mathsf{T}\in(\widehat{T},T]$ the functions ({\bf u}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{\mathsf{T}}) are a proper extension of (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}).

Let us introduce the set

[TABLE]

The calculations from the proof of Prop. 4.1 reveal that ({\bf u}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{\mathsf{T}}) fulfill the energy-dissipation inequality (2.19) (as a balance if $\widehat{\rho}$ is $1$-positively homogeneous) along any sub-interval $[s,t]\subset[0,\mathsf{T}]$. Clearly, $\widehat{T}\in\mathscr{T}$, which is thus non-empty. We aim to show that

[TABLE]

In this way, we will conclude that (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}) extends to a global solution on (0,T), fulfilling (2.19).

The proof of (5.3) is in turn split in the following steps:

**Step $1$: **

we show that the basic energy estimates for approximable solutions ({\bf u}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{\mathsf{T}}) hold with constants independent of the final time $\mathsf{T}$;

**Step $2$: **

we deduce that the local solution (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}) admits an extension ({\bf u}^{*},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}) on $(0,T^{*})$, with ${\bf u}^{*}\in H^{1}(0,T^{*};\mathbf{V})$ and {\hbox{\raise 1.94444pt\hbox{\chi}}}\in L^{2}(0,T^{*};W)\cap L^{\infty}(0,T^{*};V)\cap H^{1}(0,T^{*};H);

**Step $3$: **

we show that ({\bf u}^{*},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}) is a solution of Problem 2.2 (in the sense of (5.1)) on $(0,T^{*})$;

**Step $4$: **

we conclude (5.3) via a contradiction argument.

We conclude this section by addressing the single steps with slightly more detail.

Step $1$:

We have the following result.

Lemma 5.2**.**

Assume (2.1), (2.4), (2.7)–(2.11), and (2.13). Then, there exists a positive function $Q_{4}$ such that, for any $\mathsf{T}>0$ and any solution ({\bf u},{\hbox{\raise 1.94444pt\hbox{\chi}}}) of Problem 2.2 on $(0,\mathsf{T})$ there holds

[TABLE]

Sketch of the proof.

We consider the energy-dissipation inequality (2.19) on the interval $(0,\mathsf{T})$ and observe that the last integral term on the right-hand side can be absorbed into the dissipative term $\int_{0}^{\mathsf{T}}b({\bf u}_{t},{\bf u}_{t})\,\mathrm{d}r$ on the left-hand side, cf. (3.11). Therefore, taking into account that

[TABLE]

from (2.19) we gather the bounds

[TABLE]

whence the estimates for ${\bf u}$ and $\chi$ in $H^{1}(0,\mathsf{T};\mathbf{V})$ and $L^{\infty}(0,\mathsf{T};V)\cap H^{1}(0,\mathsf{T};H)$. The estimate for $\chi$ in $L^{2}(0,\mathsf{T};W)$ follows from a comparison for -\Delta{\hbox{\raise 1.94444pt\hbox{\chi}}} in the flow rule for $\chi$, cf. the proof of Lemma 3.6. ∎

Step $2$:

We consider a family ({\bf u}^{\mathsf{T}},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{\mathsf{T}})_{\mathsf{T}\in\mathscr{T}} of approximable solutions, and define

[TABLE]

We consider a sequence $(\mathsf{T}_{m})_{m}\subset\mathscr{T}$ with $\mathsf{T}_{m}\uparrow T^{*}$ and extract a (not) relabeled subsequence (\widetilde{{\bf u}}_{\mathsf{T}_{m}},\widetilde{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{\mathsf{T}_{m}})_{m} suitably converging to a pair ({\bf u}^{*},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}) defined on $[0,T^{*}]$, which extends (\widehat{{\bf u}},\widehat{{\hbox{\raise 1.94444pt\hbox{\chi}}}}).

Step $3$:

We show that {\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}\in L^{\infty}(0,T^{*};W)\cap H^{1}(0,T^{*};V)\cap W^{1,\infty}(0,T^{*};H) and that ({\bf u}^{*},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}) is a solution of Problem 2.2, in the sense that

[TABLE]

The proof of enhanced regularity for {\hbox{\raise 1.94444pt\hbox{\chi}}}^{*} and of the existence of the triple ({\mbox{\boldmath\zeta}}^{*},\omega^{*},\xi^{*}) relies on the very notion of approximable solution. For each element of the sequence (\widetilde{{\bf u}}_{\mathsf{T}_{m}},\widetilde{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{\mathsf{T}_{m}})_{m} from Step 2 we pick a sequence of solutions to the approximate Problem 3.1 on $(0,\mathsf{T}_{m})$, converging to (\widetilde{{\bf u}}_{\mathsf{T}_{m}},\widetilde{{\hbox{\raise 1.94444pt\hbox{\chi}}}}_{\mathsf{T}_{m}}) for fixed $m\in\mathbb{N}$. With a diagonalization procedure we extract a subsequence converging to ({\bf u}^{*},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}). We perform the regularity estimates leading to the enhanced properties of {\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}, and to the existence of the triple ({\mbox{\boldmath\zeta}}^{*},\omega^{*},\xi^{*}), on the level of the approximate system, by repeating the very same calculations in the proof of Proposition 4.1. We refer the reader to ****[1, Sec. 5]**** for all the details of the argument.

Step $4$:

Suppose that $T^{*}<T$. To obtain a contradiction, it is sufficient to extend the approximable solution ({\bf u}^{*},{\hbox{\raise 1.94444pt\hbox{\chi}}}^{*}), defined on the interval $[0,T^{*}]$, to an approximable solution on the interval $[0,T^{*}+\eta]$ for some $\eta>0$. The argument for this is completely analogous to the one developed in ****[1, Sec. 5]**** for the adhesive contact system without nonlocal effects. We once again refer to ****[1]**** for all details.

**In this way, we deduce (5.3) and thus conclude the proof of Theorem 2.1. **** **