aa r X i v : . [ m a t h . A P ] N ov Nonlinear Quasi-static Poroelasticity
Lorena Bociu ∗ Justin T. Webster † November 26, 2020
Abstract
We analyze a quasi-static Biot system of poroelasticity for both compressible and incompressibleconstituents. The main feature of this model is a nonlinear coupling of pressure and dilationthrough the system’s permeability tensor. Such a model has been analyzed previously fromthe point of view of constructing weak solutions through a fully discretized approach. In thistreatment, we consider simplified Dirichlet type boundary conditions in the elastic displacementand pressure variables and give a full treatment of weak solutions. Our construction of weaksolutions for the nonlinear problem is natural and based on a priori estimates, a requisite featurein addressing the nonlinearity. This is in contrast to previous work which exploits linearity ormonotonicity in the permeability, both of which are not available here. We utilize a spatialsemi-discretization and employ a multi-valued fixed point argument in for a clear constructionof weak solutions. We also provide regularity criteria for uniqueness of solutions.Keywords: poroelasticity, nonlinear coupling, implicit evolution equations, fixed point methods : 74F10, 76S05, 35M13, 35A01, 35B65, 35Q86, 35Q92Acknowledged Support: L. Bociu was partially supported by NSF-DMS 1555062 (CAREER).J.T. Webster was partially supported by NSF-DMS 1907620.
Poro-elasticity refers to fluid flow (Darcy flow) within a deformable, porous medium. The de-velopment of this field has been inspired by geophysics and petroleum engineering problems, inparticular reservoir, environmental and earthquake engineering. Mathematically, the subject wasinitiated by the 1D work of Terzaghi in the 1920s, and the groundbreaking consolidation theorydeveloped by Biot in the 1940–50s [3], which started the rapid development and progress of thisfield. The related literature is now abundant, and we only list here representative fundamentaltreatments: [2, 6, 10, 11, 17, 20, 25]. In all of the works motivated by geophysical applications, theporoelastic structures under consideration are typically soil and/or rock (for instance, in most ofthe aforementioned references). However, cartilages, bones, as well as brain, heart, and liver tissueetc., are also examples of poroelastic structures. Therefore, the theory of poroelasticity can beused and applied to fluid flows inside cartilages, bones, and engineered tissue scaffolds, as well asin perfusion in the optic nerve head—see [4, 5, 24] and references and discussion therein.From a mathematical point of view, poroelastic systems constitute coupled systems of a (possi-bly degenerate) parabolic fluid pressure and a hyperbolic (inertial) or elliptic (non-inertial) system ∗ [email protected] † [email protected]
1f elasticity for the displacement of the porous matrix containing the fluid. The saturated elasticmatrix is modeled through homogenization, in the sense that the pressure and displacement aredistributed quantities throughout the physical domain. In this treatment, we focus on poroelasticmodels with specific applications in biomechanics (in contrast to those tailored to geomechanicalsystems). Thus we work under the assumptions of full saturation, negligible inertia, small defor-mations, and (possibly) compressible mixture components . The applications of interest give rise toa Biot permeability taken as a nonlinear function of the fluid content (a particular linear combina-tion of pressure and dilation). This type of nonlinear coupling introduces a variety of complicationsdetailed below, and in particular destroys the monotone nature of the problem.Such a nonlinear poroelastic model was first considered—from a mathematical point—in [6],and shortly after in [4]. The former reference [6] (and latter [7]) focus on the compressible Biotmodel and constructs weak solutions through a full spatio-temporal discretization, in the mathe-matically simplified framework of homogeneous Dirichlet boundary conditions for both fluid pres-sure and solid displacement. This reference takes the linear theory developed by Showalter [19, 20]as its primary motivation, and uses Brouwer’s fixed point at the level of the fully discretized prob-lem. The latter reference [4] focuses on
Biot models with incompressible constituents and constructsweak solutions (also using discretizations in both time and space [25]) for both poroelastic andporo-viscoelastic systems with non-homogeneous mixed boundary conditions that are physicallyrelevant to opthalmological applications. A key theme in that work is the careful analysis of therequisite boundary and source regularity for the construction of weak solutions, as this aspectis crucial in understanding the mechanisms leading to tissue damage in the optic nerve head,and consequent vision loss, associated with glaucoma. Both [4, 6] obtain a priori estimates in thefully discretized setting and much of the challenge lies in adequately addressing the nonlinearand non-monotone coupling to obtain a weak solution in the limit. The reference [6] providesa straightforward regularity criterion for uniqueness of solutions, but does not actually considersmooth solutions, nor address the permissibility of requisite multipliers used to obtain estimates.In this treatment, we aim to provide a careful and precise mathematical construction of weaksolutions using only semi-discretization in space, in the setting of fully homogeneous boundaryconditions. One primary goal is to clearly elucidate the challenges introduced into the Biot problemby the inclusion of non-monotone nonlinear coupling. We also include a new, sharper uniquenesscriterion for solutions of sufficient smoothness. Our approach is based on rigorously obtaineda priori estimates for the time-dependent linearization, from which we construct a fixed pointcorrespondence. An interesting feature of this approach is that we cannot appeal to uniquenessof solutions for the aforementioned linear problem. Indeed, weak solutions themselves are notpermissible test functions, presenting a great hurdle in the analysis. To address this issue, weutilize a multi-valued fixed point approach, along with a careful construction of the correspondencebetween the permeability function and the resulting fluid content.In summary, this paper addresses existence of weak solutions to a nonlinear Biot system basedon a natural fixed point approach which circumvents the lack of monotonicity in the system’snonlinear coupling. We utilize a semi-discretization approach (in space), since a relevant time-dependent, nonlinear implicit evolution framework does not seem to be readily available. We believethat the construction given here is the most natural and most illustrative of the complications inthe analysis introduced by the presence of nonlinearity and its interaction with the boundaryconditions. We note that in [21] a nonlinear version of the Biot problem is considered, but the structure of the nonlinearitythere is monotone in nature and different from the physical nonlinearity presented here for biological applications. .1 PDE Model Let Ω be an open, bounded subset of R representing the spatial domain occupied by the fluid-solidmixture, with smooth boundary Γ = ∂ Ω. Let x be the position vector of each point in the bodywith respect to a fixed Cartesian reference frame. The symbol n will be used to denote the unitoutward normal vector to Ω. Let V f ( x , t ) be the volume occupied by the fluid component in arepresentative volume V ( x , t ) element centered at x ∈ Ω at time t . Then the porosity φ and thefluid content ζ are given by φ ( x , t ) = V f ( x , t ) /V ( x , t ) and ζ ( x , t ) = φ ( x , t ) − φ ( x ), where φ isa baseline (local) value for the porosity.Balance Equations: Under the assumptions of small deformations, full saturation of the mixture,and negligible inertia, we can write the balance of linear momentum for the mixture and thebalance of mass for the fluid component as ∂ t ζ + ∇ · v = S ( x , t ) and − ∇ · T + F ( x , t ) = in Ω × (0 , T ) (1.1)where T is the total stress, v is the discharge velocity, F is a body force per unit of volume, and S is a net volumetric fluid production rate.Constitutive Equations: We complement the balance equations with the following constitutiveequations.The total stress of the mixture is given by T = T e − αp I = 2 µ e ε ( u ) + λ e ( ∇ · u ) I − αp I , (1.2)where u is the solid displacement, the symmetrized gradient ε ( u ) = ( ∇ u + ∇ u T ) / α is the Biot-Willis constant, p is the Darcy fluid pressure, I is the identity tensor, and λ e and µ e are the Lam´e elastic parameters.The discharge velocity has the following formula via Darcy’s law [20]: v = − k ( φ ) I ∇ p. (1.3)The particular form of the relationship between the permeability k and the porosity φ depends onthe geometrical architecture of the pores inside the matrix and the physical properties of the fluid.We allow for k to be a general continuous function, assuming only that it is bounded above andbelow (as discussed below, in Assumption 1.1, and consistent with [4, 6]).The fluid content is given by ζ = c p + α ∇ · u , (1.4)where c is the constrained specific storage coefficient [11]. Using the relation between porosityand fluid content, as well as the definition of permeability, we can see that permeability in thesystem depends nonlinearly on the fluid content. In the special case of incompressible constituents,due to the fact that the constrained storage coefficient c = 0 and α = 1, the permeability be-comes a nonlinear function of dilation alone. This is the scenario that is specifically addressed in [4].Boundary Conditions: We consider homogeneous Dirichlet boundary conditions for both the struc-tural displacement u (and hence u t , when defined) and the fluid pressure p . u = , p = 0 on Γ . (1.5)3his choice is in line with the model considered in [6, 15]. In our previous work [4], we consideredmore complex physical configurations, incorporating both Dirichlet and Neumann boundary con-ditions for the elastic displacement and fluid pressure. These physical mixed boundary conditionscould be incorporated here, and existence and uniqueness of weak solution would follow similarly.Initial Conditions: ζ ( x ,
0) = d in Ω . Remark . In discussing stronger notions of solutions (as in Section 6), one can find the require-ment that d = ζ (0) = [ c p + α ∇ · u ](0) for some u ( t = 0) = u specified independently of d ,taken in an appropriate space (see [25], as well as [4]). In these works, a different construction forsolutions is utilized. We do note that for the linear case, in the most general “weak” setting [20],only d is needed. In this weak situation, the construction is done independent of a priori esti-mates obtained in standard Hilbert spaces such as L (Ω) and H (Ω). In [15], solutions are alsoconstructed in the linear case, but the initial data is taken to be smoother than “finite energy”considerations would require.PDE System: To summarize, below is the PDE nonlinear coupling that we are considering: −∇ · [2 µ e ε ( u ) + λ e ( ∇ · u ) I − αp I ] = F in Ω × (0 , T ) (1.6) ζ t − ∇ · (cid:2) k ( ζ ) ∇ p (cid:3) = S in Ω × (0 , T ) (1.7) ζ = c p + α ∇ · u in Ω × (0 , T ) (1.8) u = and p = 0 on Γ × (0 , T ) (1.9) ζ (0) = d in Ω , for t = 0 (1.10)Note that we can write (1.6) equivalently as − µ ∆ u − ( λ + µ ) ∇ ( ∇ · u ) + α ∇ p = F , where the Laplacian above is interpreted component-wise. Assumption 1.1. [Assumptions on the Permeability Function] We assume that the permeabilityfunction k : R → R is continuous and that there exist constants k > and k > s.t. < k ≤ k ( x ) ≤ k , ∀ x ∈ R . With a slight abuse of notation, we denote by k (Ψ( · , t )) the Nemytskii operator associated with k .Using our assumptions on the function k , and the theory of superposition operators [18, 23], wehave that the operator k is bounded and continuous from L (Ω × (0 , T )) into L (Ω × (0 , T )) .Remark . In order to obtain uniqueness of solution in Section 6, we will further assume that k is a globally Lipschitz function, i.e., k ∈ Lip ( R ). Assumption 1.2.
In what follows, for simplicity, we set to unity non-essential (from the math-ematical point of view) parameters. This is to say, we take λ e = µ e = α = 1 . The parameter c is retained, with no dependence on the other parameters, as we will consider taking c ց in theconstruction of weak solutions for the case of incompressible constituents. Main Results
We make the following conventions for the rest of the paper. Norms k · k D are taken to be L ( D )for a domain D . Inner products in L ( D ) are written as ( · , · ) D , where the subscript will be omittedwhen the context is clear. The standard Sobolev space of order s defined on a domain D [14] willbe denoted by H s ( D ), with H s ( D ) denoting the closure of C ∞ ( D ) in the H s ( D ) norm (whichwe denote by k · k H s ( D ) or k · k s ). Vector valued spaces will be denoted as L (Ω) ≡ [ L (Ω)] n and H s (Ω) = [ H s (Ω)] n . We make use of the standard notation for the trace of functions γ : H ( D ) → H / ( ∂D ) which generalizes restriction to a lower dimensional manifold. We will make use ofthe spaces L (0 , T ; U ) and H s (0 , T ; U ), where U is a topological vector space. These norms (andassociated inner products) will be denoted with the appropriate subscript, e.g., || · || L (0 ,T ; U ) . Weutilize the Frobenius scalar product for tensors with the Einstein summation convention:( A , B ) = Z Ω ( A ij B ij ) d Ω , sometimes also denoted by R Ω A : B d Ω. Notice that, when A = B , we write( A , A ) = Z Ω A : A d Ω = X i,j ( A ij , A ij ) = || A || , the latter norm taken in the Frobenius sense.The primary spaces in our analysis below are V ≡ H (Ω) , V ≡ ( H (Ω)) , (2.1)for the pressure p and elastic displacement from equilibrium u , respectively. The norms in thesespaces are taken in the natural sense, respectively, accounting for Poincar´e’s and Korn’s inequalities[14]. For V , we take the gradient norm (as is standard on H (Ω)): || v || V = || v || H (Ω) = ||∇ v || L (Ω) .We will frequently need to denote the duality pairing between V and V ′ or V and V ′ , for whichwe will use the generic notation h· , ·i . (For more general spaces B and B ′ , we may write h· , ·i B ′ × B for clarity.)We utilize the notation ∇ u as the Jacobian matrix of u and the associated symmetric gradient ε ( u ), yielding the following definitions and formal identities: ∇ u = ( ∂ j u i ) , ∇ u T = ( ∂ i u j ); ε ( u ) = 12 [ ∇ u + ∇ u T ] (2.2)( ∇ u , ∇ w ) Ω = Z Ω [ ∇ u : ∇ w ] d x = tr ( ∇ u ∇ w T ) = tr ( ∇ w T ∇ u ) = ( ∇ w T , ∇ u T ) (2.3)( ε ( u ) , ∇ w ) = 12 ( ∇ u , ∇ w ) + 12 ( ∇ u T , ∇ w ) = ( ε ( u ) , ε ( w )) . (2.4)In the simplified setting, the bilinear form associated with the elasticity operator is given by a ( u , w ) = ( ∇ · u , ∇ · w ) + ( ∇ u , ∇ w ) + ( ∇ u , ∇ w T ) . (2.5)We topologize the space V via a ( · , · ), which is to say that we take the norm induced by a ( · , · ) asthe norm on V , and, via Korn’s inequality and Poincar´e, this is equivalent to the full H (Ω) normon V [14], as in [4, 6]. 5n our estimates below, we utilize the notation of Q . Q to indicate that there is a constant C depending only on non-critical quantities such that Q ≤ CQ . In general, throughout thepaper the quantity C represents a generic constant that may change from line to line. If a constantexhibits a critical dependence, this will be denoted with subscripts or in parentheses, for instance: || f || ≤ C p || g || or A = A (Ω).Finally, in this analysis we assume that the principal domain Ω is of class C [8, 14], so thatstandard elliptic regularity results apply. As one can see in [6, 15, 20], for instance, there are many different notions of strong and weaksolution to poroelastic systems. Our notion of solution is consistent with the classical one providedin [25] in the sense that the solution satisfies a weak space-time form of (1.6)–(1.10). Moreover, ourweak solutions are in line with the general notion of weak solution for the time-dependent linearproblem holding in the dual sense (in L (0 , T ; V ′ )), as presented [19]. Definition . A solution to (1.6)–(1.10) with c ≥ u ∈ L (0 , T ; V ) and p ∈ L (0 , T ; V ) , with ζ = c p + ∇ · u ∈ L (0 , T ; L (Ω)), such that:(a) the following variational forms are satisfied for any w ∈ L (0 , T ; V ), q ∈ L (0 , T ; V ): Z T a ( u , w ) dt − Z T ( p, ∇ · w ) Ω dt = Z T ( F , w ) Ω dt (2.6) Z T (cid:0) k ( ζ ) ∇ p, ∇ q (cid:1) Ω dt − Z T h ζ t , q i V ′ × V dt = Z T h S, q i V ′ × V dt (2.7)(b) for every q ∈ V , the term ( ζ ( t ) , q ) L (Ω) uniquely defines an absolutely continuous function on[0 , T ] and the initial condition (cid:0) ζ (0) , q ) L (Ω) = ( d , q ) L (Ω) is satisfied. Remark . Alternatively to (b) above, one could assume that d ∈ L (Ω) but specify that ζ ∈ C ([0 , T ]; V ′ ) and ζ ( t ) (cid:12)(cid:12) t =0 = [ c p + ∇ · u ]( t ) (cid:12)(cid:12) t =0 = d in the H − (Ω) sense. This is preciselywhat we will obtain through our constructions. Remark . In [4, 25], test functions are taken as space-time products, and all terms are definedin terms of spatial L (Ω) inner products. Our formulation is equivalent by density, as the testfunctions of the form w ( x ) f ( t ), with w ∈ V and f ∈ C ∞ (0 , T ), are dense in L (0 , T ; V ); similarly,test functions of the form q ( x ) f ( t ), with q ∈ V and f ∈ C ∞ (0 , T ), are dense in L (0 , T ; V ). Remark . We note finally that the above definition of weak solution could certainly be weakened.For instance, in the weak form of elasticity, the RHS could be replaced by R T h F , w i V ′ × V dt andthe initial condition d could be taken in V ′ (being mindful of the previous approaches in [4,6,20]).However, we use Definition 1 based on the regularity required for our construction, predominantlyinfluenced by the presence of nonlinearity in the problem and a careful treatment of the spatialregularity of ζ = c p + ∇ · u .It will be convenient in the estimates below to utilize a notation for “data” associated to apriori estimates obtained in the analysis of the pressure equation (2.7). Definition . [ Notion of Data ]DATA (cid:12)(cid:12)(cid:12) T ≡ Z T h || S ( t ) || V ′ + || F t ( t ) || V ′ + || F ( t ) || V ′ i dt (2.8)6 .3 Main Results and Comparison to Previous Literature We begin this section with the statements of our main results, and follow them with an in-depth,technical discussion of our results in relation to the literature. It is important to note the ways inwhich our contributions here represent alternative proofs for similar results in the literature, andin what ways our approaches here are novel. Indeed, there is a striking amount of subtlety alreadypresent in the analysis of the associated linear Biot system.The first auxiliary result we prove is that of existence of weak solutions to the associated linearproblem for a given z ∈ L (0 , T ; L (Ω)) and permeability k ( z ): − µ ∆ u − ( λ + µ ) ∇ ( ∇ · u ) = −∇ p + F ∈ L (0 , T ; V ′ ) ζ t − ∇ · [ k ( z ) ∇ p ] = S ∈ L (0 , T ; V ′ ) ζ = c p + ∇ · u ∈ L (0 , T ; L (Ω)) c p (0) + ∇ · w (0) = d ∈ L (Ω) . (2.9) Theorem 2.1 (Linear Weak Solution) . Let c > , and assume that the permeability k satisfiesthe hypotheses of Assumption 1.1. Let d ∈ L (Ω) , z ∈ L (0 , T ; L (Ω)) , F ∈ L (0 , T ; L (Ω)) ∩ H (0 , T ; V ′ ) , and S ∈ L (0 , T ; V ′ ) . Then (2.9) has a weak solution ( u ( z ) , p ( z ) , ζ ( z )) , where u ( z ) ∈ L (0 , T ; H (Ω) ∩ V ) , p ( z ) ∈ L (0 , T ; V ) , and ζ ( z ) ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; V ′ ) , with associatedestimates: c || p || L ∞ (0 ,T ; L (Ω)) + k p k L (0 ,T ; V ) . || d || L (Ω) + DAT A | T (2.10) || u || L (0 ,T ; H (Ω) ∩ V ) . || p || L (0 ,T,V ) + || F || L (0 ,T ; L (Ω)) (2.11) k [ c p + ∇ · u ] t k L (0 ,T ; V ′ ) . k p k L (0 ,T ; V ) + k S k L (0 ,T ; V ′ ) (2.12) Remark . We note that nothing in the above result (or its corresponding proof) changes if,instead of k = k ( z ) we have k = k ( x , t ) a given L ∞ (0 , T ; L ∞ (Ω)) - function. In this case, it alsofollows from [19, p.116] that solution is unique if k t ∈ L (0 , T ; L ∞ (Ω)).Next we present our result for the existence of weak solution to the nonlinear system: Theorem 2.2 (Nonlinear Weak Solution) . Consider the nonlinear coupled system (1.6)–(1.10)with c ≥ , permeability function k ( · ) satisfying Assumption 1.1, and distributed sources S ∈ L (0 , T ; V ′ ) and F ∈ L (0 , T ; L (Ω)) ∩ H (0 , T ; V ′ ) . For initial data ζ (0) = d ∈ L (Ω) , thereexists a weak solution u ∈ L (0 , T ; V ) and p ∈ L (0 , T ; V ) in the sense of Definition 1. Moreover, c p ∈ L ∞ (0 , T ; L (Ω)) and u ∈ L (0 , T ; H (Ω) ∩ V ) , with the same inequalities (2.10) – (2.12) holding for nonlinear weak solutions.Remark . We note from the estimates above that, in the sense of the pressure equation, thesolution is truly “weak.” However, since we have fundamentally elliptic-parabolic coupling, with theregularity assumptions placed on F , the solution is “strong” in the sense of the elastic displacement,since that equation holds a.e. x , a.e. t .We now address uniqueness of solution through the imposition of additional regularity hy-potheses. As the problem is fundamentally quasi-linear in nature, such additional regularity foruniqueness is expected. Our approach to uniqueness is rooted in multiplier estimates, which arethemselves problematic for weak solutions. Hence, we need to restrict our attention to the class ofweak solutions (for fixed data F , S, d , permeability function, and intrinsic parameters) such thatthe solution can properly be used as a test function. Beyond this, additional spatial regularity willbe needed to manipulate the nonlinear permeability term.7 efinition . Define the class of solutions W T = W T ( F , S, d , k ( · )) as consisting of weak solutionsthat have additional time regularity, i.e.: W T ≡ (cid:8) ( u , p ) is a weak solution in the sense of Definition 4 on [0 , T ] | p t ∈ L (0 , T ; L (Ω)) (cid:9) . (2.13) Theorem 2.3 (Uniqueness) . Let c ≥ . Suppose that, in addition to Assumption 1.1, we have k ∈ Lip ( R ) and assume ( u , p ) ∈ W T . • Suppose d = ∇ · u + c p ∈ L (Ω) for some u ∈ V . If additionally p ∈ L (0 , T ; W , ∞ (Ω)) ,then the solution ( u , p ) is unique in the class W T . • If c > and p ∈ L (0 , T ; W , ∞ (Ω)) ( u , p ) is unique in the class W T .In each of the above cases, if any of the weak solutions in W T and one of them has additionalspatial regularity for p , then all solutions in W T are equal.Remark . By the standard Sobolev embeddings [14], it is sufficient for the theorem above toconsider p ∈ L (0 , T ; H . δ (Ω)) for any δ > Remark . We note the two above cases sacrifice one hypothesis at the cost of another. In thesecond bullet point, the problem is required to be compressible, but no additional structure ofthe data d need be assumed. In the first bullet, we can take c = 0 but require informationabout u ( t = 0) to be independently specified. We point out here that these conditions are animprovement of what is presented in [6]; in that reference they require c > ∇ p in L ∞ (cid:0) (0 , T ) × Ω (cid:1) in terms of the intrinsic parameters. We also mentionthat, similarly to our considerations, no previous work actually construct strong solutions. Challenges and Relation to Previous Literature:
The main mathematical challenges in thisproblem are represented by (i) the implicit, degenerate evolution present in the system, as well as(ii) the nonlinear coupling (with no evident monotone structure) in the permeability—it being anonlinear function of fluid content. There is substantial mathematical literature focused on well-posedness analysis for linear poroelastic systems, where the permeability tensor is assumed to beconstant. The key references in the linear setting are [15, 20, 25].A foundational reference for all of the cited mathematical Biot studies is [25]. This paperprovides a construction of solutions in the using Rothe’s method (full temporaland spatial discretization), with the analysis based on a priori estimates. The analysis is done onthe entire ( u , p ) system, and as such requires the specification of initial displacement u (0) ∈ V and initial pressure p (0) ∈ L (Ω). In contrast, the seminal work by Showalter in [19,20] reduces thefull linear Biot system to an implicit, degenerate evolution. This allows—again in the linear case—a semigroup theory to obtain both weak and strong solutions. For the weak solutions (what [20,Section 6] calls the “holomorphic case”) only specification of the initial fluid content ζ (0) ∈ H − is needed. The implicit semigroup approach does not explicitly depend a priori estimates based onsolution multipliers; instead, quotient and seminormed spaces are invoked to reduce the implicitproblem to a regular explicit Banach-valued ODE [19, Chapter IV.6]. As such, this general approachis not immediately generalizable to time-dependent or quasilinear cases. We note that in [19,Chapter III.3], a nice time-dependent formulation for weak solutions is presented based on ageneralized version of Lax-Millgram due to Lions. Lastly, with respect to the linear analysis, themore recent [15] provides a Galerkin-based construction of solutions for the full Biot problem,making use of an explicit solver for the embedded Stokes-type problem in the dynamics. In that8ork, solutions are clearly constructed without temporal discretization, but strong assumptionsare made on the data in order to obtain good a priori estimates.In the authors’ previous work [4], the nonlinear problem presented here is addressed with c = 0and allow for the possibility of visco-elastic effects in the Biot structure. Additionally, motivatedby physical considerations, a configuration with mixed boundary conditions on a Lipschitz do-main, and non-zero boundary sources, is considered. That work is based on full spatio-temporaldiscretization (adapting the linear argument in [25]). As such we make use of the stronger as-sumption on initial data in order to obtain good estimates at the temporally discretized leveland carefully pass with the limit, invoking compactness in the fluid content derived from a smallamount of elliptic regularity. The multipliers approach works here, albeit in the discrete setting,with two subsequent limit passages required. The accompanying estimates are less natural how-ever, and the fully discretized nature is neither optimal nor natural for modern numerical analysisof the nonlinear problem.In comparison, our goal here is to provide a theory of solutions for the nonlinear poroelasticcoupling in (1.6)–(1.10). As mentioned before, we consider a similar model and set of assumptionsas the ones used in [6]. However, we permit the case of fluid-solid mixtures which may haveincompressible constituents ( c = 0), with applications to biological tissues. This degeneracy israther benign at the linear level, but presents subtle challenges for the analysis here, owing to thefact that the operator B is not invertible on L (Ω), but c I + B is. Indeed, we use critically thepresence of c > c = 0 via a singularlimit approach as c ց
0. It is also worthwhile to note that, in line with biological applications, weallow the permeability to depend on the full fluid content, i.e., k ( ζ ) for ζ = c p + α · u (also as in [4]).In [6], the construction critically requires c > thepermeability depends only on dilation , i.e., k ( ∇ · u ) [6, p.1254]; this distinction is mathematicallynon-trivial.We present here what we believe to be the most direct and illustrative approach for existenceof weak solutions. Our approach does not involve the discretization of the balance equations inboth time and space [4, 6]. We believe this is beneficial for future considerations, as full discretiza-tion is cumbersome for a sought-after construction of smooth solutions, and our semi-discretizedapproach is perhaps more amenable to numerical treatment. The work in [20] focuses on constantpermeability k (which renders a linear coupling in the system) and develops a semigroup theoryfor implicit evolution equations for both strong and weak solutions. The approach is generalizedfor the case of nonlinear permeability function dependent on pressure which preserves a monotonestructure in [21]. We note that the strategy developed in [20] can not be directly applied here, asthe model at hand does not exhibit such monotonicity properties. Rather, we build linear timedependent solutions and carefully construct a functional correspondence that leads to a fixed point.In constructing weak solutions via estimates in a fixed point argument, we hope to have provideda framework for the future construction of smooth solutions, which should be unique, accordingto the criterion given here. In this section we introduce the principal operators that are used in the proofs of the main theorems,along with their properties. We follow the abstract framework provided in [20]. In the last partof the section we provide formal “translations” of the linear and nonlinear problems that allowus to consider the problem with null distributed force in the balance of linear momentum, upontranslating the initial data and the pressure source S .9 .1 Elasticity Operator E In general, the elasticity operator associated to isotropic homogenous media is given by − ( λ e + µ e ) ∇ ( ∇ · u ) − µ e ∆ u = −∇ · [2 µ e ε ( u ) + λ e ( ∇ · u ) I ] . Since we have taken µ e = λ e = 1 here, we consider an operator E ( u ), whose action in distributionis given by: E ( u ) = −∇ · [2 ε ( u ) + ∇ · u ] = − ∇ ( ∇ · u ) − ∆ u . From this action, we can create an unbounded operator E : L (Ω) → L (Ω) encoding the homoge-neous Dirichlet boundary conditions here. This is to say, E is the operator with domain D ( E ) ≡ { u ∈ V : E ( u ) ∈ L (Ω) } . (Note that E considered from V → V ′ acts boundedly.) Indeed, E is an isomorphism [8, 20] in theappropriate senses, and the operator E : L (Ω) → L (Ω) is positive, self-adjoint associated to thesymmetric bilinear form a ( · , · ) : V × V → R .In the analysis of the momentum equation, we consider a given a p ∈ L (Ω) (and thus ∇ p ∈ V ′ [20, 22]) and produce a corresponding u ∈ V which satisfies the stationary elasticity equation,which we will frequently write as E ( u ) = −∇ p + F ∈ V ′ . (3.1)This leads directly to the following lemma: Lemma 3.1.
Given G ∈ V ′ , we can consider the elasticity problem ( E ( u ) = G ∈ V ′ u = 0 on Γ . (3.2) This problem is well-posed in the standard weak sense [8, 14], with a solution u ∈ V and stabilityestimate || u || V ≤ C w || G || V ′ , ∀ u ∈ V . Moreover, as Ω is of class C , classical elliptic regularity applies [8, 22]. Hence, if G ∈ L (Ω) ,then we have that u ∈ H (Ω) ∩ V , and || u || H (Ω) ≤ C r || G || L (Ω) . Unlike [4], we are working with a smooth boundary, composed of a single Dirichlet componentupon which both pressure p and displacement u are zero. Thus classic elliptic theory can be usedfor displacement u when p ∈ V and F ∈ L (Ω). When F = in (3.1), we have p ∈ V = ⇒ ∇ p ∈ L (Ω) = ⇒ E − ( −∇ p ) = u ∈ H (Ω) ∩ V = ⇒ ∇ · u ∈ H (Ω) . (3.3)Such regularity was not available in [4], where a more complex, physically-motivated boundaryconfiguration was considered. 10 .2 Elliptic Operator A z For a smooth z ∈ C ([0 , T ] × Ω), we define the linear operator A z : V → V ′ by A z p = −∇ · [ k ( z ) ∇ p ] , ∀ p ∈ D (Ω) , (3.4)where k ( z ) is interpreted as a Nemitskii operator for the given function z ( x, t ) as in 1.1. Remark . In practice we will consider this operator through its bilinear form (defined below)when z ∈ L (0 , T ; L (Ω)), considered a.e. t .If we assume that z ∈ L (0 , T ; H (Ω)), then we have an unbounded operator A z : L (Ω) → L (Ω) with domain D ( A z ) = H (Ω) ∩ V and action given by (3.4) with associated bilinear form A [ p, q ; z ] = ( k ( z ) ∇ p, ∇ q ) , ∀ p, q ∈ V. (3.5)As noted above, the bilinear form associated to the weak form of A z requires only that z ∈ L (0 , T ; L (Ω)), and can be obtained via density. When k ≡ const , A z is a multiple of thestandard Dirichlet Laplacian. In this setting, A z is a maximal monotone operator, the bilinearform A [ · , · ; z ] continuous and coercive on V , and A z is positive and self-adjoint as an unboundedoperator on L (Ω). The pressure to dilation map was introduced in the setting of Biot problems in [19, 20]. It allowsone to reduce the ( u , p ) system in (1.6)–(1.10) to an implicit evolution problem, such as thosestudied extensively in [19, 20]. This operator is a useful, descriptive tool in the construction ofapproximate solutions, and the subsequent analyses.Consider the map B : L (Ω) → L (Ω), defined by Bp = −∇ · E − ( ∇ p ) , (3.6)motivated by the problem above in (3.2). Indeed, we have that p ∈ H s (Ω) = ⇒ ∇ p ∈ H s − (Ω) with p
7→ ∇ p continuous in this setting [14, 22]. In the specific case when p ∈ L (Ω), ∇ p ∈ H − (Ω) = V ′ . Invoking the properties of the elliptic operator E , we see that indeed B ∈ L ( L (Ω)).Similarly, the action of B extends readily to V . Considering B as above, if p ∈ V , then as abovein (3.3), Bp ∈ H (Ω) Similarly, we obtain immediately that B ∈ L ( H s (Ω) ∩ V , H s (Ω)) for s ≥ E . Note that when F ≡ Bp = ∇ · u . Remark . Consider p ∈ V , and, as above Bp = ∇ · u ∈ H (Ω). Although it is not clear that ∇ · u ∈ V , we do know that Bp = ∇ · u ∈ div[ V ], which does carry additional information, inparticular that Bp ∈ H (Ω) ∩ L (Ω) / R —see [22]. Based on the discussions above, we have:
Lemma 3.2.
Given p ∈ V and F ∈ L (Ω) , the corresponding solver E − ( −∇ p + F ) ∈ H (Ω) ∩ V with associated continuity bound. When F ≡ and p ∈ V , we have Bp = ∇ · u ∈ H (Ω) for E ( u ) = −∇ p . From this we obtain that B : H (Ω) → H (Ω) , continuously . Let us utilize the standard (abuse) of notation for the quotient space L (Ω) / R ≡ { f ∈ L (Ω) : R Ω f = 0 . } .
11e note some kernel and range properties of the B operator (closely following [20] and utilizingthe properties of ∇ and div as in [22]). Lemma 3.3.
Considered as a mapping on H (Ω) , B is injective. Considered as a mapping on L (Ω) , ker ( B ) = { constants } , and hence B is injective on L (Ω) / R .With respect to ranges, we have the following: B ( L (Ω)) ⊆ L (Ω) / R , B ( H (Ω)) ⊆ H (Ω) / R . Finally, we have that B is a self-adjoint, monotone operator when considered on L (Ω) [4, 20]. Lemma 3.4.
Considering B ∈ L ( L (Ω)) , it is a non-negative, self-adjoint operator. By the standard construction [9, 16], the self-adjoint operator B / ∈ L ( L (Ω)) is obtainedwith characterizing property( Bp, q ) L (Ω) = ( B / p, B / q ) L (Ω) = ( p, Bq ) L (Ω) , ∀ p, q ∈ L (Ω) . A central issue in the analysis here is that B ∈ L ( L (Ω)) need not be coercive in that setting.However, in the case where we consider compressible effects, the operator c I + B : L (Ω) → L (Ω)is coercive. We will use this critically and repeatedly below. Corollary 3.5.
Let c > . Then the operator c I + B : L (Ω) → L (Ω) is an isomorphism.Proof. By positivity, we note that for p ∈ L (Ω)([ c I + B ] p, p ) L (Ω) = c || p || + ( Bp, p ) ≥ c || p || . Hence, the operator c I + B is coercive on L (Ω). Since B ∈ L ( L (Ω)), surjectivity followsimmediately from Lax-Milgram applied to the form β ( p, q ) = (cid:0) [ c I + B ] p, q (cid:1) L (Ω) = c ( p, q ) L (Ω) + ( B / p, B / q ) L (Ω) . Which is to say that β ( p, q ) = ( f, q ) Ω , ∀ q ∈ L (Ω)is uniquely solvable with associated stability bound.We conclude this section with some rather important additional remarks about the B operatorthat arise critically in the context of previous approaches to the problem at hand. • Since B implicitly invokes an elliptic solver, its behaviors on V ′ or any H − s (Ω)—regularityand continuity properties, kernel and range—are not in the realm of standard elliptic theory. • It is not clear that B ∈ L ( H − (Ω)), or that that B ( H (Ω)) ⊂ H (Ω). Indeed, characterizingthe range of B on V involves ranging the divergence operator over H (Ω).12 .4 Translation to Eliminate Momentum Source F In this subsection we provide formal “translations” to the linear and nonlinear problems that allowus to consider the problem with F ≡
0. In latter sections, to simplify the analysis, we will operateon the translated problem. After obtaining the principal result in those sections, we will refer tothis section and translate back to obtain (linear and nonlinear) results for the original system.We first note that it is sufficient to solve the linear problem —where k = k ( z ) for z ∈ L (0 , T ; L (Ω))a given function—with F ≡ E ( u ) = −∇ p ∈ L (0 , T ; V ′ ) ζ t + A z p = S ∈ L (0 , T ; V ′ ) ζ = c p + ∇ · u ∈ L (0 , T ; L (Ω)) ζ (0) = d ∈ L (Ω) (3.7)Indeed, as the elasticity equation is elliptic and F ∈ L (0 , T ; L (Ω)), for a.e. t ∈ [0 , T ] we cansimply write u F ( t ) = E − ( F ( t )) ∈ H (Ω) ∩ V . (3.8)Additionally, with the regularity hypotheses of our main theorem F ∈ L (0 , T ; L (Ω)) ∩ H (0 , T ; V ′ ),we have that u F ∈ L (0 , T ; H (Ω) ∩ V ) ∩ H (0 , T ; V ). Then, considering the variable w = u − u F ,we note that u solves (3.7) if and only if w solves E ( w ) = −∇ p ∈ L (0 , T ; V ′ ) c p t + ∇ · w t + A z p = S + ∇ · u F ,t ∈ L (0 , T ; V ′ ) c p (0) + ∇ · w (0) = d − ∇ · u F (0) ∈ L (Ω) . (3.9)Hence, by re-scaling S ∈ L (0 , T ; V ′ ) and d = ζ (0) ∈ L (Ω), we obtain an equivalent linearproblem for a given z with F ≡ A z = A ζ for ζ = c p + ∇ · u , an additional step isneeded. We note that if w = u − u F as above, the fluid content has expression ζ = c p + ∇ · u = c p + ∇ · w + ∇ · u F . Since ∇ · u F ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)), we have that ∇ · u F ∈ C ([0 , T ]; L (Ω)) [13, 19].Hence, for any k ( · ) as in (1.1), we introduce the function k F ( · ) ∈ C ( R ) representing the u F -translateof k ( · ), namely k F ( · ) = k ( · + ∇ · u F ) . (3.10)Since k ( · ) satisfies 1.1, we obtain immediately that k F ( · ) satisfies the assumption as well. Then,from the (abstract) strong form of the original problem, E ( u ) = −∇ p + F ∈ L (0 , T ; V ′ ) ζ t − ∇ · [ k ( ζ ) ∇ p ] = S ∈ L (0 , T ; V ′ ) ζ = c p + ∇ · u ∈ L (0 , T ; L (Ω)) c p (0) + ∇ · u (0) = d ∈ L (Ω) , (3.11)we can write the system for w = u − u F as follows E ( w ) = −∇ p ∈ L (0 , T ; V ′ )[ c p + ∇ · w ] t − ∇ · [ k F ( c p + ∇ · w ) ∇ p ] = S + ∇ · u F ,t ∈ L (0 , T ; V ′ ) c p (0) + ∇ · w (0) = d − ∇ · u F (0) ∈ L (Ω) . (3.12)13s in the linear case, for a fixed F and k , by re-scaling S and d accordingly, and re-labeling k k F , we again obtain an equivalent problem with F = 0. In this section we present the proof of Theorem 2.1.Recall that we explicitly assume that c >
0. In line with the translation introduced in Section3.4, we consider the problem with F ≡
0. Moreover, we retain the names for the datum d ∈ L (Ω)and source S ∈ L (0 , T ; V ′ ) (after updating them, as discussed in the previous section). Thus weconsider the linear problem (3.7) for a given z ∈ L (0 , T ; L (Ω)). Interpreting it weakly throughthe associated bilinear forms a ( · , · ) and A [ · , · ; z ] , and taking the time derivative distributionally in D ′ (0 , T ) yields an equivalent formulation to our Definition 1 (see [19]). Namely, we seek u ( z ) ∈ L (0 , T ; V ), p ( z ) ∈ L (0 , T ; V ) that solve (3.7) weakly.Using the pressure to dilation operator introduced in Section 3, we equivalently reformulate3.7 as was done in [6, 20] as the (implicit) initial boundary value problem [( c I + B ) p ] t − ∇ · [ k ( z ) ∇ p ] = S, Ω × (0 , T ) p = 0 , Γ × (0 , T )( c I + B ) p (0) = d , Ω (4.1)Recall that for z ∈ L (0 , T ; L (Ω)), we defined the bilinear form A [ · , · ; z ] : V × V → R by A [ p, q ; z ] = ( k ( z ) ∇ p, ∇ q ) Ω . (4.2)Due to Assumption 1.1 on the permeability operator k , the following Proposition is immediate. Proposition 4.1.
The bilinear form A [ · , · ; z ] satisfies the following properties:1. Continuity: ∃ M > s.t. | A ( w , w ; z ) | ≤ M k w k V k w k V , ∀ w , w ∈ V , a.e. in [0 , T ] .2. Coercivity: A ( w, w ; z ) ≥ k k w k V , for all w ∈ V . Let us clearly state a weak formulation for (4.1) above. In the remaining part of this section, thenotation f ′ will denote differentiation in time, and recall the angle brackets represent the dualitypairing between V ′ and V , i.e., h φ, g i = φ ( g ) = h φ, g i V ′ × V . Definition . Given z ∈ L (0 , T ; L (Ω)), we say that p ∈ L (0 , T ; V ) with ( c I + B ) p ∈ L (0 , T ; H (Ω))and [( c + B ) p ] ′ ∈ L (0 , T ; V ′ ) is a weak solution for (4.1) provided that1. For every q ∈ L (0 , T ; V ), Z T (cid:10) [( c I + B ) p ] ′ ( t ) , q ( t ) (cid:11) dt + Z T A [ p ( t ) , q ( t ); z ( t )] dt = Z T h S ( t ) , q ( t ) i dt (4.3)2. (cid:2) ( c I + B ) p (cid:3) (0) = d in the sense of V ′ .Note that since ( c I + B ) p ∈ L (0 , T ; H (Ω)) and [( c I + B ) p ] ′ ∈ L (0 , T ; V ′ ), we have that( c I + B ) p ∈ C ([0 , T ]; V ′ ) and thus the initial condition makes sense in V ′ .14 emark . With regard to the initial condition, although we only identify the initial conditionin the sense of V ′ (as is consistent with the general weak formulation for implicit equations) wealso need d ∈ L (Ω) for the construction at hand. Secondly, we note that it is nearly the case that[ c I + B ] p is a continuous function in time into L (Ω), but since we do not know that [ c I + B ] p ∈ V ,we cannot use the standard result [13] to obtain that fact. Remark . Sometimes it is convenient to work with the following equivalent variational formu-lation for (4.3): (cid:10) [( c I + B ) p ] ′ , q (cid:11) + A [ p, q ; z ] = h S, q i , for each q ∈ V and a.e. time t ∈ [0 , T ] . (4.4)We note that the real-valued function t
7→ h [( c I + B ) p ] ′ , q (cid:11) belongs to L loc (0 , T ) ⊂ D ′ (0 , T ), andBochner’s theorem yields (cid:10) [( c I + B ) p ] ′ , q (cid:11) = ddt ([ c I + B ] p ( t ) , q ) Ω in D ′ (0 , T ) . Therefore (4.4) can be simply written in the form ddt (cid:0) [ c I + B ] p ( t ) , q (cid:1) Ω + A [ p, q ; z ] = h S, q i , in D ′ (0 , T ) , for all q ∈ V. Now, let us construct a weak solution as in Definition 4 to the reduced problem described abovein (4.1). We state this as a lemma, as it will be used in the proof of Theorem 2.2.
Lemma 4.2.
Let z ∈ L (0 , T ; L (Ω)) , S ∈ L (0 , T ; V ′ ) and d ∈ L (Ω) . Then (4.1) has weaksolution, according to Definition 4.Proof. Construction of Approximate Solution: We use Galerkin approximations. Let { w k ( x ) } ∞ k =1 be an orthogonal basis of V , and an orthonormal basis in L (Ω). (For example, we can take { w k ( x ) } ∞ k =1 to be the complete set of appropriately normalized eigenfunctions for − ∆ in V .) Let V n = span { w , ...w n } . Note that V n satisfies the conditions V n ⊂ V n +1 and ∪ V n = V . We look forsolutions of the form: p n ( t ) = n X k =1 d kn ( t ) w k , (4.5)where the coefficients d kn ( t ) ∈ H (0 , T ) for k = 1 , ..., n . Thus we consider the following finitedimensional problem on V n :Determine p n ∈ H (0 , T ; V ) such that for every k = 1 , , .., n , ( ([ c I + B ] p ′ n , w k ) L (Ω) + A [ p n , w k ; z ] = h S, w k i , a.e. in (0 , T ) ,d kn (0) = ([ c I + B ] − d , w k ) Ω , k = 1 , , ..., n. (4.6)If the differential equation in (4.6) holds for each element of the basis w k , with k = 1 , , ..., n ,then it also holds for every w ∈ V n . Moreover, since ( c I + B ) p ′ n ∈ L (0 , T ; L (Ω)), we have fromRemark 4.2 (( c I + B ) p ′ n ( t ) , w ) Ω = h ( c I + B ) p ′ n ( t ) , w i Upon expanding p n , (4.6) becomes M ( d kn ( t )) ′ + n X k =1 A [ w l , w k ; z ( t )] d ln ( t ) = S k ( t ) ,d kn (0) = ([ c I + B ] − d , w k ) , k = 1 , , ..., n, (4.7)15here M = ([ c I + B ] w k , w k ) Ω , and S k ( t ) = h S ( t ) , w k i , k = 1 , , ..., n. Since ( c I + B ) is invertible on L (Ω), we have that { [ c I + B ] w k } ∞ k =1 is linearly independentin L (Ω). Therefore we can find a permutation α ( i ) of the basis { w k } such that for all m ∈ N , thematrix (cid:8)(cid:0) [ c I + B ] w j , w α ( i ) (cid:1) Ω (cid:9) mi,j =1 is nonsingular (see Lemma 2.3 in [15]). Remark . We note here that in order to construct solutions invoking ODE theory and obtainthe subsequent energy estimates below, we require the initial condition d ∈ L (Ω); if d ∈ V ′ ,additional information about the continuity, adjoint, and invertibility of B on V ′ would be needed.By standard existence theory for ordinary differential equations, there exists a unique, ab-solutely continuous function d n ( t ) = [ d kn ( t )] nk =1 that solves (4.7). Therefore p n ( t ) ∈ H (0 , T ; V )defined in (4.5) is a solution for (4.6) for a.e. t ∈ [0 , T ].Energy Estimates: We can interpret (4.3) a.e. s ∈ [0 , T ] and let q = p n ∈ H (0 , T ; V ) in (4.3) toobtain h [ c I + B ] p ′ n ( s ) , p n ( s ) i + A [ p n ( s ) , p n ( s ); z ( s )] = h S ( s ) , p n ( s ) i Due to the fact that B is self-adjoint on L (Ω) and p n ( t ) ∈ H (0 , T ; V ), we have that h [ c I + B ] p ′ n ( s ) , p n ( s ) i = 12 dds (cid:0) [ c I + B ] p n ( s ) , p n ( s ) (cid:1) Ω Moreover, with k as the lower bound on k , i.e., the coercivity parameter for A in Remark (4.1),we have |h S ( s ) , p n ( s ) i| ≤ k k S ( s ) k V ′ + k k p n ( s ) k V Thus, with the coercivity assumed in Remark (4.1), we obtain12 dds (cid:0) [ c I + B ] p n ( s ) , p n ( s ) (cid:1) Ω + k k p n ( s ) k V ≤ k k S ( s ) k V ′ We integrate over (0 , t ) and obtain (cid:0) [ c I + B ] p n ( t ) , p n ( t ) (cid:1) Ω + k Z t k p n ( s ) k V ds ≤ ([ c I + B ] p n (0) , p n (0)) L (Ω) + 1 k Z t k S ( s ) k V ′ ds Using the properties of the operator B and B / (as in Lemma 3.4 and the discussion followingit), we obtain from this estimates point wise (in time) control of || p n ( t ) || L (Ω) and || B / p n ( t ) || L (Ω) for each t ∈ [0 , T ], as well as p n , B / p n , ( c + B ) / p n ∈ L ∞ (0 , T ; L (Ω)) , p n ∈ L (0 , T ; V ) . Thus k p n k L ∞ (0 ,T ; L (Ω)) + || B / p || L ∞ (0 ,T ; L (Ω)) ≤ ( d , ([ c I + B ] − d ) Ω + 1 k Z t k S ( s ) k V ′ ds and k p n k L (0 ,T ; V ) ≤ ( d , [ c I + B ] − d ) Ω + 1 k Z t k S ( s ) k V ′ ds B is continuous from V into H (Ω), we obtain that Bp n ∈ L (0 , T ; H (Ω)), and thus wehave k ( c I + B ) p n ( t ) k L (0 ,T ; H (Ω)) ≤ C k p n ( t ) k L (0 ,T ; V ) (4.8)Now, directly from the (4.4), using the characterization of the norm in V ′ = H − (Ω), we obtain k [( c I + B ) p n ] ′ ( s ) k V ′ ≤ M k p n ( s ) k V + k S ( s ) k V ′ , which implies that [( c I + B ) p n ] ′ ∈ L (0 , T ; V ′ )with Z t k [( c I + B ) p n ] ′ ( s ) k V ′ ≤ M Z t k p n ( s ) k V + Z t k S ( s ) k V ′ . ( d, [ c I + B ] − d ) Ω + Z t k S ( s ) k V ′ Existence: Since { p n } is bounded in L (0 , T ; V ), we can extract a weakly convergent subsequence p n k . If we call the weak limit p , then we have that p n k ⇀ p in L (0 , T ; V ) (4.9)Using the continuity of the operator B : V → H (Ω), we obtain that( c I + B ) p n k ⇀ ( c I + B ) p in L (0 , T ; H (Ω)) (4.10)According to the energy estimates above, we have that the subsequence { [( c + B ) p n k ] ′ } is boundedin L (0 , T ; V ′ ). Consequently, we obtain on a new subsequence (retaining the subscript n k ) that[( c + B ) p n k ] ′ ⇀ [( c + B ) p ] ′ in L (0 , T ; V ′ ) (4.11)Now invoking (4.3) we can write Z T h [( c I + B ) p n k ] ′ ( t ) , q ( t ) i dt + Z T A [ p n k ( t ) , q ( t ); z ( t )] dt = Z T h S, q i dt (4.12)for every q ∈ L (0 , T ; V n k ). Choose N such that N ≤ n k . In (4.12), let q = wϕ , with w ∈ V N and ϕ ∈ D (0 , T ), and let n k → ∞ . Thanks to (4.9) and (4.11) and the continuity of the bilinear form A we infer that Z T n(cid:10) [( c I + B ) p ] ′ ( t ) , w (cid:11) + A [ p ( t ) , w ; z ( t )] − h S ( t ) , w i o ϕ ( t ) dt = 0 (4.13)Letting N → ∞ and using the fact that ϕ is arbitrary, we obtain that (cid:10) [( c I + B ) p ] ′ ( t ) , w (cid:11) + A [ p ( t ) , w ; z ( t )] = h S ( t ) , w i , for a.e. t ∈ (0 , T ) , and for all w ∈ V, from which (4.3) follows.It remains to check that p satisfies the initial condition [ c I + B ] p (0) = d . We use (4.13) with ϕ ∈ C ([0 , T ]) that satisfies ϕ (0) = 1 and ϕ ( T ) = 0, and integrate by parts in the fist term. Weobtain Z T n − (cid:0) [ c I + B ) p ( t ) , w (cid:1) Ω ϕ ′ ( t ) + A [ p ( t ) , w ; z ( t )] ϕ ( t ) − h S ( t ) , w i ϕ ( t ) o dt = ([ c I + B ] p (0) , w ) Ω (4.14)17imilarly, we use q ( t ) = ϕ ( t ) w with w ∈ V n in (4.12), and integrate by parts in the first term. Weobtain Z T n − (cid:0) [ c I + B ] p n k ( t ) , w ) Ω ϕ ′ ( t )+ A [ p n k ( t ) , w ; z ( t )] ϕ ( t ) −h S ( t ) , w i ϕ ( t ) o dt = ([ c I + B ] p n k (0) , w ) Ω (4.15)If we let n k → ∞ in (4.15), the LHS converges to the LHS of (4.14) due to (4.10), and theRHS (( c I + B ) p n k (0) , w ) Ω → ( d , w ) Ω . Therefore we obtain that (( c I + B ) p (0) , w ) Ω = ( d , w ) Ω ,and using the density of V into Ω we have that [ c I + B ] p (0) = d as desired.Finally, we also note that from (4.10) and (4.11) we obtain that( c I + B ) p n k → ( c I + B ) p in L (0 , T ; L (Ω)) (4.16)Note that, through the limit point construction, we obtain the estimates below on the con-structed solutions by the weak lower semicontinuity of the norm. Corollary 4.3.
The weak solution constructed in Lemma 4.2 satisfies for a.e. t ∈ [0 , T ] theestimates c || p ( t ) || L (Ω) + || B / p ( t ) || L (Ω) + k p k L (0 ,T ; V ) . || d || L (Ω) + k S k L (0 ,T ; V ′ ) (4.17) k [( c I + B ) p ] ′ k L (0 ,T ; V ′ ) . k p k L (0 ,T ; V ) + k S k L (0 ,T ; V ′ ) (4.18) k ( c I + B ) p k L (0 ,T ; H (Ω)) . k p k L (0 ,T ; V ) . (4.19)We are now in position to conclude the proof of Theorem 2.1. Proof of Theorem 2.1.
Given the lemma above, for a given z ∈ L (0 , T ; L (Ω)), we have obtainedthe functions p ( z ) ∈ L (0 , T ; V ) , ζ ( z ) = c p ( z ) + Bp ( z ) ∈ L (0 , T ; H (Ω)) , where we have denoted the dependence of the solution on the given function z . Since ∇ p ( z ) ∈ L (Ω)a.e. t , we can invoke the elasticity isomorphism as in Section 3.1 to obtain u ( z ) = E − ( −∇ [ p ( z ))]) ∈ H (Ω) ∩ V , a.e. t. By the injectivity of B on V = H (Ω), we can identify B [ p ( z )] = ∇ · [ u ( z )] (as in (3.7)). Hence,we obtain a solution to (3.7).Finally, with the hypotheses on F and d , we can translate back to the original case as in (3.9)to immediately obtain a weak solution for E ( u ) = −∇ p + F ∈ L (0 , T ; V ′ ) ζ t − ∇ · [ k ( z ) ∇ p ] = S ∈ L (0 , T ; V ′ ) ζ = c p + ∇ · u ∈ L (0 , T ; L (Ω)) c p (0) + ∇ · u (0) = d ∈ L (Ω) . (4.20)18 Nonlinear Problem - Existence of Solutions
This section contains the proof of Theorem 2.2. We divide the proof into two parts. First, we focuson the case of compressible constituents, i.e., c >
0. Our strategy in this scenario is to show thatthe map z ζ provided by Theorem 2.1 has a fixed point. In part two of the proof, we obtainexistence of solutions for the case of incompressible mixture constituents using a limiting process c ց c > We begin by considering the general translated problem with F ≡
0. By Theorem 2.1, given z ∈ L (0 , T ; L (Ω)), the problem (3.7) (with associated regularity of data) has a weak solution , writtenas ( u ( z ) , ζ ( z ) , p ( z )), satisfying the estimates in (4.17)–(4.19). We note that since the solution tothe linear problem provided by Theorem 2.1 is not necessarily shown to be unique, we must allowthe possibility that the solution mapping is multi-valued (in the sense of the Appendix 7). Thuswe consider the reduced problem ζ t − ∇ · [ k ( z ) ∇ p ] = S ∈ L (0 , T ; V ′ ) ζ = c p + Bp ∈ L (0 , T ; L (Ω)) ζ (0) = d ∈ L (Ω) , (5.1)and define a correspondence between the given permeability argument z ∈ L (0 , T ; L (Ω)) and theresulting fluid contents ζ taken from weak solutions (in the sense of Definition 4) correspondingto given z satisfying the a priori estimates in (4.17)–(4.19). Definition . For fixed “data” d and S as above, and given z ∈ L (0 , T ; L (Ω), we define thecorrespondence F : L (0 , T ; L (Ω)) ։ L (0 , T ; L (Ω)) , by F ( z ) = (cid:8) ζ = [ c I + B ] p : ( p, ζ ) is a weak solution of (5.1) that satisfies (4.17)–(4.19) (cid:9) Clearly, using Lemma 4.2 and Corollary 4.3, we have that the set F ( z ) = ∅ . Moreover, the factthat the range of the correspondence R ( F ) ⊆ L (0 , T ; L (Ω)) is immediate, since all the elementsin the set p ( z ) belong to L (0 , T ; L (Ω)) trivially, and hence by the boundedness of B ∈ L ( L (Ω))we have that [ c I + B ] p ∈ L (0 , T ; L (Ω)) , ∀ p ∈ p ( z ) . Note here that by the definition of the correspondence, the satisfaction of the initial conditionis included in the definition of F . Also, note that passing between a ζ ( z ) and a p ( z ) simply usesthe invertibility of [ c I + B ] on L (0 , T ; L (Ω)), as in Corollary 3.5.We will use the Bohnenblust-Karlin Fixed Point Theorem for correspondences [1] to obtainthe existence of (at least) one fixed point for F . The statement of the theorem, along with therelevant background definitions can be found in the Appendix 7. We have the following theorem: Theorem 5.1.
The correspondence F : L (0 , T ; L (Ω)) ։ L (0 , T ; L (Ω)) defined above has afixed point. The set of fixed points of F is compact in L (0 , T ; L (Ω)) . roof of Theorem 5.1. Let d ∈ L (Ω) and S ∈ L (0 , T ; V ′ ) be given. We consider the correspon-dence F : L (0 , T ; L (Ω)) ։ L (0 , T ; L (Ω)) defined above in Definition 5. By the construction of linear solutions as given in Lemma 4.2 and the corresponding estimates in Corollary 4.3, we havethat for each z ∈ L (0 , T ; L (Ω)), the set F ( z ) = ∅ . Moreover, for each element ζ ∈ F ( z ), via 4.2,we have that ζ ∈ L (0 , T ; V ) , and ζ t ∈ L (0 , T ; V ′ ) , with associated estimates in (4.17)–(4.19).Step I. First, we show that the correspondence is convex- and closed-valued, i.e., F ( z ) is convexand closed, for each z ∈ L (0 , T ; L (Ω)). Thus let z ∈ L (0 , T ; L (Ω)) and let ζ , ζ ∈ F ( z ). Thismeans that the pairs (cid:0) p i , ζ i = [ c I + B ] p i (cid:1) satisfy the definition provided in Definition 4 for i = 1 , q ∈ L (0 , T ; V ), we have Z T (cid:10) [ c I + B ] p ′ i ( t ) , q ( t ) (cid:11) dt + Z T A [ p i ( t ) , q ( t ); z ( t )] dt = Z T h S ( t ) , q ( t ) i dt (5.2)and (cid:2) ( c I + B ) p i (cid:3) (0) = d .Since the problem is linear in k ( z ) with A [ p, q ; z ] = (cid:0) k ( z ) ∇ p, ∇ q ) Ω , convexity in the weak formof solutions and initial conditions is immediate by taking the appropriate linear combination ofthe above equalities. For the associated inequalities in (4.17)–(4.19), the convexity of norms issufficient. Indeed, we show this for (4.17): Suppose for p i ( z ), i = 1 , c || p i ( z ) || L ∞ (0 ,T ; L (Ω)) + || B / p i ( z ) || L ∞ (0 ,T ; L (Ω)) + k p i ( z ) k L (0 ,T ; V ) ≤ C [ || d || L (Ω) + || S || L (0 ,T ; V ′ ) ] . (5.3)Then || αp + (1 − α ) p || ≤ α || p || + (1 − α ) || p || , and hence by multiplying the inequality (5.3) by α when i = 1, and again multiplying by (1 − α )when i = 2, then adding the results, yields that the function αp + (1 − α ) p satisfies (2.10) forany α ∈ [0 , with the same constant C associated to the RHS .To show that F ( z ) is closed, consider a sequence ζ n ∈ F ( z ) such that ζ n → ζ ∈ L (0 , T ; L (Ω)).Using the invertibility of the linear operator c I + B on L (0 , T ; L (Ω)), we have that p n → ( c I + B ) − ζ ∈ L (0 , T ; L (Ω)), so we let p = ( c I + B ) − ζ ∈ L (0 , T ; L (Ω)). Since the pair ( p n , ζ n )satisfies the estimates (4.17)–(4.19), we know that p n is uniformly-in- n bounded in L (0 , T ; V ).Therefore p n has a weakly convergent subsequence, whose limit is identified with p by uniquenessof limits. This yields that p lies in L (0 , T ; V ).Passing to the limit, then, in the weak form in (4) is immediate, since the problem is linearin k ( z ). The estimates in (4.17)–(4.19) on p ( z ) and ζ ( z ) = [ c I + B ] p ( z ) follow from weak lowersemicontinuity of norms, since each ( p n , ζ n ) satisfies them by hypothesis. Lastly, obtaining theinitial condition is immediate, since each ζ n corresponds to a solution with the same initial condition ζ n ( t = 0) = d . The estimates on solutions in (4.17)–(4.19) ensure that ζ n , ζ ∈ H (0 , T ; V ′ ), andhence we have that ζ ( t = 0) = d as the limit point of ζ n (0) (in the sense of V ′ ).Step II. Next, we show the sequential criterion for UHC of the correspondence F , as in Theorem7.3. To that end, let { ( z n , ζ n ) } ⊆ G ( F ). Suppose further that z n → z ∈ L (0 , T ; L (Ω)). We wantto conclude that ζ n has a (strong) limit point ζ ∈ F ( z ).First, by Assumption 1.1, the function k ( · ) considered as Nemytskii operator, has the propertythat k ( z n ) → k ( z ) ∈ L (0 , T ; L (Ω)). Now, since ζ n ∈ F ( z n ), for the unique p n = [ c I + B ] − ζ n we20ave by definition of F the estimate (4.17) and that the weak form of the equation is satisfied(as in Definition 4). The estimate (4.17) (with fixed RHS in terms of data) yields a uniform-in- n bound on || p n || L (0 ,T ; V ) , || p n || L ∞ (0 ,T ; L (Ω)) , || B / p n || L ∞ (0 ,T ; L (Ω)) . From the bound on p n in L (0 , T ; V ) we extract a weak subsequential limit point, i.e., p n k ⇀ p ∈ L (0 , T ; V ) . From this and the continuity of [ c I + B ] ∈ L ( L (0 , T ; L (Ω))) , we obtain immediatelythat ζ n k = [ c I + B ] p n k ⇀ [ c I + B ] p. We define this latter quantity as ζ ≡ [ c I + B ] p, and hence ζ n k ⇀ ζ. In addition, the estimate (4.19) in the definition of F and the uniqueness of limits ensurethat (perhaps passing to a further subsequence with the same label) ζ n k ⇀ ζ ∈ H (0 , T ; V ′ ) . Wewant to show that ζ ∈ F ( z ), and this is accomplished by passing with the limit on the subsequence n k in the weak formulation (4.3). To that end, let us again consider the weak form evaluated on n k , and restrict our spatial test functions to q ∈ L (0 , T ; V ) ∩ L ∞ (0 , T ; W , ∞ (Ω)): Z T (cid:10) ζ ′ n k ( t ) , q ( t ) (cid:11) dt + Z T A [ p n k ( t ) , q ( t ); z n k ( t )] dt = Z T h S ( t ) , q ( t ) i dt (5.4)Limit passage on the first term on the LHS is immediate identifying weak limits in this weak form.For the second term, more care must be taken. Consider: Z T (cid:0) k ( z n k ) ∇ p n k , ∇ q ( t ) (cid:1) dt = Z T (cid:0) [ k ( z n k ) − k ( z )] ∇ p n k , ∇ q ( t ) (cid:1) dt + Z T ( k ( z ) ∇ p n k , ∇ q ( t )) dt. (5.5)The first term on the RHS is handled through the Nemytskii property of k ( · ): Z T ([ k ( z n k ) − k ( z )] ∇ p n k , q ( t )) dt ≤ C ( || q || L ∞ (0 ,T ; W , ∞ (Ω)) ) || k ( z n k ) − k ( z ) || L (0 ,T ; L (Ω)) || p n k || L (0 ,T ; V ) ≤ C ( q, || p || L (0 ,T ; V ) ) || k ( z n k ) − k ( z ) || L (0 ,T ; L (Ω)) → , by the uniform bound on p n k in L (0 , T ; V ). Convergence of the second term in (5.5) is imme-diate, since by the boundedness of k we have k ( z ) ∇ q ∈ L (0 , T ; L (Ω)); thence, ∇ p n k ⇀ ∇ p ∈ L (0 , T ; L (Ω)).Thus, we have shown that for q ∈ L (0 , T ; V ) ∩ L ∞ (0 , T ; W , ∞ (Ω)) Z T ( k ( z n k ) ∇ p n k , ∇ q ( t )) dt → Z T ( k ( z ) ∇ p, ∇ q ( t )) dt, and hence, passing to the limit as k → ∞ in (5.4) we obtain for ζ = ζ ( z ) the identity Z T h ζ t , q i dt + Z T ( k ( z ) ∇ p, ∇ q ( t )) dt = Z T h S, q ( t ) i dt (5.6)for all q ∈ L (0 , T ; V ) ∩ L ∞ (0 , T ; W , ∞ (Ω)), the latter being dense in L (0 , T ; V ). Thus we haveshown that ( ζ ( z ) , p ( z )) satisfies the weak form of the pressure equation and hence we have con-structed a weak solution ( ζ ( z ) , p ( z )) for z ∈ L (0 , T ; L (Ω)). The requisite estimates in the defini-tion of F ((2.10)–(2.12)) hold immediately on the weak subsequential limit points ( ζ ( z ) , p ( z )) (inthe relevant topologies) by weak lower semicontinuity of norms. Obtaining the initial condition isalso immediate from the definition of F . Hence ζ n has a weak subsequential limit point ζ ∈ F ( z ).To conclude the UHC property of the correspondence F , we must improve the convergenceof ζ n k → ζ strongly in L (Ω). This is done via the Lions-Aubin compactness theorem (see, for21nstance, [19]). In addition to the estimate (4.17) for the sequence p n k , we obtain two additionaluniform-in- k estimates from continuity of B : V → H (Ω) and from satisfying the weak form ofthe pressure equation, namely: || ζ n k || L (0 ,T ; H (Ω)) = || c p n k + Bp n k || L (0 ,T ; H (Ω)) . || p || L (0 ,T ; V ) (5.7) k [ ζ n k ] ′ k L (0 ,T ; V ′ ) = k [( c I + B ) p n k ] ′ k L (0 ,T ; V ′ ) . k p k L (0 ,T ; V ) + k S k L (0 ,T ; V ′ ) (5.8)By possibly passing to a further subsequence n k m (not affecting the previous steps in estab-lishing the weak solution or associated estimates), we improve the convergence of ζ n km → ζ ∈ L (0 , T ; L (Ω)). Applying Theorem 7.3, we obtain that F : L (0 , T ; L (Ω)) ։ L (0 , T ; L (Ω)) isUHC as well as compact-valued. Subsequently, from Theorem 7.2, we have that F is a closedcorrespondence.Step III. Lastly, to invoke the Bohnenblust-Karlin Fixed Point Theorem, we must show that therange of F is relatively compact in L (0 , T ; L (Ω)). But, as in the previous step, this will followfrom the Lions-Aubin compactness criterion. Indeed, for any ζ ∈ R ( F ), ζ corresponds to a weaksolution satisfying the estimate (4.17), and subsequently (4.18)–(4.19). In particular, we obtain forany such ζ there is an associated p = [ c I + B ] − ζ such that: || ζ || L (0 ,T ; H (Ω)) ≤ C || p || L (0 ,T ; V ) ≤ C (cid:2) || d || L (Ω) + || S || L (0 ,T ; V ′ ) ] (5.9) k ζ ′ k L (0 ,T ; V ′ ) ≤ C (cid:2) k p k L (0 ,T ; V ) + k S k L (0 ,T ; V ′ ) (cid:3) ≤ C (cid:2) || d || L (Ω) + || S || L (0 ,T ; V ′ ) (cid:3) (5.10)A subset of L (0 , T ; L (Ω)) which is bounded as in the previous two estimates is relatively compactby the Lions-Aubin criterion, and hence ζ ∈ R ( F ) lies in a compact set. This is the final hypothesisto be satisfied for applying the fixed point result, Theorem 7.4.Applying the fixed point theorem yields the existence of a function z ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; V ′ ) and an associated weak solution ( ζ ( z ) , p ( z )) for which z ∈ F ( z ). Remark . We again note that, owing the presence of the nonlinearity, regularity of the solution ζ —in particular of ∇ · u —needs to be better than L (0 , T ; L (Ω)). This is because we must obtaincompactness in ζ to utilize the Nemytskii property of k ( · ). Moreover, if d ∈ V ′ only, this wouldpreclude our ability to obtain such regularity, as this would seem to lower the evolution of Bp = ∇· u to the regularity of V ′ .Now we proceed to obtain a weak solution as in Definition 1 to the original problem, which isstated in strong form in Section 3.4 as (3.11). By applying the fixed point result from the previoussection, Theorem 5.1, with appropriately scaled/modified data S, d , k ( · ) we will finally prove themain result Theorem 2.2. Proof of Theorem 2.2.
Considering the strong form of the nonlinear dynamics in (3.11), we applythe translation as in Section 3.4. For the new variable w = u − u F , where u F is defined by (3.8),we obtain the system (3.12), which we restate here for clarity: E ( w ) = −∇ p ∈ L (0 , T ; V ′ )[ c p + ∇ · w ] t − ∇ · [ k F ( c p + ∇ · w ) ∇ p ] = S + ∇ · u F ,t ∈ L (0 , T ; V ′ ) c p (0) + ∇ · w (0) = d − ∇ · u F (0) ∈ L (Ω) . (5.11)Under the regularity assumptions on F , we have u F ∈ L (0 , T ; H (Ω) ∩ V ) ∩ H (0 , T ; V ). Hence,in the above strong form, we have ∇ · u F ,t ∈ L (0 , T ; L (Ω)) ⊂ L (0 , T ; V ′ ). Additionally, ∇ · F ∈ C ([0 , T ]; L (Ω)) [13, 19], so we can extract its time trace at t = 0. Finally, since ∇ · u F ∈ C ([0 , T ]; L (Ω)), we observe k F ( · ) = k ( · + ∇ · u F ) satisfies the hypotheses of Assumption 1.1 since k ( · ) is assumed to satisfy them. We can then reduce the above system to a version of (5.1), wherethe corresponding “data” satisfies the hypotheses of Theorem 5.1, and we can apply the result ofthe fixed point theorem.Doing so, we obtain a function p ∈ L (0 , T ; V ) that satisfies: • For every q ∈ L (0 , T ; V ), Z T (cid:10) [( c I + B ) p ] ′ , q (cid:11) dt + Z T (cid:0) k ( c p + Bp + ∇· u F ) ∇ p, ∇ q ) dt = Z T h S, q i dt + Z T ( ∇· u F ,t , q ) dt. • ζ = c p + Bp a.e. t and a.e. x . • (cid:2) ( c I + B ) p (cid:3) (0) = d in the sense of V ′ . • The following estimates hold: c || p ( t ) || L (Ω) + || B / p ( t ) || L (Ω) + k p k L (0 ,T ; V ) . || d || L (Ω) + ||∇ · u F (0) || L (Ω) + k S k L (0 ,T ; V ′ ) k [( c I + B ) p ] ′ k L (0 ,T ; V ′ ) . k p k L (0 ,T ; V ) + k S k L (0 ,T ; V ′ ) + ||∇ · u F ,t || L (0 ,T ; V ′ ) k ( c I + B ) p k L (0 ,T ; H (Ω)) . k p k L (0 ,T ; V ) . From this we can obtain a w ∈ L (0 , T ; H (Ω) ∩ V ) and then a u = w + u F , resulting in u = E − ( −∇ p + F ) . This u has the necessary property that it can be identified via the relation ∇ · u = Bp + ∇ · u F in a point-wise sense. The above is sufficient to conclude the weak form of the elasticity equation,(2.6). Using the fact that || u F || V . || F || V ′ and the embedding H (0 , T ; L (Ω)) ֒ → C ([0 , T ]; L (Ω)),we re-interpret the estimates above as c || p || L ∞ (0 ,T ; L (Ω)) + k p k L (0 ,T ; V ) . || d || L (Ω) + DAT A (cid:12)(cid:12) T (5.12) || u || L (0 ,T ; H ∩ V ) . || p || L (0 ,T ; V ) + || F || L (0 ,T ; L (Ω)) (5.13) k [ c p + ∇ · u ] ′ k L (0 ,T ; V ′ ) . k p k L (0 ,T ; V ) + DAT A (cid:12)(cid:12) T (5.14) k c p + ∇ · u k L (0 ,T ; H (Ω)) . k p k L (0 ,T ; V ) . (5.15)(Note that in these final estimates for the original problem we have omitted references to the B operator, in doing so, discarding the information on B / p ∈ L ∞ (0 , T ; L (Ω)) . )This finally concludes the proof of Theorem 2.2. c = 0 Consider a sequence or real numbers { c m } ∞ such that c m <
1, for all m ≥
1, and c m ց m → ∞ . To each c m we attribute a particular weak solution ( p m , u m ) ∈ L (0 , T ; V ) × L (0 , T ; V )to (3.11) (i.e., a solution in the sense of Definition 1) with a fixed initial condition d ∈ L (Ω) for23ll m (the intended initial condition when c = 0). Such a solution has, by construction in theproof of Theorem 2.2, the energy estimate: c m || p m || L ∞ (0 ,T ; L (Ω)) + k Z T ||∇ p m || dτ . || d || + DAT A (cid:12)(cid:12) T . Note, the bound on the RHS is uniform in m . This provides a weak-* subsequential limit point for p c m p m ∈ L ∞ (0 , T ; L (Ω))), and a weak subsequential limit point labeled p for the sequence p m ∈ L (0 , T ; V ). From the elasticity equation, we infer that the associated u m ∈ L (0 , T ; H (Ω) ∩ V )has the bound Z T || u m || H (Ω) ∩ V dt . Z T ||E ( u m ) || dt . Z T ||∇ p m || dt + Z T || F || L (Ω) dt, and taken in conjunction with the estimate above, leads to a uniform-in- m estimate. To the se-quence u m we also associate a weak subsequential limit point (with the same subsequence associ-ated to p , perhaps upon re-indexing): u m ⇀ u ∈ L (0 , T ; V ) . From the construction of the weak solution, we have that p m satisfies for all q ∈ L (0 , T ; V ): Z T h ζ mt ( t ) , q ( t ) i dt + Z T ( k ( ζ m ( t )) ∇ p m ( t ) , ∇ q ( t )) dt = Z T h S ( t ) , q ( t ) i dt, (5.16)with ζ m = c m p m + ∇ · u m = [ c m I + B ] p m + ∇ · u F ∈ L (0 , T ; L (Ω)) and the associated bound || ζ m || L (0 ,T ; L (Ω)) = || c m p m + ∇ · u m || L (0 ,T ; L (Ω)) (5.17) ≤ c m || p m || L (0 ,T ; L (Ω)) + ||∇ · u m || L (0 ,T ; L (Ω)) ≤ C (cid:2) || p || L (0 ,T ; V ) + || u || L (0 ,T, V ) (cid:3) . || d || + DAT A (cid:12)(cid:12) T + || F || L (0 ,T ; L (Ω)) (5.18)by lower weak semicontinuity of the norm, Poincar´e, and the boundedness of the sequence { c m } .Hence there is ζ ∈ L (0 , T ; L (Ω)) so that ζ m ⇀ ζ ∈ L (0 , T ; L (Ω)) . As in the construction of the solution, satisfying the equation (5.16) is also sufficient to deducethat ζ mt ∈ L (0 , T ; V ′ ) (with associated uniform-in- m bound as above in (5.14)). Again, by such auniform-in- m bound on ζ mt ∈ L (0 , T ; V ′ ) we can extract a weak subsequential limit in that space,and identify in the standard way for the distributional derivative [14] ζ mt ⇀ ζ t ∈ L (0 , T ; V ′ ) . Additionally, the sequence p m itself is bounded as || p m || L (0 ,T ; L (Ω)) ≤ C || p m || L (0 ,T ; V ) . || p || L (0 ,T ; V ) . This implies that the sequence [ c m p m ] → ∈ L (0 , T ; L (Ω)), as well as, for any φ ∈ V and f ∈ D (0 , T ), c m Z T ( p m ( t ) , φ ) Ω f ′ ( t ) dt → , c m →
0. From which we deduce by the definition of the distributional derivative in t (see Remark4.2) that c m Z T h p mt ( t ) , q ( t ) i V ′ × V dt → , ∀ q ∈ L (0 , T ; V ) . (5.19)Now, it is immediate from the previously established convergences that ζ m = c m p m + ∇ · u m ⇀ ∇· u ∈ L (0 , T ; L (Ω)), and by uniqueness of limits, we must have that ζ = ∇· u ∈ L (0 , T ; L (Ω)),possibly passing to a subsequence. Moreover, since ζ mt ∈ L (0 , T ; V ′ ) with c m p mt ⇀
0, we obtainthat Z T h ζ mt , q i dt → Z T h ζ t , q i , Z T h ζ mt , q i dt = Z T (cid:2) h c m p mt , q i + h∇ · u mt , q i (cid:3) dt. But we know, again by uniqueness of limits, since we have (5.19), that Z T h ζ mt , q i → Z T h ζ t , q i dt = Z T h∇ · u t , q i dt. Lastly, we need to show the nonlinear term exhibits convergence; namely, we want to show Z T ( k ( ζ m ) ∇ p m , ∇ q ) dt → Z T ( k ( ∇ · u ) ∇ p, ∇ q ) dt. (5.20)This will follow immediately as in Step II in the proof of Theorem 5.1, using the Nemytskii propertyof k ( · ). Indeed, we first improve the bounds on ζ m : || ζ m || L (0 ,T ; H (Ω)) = || c m p m + ∇ · u m || L (0 ,T ; H (Ω)) . c m || p m || L (0 ,T ; V ) + || u m || H (Ω) . || p || L (0 ,T ; V ) + || u || L (0 ,T ; V ∩ H (Ω)) (5.21)This again results in uniform-in- m boundedness, and we note that (perhaps on a subsequence),again by The Lions-Aubin criterion and the uniform boundedness of { ζ m } ⊂ L (0 , T ; H (Ω)) and { ζ mt } ⊂ L (0 , T ; V ′ ), we can improve the convergence of ζ m → ζ ∈ L (0 , T ; L (Ω)) to strong convergence. At this point, k ( ζ m ) → k ( ζ ) ∈ L (0 , T ; L (Ω)), and we invoke the identification ζ = ∇ · u . From this, (5.20) follows.Hence, upon taking the limit as c m → p, u ) ∈ L (0 , T ; V × V ) satisfies: Z T h∇ · u t ( t ) , q i dt + Z T (cid:0) k ( ζ ( t )) ∇ p ( t ) , ∇ q (cid:1) f ( t ) dt = Z T h S ( t ) , q i f ( t ) dt, (5.22)for any q ∈ L (0 , T ; V ).As in the c > E ( u ) = −∇ p + F , a.e. x , t, which of course yields the weak form in Definition (4).Finally, by the equation, we again have that ∇ · u ∈ H (0 , T ; V ′ ) ∩ L (0 , T ; H (Ω)), so ∇ · u ∈ C ([0 , T ]; V ′ ), which permits the initial condition for the quantity ∇ · u (0) = d in the V ′ sense,though d ∈ L (Ω). This ensures that ( p, u ) ∈ L (0 , T ; V × V ) is in fact a weak solution with c = 0, in the sense of Definition 1 (see Remark 2.1).25 Uniqueness - Proof of Theorem 2.3
In this section we divide our considerations into two approaches, depending on which terms havepoint-wise-in- t control: (i) the first one considers a weak solution to the full dynamical system inboth dependent variables ( u , p ); (ii) the second approach considers the reduced system phrased interms of p and Bp . There are subtle differences in the approaches and in the requisite hypothesesto obtain unique solutions. We point out that—since we do not construct strong solutions in thispaper—we phrase our results below as: If weak solutions exhibit additional regularity properties,then among such weak solutions there is uniqueness . We recall the space of weak solutions (forgiven data d , S, F , k ( · )) with the additional regularity p t ∈ L (0 , T ; L (Ω)) denoted by W T —see(2.13). In particular, our result says: If one weak solution in W T has additional spatial regularity,then all weak solutions in W T are equal to it. Let us introduce both types of formal energy identities utilized later in the proof of Theorem 2.3for c ≥
0. Consider ( u , p ) and ( u , p ) two weak solutions coming from W T . Then, we subtractthe weak forms of the equations as in Definition 4 and test the pressure equation with p = p − p and the elasticity equation with u t = u t − u t . The latter is justified, as u it = E − (cid:0) − ∇ p it + F t (cid:1) ∈ L (0 , T ; V ) since F t ∈ L (0 , T ; V ′ ) by our regularity hypotheses on F and ∇ p it ∈ L (0 , T ; V ′ ) since p it ∈ L (0 , T ; L (Ω)).This yields the unsimplified identities: Z T hE ( u ) , u t i V × V ′ dt − Z T ( p, ∇ · u t ) Ω dt = 0 (6.1) Z T (cid:2) c ( p t , p ) Ω + ( ∇ · u t , p ) Ω (cid:3) dt + Z T (cid:0) k ( ζ ) ∇ p − k ( ζ ) ∇ p , ∇ p (cid:1) Ω dt = 0 , (6.2)where we denote ζ i = c p i + ∇ · u i .Now, let us consider the formal energy relation for the (partial, omitting the equation for u )reduced formulation making use of the B operator, again, with no simplifications: (cid:0) [ c I + B ] p t , p (cid:1) Ω + (cid:0) k ( ζ ) ∇ p − k ( ζ ) ∇ p , ∇ p ) Ω = 0 . (6.3)We have replaced the V ′ × V duality pairings in the pressure equations above through theassumption that p it ∈ L (0 , T ; L (Ω)). Both approaches to uniqueness hinge on the analysis of thenonlinear term. The goal is to apply a version of Gr¨onwall to the formal estimates.Let us begin by estimating directly this nonlinear term above. Z T (cid:0) [ k ( t ) ∇ p ( t ) − k ( t ) ∇ p ( t )] , ∇ p (cid:1) dt = Z T (cid:0) [ k ( t ) − k ( t )] ∇ p ( t ) , ∇ p ( t ) (cid:1) dt (6.4)+ Z T (cid:0) k ( t ) ∇ p ( t ) , ∇ p ( t ) (cid:1) dt, where we have used the shorthand k i ( t ) ≡ k ( c p i + ∇· u i ). Using the lower bound on the permeabilityfunction 0 < k ≤ k ( · ) from Assumption 1.1 the second term above will serve as dissipation tohelp with further estimation k || p || L (0 ,T ; V ) ≤ Z T (cid:0) k ( t ) ∇ p ( t ) , ∇ p ( t ) (cid:1) dt. The remaining nonlinear term on the RHS can be estimated in two ways, yielding the twodistinct hypotheses. With the supplemental hypothesis that k ∈ Lip ( R ) with L k > | k ( t ) − k ( t ) | = (cid:12)(cid:12) k ( ζ ) − k ( ζ ) | ≤ L k | ζ | , again where ζ = ζ − ζ , and ζ i = c p i + ∇ · u i . Using Cauchy-Schwartz, then, we have Z T (cid:0) [ k ( t ) − k ( t )] ∇ p ( t ) , ∇ p ( t ) (cid:1) dt ≤ L k Z T ||∇ p || L ∞ (Ω) || ζ || L (Ω) ||∇ p || L (Ω) dt. We proceed straightforwardly, retaining the supremum term under the integration: Z T (cid:0) [ k ( t ) − k ( t )] ∇ p ( t ) , ∇ p ( t ) (cid:1) dt ≤ Z T (cid:2) L k ||∇ p || L ∞ (Ω) (cid:3) || ζ ( t ) || L (Ω) ||∇ p || L (Ω) dt ≤ Z T L k ǫ ||∇ p ( t ) || L ∞ (Ω) (cid:2) || ζ ( t ) || L (Ω) (cid:3) dt (6.5)+ ǫ Z T ||∇ p || L (Ω) dt, ∀ ǫ > . Remark . One can also pull the supremum term outside the integral; this is akin to the approachtaken in [6]. Z T (cid:0) [ k ( t ) − k ( t )] ∇ p ( t ) , ∇ p ( t ) (cid:1) dt ≤ L k ǫ ||∇ p || L ∞ (0 ,T ; L ∞ (Ω)) Z T || ζ ( t ) || L (Ω) dt + ǫ Z T ||∇ p || L (Ω) dt, ∀ ǫ > . Anticipating the use of Gr¨onwall below, we note that ∇ p ∈ L ∞ (Ω) is obtained through theSobolev embeddings in 3-D if for instance, if p ∈ H (Ω) (or any Sobolev index above 2 . c ≥ In working with the full system, we can exploit cancellation in the structure of the Biot systemto obtain more explicit energy estimates to be used for uniqueness. In this framework, as weshall see, we need to specify the initial displacement u ∈ V independently of c p , recalling that ζ (0) = d = [ c p + ∇ · u ](0) . Then, we consider (6.1)–(6.2) and add the two equations, cancelling cross-terms on the RHSand simplifying by integration by parts. This yields: a ( u ( t ) , u ( t )) + c || p ( t ) || + 2 Z t ( k ( ζ ) ∇ p − k ( ζ ) ∇ p , ∇ p ) dt = a ( u , u ) + c || p || . (6.6)We recall the estimate on a single trajectory: || u || L ∞ (0 ,T ; V ) + c || p || L ∞ (0 ,T ; L (Ω)) + k || p || L (0 ,T ; V ) ≤ C ( u , p ) + DAT A (cid:12)(cid:12) T . (6.7)The resulting estimate on ( u , p ) as above in (6.5) is || u || L ∞ (0 ,T ; V ) + c || p || L ∞ (0 ,T ; L (Ω)) + k || p || L (0 ,T ; V ) . C ( u , p ) + Z T (cid:0) [ k ( t ) − k ( t )] ∇ p , ∇ p (cid:1) dt, . L k ǫ Z T ||∇ p ( t ) || L ∞ (Ω) (cid:2) || ζ ( t ) || L (Ω) (cid:3) dt + ǫ Z T ||∇ p || L (Ω) dt (6.8)27e then note that ζ = c p + ∇ · u , and hence || ζ || . c || p || + || u || V . Absorbing on the RHS by choosing, e.g., ǫ = k , we obtain: || u || L ∞ (0 ,T ; V ) + c || p || L ∞ (0 ,T ; L (Ω)) . L k k Z T ||∇ p ( t ) || L ∞ (Ω) (cid:2) c || p || + || u || V (cid:3) dt. (6.9)If c <
1, then we have || u || L ∞ (0 ,T ; V ) + c || p || L ∞ (0 ,T ; L (Ω)) . L k k Z T ||∇ p ( t ) || L ∞ (Ω) (cid:2) c || p || + || u || V (cid:3) dt (6.10)If c >
1, then we have || u || L ∞ (0 ,T ; V ) + c || p || L ∞ (0 ,T ; L (Ω)) . L k c k Z T ||∇ p ( t ) || L ∞ (Ω) (cid:2) c || p || + || u || V (cid:3) dt (6.11)From here, we may invoke L -kernel version of Gr¨onwall as in [12, Theorem 9], and uniqueness ofsolutions is deduced in the standard way. Remark . Recalling the space H = ∇ H (Ω) = ∇ V (from [22]), wenote that for weak solutions we have that u t ∈ L (0 , T ; [ H ] ′ ), by satisfying the weak form ofthe pressure equation (since it can act on ∇ q ∈ ∇ V ). On the other hand, since p ∈ L (0 , T ; V ),we have that ∇ p ∈ L (0 , T ; H ). Thus, the “cross-terms” h∇ · u t , p i and h∇ p, u t i can both beinterpreted in the [ H ] ′ × H sense. In addition, through satisfying the elasticity equation, we havethat E ( u ) = ∇ p ∈ H , and hence the pairing hE ( u ) , u t i is also defined similarly. Hence, we couldreduce our uniqueness assumption on the solutions to: for a weak solution ( u , p ), the [ H ] ′ × H duality pairing below has the property that hE ( u ) , u t i H × [ H ] ′ = 12 ddt a ( u , u ) , which may not be the case in general. c > In this section we consider working with the reduced equation directly. We assume only that d ∈ L (Ω), forgoing any assumptions on u ( t = 0). As we will see, we need to assume c > F and S as above, let us consider two weak solutions p i ( t ) ∈ L (0 , T ; V ) ∩ H (0 , T ; L (Ω))(this follows, for instance, if ( u , p ) ∈ W T and the problem is reduced through the B operator) to[ c I + B ] p t − ∇ · k ( ζ ) ∇ p = S + ∇ · u F ,t ∈ L (0 , T ; V ′ ) , using the notation from Section 3.4. We will denote ζ = c p + Bp + ∇ · u F here for the fluid content. Remark . Here, the main regularity we need is to be able to interpretthe pairing h [ c I + B ] p t , p i in some sense. The challenge is that the properties of B in both V and V ′ are not clear (e.g., self-adjointness), and for p ∈ L (0 , T ; V ), it is not clear that Bp ∈ L (0 , T ; V ).28et p = p − p as before, and hence ζ = c p + Bp . Then the straightforward energy relationin (6.3) simplifies to 12 ddt (cid:2) c || p || + ( Bp, p ) (cid:3) + ( k ( ζ ) ∇ p − k ( ζ ) ∇ p , ∇ p ) = 0 . Add and subtract, anticipating using the Lipschitz property of k : c || p ( t ) || + || B / p ( t ) || + 2 Z t ( ∇ p [ k ( ζ ) − k ( ζ )] , ∇ p ) + ( k ( ζ ) ∇ p, ∇ p ) dt = ( d , p (0))Since c >
0, we can recover p (0) = p = [ c I + B ] − d . Estimating as in the previous section andinvoking the assumptions on k ( · ), we obtain c || p || L ∞ (0 ,T ; L (Ω)) + || B / p || L ∞ (0 ,T ; L (Ω)) + k || p || L (0 ,T ; V ) . || d || + L k k Z T ||∇ p || L ∞ (Ω) || ζ || L (Ω) dτ. To proceed as before with Gr¨onwall, it is imperative here that c > B or B / is coercive. We then estimate ζ carefully: || ζ || = || c p + Bp || ≤ C || p || , where all norms are taken in the L (Ω) sense. Simplifying the above inequality, and invoking thisestimate, we obtain c || p || L ∞ (0 ,T ; L (Ω) . || d || L (Ω) + L k k Z T ||∇ p || L ∞ (Ω) || p || L (Ω) dt. (6.12)Since p ∈ H (0 , T ; L (Ω)) in this case, p ∈ C ([0 , T ]; L (Ω)) and at this point, Gr¨onwall can beapplied as before to obtain uniqueness in p a.e. t and x , which can then be transferred throughthe elasticity isomorphism E to p . This results in uniqueness of the weak solution ( u , p ) ∈ W T . We begin with a handful of definitions and straightforward theorems that will be relevant to thefixed point we are using in the construction of weak solutions. All of these considerations are takenfrom [1].The basic setting considers φ : X ։ Y as a correspondence, where, for each x ∈ X , φ ( x )represents a subset of Y . (We do not use the equivalent point of view that φ : X → Y .) The ։ notation indicates that φ need not be a function, but is thought of as a “multi-valued function.” Definition . A correspondence φ : X ։ Y betweentopological spaces is closed-valued if φ ( x ) is a closed set for each x ∈ X . The analogous definitionis used for a compact-valued correspondence.A correspondence φ : X ։ Y between topological spaces is closed (or has a closed graph ) if G ( φ ) ≡ { ( x, y ) ∈ X × Y : y ∈ φ ( x ) } is closed as a subset of X × Y . 29 efinition . A correspondence φ : X ։ Y between topological spaces is called upper hemiconti-nous (or UHC) at the point x ∈ X if for every neighborhood U ∋ x there is a neighborhood V ∋ x such that z ∈ V = ⇒ φ ( z ) ⊆ U. We say that φ is UHC on X if it is UHC at each x ∈ X .The next theorem provides the relationship between graph closedness and UHC. (We do notexplicitly use this version in the body of the paper.) Theorem 7.1.
Suppose φ : X ։ Y is closed-valued. If φ is UHC at x , then for all x n ∈ X , y ∈ Y ,and y n ∈ φ ( x n ) x n → x and y n → y = ⇒ y ∈ φ ( x ) . If φ is closed-valued and the range of φ is compact, then the converse holds. Alternatively, the following is the criteria we invoke in the proof of our main result:
Theorem 7.2.
Suppose φ : X ։ Y is an UHC correspondence. If φ is closed-valued (and Y isregular) OR φ is compact-valued (and Y is Hausdorff ), then φ is closed. The next theorem is a subtle variation on the previous sequential criteria for upper-hemicontinuity.
Theorem 7.3.
Assume that a topological space X is first countable and Y is metrizable. Then fora correspondence φ : X ։ Y and a point x ∈ X TFAE: • φ is UHC at x and φ ( x ) ⊂⊂ Y . • If a sequence { ( x n , y n ) } in G ( φ ) satisfies x n → x then { y n } has a limit point in φ ( x ) . Finally, we are in a position to state the multi-valued fixed point theorem employed in our con-structions above, the
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