Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations
aa r X i v : . [ m a t h . A P ] J a n DISCRETE DUALITY FINITE VOLUME SCHEMESFOR DOUBLY NONLINEAR DEGENERATEHYPERBOLIC-PARABOLIC EQUATIONS
B. ANDREIANOV, M. BENDAHMANE, AND K. H. KARLSEN
Abstract.
We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for whichwe first establish the existence and uniqueness of entropy solutions. We thenturn to the construction and analysis of discrete duality finite volume schemes(in the spirit of Domelevo and Omn`es [41]) for these problems in two and threespatial dimensions. We derive a series of discrete duality formulas and entropydissipation inequalities for the schemes. We establish the existence of solutionsto the discrete problems, and prove that sequences of approximate solutionsgenerated by the discrete duality finite volume schemes converge strongly tothe entropy solution of the continuous problem. The proof revolves aroundsome basic a priori estimates, the discrete duality features, Minty-Browdertype arguments, and “hyperbolic” L ∞ weak- ⋆ compactness arguments (i.e.,propagation of compactness along the lines of Tartar, DiPerna, . . . ). Ourresults cover the case of non-Lipschitz nonlinearities. Contents
1. Introduction 22. Notions of solution and well-posedness 53. Discrete duality finite volume (DDFV) schemes 133.1. Construction of “double” conformal meshes 143.2. Mesh parameters and regularity of meshes 163.3. Discrete gradient and divergence operators 173.4. Penalization operator 193.5. Discrete convection operator 203.6. Projection operators and test functions 213.7. Dependency on t and further notation 213.8. The finite volume scheme 224. Elements of discrete calculus for DDFV schemes 23 Date : November 15, 2018.2000
Mathematics Subject Classification.
Primary 35K65, 74S10; Secondary 35A05, 65M12.
Key words and phrases.
Degenerate hyperbolic-parabolic equation, conservation law, Leray-Lions type operator, non-Lipschitz flux, entropy solution, existence, uniqueness, finite volumescheme, discrete duality, convergence.The work of M. Bendahmane was supported by the FONDECYT project 1070682. The work ofK. H. Karlsen was supported by the Research Council of Norway through an Outstanding YoungInvestigators Award. A part of this work was done while B. Andreianov enjoyed the hospitality ofthe Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway. This articlewas written as part of the the international research program on Nonlinear Partial DifferentialEquations at the Centre for Advanced Study at the Norwegian Academy of Science and Lettersin Oslo during the academic year 2008–09.
Introduction
In this paper we consider degenerate hyperbolic-parabolic problems of the form(1) ∂ t u + div f ( u ) − div a ( ∇ A ( u )) = S , in Q := (0 , T ) × Ω ,u | t =0 = u , in Ω u = 0 , on Σ = (0 , T ) × ∂ Ω , where u : ( t, x ) ∈ Q → R is the unknown function, T > ⊂ R d is a bounded domain with polygonal boundary ∂ Ω and outward unit normal n . Weconsider the cases d = 2 and d = 3. The initial data u : Ω → R are assumed to bea bounded measurable function, i.e., u ∈ L ∞ (Ω) , while the source S : Q → R is assumed to be a measurable function for which S ( t, · ) ∈ L ∞ (Ω) for a.e. t ∈ (0 , T ) and R T k S ( t, · ) k L ∞ (Ω) dt < ∞ ; we abusivelydenote it by(2) S ∈ L (0 , T ; L ∞ (Ω)) . The function a : R N → R N is taken under the form a ( ξ ) = k ( ξ ) ξ, where k is a scalar function. The function a is assumed to be continuous and strictlymonotone. We assume that there exist p ∈ (1 , + ∞ ) and C > C | ξ | p − ≤ k ( ξ ) ≤ C | ξ | p − , ∀ ξ ∈ R d \ { } . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 3
In particular, the associated operator w
7→ − div (cid:0) k ( | ∇ w | ) ∇ w (cid:1) is a Leray-Lionsoperator acting from W ,p (Ω) to W − ,p ′ (Ω) with p ′ = pp − . A prototype exampleis the p -laplacian, which corresponds to k ( ξ ) = | ξ | p − .We assume that the diffusion function A ( · ) satisfies A ( · ) is continuous and nondecreasing, normalized by A (0) = 0 , while the convective flux function f ( · ) satisfies f = ( f , . . . , f d ) : Q × R → R d is continuous and normalized by f (0) = 0.We emphasize that the fluxes f , A are not necessarily locally Lipschitz continuous.Problems more general than (1), for which our results can be extended, will bediscussed in Section 8.The class (1) of nonlinear partial differential equations includes several importantparticular cases. The hyperbolic conservation law ∂ t u + div f ( u ) = 0is a special case of (1). The celebrated theory of L ∞ entropy solutions for scalarconservation laws in R d was developed by Kruzhkov [63], while the BV theorywas set up by Vol’pert [77]. The extensions for the Dirichlet problem in boundeddomains are due to Bardos, LeRoux, N´ed´elec [15] (for the BV setting) and Otto[69] (for the L ∞ setting). Note that the boundary condition is only verified in somegeneralized sense (see [15, 69, 65, 29, 73, 66, 79, 49, 67, 5]).Many other well-known partial differential equations (usually possessing moreregular solutions) are also special cases of (1). Let us mention the heat and porousmedium equations ∂ t u = ∆ u, ∂ t u = ∆ u m , m > , and more generally degenerate convection-diffusion equations of the type(3) ∂ t u + div f ( u ) = ∆ A ( u ) . Degenerate parabolic equations like (3) occur in theories of flow in porous media(see discussion and references [43]) and sedimentation-consolidation processes [27].As other famous representatives of the class of equations that is consideredherein, we mention the p -Laplace equation ∂ t u = div (cid:0) |∇ u | p − ∇ u (cid:1) , p > , which arises in the theory of non-Newtonian filtration. Also well known is the moregeneral polytropic filtration equation ∂ t u = div (cid:18)(cid:12)(cid:12)(cid:12) ∇ (cid:16) | u | m − u (cid:17)(cid:12)(cid:12)(cid:12) p − ∇ (cid:16) | u | m − u (cid:17)(cid:19) , m, p > . A related class of equations consists of the so-called elliptic-parabolic equations ∂ t b ( v ) = div a ( v, ∇ v ) , where b : R → R is continuous nondecreasing, and a ( r, ξ ) : R × R N → R N givesrise to a Leray-Lions operator. We refer to [4, 20, 70, 30, 6] and the references citedtherein for more information on elliptic-parabolic equations.A chief goal of this paper is to propose and analyze a specific class of finitevolume schemes for the problem (1). Note that finite volume schemes are well suitedfor approximation of equations in divergence form, such as (1). Discretization of B. ANDREIANOV, M. BENDAHMANE, AND K. H. KARLSEN the aforementioned hyperbolic, porous medium, convection-diffusion, and elliptic-parabolic equations by finite volume methods is quite standard by now and oftenused in engeneering practice. We refer to [48, 31, 3, 44, 45, 61, 57, 68, 79, 49, 67,12, 10, 11, 42] and references therein for different convergence results and numericalexperiments. For related works on linear elliptic problems, see [2, 1, 57, 41, 23, 58,50, 51, 53, 52] and the discussion in Section 8. Alternative numerical approacheshave also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14, 22, 55] and references therein)and operator splitting schemes (see [43]).Having said that, we are not aware of any papers that construct convergentnumerical schemes for mixed type equations of the generality considered herein.Indeed, they combine a number of difficulties such as nonlinear convection, doublynonlinear diffusion, strong degeneracy, and shocks, which in turn necessitates theuse of a suitable framework of discontinuous entropy solutions. Furthermore, in theabsence of the Lipschitz continuity assumption on the convective flux f ( · ), the CFLcondition does not make sense; therefore we have to discretize the convective termwith a time-implicit scheme.We begin by providing the entropy solution framework for (1); this is the topicof Section 2 and Appendix A. Due to the nonlinearity of f ( · ) and the possibledegeneracy of A ( · ), the problem (1) will in general possess shock wave solutions,a feature that can reflect the physical phenomenon of breaking of waves. Thisis well known in the context of conservation laws. Also the boundary conditioncannot be prescribed pointwise on the whole boundary Σ when A is not strictlyincreasing. Due to this loss of regularity, it is necessary to work with weak solutions;moreover, to single out a physically relevant and unique weak solution, we need toimpose additional “entropy inequalities”, in the spirit of Kruzhkov [63]. Earlyresults on hyperbolic-parabolic equations were obtained by Volpert, Hudjaev [78];see also [80, 82, 81], [74], [28], [19] and references cited therein, and [76], [37],[32]. L entropy techniques for degenerate convection-diffusion equations like (3),which take into account both hyperbolic and parabolic features, were developed byCarrillo [29] for the homogeneous Dirichlet problem in bounded domains. Sincethen, many authors extended the Carrillo results in various directions (see e.g.[30, 59, 66, 73, 25, 46, 60, 62, 67, 49, 5, 13]). Some additional techniques arerequired for anisotropic diffusion problems, where a kinetic approach (see Chen,Perthame [35]) and an accurate entropic approach (see Bendahmane, Karlsen [17,18]) were developed in the few last years; see also Souganidis, Perthame [75] andand Chen, Karlsen [34]. In this paper, we use a variant of the Carrillo entropicapproach. Following the Tartar-DiPerna idea of measure-valued solutions and usingthe techniques of Eymard, Gallou¨et, Herbin [48], we introduce a notion of entropyprocess solution for (1), and establish the related identification and uniquenessresults. More exactly, we work with entropy double-process solutions arising in theparticular context of discrete duality finite volume schemes. In Section 2, we showthe existence result for (1) and state uniqueness; an adaptation of the standarduniqueness (and, more generally, L contraction and comparison principle) proof isgiven in Appendix A.In Section 3 we construct discrete duality finite volume (DDFV) schemes for(1) in two and three spatial dimensions (some other schemes are briefly discussedin Section 8). We adapt the approximations used by Eymard, Gallou¨et, Herbin V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 5 [48] (see also [31, 79, 49, 67]) for the nonlinear convection term, and those used byHermeline [57, 58], Domelevo, Omn`es [41] and Andreianov, Boyer, Hubert [11] forthe doubly nonlinear diffusion term. In 3D, we propose new DDFV schemes thatpossess convenient discrete duality properties.Our 3D scheme is a very particular case of the schemes introduced and studiednumerically by Hermeline in [58]. In passing, we mention that different kinds of3D discrete duality schemes were constructed in [72, 39] and in [38]. Appendix B(see also [8, 7]) is devoted to an elementary reconstruction lemma which underliesour DDFV schemes in 3D. In contrast to [41, 11], we are led to penalize our DDFVschemes to ensure that the two approximations of A ( u ) actually converge to thesame limit (see Section 3.4). The DDFV schemes constructed in Section 3 possessseveral convenient discrete calculus formulas that we collect in Section 4. Relatedconsistency estimates and properties of the associated spaces of discrete functionsare given in Section 5. The (few) available a priori estimates for the discrete so-lutions are collected in In Section 6. In the same section, the existence of discretesolutions is shown. Furthermore, we establish that, up to an error term in the equa-tion depending on the discretization parameter, discrete solutions can be consideredas entropy solutions of (1). In Section 7 we prove that discrete solutions converge,as the discretization parameter tends to zero, to an entropy double-process solutionthat turns out to be the (unique) entropy solution of (1). It should emphasizedthat we obtain strong convergence of both convective and diffusive fluxes, in spiteof the double nonlinearity of the problem (1). Section 8 contains references to someknown finite volume schemes for nonlinear diffusion-convection equations, and dis-cusses the extension of our results to different generalizations of problem (1).2. Notions of solution and well-posedness
As it was explained in the introduction, we need the notion of weak solutionfor (1) with additional “entropy” conditions. In order to use entropy conditionsin the interior of Q and, moreover, take into account the homogeneous Dirichletboundary condition on Σ, following Carrillo [29] we will work with the so-called“semi-Kruzhkov” entropy-entropy flux pairs ( η ± c , q ± c ) for each c ∈ R ; they aredefined as η + c ( z ) = ( z − c ) + , η − c ( z ) = ( z − c ) − , q + c ( z ) = sign + ( z − c ) ( f ( z ) − f ( c )) , q − c ( z ) = sign − ( z − c ) ( f ( z ) − f ( c )) . By convention, we assign ( η ± c ) ′ ( c ) to be zero. Here ( z − c ) ± denote the nonnegativequantities satisfying z − c = ( z − c ) + − ( z − c ) − ; moreover, we use the notationsign + ( z − c ) = ( η + c ) ′ ( z ) = ( , z > c , z ≤ c, sign − ( z − c ) = ( η − c ) ′ ( z ) = ( , z ≥ c, − , z < c. At certain points, we will also need smooth regularizations of the semi-Kruzhkoventropy-entropy flux pairs; it is sufficient to consider regular “boundary” entropypairs ( η ± c,ε , q ± c,ε ) (cf. Otto [69] and the book [65]), which are W , ∞ pairs with thesame support as ( η ± c , q ± c ), converging pointwise to ( η ± c , q ± c ) as ε →
0. Specifically,
B. ANDREIANOV, M. BENDAHMANE, AND K. H. KARLSEN the functionssign + ε ( z ) = 1 ε min { z + , ε } , sign − ε ( z ) = 1 ε max {− z − , − ε } will be used to approximate sign ± ( · ) = ( η ± ) ′ ( · ).In view of the monotonicity of A : R → R , the following definition is meaningful. Definition 2.1.
For any locally bounded piecewise continuous function θ : R → R ,we define (using, e.g., the Stieltjes integral) the function A θ : R → R by (4) A θ ( z ) = Z z θ ( s ) dA ( s ) . The ensuing lemma shows that there exists a continuous function e A θ such that A θ ( z ) = e A θ ( A ( z )). We prove this lemma under rather strong assumptions, butthey are still sufficient for our needs. Lemma 2.1. (i) Let θ, A θ be a couple of functions as introduced in Definition .Then there exists a continuous function e A θ : A ( R ) → R such that A θ ( z ) = e A θ ( A ( z )) , ∀ z ∈ R . Moreover, e A θ is Lipschitz continuous.(ii) Assume additionally that θ ∈ W , ∞ ( R ) , and let ( A ρ ) ρ be a sequence ofnondecreasing continuous surjective functions converging to A pointwise on R as ρ → . Define e A ρθ , A ρθ by (i) and (4) with A ρ replacing A . Then e A ρθ converges to e A θ uniformly on compact subsets of A ( R ) .Proof. (i) For b ∈ A ( R ), we can define e A θ by e A θ ( b ) = A θ ( z ) for some z ∈ A − ( b ).If A ( z ) = A (ˆ z ), then the measure dA ( s ) vanishes between z and ˆ z ; thus A θ ( z ) − A θ (ˆ z ) = Z z ˆ z θ ( s ) dA ( s ) = 0 , and e A θ is well-defined. For all b, ˆ b ∈ A ( R ), e A θ ( b ) − e A θ (ˆ b ) = A θ ( z ) − A θ (ˆ z ) = Z z ˆ z θ ( s ) dA ( s ) , z ∈ A − ( b ) , ˆ z ∈ A − (ˆ b ) . Consequently, (cid:12)(cid:12)(cid:12) e A θ ( b ) − e A θ (ˆ b ) (cid:12)(cid:12)(cid:12) ≤ k θ k L ∞ | A ( z ) − A (ˆ z ) | = k θ k L ∞ (cid:12)(cid:12)(cid:12) b − ˆ b (cid:12)(cid:12)(cid:12) . (ii) Since the functions e A ρθ are monotone, by the Dini theorem it is sufficient toprove the pointwise convergence. By the same argument, the convergence of A ρ to A is actually uniform on compact subsets of R . Take b ∈ A ( R ) and z ∈ A − ( b ).Set b ρ = A ρ ( z ); we have b ρ → b as ρ →
0. Using (i) and the integration-by-partsformula for the Stieltjes integral, we get (cid:12)(cid:12)(cid:12) e A ρθ ( b ) − e A θ ( b ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) e A ρθ ( b ) − e A ρθ ( b ρ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) e A ρθ ( b ρ ) − e A θ ( b ) (cid:12)(cid:12)(cid:12) ≤ k θ k L ∞ | b − b ρ | + (cid:12)(cid:12)(cid:12)(cid:12)Z z θ ( s ) d ( A ρ ( s ) − A ( s )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k θ k L ∞ | b − b ρ | + (cid:12)(cid:12)(cid:12)(cid:12)Z z ( A ρ ( s ) − A ( s )) θ ′ ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) . The right-hand side converges to zero as ρ →
0. Thus the claim follows. (cid:3)
V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 7
We have now came to the definition of an entropy solution. Here and in thesequel, R ± denote { k ∈ R | ± k ≥ } , respectively. Definition 2.2 (entropy solution) . An entropy solution of the initial-boundaryvalue problem (1) is a measurable function u : Q T → R satisfying ( D .1) u ∈ L ∞ ( Q ) and w = A ( u ) ∈ L p (0 , T ; W ,p (Ω)) ; ( D .2) for all ψ ∈ D ([0 , T ) × Ω) , Z Q (cid:18) u∂ t ψ + f ( u ) · ∇ ψ − k ( ∇ w ) ∇ w · ∇ ψ (cid:19) dx dt + Z Ω u ψ (0 , · ) dx + Z Q S ψ dx dt = 0;( D .3) for all pairs ( c, ψ ) ∈ R ± × D ([0 , T ) × Ω) , ψ ≥ , and also for all pairs ( c, ψ ) ∈ R × D ([0 , T ) × Ω) , ψ ≥ , Z Q (cid:18) η ± c ( u ) ∂ t ψ + q ± c ( u ) · ∇ ψ − k ( ∇ w ) ∇ e A ( η ± c ) ′ ( w ) · ∇ ψ (cid:19) dx dt + Z Ω η ± c ( u ) ψ (0 , · ) dx + Z Q ( η ± c ) ′ ( u ) S ψ dx dt ≥ . For the convergence proof we need the notion of entropy double-process solutions;we adapt this notion from [48, 31, 54, 49], where entropy process solutions havebeen introduced for hyperbolic problems and degenerate parabolic problems withlinear diffusion. This definition is based upon the so-called “nonlinear L ∞ weak- ⋆ convergence” property, which is well-known in the equivalent framework of measure-valued solutions developed earlier by Tartar and DiPerna:(5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) each sequence ( u ρ ) bounded in L ∞ ( Q ) admits a subsequence such that ∀ F ∈ C ( R ), F ( u ρ ( · , · )) → Z F ( µ ( · , · , α )) dα in L ∞ ( Q ) weak- ⋆, where the function µ ∈ L ∞ ( Q × (0 , µ, µ ∗ , both corresponding to the single unknown function u , isthat it permits us to handle the double approximation of u by pairs u M , u M ∗ in theframework of DDFV schemes (see Section 3). Definition 2.3 (entropy double-process solution) . A triplet ( µ, µ ∗ , w ) of measur-able functions, with µ, µ ∗ : Q × (0 , → R and w : Q → R , is called an entropydouble-process solution of the initial-boundary value problem (1) if the followingconditions are met: ( D’ .1) µ, µ ∗ ∈ L ∞ ( Q × (0 , , w ∈ L p (0 , T ; W ,p (Ω)) , and A ( µ ( t, x, α )) ≡ w ( t, x ) ≡ A ( µ ∗ ( t, x, α )) , for a.e. ( t, x, α ) ∈ Q × (0 , . B. ANDREIANOV, M. BENDAHMANE, AND K. H. KARLSEN ( D’ .2) For all ψ ∈ D ([0 , T ) × Ω) , Z Z Q (cid:18) d (cid:0) µ + ( d − µ ∗ (cid:1) ∂ t ψ + 1 d (cid:0) f ( µ ) + ( d − f ( µ ∗ ) (cid:1) · ∇ ψ (cid:19) dx dt dα − Z Q k ( ∇ w ) ∇ w · ∇ ψ dx dt + Z Ω u ψ (0 , · ) dx + Z Q S ψ dx dt = 0 . ( D’ .3) For all pairs ( c, ψ ) ∈ R ± × D ([0 , T ) × Ω) , ψ ≥ , and also for all pairs ( c, ψ ) ∈ R × D ([0 , T ) × Ω) , ψ ≥ , Z Z Q (cid:18) d (cid:0) η ± c ( µ ) + ( d − η ± c ( µ ∗ ) (cid:1) ∂ t ψ + 1 d (cid:0) q ± c ( µ ) + ( d − q ± c ( µ ∗ ) (cid:1) · ∇ ψ (cid:19) dx dt dα − Z Q k ( ∇ w ) ∇ e A ( η ± c ) ′ ( w ) · ∇ ψ dx dt + Z Ω η ± c ( u ) ψ (0 , · ) dx + Z Z Q d (cid:0) ( η ± c ) ′ ( µ ) + ( d − η ± c ) ′ ( µ ∗ ) (cid:1) S ψ dx dt dα ≥ . Remark 2.1.
Since ∇ w = 0 a.e. on { ( t, x ) ∈ Q ) | w ( t, x ) = A ( c ) } for any c ∈ R ,the term k ( ∇ w ) ∇ e A ( η ± c ) ′ ( w ) in the above definitions can be rewritten as(6) ( η ± c ) ′ ( z ) a ( ∇ w ) for any z ∈ A − ( w ), and also as sign ± ( w − A ( c )) a ( ∇ w ).The form used in ( D .3) and ( D’ .3) is convenient for expressing the approximateentropy inequalities at the discrete level; the equivalent form (6) is used in theuniqueness proof. Both forms are exploited in the existence proof below. Remark 2.2.
Let u be an entropy solution of (1). Then the triplet ( µ, µ ∗ , w )defined by µ ( t, x, α ) = µ ∗ ( t, x, α ) = u ( t, x ) for a.e. ( t, x, α ) ∈ Q × (0 , ,w ( t, x ) = A ( u ( t, x )) for a.e. ( t, x ) ∈ Q. is an entropy double-process solution of (1).Conversely, if ( µ, µ ∗ , w ) is an entropy double-process solution of (1) for which µ ( t, x, α ) = µ ∗ ( t, x, α ) = u ( t, x ) a.e. on Q × (0 ,
1) for some function u : Q → R ,then this u is an entropy solution of (1).Note that in Definition 2.2, we have only considered α -independent data u , f .In this case, the notion of entropy double-process solution is just a technical toolthat permits to bypass the lack of strong compactness of sequences of approximatesolutions. As a first illustration of this, we pass to the limit in vanishing viscosityapproximations (without BV estimates) to prove the existence of an entropy double-process solution such that µ ≡ µ ∗ . Theorem 2.1.
Under the assumptions stated in Section , there exists an entropydouble-process solution to the initial-boundary value problem (1) for which µ ≡ µ ∗ . Notice that the above result holds for any Lipschitz domain Ω in any spacedimension. In passing, we also mention that the existence result of Theorem 2.1has recently been generalized by Ouaro and the authors [9] to the case of a triplynonlinear degenerate diffusion equation.
V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 9
Proof.
The proof is divided into several steps. (i)
We approximate problem (1) by regular problems (1) ρ with f , A replaced by f ρ , A ρ such that f ρ , A ρ , [ A ρ ] − are Lipschitz continuous on R and f ρ , A ρ converge to f , A , respectively, uniformly on compacts sets as ρ → u ρ ∈ L p (0 , T ; W ,p (Ω)) to problem (1) ρ in thefollowing sense:(7) ( ∂ t u ρ + div f ρ ( u ρ ) = div a ( ∇ A ρ ( u ρ )) + S in L p ′ (0 , T ; W − ,p ′ (Ω))+ L ( Q ) , u ρ | t =0 = u . Moreover, since f ρ ◦ A − ρ is Lipschitz continuous, the L contraction property andcomparison principle for weak solutions can be verified. It can be obtained either bythe technique of Otto [70] (doubling the time variable) or using the theory of integralsolutions and nonlinear semigroup methods, consult for example [30]. Besides, u ρ verifies the entropy formulation of Definition 2.2 with fluxes f ρ , A ρ , where η ± c canbe replaced by regular “boundary” entropies η ± c,ε , whenever we prefer to do so. (ii) We claim that the following quantities are uniformly bounded in ρ : • k u ρ k L ∞ (Ω) and k A ρ ( u ρ ) k L p (0 ,T ; W ,p (Ω)) ; • space translates of A ρ ( u ρ ) in L ( Q ) (consequence of previous estimate); • time translates of A ρ ( u ρ ) in L ( Q ).Indeed, for the first point consider the function M ( t ) = k u k L ∞ (Ω) + Z t k S ( τ, · ) k L ∞ (Ω) dτ, which is a solution of (1) ρ with x -constant data k u k L ∞ (Ω) , k S ( t, · ) k L ∞ (Ω) . Thecomparison principle mentioned in (i) ensures that a.e. on Q , − M ( T ) ≤ − M ( t ) ≤ u ρ ( t, x ) ≤ M ( t ) ≤ M ( T ) . Next, we employ A ρ ( u ρ ) as a test function in (7). The product between ∂ t u ρ and A ρ ( u ρ ) is handled using the usual chain rule argument (see, e.g., [4, 70, 30]), wherethe relevant duality is between the space E := L p (0 , T ; W ,p (Ω)) ∩ L ∞ ( Q ) andthe space L p ′ (0 , T ; W − ,p ′ (Ω)) + L ( Q ) ⊂ E ∗ . Here we are also exploiting the L ∞ bound on f ρ ( u ρ ) in a straightforward fashion to treat the term f ρ ( u ρ ) · ∇ A ρ ( u ρ ); butnotice that using the Green-Gauss trick (16) below, we can supply a finer analysisof this term.For the third bullet point, we first use (7) to get, for a.e. t, t + ∆ ∈ (0 , T ), Z Ω ( u ρ ( t + ∆ ) − u ρ ( t )) ξ = Z t +∆ t Z Ω (cid:2) (cid:0) − f ρ ( u ρ ) + a ( ∇ A ρ ( u ρ )) (cid:1) · ∇ ξ + S ξ (cid:3) for all ξ ∈ W ,p (Ω) ∩ L ∞ (Ω). Taking ξ = A ρ ( u ρ ( t + ∆ )) − A ρ ( u ρ ( t )) and integratingin t , using the two previously obtained estimates, we deduce that(8) Z Z Q | u ρ ( t + ∆ ) − u ρ ( t ) | | A ρ ( u ρ ( t + ∆ )) − A ρ ( u ρ ( t )) | ≤ Const | ∆ | . Now, let π be a (common for all ρ ) concave modulus of continuity for A ρ on[ − M ( T ) , M ( T )], Π be its inverse, and set ˜Π( r ) = r Π( r ). Let ˜ π be the inverse of ˜Π. Note that ˜ π is concave, continuous, and ˜ π (0) = 0. Set v ( t, x ) = u ρ ( t + ∆ , x )and y ( t, x ) = u ρ ( t, x ). We have Z Q | A ρ ( v ) − A ρ ( y ) | = Z Q ˜ π (cid:18) ˜Π( | A ρ ( v ) − A ρ ( y ) | ) (cid:19) ≤ | Q | ˜ π (cid:18) | Q | Z Q ˜Π( | A ρ ( v ) − A ρ ( y ) | ) (cid:19) . Since | A ρ ( v ) − A ρ ( y ) | ≤ π ( | v − y | ), we have Π( | A ρ ( v ) − A ρ ( y ) | ) ≤ | v − y | and˜Π( | A ρ ( v ) − A ρ ( y ) | ) = Π( | A ρ ( v ) − A ρ ( y ) | ) | A ρ ( v ) − A ρ ( y ) |≤ | v − y | | A ρ ( v ) − A ρ ( y ) | . Therefore, estimate (8) implies Z Q | A ρ ( u ρ ( t + ∆ , x )) − A ρ ( u ρ ( t, x )) |≤ | Q | ˜ π (cid:18) | Q | Z Q | v − y | | A ρ ( v ) − A ρ ( y ) | (cid:19) = | Q | ˜ π (cid:18) | Q | J ( ∆ ) (cid:19) ≤ C ˜ π ( C ∆ ) =: ω A ( ∆ ) , (9)where ω A ∈ C ( R + , R + ), ω A (0) = 0. (iii) Thanks to the estimates in (ii) and standard compactness results, thereexists a (not labelled) sequence ρ → • w ρ = A ρ ( u ρ ) converges strongly in L ( Q ) and pointwise a.e. on Q ; • ∇ w ρ converges weakly in L p ( Q ); • a ( ∇ w ρ ) converges weakly in L p ′ ( Q ) to some limit χ ; • u ρ converges to µ : Q × (0 , ∈ R in the sense of (5).Let us introduce the function(10) u ( t, x ) = Z µ ( t, x, α ) dα, for a.e. ( t, x ) ∈ Q. Thanks to the convergence of A ρ to A , we can identify the limit of w ρ ( · , · ) with R A ( µ ( · , · , α )) dα . Moreover, since w ρ is converging strongly, A ( µ ( · , · , α )) is actuallyindependent of α ∈ (0 ,
1) and equals A ( u ( · , · )). Using distributional derivatives, wealso identify the limit of ∇ w ρ with ∇ A ( u ). (iv) We have now come to the main step of the proof, namely to improve theweak convergence of ∇ A ρ ( u ρ ) to strong convergence, and to identify the weaklimit of a ( ∇ A ρ ( u ρ )) with a ( ∇ A ( u )), where u is defined in (10); of course, the chiefdifficulty comes from the lack of strong convergence of u ρ .We begin by specifying the test function in (7) as w ρ ζ , yielding(11) Z T h ∂ t u ρ , w ρ ζ i | {z } I ,ρ − Z Q f ρ ( u ρ ) · ∇ w ρ ζ | {z } I ,ρ + Z Q a ( ∇ w ρ ) · ∇ w ρ ζ − Z Q S w ρ ζ | {z } I ,ρ = 0 , where w ρ = A ρ ( u ρ ) and ζ ∈ D ([0 , T )) is nonincreasing with ζ (0) = 1. Next, wepass to the limit into the weak formulation (7), obtaining(12) ( ∂ t u + div R f ( µ ) dα = div χ + S in L p ′ (0 , T ; W − ,p ′ (Ω))+ L ( Q ) , u ρ | t =0 = u . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 11
In (12), we take w ζ as test function, where w = A ( u ), u is defined in (10), and ζ is as specified above. The result is(13) Z T h ∂ t u, w ζ i | {z } I − Z Q Z f ( µ ) · ∇ w ζ | {z } I + Z Q χ · ∇ A ( u ) ζ − Z Q S w ζ | {z } I = 0 . In order to later use the Minty-Browder trick, we shall combine (13) and the“ ρ →
0” limit of (11) to conclude the validity of the following inequality:(14) Z Q χ · ∇ A ( u ) ≥ lim inf ρ → Z Q a ( ∇ w ρ ) · ∇ w ρ . A crucial role is played by the following calculation, which reveals that thelack of strong convergence of f ρ ( u ρ ) is not an obstacle. Indeed, a componentwiseapplication of Lemma 2.1 (i) yields the existence of a Lipschitz continuous vector-valued function e A f such that(15) Z z f ( s ) dA ( s ) = e A f ( A ( z )) . Hence, by the chain rule and the Green-Gauss formula, we can calculate as follows: Z Q Z f ( µ ) · ∇ A ( u ) = Z Q Z f ( µ ) · ∇ A ( µ ) = Z Z T Z Ω div e A f ( A ( µ ))= Z T Z ∂ Ω e A f ( A ( u )) · n = 0 , because for a.e. α ∈ (0 , A ( µ ( · , · , α )) = A ( u ( · , · )) ∈ L p (0 , T ; W ,p (Ω)) . By similar (simpler) arguments and u ρ ∈ L p (0 , T ; W ,p (Ω)), we also have Z Q f ρ ( u ρ ) · ∇ A ρ ( u ρ ) = Z T Z Ω div (cid:18)Z u ρ f ρ ( s ) dA ρ ( s ) (cid:19) = 0 . (16)Consequently, we can make I and I ,ρ (for each ρ >
0) vanish.Next, let us prove that I ≤ lim ρ → I ,ρ . As above, the duality products h ∂ t u ρ , A ρ ( u ρ ) i , h ∂ t u, A ( u ) i are treated via the chain rule argument (cf. [4]). Set B ( z ) = R z A ( s ) ds , B ρ ( z ) = R z A ρ ( s ) ds , and note that these functions are convex.Also, B ρ → B uniformly on compact subsets of R . With the help of Jensen’s inequality, I = Z T h ∂ t u, A ( u ) ζ i = − Z Q B ( u ) ζ ′ − Z Ω B ( u )= Z Q B (cid:18)Z µ ( t, x, α ) dα (cid:19) ( − ζ ′ ) − Z Ω B ( u ) ≤ Z Q Z B ( µ ( t, x, α )) dα ( − ζ ′ ) − Z Ω B ( u )= lim ρ → (cid:18) − Z Q B ρ ( u ρ ) ζ ′ − Z Ω B ρ ( u ) (cid:19) = lim ρ → Z T h ∂ t u ρ , A ρ ( u ρ ) ζ i = lim ρ → I ,ρ . Finally, it is clear that I ,ρ → I as ρ →
0. Letting ζ tend to 1l [0 ,T ) , the desiredinequality (14) follows from subtracting the “ ρ →
0” limit of (11) from (13) andthe above calculations.Starting off from (14), we can use the Minty-Browder trick (see, for example,[64, 24, 4, 21] and the proof of Theorem 7.1 in Section 7) to deduce that(17) a ( ∇ w ρ ) − a ( ∇ A ( u )) → L p ′ ( Q ) as ρ → . Thus χ = a ( ∇ A ( u )). Simultaneously, from the strict monotonicity of a ( · ) wededuce that, firstly, the convergence in (17) also takes place a.e. in Q ; secondly,that (14) actually holds with an equality sign. Next, we consider the functions g ρ := a ( ∇ w ρ ) · ∇ w ρ ≥ g := a ( ∇ A ( u )) · ∇ A ( u ) ≥
0, and observe that g ρ → g a.e. in Q, Z Q g ρ → Z Q g as ρ → . Hence, we deduce that a subsequence of ( g ρ ) ρ converges to g strongly in L ( Q ),cf. [24], [21, Lemma 5], [42, Lemma 8.4]. Due to the coercivity of a ( · ), (cid:0) | ∇ w ρ | p (cid:1) ρ is equi-integrable, so the Vitali theorem yields the strong L p convergence of ∇ w ρ ,along a subsequence if necessary, to a limit already identified as ∇ w , w = A ( u ). (v) By (12), we readily conclude that ( µ, µ, w ) verifies ( D’ .2). Now we can passto the limit in the entropy inequalities corresponding to (1) ρ and deduce ( D’ .3).Let us first show that ∇ e A ρ ( η ± c,ε ) ′ ( w ρ ) converges weakly to ∇ e A ( η ± c,ε ) ′ ( w ) in L p ( Q ).By Lemma 2.1 (i), A ρ ( η ± c,ε ) ′ ( · ) are uniformly Lipshitz continuous functions. Thus ∇ e A ρ ( η ± c,ε ) ′ ( w ρ ) are uniformly bounded and weakly compact in L p ( Q ). Moreover, e A ρ ( η ± c,ε ) ′ ( w ρ ) converges to e A ( η ± c,ε ) ′ ( w ) by Lemma 2.1 (ii) and because of the pointwiseconvergence of w ρ to w . Using the distributional convergence, we eventually workout our claim.Now note that if p >
2, then k is continuous. By the last result of (iv) , wecan assume without loss of generality that k ( ∇ w ρ ) converges to k ( ∇ w ) a.e. in Q .Moreover, ( k ( ∇ w ρ )) is bounded in L pp − ( Q ), since ( ∇ w ρ ) is bounded in L p ( Q ).Applying the Egorov theorem and H¨older’s inequality with exponents p ′ , p in theproduct (cid:0) k ( ∇ w ρ ) ∇ ψ (cid:1) · ∇ e A ( η ± c,ε ) ′ ( w ρ ), we deduce that(18) lim ρ → Z Q k ( ∇ w ρ ) ∇ e A ( η ± c,ε ) ′ ( w ρ ) · ∇ ψ = Z Q k ( ∇ w ) ∇ e A ( η ± c,ε ) ′ ( w ) · ∇ ψ. V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 13 If p ≤
2, we fix a small δ > k ( · ) in the δ -neigbourhood of the origin(if k ( · ) is replaced by k δ ( · ) = min { k ( · ) , min | ξ |≤ δ k ( ξ ) } , the argument used for p > { ( t, x ) | | ∇ w ρ ( t, x ) | < δ } . On this set, k ( ∇ w ρ ) (cid:12)(cid:12) ∇ e A ( η ± c,ε ) ′ ( w ρ ) (cid:12)(cid:12) ≤ C k ( η ± c,ε ) ′ k ∞ | ∇ w ρ | p − ≤ Const δ p − , uniformly in ρ . To conclude that (18) still holds, we first pass to the limit as ρ → δ >
0, and then send δ → η ± c,ε such that ( η ± c,ε ) ′ approximate( η ± c ) ′ (extended by zero at the point c , by our convention), pointwise a.e. in R as ε →
0. We use (18) to pass to the limit in the entropy inequality correspondingto (1) ρ . We pass to the limit in the remaining terms in this entropy inequalityusing the continuity of η ± c,ε , q ± c,ε , ( η ± c,ε ) ′ and the nonlinear L ∞ weak- ⋆ convergenceproperty (5). Finally, we pass to the limit as ε →
0, rewriting ∇ e A ( η ± c,ε ) ′ ( w ) as( η ± c,ε ) ′ ( u ) ∇ w (consult Remark 2.1) and using the Lebesgue dominated convergencetheorem and the pointwise convergences of η ± c,ε , q ± c,ε , ( η ± c,ε ) ′ . The passage to thelimit in the weak formulation is similar. (vi) We conclude that ( µ, µ ∗ , A ( u )) is an entropy double-process solution of (1)such that µ ∗ = µ . (cid:3) Given Theorem 2.1, the uniqueness of an entropy double-process solution can beestablished using Kruzhkov’s method, along the lines of Carrillo [29].
Theorem 2.2.
Suppose the assumptions stated in Section hold. Let ( µ, µ ∗ , w ) bean entropy double-process solution of the initial-boundary value problem (1) . Thenit is unique. Moreover, there exists a function u ∈ L ∞ ( Q ) such that µ ( t, x, α ) = u ( t, x ) = µ ∗ ( t, x, α ) for a.e. ( t, x, α ) ∈ Q × (0 , . We refer to Appendix A for a sketch of the proof.Theorems 2.1 and 2.2 as well as the arguments of Appendix A imply
Corollary 2.1 (well-posedness) . Under the assumptions stated in Section , thereexists a unique entropy solution of the initial-boundary value problem (1) . Let u and v be two entropy solutions of (1) with initial data u | t =0 = u ∈ L ∞ (Ω) and v | t =0 = v ∈ L ∞ (Ω) and source terms S and T of the kind (2) , respectively. Fora.e. t ∈ (0 , T ) , we have Z Ω ( u ( t, x ) − v ( t, x )) + dx ≤ Z Ω ( u − v ) + dx + Z t Z Ω ( S − T ) + . Consequently, if u ≤ v a.e. in Ω and S ≤ T a.e. on Q , then u ≤ v a.e. in Q .Finally, if u = v a.e. in Ω and S = T a.e. on Q , then u = v a.e. in Q . The upcoming sections are concerned with the construction of finite volumeschemes for which the corresponding discrete solutions converge to the unique en-tropy solution of (1) as the discretization parameter (mesh size) tends to zero. Theconvergence proof will attempt to mimic the proof of Theorem 2.1.3.
Discrete duality finite volume (DDFV) schemes
Let Ω be a polygonal (respectively, polyhedral) open bounded subset of R d , d = 2 (respectively, d = 3). In what follows, we introduce most of the notationrelated to DDFV schemes; each piece of new notation is given in italic script. Construction of “double” conformal meshes. • A partition of Ω is a finite set of disjoint open polygonal (respectively, poly-hedral) subsets of Ω such that Ω is contained in their union, up to a set of zero d -dimensional measure.Following Hermeline [57], Domelevo, Omn`es [41] and Andreianov, Boyer, Hubert[11], we consider a DDFV mesh which is a triple T = (cid:0) M , M ∗ , S (cid:1) described below. • We let M be a partition of Ω into triangles (respectively, tetrahedra); a moregeneral case is discussed in Section 8 . We assume that the mesh satisfies theDelaunay condition (see, e.g., [48]); for simplicity of the representation, the readermay assume that each triangle (respectively, tetrahedron) contains the centre if itscircumscribed circle (respectively, ball). We assume in addition(19) (cid:12)(cid:12)(cid:12)(cid:12) if d = 3, each face of each tetrahedron of M contains the centre of its circumscribed circle.Although the definition of the scheme does not require condition (19) (see Re-mark 3.1 below), we do need this condition in order to deduce the discrete entropyinequalities and to prove that the scheme converges.Each control volume K ∈ M is supplied with a centre x K that we choose to bethe centre of the circle (respectively, ball) circumscribed around K . We call ∂ M the set of all edges (respectively, faces) of control volumes that are included in ∂ Ω. These edges (respectively, faces) are considered as boundary control volumes ;for K ∈ ∂ M , we choose the middle of K (respectively, the centre of the circlecircumscribed around K ) for the centre x K . We denote by M the union M ∪ ∂ M .We call vertex (of M ) any vertex of any control volume K ∈ M . • (see Figure 1) We take M ∗ as the partition of Ω into dual control volumes K ∗ ,supplied with dual centres x K ∗ , such that x K ∗ is a vertex of M and K ∗ is the subsetof points of Ω that are closer to x K ∗ than to any other vertex of M . In otherwords, M ∗ is the Vorono¨ı mesh constructed from the vertices of M . If x K ∗ ∈ Ω,we say that K ∗ is a dual control volume and write K ∗ ∈ M ∗ ; and if x K ∗ ∈ ∂ Ω,we say that K ∗ is a boundary dual control volume and write K ∗ ∈ ∂ M ∗ . Thus M ∗ = M ∗ ∪ ∂ M ∗ . We call dual vertex (of M ∗ ) any vertex of any dual controlvolume K ∗ ∈ M ∗ . Note that by the choice of x K , the set of centres coincides withthe set of dual vertices, and the set of vertices coincides with the set of dual centres.In other words, M and M ∗ are finite volume meshes that are dual each one to theother. • We call neighbours of K , all control volumes L ∈ M such that K and L have acommon edge (respectively, common face). The set of all neighbours of K is denotedby N ( K ). Note that if L ∈ N ( K ), then K ∈ N ( L ); in this case we simply say that K and L are (a couple of) neighbours. • (see Figures 1 and 2(b)) If K and L are neighbours, we denote by K | L the interface ∂ K ∩ ∂ L between K and L . The set of all interfaces is denoted by E . In particular, in the two dimensional case we can partition Ω into polygons that admit acircumscribed circle. In the three dimensional case , we can partition Ω in polyhedra that havetriangular faces and admit a circumscribed ball. in order to avoid pathological situations which could appear in non-convex domains, e.g., indomains with cracks, here the distance between two points x, y of Ω is understood as the lengthof the shortest path which connects x with y and which lies within Ω V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 15 volume x K volumeprimal x K ∗ dual KK ∗ diamond D subdiamond S x K boundary primal“volume” K volume K ∗ boundary dual x K ∗ Figure 1.
2D primal and dual meshes; diamond (subdiamond). • In the same way, we denote by
N ∗ ( K ∗ ) the set of (dual) neighbours of a dualcontrol volume K ∗ , and by K ∗ | L ∗ , the (dual) interface ∂ K ∗ ∩ ∂ L ∗ between dual neigh-bours K ∗ and L ∗ . The set of all dual interfaces is denoted by E ∗ . • (see Figure 2) The meshes M and M ∗ induce partitions of Ω into diamondsand subdiamonds. Let us describe them separetely for d = 2 and d = 3.For d = 2 (see Figure 2(a)), if K , L ∈ M are neighbours, then there exists aunique couple of dual neighbours { K ∗ , L ∗ } such that the interface K | L is the segmentwith summits x K ∗ and x L ∗ . Then the quadrilateral D K , LK ∗ , L ∗ which is either the union(if x K , x L lie on different sides from K | L ) or the difference (if x K , x L lie on the sameside from K | L ) of the triangles x K x K ∗ x L ∗ , x L x K ∗ x L ∗ is called a diamond ; it is alsounambiguously denoted by D K , L .For d = 2, every diamond is also called a subdiamond ; the subdiamond whichcoincides with a diamond D K , L is denoted by S K , LK ∗ , L ∗ .For d = 3 (see Figure 2(b)), if K , L ∈ M are neighbours, then there exists aunique triple of dual neighbours { K ∗ , L ∗ , M ∗ } (which are neighbours pairwise) suchthat the interface K | L is the triangle with summits x K ∗ , x L ∗ and x M ∗ . Then thepolyhedron D K , LK ∗ , L ∗ , M ∗ which is either the union (if x K , x L lie on different sides from K | L ) or the difference (if x K , x L lie on the same side from K | L ) of the pyramids x K x K ∗ x L ∗ x M ∗ , x L x K ∗ x L ∗ x M ∗ is called a diamond ; it is also unambiguously denotedby D K , L . Each diamond is split into three subdiamonds; e.g., the subdiamond S K , LK ∗ , L ∗ is the convex hull of x K , x K ∗ , x L , x L ∗ .We denote by D , S the sets of all diamonds and the set of all subdiamonds,respectively. Generic elements of D , S are denoted by D , S , respectively. Remark 3.1.
If we drop condition (19), the orthogonal projection of x K (whichcoincides with the projection of x L ) on K | L may not be contained within K | L . Tocope with this problem, one could consider subdiamonds of signed volume, notnecessarily contained within the corresponding diamonds. Up to a permutationof the subscripts K ∗ , L ∗ , M ∗ , we have instead of the decomposition D K , LK ∗ , L ∗ , M ∗ = S K , LK ∗ , L ∗ ∪ S K , LL ∗ , M ∗ ∪ S K , LM ∗ , K ∗ , the decomposition D K , LK ∗ , L ∗ , M ∗ = (cid:0) S K , LK ∗ , L ∗ ∪ S K , LL ∗ , M ∗ (cid:1) \ S K , LM ∗ , K ∗ ; in this case the volume of S K , LM ∗ , K ∗ will be taken with the sign “minus”. Under thisconvention, Lemma 3.1 below holds true, so that formulas (22), (23)-(25) below stillyield consistent discrete gradient and discrete divergence operators which enjoy thediscrete duality property [7]. But the discrete entropy dissipation inequalities ofProposition 4.2 would fail, which undermines the subsequent convergence analysis. • For all bounded set E ⊂ R d , set diam ( E ) = sup x, ˆ x ∈ E | x − ˆ x | . • We denote by m E the measure of an object E in its natural dimension (i.e.,the d -dimensional measure, if E is a control volume, a dual control volume, asubdiamond or a diamond; and the ( d − E is an interfaceor a part of an interface). According to Remark 3.1, for the definition of the schemewe could drop (19), in which case for a subdiamond S K , LK ∗ , L ∗ such that S K , LK ∗ , L ∗ ∩ D K , L =Ø its volume is taken with the sign “minus”.3.2. Mesh parameters and regularity of meshes. • We define the size of the mesh by size( T ) = max E ∈ M ∪ M ∗ ∪ D diam ( E ). • Following [11], we call the maximum amongmax K ∗ card( N ∗ ( K ∗ )) , max K (diam ( K )) d m K , max K ∗ (diam ( K ∗ )) d m K ∗ , max K ∩ D =Ø (cid:18) diam ( K )diam ( D ) + diam ( D )diam ( K ) (cid:19) , max K ∗ ∩ D =Ø (cid:18) diam ( K ∗ )diam ( D ) + diam ( D )diam ( K ∗ ) (cid:19) , (where the maximums are taken over all K ∈ M , K ∗ ∈ M ∗ , D ∈ D ) the regularityconstant of the mesh and we denote it by reg( T ). Roughly speaking, this constantcontrols the ratio of dimensions of neighbouring control volumes, diamonds anddual control volumes, as well as the proportions of each volume.In all the discrete estimates and convergence results stated below, we requirethe family of meshes ( T h ) h to have regularity constants reg( T h ) that are uniformlybounded in h . In the sequel, whenever there is a dependency of various constantson reg( T ), we tacitly assume that this dependency is increasing. primalinterface K | L dualinterface K ∗ | L ∗ x K x L ∗ x L x K ∗ ≡ Subdiamond S K , LK ∗ , L ∗ Diamond D K , L (a) 2D (sub)diamond. x K x K ∗ primal x K K | L x M ∗ x M ∗ x K ∗ interface x L x K ∗ x L ∗ x K x L Subdiamond S K,LK ∗ ,L ∗ x L ∗ x L ∗ x L Diamond D K,L volume K primalvolume L x L (b) 3D primal volumes, diamond, subdiamond. Figure 2.
Diamonds and subdiamonds.
V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 17
Discrete gradient and divergence operators.
Diamonds permit to definethe discrete gradient operator, while subdiamonds permit to define the discretedivergence operator (see (20), (21), (22) and (23), (24), (25) below, respectively).Both are needed to discretize the second order “diffusion” operator in equation (1).But first we need to introduce some more notation. • (see Figure 3) For a subdiamond S = S K , LK ∗ , L ∗ , we denote by σ = σ S , σ ∗ = σ ∗ S the (parts of the) interfaces S ∩ K | L and S ∩ K ∗ | L ∗ , respectively, and by ν S , ν ∗ S , unitnormal vectors to σ S and σ ∗ S , respectively (their orientation is chosen arbitrarily). • For a diamond D = D K , L , we denote by Proj D , Proj ∗ D the operators of orthogo-nal projection of R d on the subspaces < −−−→ x K x L > and on < −−−→ x K x L > ⊥ , respectively.One should note that we have < ν S > = < −−−→ x K x L > and < ν ∗ S > ⊂ < −−−→ x K x L > ⊥ for all S ∈ S such that S ⊂ D . x K x K ∗ x L x L ∗ x M ∗ ν ∗ S ν S σ ∗ S interface interface σ S ν ∗ S ν S interface σ ∗ S x K x L ∗ x L x K ∗ interface σ S Figure 3.
Notation in a subdiamond (2D and 3D). • For a couple of neighbours K , L ∈ M , denote by d KL , d K , K | L , and ν K , L thedistance between x K and x L , the distance from x K to K | L , and the unit normalvector to K | L pointing from K to L , respectively. More generally, if K ∈ M , then ν K denotes the exterior unit normal vector to ∂ K . In the same way, for neighbours K ∗ , L ∗ ∈ M ∗ we define d K ∗ L ∗ , d K ∗ , K ∗| L ∗ , and ν K ∗ , L ∗ ; for K ∗ ∈ M ∗ , we define ν K ∗ . Remark 3.2.
Note that by construction both meshes M , M ∗ are conformal (or-thogonal) in the sense if [48]); combined with the Delaunay condition, this meansthat ν K , L · −−−→ x K x L = d KL , ν K ∗ , L ∗ · −−−−→ x K ∗ x L ∗ = d K ∗ L ∗ for all neighbours K , L and K ∗ , L ∗ ,respectively.The conformity property is particularly important for our L framework imposedby the possible degeneracy of the diffusion term and the presence of the hyperbolicconvective term. On the other hand, if this term is dropped, non-conformal doublemeshes can be considered for d = 2 (see [57, 41, 11]) and d = 3 (see [72, 39, 58, 8,7, 38]) within the variational framework. • A discrete function on Ω is a set w T = (cid:0) u M , u M ∗ (cid:1) consisting of two sets of realvalues w M = ( w K ) K ∈ M and w M ∗ = ( w K ∗ ) K ∗ ∈ M ∗ . The set of all such functions isdenoted by R T .A discrete function on Ω is a set w T = (cid:0) w M , w M ∗ , w ∂ M , w ∂ M ∗ (cid:1) ≡ (cid:0) w T , w ∂ M , w ∂ M ∗ (cid:1) consisting of four sets of real values w M = ( w K ) K ∈ M , w M ∗ = ( w K ∗ ) K ∗ ∈ M ∗ , w ∂ M = ( w K ) K ∈ ∂ M , w ∂ M ∗ = ( w K ∗ ) K ∗ ∈ ∂ M ∗ . The set of all such functions is denoted by R T . In case all the components of w ∂ M and of w ∂ M ∗ are zero, we write w T ∈ R T . • A discrete field on Ω is a set F T = (cid:0) F D (cid:1) D ∈ D of vectors of R d . The set of allsuch functions is denoted by ( R d ) D . • On the set R T of discrete functions w T on Ω, we define the discrete gradient operator ∇ T [ · ] by(20) ∇ T : w T ∈ R T
7→ ∇ T w T = (cid:0) ∇ D w T (cid:1) D ∈ D ∈ ( R d ) D where ∇ T w T is the discrete field on Ω with valuesfor d = 2:(21) ∇ D w T = w L − w K d KL ν K , L + w L ∗ − w K ∗ d K ∗ L ∗ ν K ∗ , L ∗ for D = D K , L = S K , LK ∗ , L ∗ ;for d = 3:(22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ D w T = w L − w K d KL ν K , L + 2 m D (cid:18) m S K,LK ∗ ,L ∗ w L ∗ − w K ∗ d K ∗ L ∗ ν K ∗ , L ∗ + m S K,LL ∗ ,M ∗ w M ∗ − w L ∗ d K ∗ L ∗ ν L ∗ , M ∗ + m S K,LM ∗ ,K ∗ w K ∗ − w M ∗ d M ∗ K ∗ ν M ∗ , K ∗ (cid:19) for D = D K , LK ∗ , L ∗ , M ∗ = S K , LK ∗ , L ∗ ∪ S K , LL ∗ , M ∗ ∪ S K , LM ∗ , K ∗ . Remark 3.3.
Formulas (21) and (22) have the following common meaning. Thevector ∇ D w T is the unique element of R d such that Proj D ( ∇ D w T ) = w L ∗ − w K ∗ d K ∗ L ∗ ν K ∗ , L ∗ .Further, for d = 2, Proj ∗ D ( ∇ D w T ) is the gradient of the (unique) affine functionon the interface K | L (which is a segment with summits x K ∗ , x L ∗ ) that takes thevalues w K ∗ , w L ∗ at the points x K ∗ and x L ∗ , respectively. Similarly, for d = 3,Proj ∗ D ( ∇ D w T ) is the gradient of the (unique) affine function on the interface K | L (which is a triangle with summits x K ∗ , x L ∗ , x M ∗ ) that takes the values w K ∗ , w L ∗ , w M ∗ at the points x K ∗ , x L ∗ , x M ∗ , respectively.Thus, the primal mesh M serves to reconstruct one component of the gradient,which is the one in the direction −−−→ x K x L . The dual mesh M ∗ serves to reconstructthe ( d −
1) other components which are the components in the ( d − K | L and is orthogonal to −−−→ x K x L .The first and second assertions of Remark 3.3 are evident. Note that formula(21) easily generalizes to quite arbitrary non conformal double meshes (see [11,Lemma 2.4]). The third assertion is a direct consequence of the 2D reconstructionresult of Lemma 9.6 given and proved in Appendix B (see also [8, 7]). Remark 3.4.
The discrete gradient is exact on affine functions. More precisely, let D be a diamond ( D = D K , LK ∗ , L ∗ , if d = 2; D = D K , LK ∗ , L ∗ , M ∗ , if d = 3). Let w ( x ) := w + r · x , w , r ∈ R d , be an affine function. If w T is a discrete function with values w K = w ( x K ) , w L = w ( x L ); w K ∗ = w ( x K ∗ ) , w L ∗ = w ( x L ∗ ) (and w M ∗ = w ( x M ∗ ) if d = 3) , then ∇ D w T = r ≡ ∇ w . This property follows by a straightforward comparison ofthe formulas (21) and (22) for the discrete gradient with the reconstruction formulasof the next lemma. Lemma 3.1.
Consider D = D K , L ∈ D . With the notation above, for all r ∈ R d one has the following reconstruction properties: V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 19 for d = 2 , r = ( r · ν K , L ) ν K , L + ( r · ν K ∗ , L ∗ ) ν K ∗ , L ∗ ; for d = 3 , r = ( r · ν K , L ) ν K , L + 2 m D (cid:18) m S K,LK ∗ ,L ∗ ( r · ν K ∗ , L ∗ ) ν K ∗ , L ∗ + m S K,LL ∗ ,M ∗ ( r · ν L ∗ , M ∗ ) ν L ∗ , M ∗ + m S K,LM ∗ ,K ∗ ( r · ν M ∗ , K ∗ ) ν M ∗ , K ∗ (cid:19) . Proof.
For d = 2, the claim is a straightforward consequence of the conformityof the meshes (see Remark 3.2); ν K , L , ν K ∗ , L ∗ form an orthonormal basis of R .When d = 3, the claim follows from the orthogonality of ν K , L to K | L and from the2D reconstruction property of Lemma 9.6 (cf. Appendix B) applied in the planecontaining K | L . (cid:3) Remark 3.5.
The fourth assertion of Remark 3.3 indicates possible generalizationsto the multi-dimensional case. Unfortunately, it can be shown that if d ≥
4, thedirect generalization of the reconstitution formula of Lemma 3.1 holds only formeshes M with very special geometries, such as the uniform simplicial meshes(see Remark 9.2, which has to be combined with an induction argument on thedimension d in order to link the weighted projections on the edges appearing inLemma 3.1 with the weighted projections on the faces appearing in Lemma 9.6). • For S ∈ S such that S ⊂ D with D ∈ D , we assign ∇ S u T = ∇ D u T . Moregenerally, if F T is a discrete field on Ω, we assign F S = F D for S ⊂ D .For K ∈ M , we denote by V ( K ) the set of all subdiamonds S ∈ S such that K ∩ S = ∅ . In the same way, for K ∗ ∈ M ∗ we define the set V∗ ( K ∗ ) of subdiamondsintersecting K ∗ . • On the set ( R d ) D of discrete fields F T , we define the discrete divergence oper-ator div T [ · ] by(23) div T : F T ∈ ( R d ) D v T = div T [ F T ] ∈ R T , where the discrete function v T = (cid:0) v M , v M ∗ (cid:1) on Ω is given by(24) v M = ( v K ) K ∈ M with v K = 1 m K X S ∈ V ( K ) m σ S F S · ν K , where ν K = ν K | S ;(25) v M ∗ = ( v K ∗ ) K ∗ ∈ M ∗ , v K ∗ = 1 m K ∗ X S ∈ V∗ ( K ∗ ) m σ ∗ S F S · ν K ∗ , ν K ∗ = ν K ∗ | S . In (24),(25) for S given, ν K = ν K | S denotes the restriction on σ S of the unit normalvector ν K to ∂K exterior to K ; therefore it means the one of the vectors ν S , − ν S that is exterior to K (see Figure 3). Similarly, ν K ∗ = ν K ∗ | S is the one of the vectors ν ∗ S , − ν ∗ S that is exterior to K ∗ .In fact, formulas (24), (25) can be conveniently expressed in terms of vectorproducts involving the discrete field F S and specific geometric objects depicted inFigure 3 (see [8, 7]).3.4. Penalization operator.
On the set R T of discrete functions w T on Ω, wedefine the operator P T [ · ] of double mesh penalization by P T : w T ∈ R T v T = P T [ w T ] ∈ R T , where the discrete function v T = (cid:0) v M , v M ∗ (cid:1) on Ω is given by(26) v M = ( v K ) K ∈ M with v K = ( d −
1) 1size( T ) 1 m K X K ∗ ∈ M ∗ m K ∩ K ∗ ( w K − w K ∗ );(27) v M ∗ = ( v K ∗ ) K ∗ ∈ M ∗ with v K ∗ = 1size( T ) 1 m K ∗ X K ∈ M m K ∩ K ∗ ( w K ∗ − w K ) . The penalization is needed in order to ensure (without using the strong convergenceof ∇ T w T , cf. the proof of [11, Theorem 5.1]), that the two components of a discrete“double” function w T converge to the same limit. Remark 3.6.
The choice of penalization operator we propose here is just thesimplest possibility. In (26),(27), the difference ( w K ∗ − w K ) could be replaced by | w K ∗ − w K | p − ( w K ∗ − w K ), which seems more natural with respect to the assump-tions on a ; the power of size( T ) in the denominator can be chosen arbitrarily. Theconvergence of the scheme would remain true. The question of optimal choice ofthe penalization operator is beyond the scope of this paper.3.5. Discrete convection operator.
Let f : R → R d be continuous. Denote by ω M ( · ) a modulus of continuity of f on [ − M, M ], i.e., a continuous concave functionon [0 , M ] with ω M (0) = 0 andmax a,b ∈ [ − M,M ] , | a − b |≤ r k f ( a ) − f ( b ) k ≤ ω M ( r ) . Note that we can always choose ω M strictly increasing, upon replacing ω M by ω M + Id if needed.Following Eymard, Gallou¨et, Herbin [48], we now define discrete convectionfluxes, separately for each of the meshes M , M ∗ . This will allow to discretizethe convective part of equation (1). • Let K | L ∈ E . To approximate f ( u ) · ν K , L by means of the two values u K , u L thatare available in the neighbourhood of the interface K | L , let us use some function g K , L of the couple ( u K , u L ) ∈ R . More exactly, take a collection of numerical convectionflux functions ( g K , L ) K | L ∈ E , g K , L ∈ C ( R , R ), with the following properties:(28) (a) g K , L ( · , b ) is nondecreasing for all b ∈ R ,and g K , L ( a, · ) is nonincreasing for all a ∈ R ;(b) g K , L ( a, a ) = f ( a ) · ν K , L for all a ∈ R ;(c) g K , L ( a, b ) = − g K , L ( b, a ) ∀ a, b ∈ R , for all neighbours K , L ∈ M ;(d) g K , L has the same modulus of continuity as f , i.e.,there exists C independent of K | L such that ∀ a, b, c, d ∈ [ − M, M ], | g K , L ( a, b ) − g K , L ( c, d ) | ≤ C ( ω M ( | a − c | ) + ω M ( | b − d | ) ) . These assumptions (see [48]) are by now standard. Note that the assumption(28)(d) usually states that f , g K , L are Lipschitz continuous, with the same Lipschitzconstant; here, we adapt it to the case of general continuous function f .Note that (28) (b) and (c) are compatible. Also note that the consistency re-quirement (28)(b) together with the Green-Gauss formula imply(29) X L ∈ N ( K ) m K | L g K , L ( a, a ) = f ( a ) · Z ∂K ν K = 0 for all a ∈ R , for all K ∈ M . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 21
Practical examples of numerical convective flux functions can be found in [48].These include the Godunov, Lax-Friedrichs, Engquist-Osher and Rusanov fluxes asparticular cases. • Numerical convective flux functions g K ∗ , L ∗ , K ∗ | L ∗ ∈ E ∗ , are defined similarly. • On the set R T of discrete functions u T on Ω, we define the operator (div c f ) T [ · ]of discrete convection by(div c f ) T : u T ∈ R T v T = (div c f ) T [ u T ] ∈ R T , where the discrete function v T = (cid:0) v M , v M ∗ (cid:1) on Ω is given by v M = ( v K ) K ∈ M with v K = 1 m K X L ∈ N ( K ) m K | L g K , L ( u K , u L ); v M ∗ = ( v K ∗ ) K ∗ ∈ M ∗ , v K ∗ = 1 m K ∗ X L ∗ ∈ N∗ ( K ∗ ) m K ∗| L ∗ g K ∗ , L ∗ ( u K ∗ , u L ∗ ) . Projection operators and test functions. • On L (Ω), we define the mesh projection operator P T [ · ] on the space of discretefunctions on Ω by P T : S ∈ L (Ω) S T = P T [ S ] ∈ R T , where the discrete function S T = (cid:0) S M , S M ∗ (cid:1) on Ω is given by(30) S M = ( S K ) K ∈ M with S K = 1 m K Z K S ( x ) dx ; S M ∗ = ( S n K ∗ ) K ∗ ∈ M ∗ with S K ∗ = 1 m K ∗ Z K ∗ S ( x ) dx. • For a sufficiently regular function ψ on Ω, we will often employ the notations ψ T = P T [ ψ ] and ( ∇ ψ ) T = P T [ ∇ ψ ] ( ∇ ψ being R d -valued, the projection is takencomponent per component). Further, for K | L ∈ E and K ∗ | L ∗ ∈ E ∗ , we introduce(31) ψ K | L = 1 m K | L Z K | L ψ, ψ K ∗| L ∗ = 1 m K ∗| L ∗ Z K ∗| L ∗ ψ. For L ∈ ∂ M , there exists K | L ⊂ ∂ Ω that coincides with L ; in this case we assign ψ L = ψ K | L . If ψ | ∂ Ω = 0, we have ψ L = 0 for all L ∈ ∂ M . For L ∗ ∈ ∂ M ∗ , we assign ψ L ∗ = m L ∗ R L ∗ ψ . If ψ has a compact support in Ω and size( T ) is small enough, wehave ψ L ∗ = 0 for all L ∗ ∈ ∂ M ∗ .Combining the above notation, we write ψ T = (cid:0) P [ ψ ] , ( ψ K ) K ∈ ∂ M , ( ψ K ∗ ) K ∗ ∈ ∂ M ∗ (cid:1) for the projection of a sufficiently regular function ψ on the space R T , and denotethe corresponding projection operator by P T .3.7. Dependency on t and further notation. • Let T be a DDFV mesh as described above. Let ∆ t > h = max { size( T ) , ∆ t } . By convention, we will use h as the parameterfor a sequence of finite volume schemes; our interest lies in studying convergence ofcorresponding discrete solutions as h ↓ N the integer part of T / ∆ t . In the sequel, in our notation we omitthe dependency of N , T and ∆ t on h . • For a functional space X on Ω, we denote by S ∆ t the projection operator(32) S ∆ t : S ∈ L (0 , T ; X ) ( S n ) n =1 ,...,N ∈ ( X ) N , S n = 1 ∆ t Z n ∆ t ( n − t S ( t ) dt. • A discrete function on Q is a set u T , ∆ t = ( u T ,n ) n =1 ,...,N , where for each n , u T ,n is a discrete function on Ω. The set of all such functions is denoted R N × T .A discrete function on Q is a set u T , ∆ t = ( u T ,n ) n =0 ,...,N , where for each n , u T ,n is a discrete function on Ω. The set of all such functions is denoted by R ( N +1) × T .We also use discrete functions u T , ∆ t ∈ R N × T and u T , ∆ t ∈ R ( N +1) × T . Each of u T , ∆ t , u T , ∆ t , u T , ∆ t is therefore a restriction of u T , ∆ t . The entries of u T ,n are denotedby u n K (respectively, u n K ∗ ) for K ∈ M ∪ ∂ M (respectively, for K ∗ ∈ M ∗ ∪ ∂ M ∗ ).A discrete field on Q is a set F T , ∆ t = (cid:0) F T ,n (cid:1) n =1 ,...,N where for each n , F T ,n isa discrete field on Ω. The set of all such fields is denoted by ( R d ) N × D . • Any discrete function can be composed with a mapping A : R → R m , m ∈ N ;for instance, A ( u T , ∆ t ) stands for w T , ∆ t with values w nK = A ( u nK ) for K ∈ M and w nK ∗ = A ( u nK ∗ ) for K ∗ ∈ M ∗ , for n = 0 , . . . , N . Similarly, any discrete field can becomposed with a mapping ϕ : R d → R m ; one has ϕ ( F T ) = (cid:0) ϕ ( F D ) (cid:1) D ∈ D . • We say that a discrete function is nonnegative ( respectively, nonpositive), ifall its entries are nonnegative (respectively, nonpositive); e.g., for v T ∈ R T thenotation v T ≥ v K ≥ K ∈ M and v K ∗ ≥ K ∗ ∈ M ∗ .3.8. The finite volume scheme.
With the notation introduced above, the finitevolume discretization of problem (1) takes the following compact form:find a discrete function u T , ∆ t on Q satisfying for n = 1 , . . . , N the equations(33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u T ,n − u T , ( n − t + (div c f ) T [ u T ,n ] − div T [ a ( ∇ T w T ,n )]+ P T [ w T ,n ] = P T ( S ∆ t [ S ]) n ,w T ,n = A ( u T ,n ) , together with the boundary and initial conditions(34) for all n = 1 , . . . , N , ( u n K = 0 for all K ∈ ∂ M u n K ∗ = 0 for all K ∗ ∈ ∂ M ∗ ;(35) ( u K = m K R K u for all K ∈ M u K ∗ = m K ∗ R K ∗ u for all K ∗ ∈ M ∗ . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 23
Let us state (33) in a more explicit form: for all n = 1 , . . . , N,m K u nK − u ( n − K ∆ t + P L ∈ N ( K ) m K | L g K , L ( u n K , u n L ) − P S ∈ V ( K ) m σ S a ( ∇ S A ( u T ,n )) · ν K + d − T ) X K ∗ ∈ M ∗ m K ∩ K ∗ ( A ( u K ) − A ( u K ∗ )) = m K S n K , for all K ∈ ∂ M ,m K ∗ u nK ∗ − u ( n − K ∗ ∆ t + P L ∗ ∈ N∗ ( K ∗ ) m K ∗| L ∗ g K ∗ , L ∗ ( u n K ∗ , u n L ∗ ) − P S ∈ V∗ ( K ∗ ) m σ ∗ S a ( ∇ S A ( u T ,n )) · ν K ∗ + 1size( T ) X K ∈ M m K ∩ K ∗ ( A ( u K ∗ ) − A ( u K )) = m K ∗ S n K ∗ , for all K ∗ ∈ ∂ M ∗ . Here S n K , S n L are given by (30),(32); g K , L , g K ∗ , L ∗ are some numerical convection fuxessatisfying (28); ν K , ν K ∗ for S given have the same meaning as in (24),(25); finally,for S given such that S ⊂ D ∈ D , ∇ S A ( u T ,n ) is the vector of R d constructed fromthe values w K = A ( u n K ), w K ∗ = A ( u n K ∗ ) by formulas (21) (for d = 2) or (22) (for d = 3), i.e., in the way indicated in Remark 3.3.4. Elements of discrete calculus for DDFV schemes
In this section, we list convenient formulations of various summation-by-partsformulas and chain rules needed for the analysis of the discrete problem (33).4.1.
Discrete duality formulas for the diffusion terms. • Recall that R T is the space of all discrete functions on Ω. For m ∈ N and w T , v T ∈ (cid:0) R T (cid:1) m , set(36) hh w T , v T ii = 1 d X K ∈ M m K w K · v K + d − d X K ∗ ∈ M ∗ m K ∗ w K ∗ · v K ∗ (here · denotes the scalar product in R m ); it is clear that hh · , · ii is a scalar producton (cid:0) R T (cid:1) m . We will use it for m = 1 or m = d . • Recall that ( R d ) D is the space of all discrete fields on Ω. For F T , G T ∈ ( R d ) D ,set(37) nn F T , G T oo = X D ∈ D m D F D · G D ;it is clear that nn · , · oo is a scalar product on ( R d ) D .A key property of DDFV schemes (see [41, 11]) is the following discrete analogueof the duality between the − div [ · ] and the ∇ [ · ] operators; it is sometimes calledthe discrete duality property for finite volumes. Proposition 4.1.
Let v T ∈ R T and F T ∈ ( R d ) D . Then hh − div T [ F T ] , v T ii = nn F T , ∇ T v T oo . Proof.
The proof is straightforward, using the summation-by-parts procedure. Letus give it for the case d = 2. Note that for D = S = S K , LK ∗ , L ∗ , m S = m K | L d KL = m K ∗| L ∗ d K ∗ L ∗ . By (36), by (24),(25), and finally by (21),(37), we get hh − div T [ F T ] , v T ii = − X K ∈ M (cid:18) X S ∈ V ( K ) m σ S F S · ν K (cid:19) v K − X K ∗ ∈ M ∗ (cid:18) X S ∈ V∗ ( K ∗ ) m σ ∗ S F S · ν K ∗ (cid:19) v K ∗ = − X K ∈ M (cid:18) X S ∈ V ( K ) m σ S F S · ν K (cid:19) v K − X K ∗ ∈ M ∗ (cid:18) X S ∈ V∗ ( K ∗ ) m σ ∗ S F S · ν K ∗ (cid:19) v K ∗ = 12 X S ∈ S , S = SK,LK ∗ ,L ∗ F S · (cid:0) m K | L ( v L − v K ) ν K , L + m K ∗| L ∗ ( v L ∗ − v K ∗ ) ν K ∗ , L ∗ (cid:1) = X S ∈ S , S = SK,LK ∗ ,L ∗ m S F S · (cid:0) v L − v K d KL ν K , L + v L ∗ − v K ∗ d K ∗ L ∗ ν K ∗ , L ∗ (cid:1) = X S ∈ S m S F S · ∇ S v T = X D ∈ D m D F D · ∇ D v T = nn F T , ∇ T v T oo . (cid:3) Furthermore, we have the following “entropy dissipation” inequalities:
Proposition 4.2.
Let u T ∈ R T and ψ ∈ D (Ω) , ψ ≥ . Let θ : R → R be anondecreasing function. Assume that (38) either θ (0) = 0 , or ψ ∈ D (Ω) and size ( T ) is small enough . Denote ψ T = P T [ ψ ] . Then (39) hh div T (cid:2) k ( ∇ T A ( u T )) ∇ T A ( u T ) (cid:3) , θ ( u T ) ψ T ii ≤ − nn k (cid:0) ∇ T A ( u T ) (cid:1) ∇ T A θ ( u T ) , ∇ T ψ T oo . Remark 4.1.
Note that the conformity of the meshes (see Remark 3.2) is essentialfor this result, as well as the particular form of a and (for d = 3) condition (19). Proof.
Let us treat the left-hand side of (39) term by term. It is the sum of genericterms of the form T K , S , T ∗ K ∗ , S ; here T K , S = 1 d m K m K m K | L k ( ∇ S A ( u T )) ∇ S A ( u T ) · ν K , L θ ( u K ) ψ K = 1 d m K | L k (cid:0) ∇ S A ( u T ) (cid:1) ∇ S A ( u T ) · ν K , L θ ( u K ) ψ K with S = S K , L ∈ V ( K ). The notation T ∗ K ∗ , S stands for analogous terms involving K ∗ and S ∈ V∗ ( K ∗ ). Notice that thanks to assumption (38), θ ( u T ) ψ T ∈ R T , sothat we can also add the terms T K , S , T ∗ K ∗ , S corresponding to K ∈ ∂ M , K ∗ ∈ ∂ M ∗ ,respectively. The summation of T K , S , T ∗ K ∗ , S therefore runs on all subdiamonds S = S K , LK ∗ , L ∗ ∈ S , with the associated K , L ∈ M , K ∗ , L ∗ ∈ M ∗ .The convexity argument yields(40) ( A ( z ) − A (ˆ z )) θ (ˆ z ) ≤ A θ ( z ) − A θ (ˆ z ) for all z, ˆ z ∈ R . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 25
By (20), using the positivity of ψ K and applying inequality (40), we get(41) T K , S = 1 d m K | L k (cid:0) ∇ S A ( u T ) (cid:1) A ( u L ) − A ( u K ) d KL θ ( u K ) ψ K ≤ d m K | L k (cid:0) ∇ S A ( u T ) (cid:1) A θ ( u L ) − A θ ( u K ) d KL ψ K . The terms T K ∗ , S are treated in the same way. Now by the same computation asin the proof of Proposition 4.1, one shows that the right-hand sides of (41) and ofthe corresponding inequality for T K ∗ , S sum up to yield the right-hand side of (39).This concludes the proof. (cid:3) Summation formulas for the penalization terms.
For the penalizationoperator P T , we have the following summation formulas. Lemma 4.1.
Let w T ∈ R T and ψ T ∈ R T . Then (42) hh P [ w T ] , ψ T ii = d − d X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ ( w K − w K ∗ )( ψ K − ψ K ∗ ) size ( T ) . Further, let
A, θ : R → R be nondecreasing. Assume u T ∈ R T is such that A ( u T ) belongs to R T . Let ψ ∈ D (Ω) , ψ ≥ ; denote ψ T = P T [ ψ ] . Assume (38) . Then (43) hh P [ A ( u T )] , θ ( u T ) ψ T ii ≥ d − d X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ θ ( u K ) ( A ( u K ) − A ( u K ∗ ))( ψ K − ψ K ∗ ) size ( T ) . In both formulas (42) , (43) , the values ψ K , ψ K ∗ for K ∈ M , K ∗ ∈ M ∗ are those ofthe corresponding discrete function ψ T . The proof is straightforward from the definitions of hh · , · ii and P T , using thesummation-by-parts procedure.4.3. Discrete duality formulas for the evolution terms.Lemma 4.2.
Let θ : R → R be a nondecreasing function, and η = R θ ( s ) ds beits primitive. Let ψ ∈ D ( Q ) , ψ ≥ . Denote ψ T , ∆ t = P T ◦ S ∆ t [ ψ ] . Then for all u T , ∆ t ∈ R ( N +1) × T one has N X n =1 ∆ t hh u T ,n − u T , ( n − t , θ ( u T ,n ) ψ T ,n ii ≥ − N − X n =1 ∆ t hh η ( u T ,n ) , ψ T , ( n +1) − ψ T ,n ∆ t ii + hh η ( u T ,N ) , ψ T ,N ii − hh η ( u T , ) , ψ T , ii . Proof.
The formula follows by the Abel transformation combined with the convexityinequality: ( z − ˆ z ) θ ( z ) ≥ η ( z ) − η (ˆ z ) for all z, ˆ z ∈ R . (cid:3) Discrete duality formulas for the convection terms.
For the convectionterms, we have a more involved “entropy dissipation” duality formula. For lateruse, we state it in the double framework, although each of the meshes M , M ∗ istreated separately in the proof. Proposition 4.3.
Let u T ∈ R T and ψ ∈ D (Ω) . Let θ : R → R be a nondecreasingfunction. Assume (38) . Consider the associated entropy-flux pair η = Z θ ( s ) ds, q = θ f − Z f ( s ) dθ ( s ) . Denote ψ T = P T [ ψ ] and ( ∇ ψ ) T = P T [ ∇ ψ ] . One has hh (div c f ) T [ u T ] , θ ( u T ) ψ T ii = − hh q ( u T ) , ( ∇ ψ ) T ii + I θ [ u M , ψ ] + R θ [ u M , ψ ] + I ∗ θ [ u M ∗ , ψ ] + R ∗ θ [ u M ∗ , ψ ] , (44) where (45) I θ [ u M , ψ ] = 1 d X K | L ∈ E m K | L I K | L θ ψ K | L , I ∗ θ [ u M ∗ , ψ ] = d − d X K ∗| L ∗ ∈ E∗ m K ∗| L ∗ I K ∗| L ∗ θ ψ K ∗| L ∗ with (46) I K | L θ = Z u L u K ( g K , L ( s, s ) − g K , L ( u K , u L )) dθ ( s ) ,I K ∗| L ∗ θ = Z u L ∗ u K ∗ ( g K ∗ , L ∗ ( s, s ) − g K ∗ , L ∗ ( u K ∗ , u L ∗ )) dθ ( s ) . Further, one has I K | L θ ≥ for all K | L ∈ E , and the remainder term R θ satisfies (47) (cid:12)(cid:12) R θ [ u M , ψ ] (cid:12)(cid:12) ≤ (max K ∈ M | θ ( u K ) | ) X K | L ∈ E m K | L (cid:0) R K | LK + R K | LL (cid:1) (cid:0) | ψ K − ψ K | L | + | ψ L − ψ K | L | (cid:1) , (48) R K | LK = | g K , L ( u K , u K ) − g K , L ( u K , u L ) | , R K | LL = | g K , L ( u L , u L ) − g K , L ( u K , u L ) | . Similarly, one has I K ∗| L ∗ θ ≥ for all K ∗ | L ∗ ∈ E ∗ , and the remainder term R ∗ θ satisfies the analogue of (47) , (48) with K , L , M , E replaced by K ∗ , L ∗ , M ∗ , E ∗ . Note that our notation is consistent: we have I K | L θ = I L | K θ , R K | LK = R L | KL for allneighbours K , L (for dual neighbours K ∗ , L ∗ , similar identities hold). Proof.
We exploit the ideas of [48] and [29].Thanks to (38) and because u T is zero on boundary volumes, we have(49) η ( u L ) ψ L = 0 for all L ∈ ∂ M ; η ( u L ∗ ) ψ L ∗ = 0 for all L ∗ ∈ ∂ M ∗ . Separating the contributions of M and M ∗ , we write the left-hand side of (44)as d I + d − d I ∗ , where(50) I := X K ∈ M m K (cid:0) m K X L ∈ N ( K ) m K | L g K , L ( u K , u L ) (cid:1) θ ( u K ) ψ K . Applying (28)(c) and (29), using (49) in the summation-by-parts procedure, we get I = X K | L ∈ E m K | L (cid:18) θ ( u L ) (cid:0) g K , L ( u L , u L ) − g K , L ( u K , u L ) (cid:1) ψ L − θ ( u K ) (cid:0) g K , L ( u K , u K ) − g K , L ( u K , u L ) (cid:1) ψ K (cid:19) . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 27
Hence, choosing ψ K | L as defined in (31), we have I = X K | L ∈ E m K | L (cid:18) θ ( u L ) (cid:0) g K , L ( u L , u L ) − g K , L ( u K , u L ) (cid:1) − θ ( u K ) (cid:0) g K , L ( u K , u K ) − g K , L ( u K , u L ) (cid:1)(cid:19) ψ K | L + X K | L ∈ E m K | L (cid:18) θ ( u L ) (cid:0) g K , L ( u L , u L ) − g K , L ( u K , u L ) (cid:1) ( ψ L − ψ K | L ) − θ ( u K ) (cid:0) g K , L ( u K , u K ) − g K , L ( u K , u L ) (cid:1) ( ψ K − ψ K | L ) (cid:19) . Now recall that g = g K , L satisfies (28)(b). Thus the following integration-by-partsformula holds true:( q ( b ) − q ( a )) · ν K , L = (cid:0) θ ( b ) f ( b ) − θ ( a ) f ( a ) − Z ba f ( s ) dθ ( s ) (cid:1) · ν K , L = θ ( b )( g ( b, b ) − g ( a, b )) − θ ( a )( g ( a, a ) − g ( a, b )) − Z ba ( g ( s, s ) − g ( a, b )) dθ ( s ) . We deduce I = J + I θ + R θ , where J = X K | L ∈ E m K | L (cid:0) q ( u K ) − q ( u L ) (cid:1) · ν K , L ψ K | L = X K ∈ M q ( u K ) · (cid:0) X L ∈ N ( K ) m K | L ψ K | L ν K , L (cid:1) = X K ∈ M q ( u K ) · Z ∂ K ψ ν K = X K ∈ M Z K div (cid:0) q ( u K ) ψ (cid:1) = X K ∈ M Z K q ( u K ) · ∇ ψ = X K ∈ M m K q ( u K ) · ( ∇ ψ ) K , and I θ = X K | L ∈ E m K | L (cid:0)Z u L u K ( g K , L ( s, s ) − g K , L ( u K , u L )) dθ ( s ) (cid:1) ψ K | L , | R θ | ≤ X K | L ∈ E m K | L (cid:18) | g K , L ( u K , u K ) − g K , L ( u K , u L ) | + | g K , L ( u L , u L ) − g K , L ( u K , u L ) | (cid:19) × (cid:18) | ψ K − ψ K | L | + | ψ L − ψ K | L | (cid:19) × (max K ∈ M | θ ( u K ) | ) . In the same way, I ∗ = J ∗ + I ∗ θ + R ∗ θ with analogous estimates. We have the equality d J + d − d J ∗ = hh q ( u T ) , ( ∇ ψ ) T ii . With the notation of (45)-(48) the result of theproposition follows. (cid:3) Properties of discrete operators and functional spaces
In this section we state important embedding and compactness properties ofspaces of discrete functions, as well as the asymptotic (as h →
0) properties ofvarious discrete operators.
Discrete functions and fields as elements of Lebesgue spaces.
For any E ⊂ Q , denote by 1l E its characteristic function.For n = 1 , . . . , N , set Q n K = [( n − ∆ t, n ∆ t [ × K , for K ∈ M ; Q n K ∗ = [( n − ∆ t, n ∆ t [ × K ∗ , for K ∗ ∈ M ∗ ; Q n D = [( n − ∆ t, n ∆ t [ × D , for D ∈ D . For a discrete function v T , ∆ t on Q , denote by v M , ∆ t (respectively, by v M ∗ , ∆ t ) thepiecewise constant function v M , ∆ t ( t, x ) = N X n =1 X K ∈ M u n K Q nK ( t, x ) (cid:18) respectively, v M ∗ , ∆ t ( t, x ) = N X n =1 X K ∗ ∈ M ∗ u n K ∗ Q nK ∗ ( t, x ) (cid:19) . Whenever it is convenient, we identify the discrete function v T , ∆ t ∈ R N × T withthe function on Q given by v T , ∆ t ( t, x ) = 1 d v M , ∆ t ( t, x ) + d − d v M ∗ , ∆ t ( t, x ) . In a similar way, we identify a discrete field F T , ∆ t ∈ R N × D on Q with the function F T , ∆ t ( t, x ) = N X n =1 X D ∈ D F n D Q nD ( t, x ) . Analogous conventions apply to time-independent discrete functions and discretefields, in which case we suppress the superscript ∆ t in the notation.5.2. Consistency properties of discrete operators.
In the proposition belowwe show the consistency properties of the projection and discrete gradient operatorsin Lebesgue spaces. Also note the property ( iv ), which, combined with formula (42),expresses the fact that the penalization operator introduced in Section 3.4 vanishes(in an appropriate sense) as size( T ) → Proposition 5.1.
Let T be a double mesh of Ω , ∆ t > , h = max { size ( T ) , ∆ t } ,and q ∈ [1 , + ∞ ] . Then(i) there exists a constant C that only depends on Ω , q and reg( T ) such that ∀ w ∈ L q ( Q ) , k ( P T ◦ S ∆ t w ) M , ∆ t k L q + k ( P T ◦ S ∆ t w ) M ∗ , ∆ t k L q ≤ C k w k L q , and ∀ w ∈ L q (0 , T ; W ,q (Ω))) , k ∇ T P T ◦ S ∆ t w k L q ≤ C k ∇ w k L q ;(ii) for all w ∈ L q ( Q ) , q < + ∞ , both ( P T ◦ S ∆ t w ) M , ∆ t and ( P T ◦ S ∆ t w ) M ∗ , ∆ t converge to w in L q ( Q ) as h → ;(iii) for all w ∈ L q (0 , T ; W ,q (Ω)) , q < + ∞ , the discrete fields ∇ T P T ◦ S ∆ t w converge to ∇ w in ( L q ( Q )) d as h → ; V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 29 (iv) let ψ ∈ D (Ω) , and ψ T ,n = P T ( S [ ψ ]) n , n = 1 , . . . , N . There exists a constant C that only depends on Q and reg( T ) such that N X n =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ ( ψ K − ψ K ∗ ) size ( T ) ≤ C k ∇ ψ k L ∞ × size ( T ) . Proof.
The proof of (i)-(iii) is a straightforward generalization of [11, Lemma 3.3,Proposition 3.4 and Corollary 3.5]. We need to take into account the fact that k S ∆ t w k L q ( X ) ≤ k w k L q ( X ) and (for q = + ∞ ) k S ∆ t w − w k L q ( X ) → ∆ t → w ∈ L q (0 , T ; X ), where X stands for L q (Ω) or for W ,q (Ω). Remark 3.4is important for (iii) (thus, the Delaunay property of M is used). Further, in astandard way similar to [11, Lemma 3.3] one proves that for all K ∈ M , K ∗ ∈ M ∗ such that K ∩ K ∗ = Ø, one has | ψ n K − ψ n K ∗ | ≤ C (reg( T )) k ∇ ψ k L ∞ × size( T ) for all n = 1 , . . . , N . Hence the claim (iv) follows. (cid:3) Discrete embedding and compactness results.
Next we state a versionof the Poincar´e inequality and an embedding-kind translation estimate on doublediscrete functions.
Proposition 5.2.
Assume T is a double mesh on Ω , ∆ t > . Let q ∈ [1 , + ∞ ) .There exists a constant C > that only depends on diam (Ω) and q such that(i) for all w T , ∆ t ∈ R N × T one has k w M , ∆ t k L q + k w M ∗ , ∆ t k L q ≤ C k ∇ T w T , ∆ t k q ; (ii) for all w T ∈ R T , for all ∆ ∈ R d one has k w M ( · + ∆ ) − w M ( · ) k L q + k w M ∗ ( · + ∆ ) − w M ∗ ( · ) k L q ≤ C k ∇ T w T k L q × | ∆ | /q . Proof.
The proof follows the lines of [12, Lemma 1] and [11, Lemma 3.6]. Notethat if d = 3, the fact that all interfaces K | L are triangles plays an important rolein the proof. (cid:3) Here is the asymptotic compactness result for “discrete L p (0 , T ; W ,p (Ω)” spaces. Proposition 5.3.
Let p ∈ (1 , + ∞ ) . Assume we are given a family { w T , ∆ t } h ofdiscrete functions in R N × ¯ T corresponding to a family of double meshes T suchthat reg( T ) is uniformly bounded (recall that we parametrize the meshes by h =max { size ( T ) , ∆ t } ).(i) Assume that there exists a constant C > such that k ∇ T w T , ∆ t k L p ≤ C. Then there exists a (not labelled) sequence of meshes such that as h → w T , ∆ t = 1 d w M , ∆ t + d − d w M ∗ , ∆ t converge weakly in L p ( Q ) to some limit w ;furthermore, w ∈ L p (0 , T ; W ,p (Ω)) andthe discrete fields ∇ T w T , ∆ t converge weakly in ( L p ( Q )) d to ∇ w as h → . (ii) If, in addition, N X n =1 ∆ t hh P T [ w T ,n ] , w T ,n ii ≤ C, where P T are the penalization operators introduced in Section , thenboth w M , ∆ t and w M ∗ , ∆ t converge to w weakly in L p ( Q ) as h → . Remark 5.1.
Note that upon providing uniform estimates on time translates of w T , ∆ t in L p ( Q ), strong convergence to w in L p ( Q ) holds true (see Section 7). Proof. (i) The proof is very similar to the one of [11, Lemma 3.8].First, by Proposition 5.2(i), both families { w M , ∆ t } h , { w M ∗ , ∆ t } h of components of w T , ∆ t are bounded in L p ( Q ). Therefore we can choose a common sequence such thatboth components converge weakly in L p ( Q ). Also w T , ∆ t = d w M , ∆ t + d − d w M ∗ , ∆ t converge weakly to some limit that we denote w . We can also assume that thecorresponding sequence { ∇ T w T , ∆ t } h converges weakly in ( L p ( Q )) d to some limit χ . Let us show that w ∈ L p (0 , T ; W ,p (Ω)) and χ = ∇ w .Take any field F ∈ ( L p ′ (0 , T ; W ,p ′ (Ω)) d . Denote by F T , ∆ t the discrete field on Q with entries F n D = 1 ∆ t × m D Z n ∆ t ( n − t Z D F . Denote by (div F ) T , ∆ t the discrete function P T ◦ S ∆ t [div F ] on Q , which has theentries(div F ) n K = 1 ∆ tm K Z n ∆ t ( n − t Z K div F = 1 ∆ tm K Z n ∆ t ( n − t X S ∈ V ( K ) Z σ S F · ν K , (div F ) n K ∗ = 1 ∆ tm K ∗ Z n ∆ t ( n − t Z K ∗ div F = 1 ∆ tm K ∗ Z n ∆ t ( n − t X S ∈ V∗ ( K ∗ ) Z σ ∗ S F · ν K ∗ . By Proposition 4.1, by definitions of nn · , · oo , hh · , · ii and using the notation introducedin Section 5.1, we have0 = N X n =1 ∆ t nn F T ,n , ∇ T w T ,n oo + N X n =1 ∆ t hh div T [ F T ,n ] , w T ,n ii = N X n =1 ∆ t nn F T ,n , ∇ T w T ,n oo + N X n =1 ∆ t hh (div F ) T ,n , w T ,n ii + N X n =1 ∆ t hh div T [ F T ,n ] − (div F ) T ,n , w T ,n ii = Z Q F T , ∆ t · ∇ T w T , ∆ t + Z Q (cid:0) div F (cid:1) (cid:0) d w M , ∆ t + d − d w M ∗ , ∆ t (cid:1) + N X n =1 ∆ t hh div T [ F T ,n ] − (div F ) T ,n , w T ,n ii . As in Proposition 5.1, one shows that kF T ,n − Fk L p ′ tends to zero as h → Z Q F · χ + Z Q (cid:0) div F (cid:1) w + lim h → N X n =1 ∆ t hh div T [ F T ,n ] − (div F ) T ,n , w T ,n ii . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 31
By definition of hh · , · ii , we have N X n =1 ∆ t hh div T [ F T ,n ] − (div F ) T ,n , w T ,n ii = 1 d N X n =1 ∆ t (cid:18) m K w n K ∆ tm K Z n ∆ t ( n − t X S ∈ V ( K ) (cid:0)Z σ S F − m σ S F n S ) · ν K (cid:19) + d − d N X n =1 ∆ t (cid:18) m K ∗ w n K ∗ ∆ tm K ∗ Z n ∆ t ( n − t X S ∈ V∗ ( K ∗ ) (cid:0)Z σ ∗ S F − m σ ∗ S F n S ) · ν K ∗ (cid:19) . Denote by R + R ∗ the right-hand side above. Summing by parts, we get R = 1 d N X n =1 X K | L ∈ E d KL Z n ∆ t ( n − t Z K | L (cid:18) F − m D Z D F (cid:19) · ν K , L w n K − w n L d KL where D stands for the diamond D KL containing the interface K | L . By the H¨olderinequality, we deduce that | R | is controlled by (cid:18) N X n =1 X K | L ∈ E d KL Z n ∆ t ( n − t Z K | L (cid:12)(cid:12)(cid:12)(cid:12) F − m D Z D F (cid:12)(cid:12)(cid:12)(cid:12) p ′ (cid:19) p ′ × (cid:18) N X n =1 ∆ t X K | L ∈ E m K | L d KL (cid:12)(cid:12) w n K − w n L d KL (cid:12)(cid:12) p (cid:19) p . Using standard estimates similar to [11, Lemma 3.2] and the definition of ∇ T w T , ∆ t ,we conclude that | R | ≤ C (reg( T )) × size( T ) × kFk L p ′ ( W ,p ′ ) k ∇ T w T , ∆ t k L p ≤ C (reg( T )) × h × kFk L p ′ ( W ,p ′ ) × C → h →
0. In the same way, we find | R ∗ | → h → F ∈ ( L p ′ (0 , T ; W ,p ′ (Ω)) d , the last term in (51) is zero, so that w ∈ L p (0 , T ; W ,p (Ω)) and χ = ∇ w .(ii) If also N X n =1 ∆ t hh P T [ w T ,n ] , w T ,n ii ≤ C , then by Lemma 4.1 with ψ T = w T we get d − d N X n =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ ( w n K − w n K ∗ ) ≤ Ch.
This means that k w M , ∆ t − w M ∗ , ∆ t k L → h →
0, which permits to identify theweak limits of both w M , ∆ t and w M ∗ , ∆ t with w . (cid:3) Properties of discrete solutions
A priori estimates.Proposition 6.1.
Assume we are given a family of double meshes T of Ω andassociated time steps ∆ t such that h = max { size ( T ) , ∆ t } → . Assume that reg( T ) is uniformly bounded.Let u T , ∆ t be a solution to (33) , (34) , (35) (recall that w T , ∆ t = A ( u T , ∆ t ) ). Thenthe following a priori estimates hold uniformly in h : (i) max {k u M , ∆ t k L ∞ , k u M ∗ , ∆ t k L ∞ } ≤ M := k u k L ∞ + Z T k S ( t, · ) k L ∞ dt ;(ii) there exists C > such that k ∇ T w T , ∆ t k L p ≤ C and N X n =1 ∆ t hh P T [ w T ,n ] , w T ,n ii ≤ C ; (iii) there exists C > such that (with the notation of Proposition ) N X n =1 ∆ t (cid:0) I Id [ u M ,n ,
1] + I ∗ Id [ u M ∗ ,n , (cid:1) ≤ C ; (iv) there exists a modulus of continuity ω A ( · ) such that for all ∆ > , Z Q | w M , ∆ t ( t + ∆ , x ) − w M , ∆ t ( t, x ) | + | w M ∗ , ∆ t ( t + ∆ , x ) − w M ∗ , ∆ t ( t, x ) | ≤ ω A ( ∆ ) , where w M , ∆ t , w M ∗ , ∆ t are extended by zero on ( N ∆ t, + ∞ ) × Ω .Proof. (i) Denote S i = ( S ∆ t [ S ]) i and S T ,i = P T [( S ∆ t [ S ]) i ]. For n = 0 , . . . , N , set c n = k u k L ∞ + P ni =1 ∆ t k S i k L ∞ ; note that c n ≤ k u k L ∞ + R T k S ( t, · ) k L ∞ dt = M for all n = 1 , . . . , N .Let us prove by induction that k u M ,n k L ∞ ≤ c n , k u M ∗ ,n k L ∞ ≤ c n . This claim isclear for n = 0. Assume it holds true for n = k −
1. Take the scalar product hh · , · ii of equations (33) corresponding to n = k with the discrete function θ ( u T ,k ) :=sign + ( u T ,k − c k ). We get(52) hh u T ,k − u T , ( k − t − S T ,k , θ ( u T ,k ) ii + hh (div c f ) T [ u T ,k ] , θ ( u T ,k ) ii − hh div T [ a ( ∇ T A ( u T ,k ))] , θ ( u T ,k ) ii + hh P T [ w T ,k ] , θ ( u T ,k ) ii = 0 . Let us apply to the last three terms above Proposition 4.3, Proposition 4.2 andLemma 4.1 respectively, with ψ ≡
1. Note that θ (0) = 0, so that (38) holds. Weconclude that each of the three last terms in (52) is nonnegative. Hence0 ≥ hh u T ,k − u T , ( k − K ∆ t − S T ,k , θ ( u T ,k ) ii = hh ( u T ,k − c k ) − ( u T , ( k − − c ( k − ) ∆ t + (cid:0) k S k k L ∞ − S T ,k (cid:1) , sign + ( u T ,k − c k ) ii ≥ hh ( u T ,k − c k ) − ( u T , ( k − − c ( k − ) ∆ t , sign + ( u T ,k − c k ) ii ≥ hh ( u T ,k − c k ) + − ( u T , ( k − − c ( k − ) + , T ii , where 1 T = P T [1]. By the induction hypothesis we deduce that ( u T ,k − c k ) + ≤ n = k . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 33 (ii) For n = 1 , . . . , N , take the scalar product hh · , · ii of equations (33) with thediscrete function w T ,n = A ( u T ,n ). Multiply by ∆ t and sum up in n . We get(53) N X n =1 ∆ t hh u T ,n − u T , ( n − t , A ( u T ,n ) ii + N X n =1 ∆ t hh (div c f ) T [ u T ,n ] , A ( u T ,n ) ii − N X n =1 ∆ t hh div T [ a ( ∇ T w T ,n )] , w T ,n ii + N X n =1 ∆ t hh P T [ w T ,n ] , w T ,n ii = N X n =1 ∆ t hh P T ( S ∆ t [ S ]) n , w T ,n ii . Note that with θ ( · ) = A ( · ) and ψ ≡
1, (38) holds. Applying Lemma 4.2, Proposi-tion 4.3, Proposition 4.1 and Lemma 4.1, respectively, to the terms on the left-handside of (53), we find(54) hh η ( u T ,N ) , T ii + N X n =1 ∆ t (cid:0) I A [ u M ,n ,
1] + I ∗ A [ u M ∗ ,n , (cid:1) + N X n =1 ∆ t nn a ( ∇ T w T ,n ) , ∇ T w T ,n oo + N X n =1 ∆ t hh P T [ w T ,n ] , w T ,n ii = N X n =1 ∆ t hh P T ( S ∆ t [ S ]) n , w T ,n ii + hh B A ( u T , ) , T ii , where B A ( z ) = R z A ( s ) ds and I A , I ∗ A are defined in Proposition 4.3. The first twoterms in (54) are nonnegative; the next one is lower bounded by a constant times (cid:0) k ∇ T w T , ∆ t k L p (cid:1) p due to the coercivity assumption on a . By H¨older’s inequality,Proposition 5.1(i) and Proposition 5.2(i), the first term in the right-hand side of(54) is majorated by C (reg( T )) × k f k Lp ′ × k ∇ T w T , ∆ t k L p . Finally, the last term in(54) is upper bounded by a constant times m Ω R k u k L ∞ −k u k L ∞ A ( s ) ds . Hence, (ii) follows.(iii) We proceed as in (ii), multiplying equations (33) by u T ,n instead of A ( u T ,n ).As in (54) above, taking θ = Id , ψ ≡
1, applying Proposition 4.2 instead of Propo-sition 4.1, neglecting the nonnegative terms on the left-hand side, we get N X n =1 ∆ t (cid:0) I Id [ u M ,n ,
1] + I ∗ Id [ u M ∗ ,n , (cid:1) ≤ N X n =1 ∆ t hh P T ( S ∆ t [ S ]) n , u T ,n ii + hh
12 ( u T , ) , T ii . Using the L ∞ estimate (i) of the present proposition together with Proposition 5.1(i),we finally get (iii) with the constant C = C (reg( T )) × M × k S k L + 12 m Ω × ( k u k L ∞ ) . (iv) We adapt to the discrete framework the calculation that led to estimate (8) inthe proof of Theorem 2.1(ii). Denote by J ( ∆ ) , J ∗ ( ∆ ), respectively, the integrals Z Q | u M , ∆ t ( t + ∆ , x ) − u M , ∆ t ( t, x ) | | A ( u M , ∆ t )( t + ∆ , x ) − A ( u M , ∆ t )( t, x ) | , Z Q | u M ∗ , ∆ t ( t + ∆ , x ) − u M ∗ , ∆ t ( t, x ) | | A ( u M ∗ , ∆ t )( t + ∆ , x ) − A ( u M ∗ , ∆ t )( t, x ) | . Let us first take k ∈ { , . . . , N } and estimate the quantity J ( k ) := N X n = k +1 ∆ t hh u T ,n − u T , ( n − k ) , A ( u T ,n ) − A ( u T , ( n − k ) ) ii . To do this, for n = ( k + 1) , . . . , N we take the sum in i from ( n − k + 1) to n ofequations (33) and make the scalar product hh · , · ii with the discrete functions v T ,n ,where v T ,n := A ( u T ,n ) − A ( u T , ( n − k ) ) ∈ R T for n = ( k + 1) , . . . , N . Summing in n and assigning v T ,n = 0 for n = 1 , . . . , k and n = ( N + 1) , . . . , ( N + k − J ( k ) ∆ t = N X n = k +1 hh u T ,n − u T , ( n − k ) , v T ,n ii = N X n = k +1 ∆ t hh n X i = n − k +1 u T ,i − u T , ( i − t , v T ,n ii = N X i =2 min { k,N − i +1 } X j =max { ,k − i +2 } ∆ t hh u T ,i − u T , ( i − t , v T , ( i + j − ii = k X j =1 N X i =2 ∆ t hh − (div c f ) T [ u T ,i ] + div T [ a ( ∇ T w T ,i )] −P T [ w T ,i ] + P T ( S ∆ t [ S ]) i , v T , ( i + j − ii . We claim that the right-hand side of (55) is bounded by a constant independent of h . Indeed, for each j = 1 , . . . , k , define z j T ,i = v T , ( i + j − , i = 1 , . . . , N . First, fromthe property (ii) of the present proposition and from formula (42) we deduce(56) k ∇ T z T , ∆ tj k L p ≤ C, N X i =1 ∆ t hh P T [ z T ,ij ] , z T ,ij ii ≤ C, for all j = 1 , . . . , k. In the sequel, we will omit the dependency of the entries of z T , ∆ tj on j .By definition of (div c f ) T [ · ], taking into account that z T ,nj ∈ R T and usingsummation-by-parts, we deduce that for all j = 1 , . . . , k , J ,j := (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ∆ t hh (div c f ) T [ u T ,i ] , z T ,ij ii(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ∆ t (cid:18) d X K | L ∈ E m K | L g K , L ( u i K , u i L ) ( z i K − z i L )+ d − d X K ∗| L ∗ ∈ E∗ m K ∗| L ∗ g K ∗ , L ∗ ( u i K ∗ , u i L ∗ ) ( z i K ∗ − z i L ∗ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 35
Since by (i), u M , ∆ t , u M ∗ , ∆ t are bounded by M , using property (28)(d) we boundall values of g K | L , g K ∗| L ∗ above by Cω M ( M ). It follows by Remark 3.3 that | z i K − z i L | d KL + | z i K ∗ − z i L ∗ | d K ∗ L ∗ ≤ | ∇ S z T ,i | , where S = S K , LK ∗ , L ∗ . Hence J ,j ≤ C ( d − ω M ( M ) N X i =1 ∆ t X S ∈ S m S | ∇ S z T ,i | ≤ const k ∇ T z T , ∆ tj k L . Using (56), we can uniformly bound J ,j . Further, by Proposition 4.1 and theH¨older inequality, J ,j := (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ∆ t hh div T [ a ( ∇ T w T ,i )] , z T ,ij ii(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ∆ t nn a ( ∇ T w T ,i ) , ∇ T z T ,ij oo(cid:12)(cid:12)(cid:12)(cid:12) ≤ k a ( ∇ T w T , ∆ t ) k L p ′ k ∇ T z T , ∆ tj k L p . Using the growth assumption on a together with (56) and (ii) of the present lemma,we can uniformly bound J ,j . Next, by (42) and the Cauchy-Schwarz inequality, J ,j := (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ∆ t hh P T [ w T ,i ] , z T ,ij ii(cid:12)(cid:12)(cid:12)(cid:12) ≤ d − d N X i =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ | w i K − w i K ∗ | p size( T ) | z i K − z i K ∗ | p size( T ) ≤ d − d N X i =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ | w i K − w i K ∗ | size( T ) / × N X i =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ | z i K − z i K ∗ | size( T ) / = N X i =1 ∆ t hh P T [ w T ,i ] , w T ,i ii! / N X i =1 ∆ t hh P T [ z T ,ij ] , z T ,ij ii! / . Using again (56) and (ii) of the present proposition, we can uniformly bound J ,j .Finally, like in (54), we have J ,j := (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ∆ t hh P T ( S ∆ t [ S ]) i , z T ,ij ii (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (reg( T )) × k S k L p ′ × k ∇ T z T , ∆ tj k L p , which is also uniformly bounded, thanks to (56). Gathering the estimates above,we conclude J ( k ) ≤ ∆ t k X j =1 (cid:18) J ,j + J ,j + J ,j + J ,j (cid:19) ≤ C k ∆ t. Using the definition of hh · · ii and the L ∞ estimate on u M , ∆ t , cf. (i), we get(57) 1 d J ( k ∆ t ) + d − d J ∗ ( k ∆ t ) ≤ J ( k )+ Z N ∆ t ( N − k )∆ t m Ω M max {± A ( ± M ) } ≤ C k ∆ t. Now let 0 < ∆ < T . We have ∆ / ∆ t = ( k −
1) + α for some k ∈ { , . . . , N } and α ∈ [0 , u M , ∆ t is piecewise constant in t with step ∆ t , we have J ( ∆ ) = J (( k − ∆ t + α ∆ t ) ≤ αJ ( k ∆ t ) + (1 − α ) J (( k − ∆ t ) ≤ αC k ∆ t + (1 − α ) C ( k − ∆ t ≤ C (( k −
1) + α ) ∆ t = C ∆ . (58)From (58), together with the calculation used to pass from (8) to (9) (cf. the proofof Theorem 2.1), we deduce the required estimate Z Q | A ( u M , ∆ t )( t + ∆ , x ) − A ( u M , ∆ t )( t, x ) | ≤ ω A ( ∆ ) , ∆ > . Similarly, time translates of A ( u M ∗ , ∆ t ) are controlled with J ∗ ( k ∆ t ) in (57). (cid:3) Existence of discrete solutions.Proposition 6.2.
Let T be a double mesh of Ω and ∆ t > . There exists a solution u T , ∆ t of the finite volume scheme (33) , (34) , (35) .Proof. First note that it is sufficient to prove existence of solutions u T , ∆ tρ to (33),(34), (35) with A ( · ) replaced by a strictly increasing function A ρ ( · ). Indeed, usingthe L ∞ estimate (i) of Proposition 6.1, which is independent of the choice of A ( · ),we get compactness of u T , ∆ tρ in the finite-dimensional space R ( N +1) × T . Choosinga sequence of strictly increasing functions A ρ that converges to A uniformly on allcompact of R , we pass to the limit in the scheme (33), (34), (35) written for (asubsequence of) A ρ ( · ) and u T , ∆ tρ and obtain existence for general A ( · ).Let us now assume that A ( · ) is invertible and rewrite the scheme in terms of w T , ∆ t with u T , ∆ t = A − ( w T , ∆ t ). The existence of w T ,n is shown by induction on n = 0 , . . . , N . For n = 0, solution is given by (35). Assume that w T , ( n − exists.Choose hh · , · ii as the scalar product on R T . We are looking for a solution w T ,n to L [ w T ,n ] = 0, where the operator L is given by L : z T ∈ R T A − ( z T ) − A − ( w T , ( n − ) ∆ t + (div c f ) T [ A − ( z T )] − div T [ a ( ∇ T z T )] + P T [ z T ] − P T ( S ∆ t [ S ]) n . By Proposition 4.3 with θ = Id and ψ ≡
1, by Proposition 4.1 and by Lemma 4.1,there exists a constant C = C (cid:0) k w T ,n − k R T , k P T ( S ∆ t [ S ]) n k R T , ∆ t (cid:1) such that hh L [ z T ] , z T ii ≥ nn a ( ∇ T z T ) , ∇ T z T oo − C k z T k R T . By the coercivity assumption on a and by Proposition 5.2(i) we have(59) nn a ( ∇ T z T ) , ∇ T z T oo ≥ const k ∇ T z T k pL p ≥ const (cid:0) k z M k pL p + k z M ∗ k pL p (cid:1) . Because the right-hand side of (59) is equivalent to (cid:0) k z T k R T (cid:1) p , we conclude that hh L [ z T ] , z T ii ≥ k z T k R T sufficiently large. The existence of w T ,n follows bythe standard Brouwer fixed point argument (see [64, Lemme 4.3]). (cid:3) V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 37
We point out that the uniqueness and, more generally, continuous dependencyof the discrete solutions on the data can be established as well (see [48, 49, 11] forresults of that sort). However, in view of the convergence result of Theorem 7.1and the well-posedness of the continuous problem, we view these questions to be ofless importance.6.3.
Discrete entropy inequalities.Proposition 6.3.
Let T be a double mesh of Ω and ∆ t > . Consider a solution u T , ∆ t to the scheme (33) , (34) , (35) ; recall that w T , ∆ t = A ( u T , ∆ t ) .Let ψ ∈ D ( Q ) , ψ ≥ ; set ψ T , ∆ t = P T ◦ S ∆ t [ ψ ] . Let θ : R → R be a nondecreasingfunction; assume that ψ and θ are chosen so that (38) holds; assume that ∆ t is smallenough. Then (60) − hh η ( u T ,N ) , ψ T ,N ii + N − X n =1 ∆ t hh η ( u T ,n ) , ψ T , ( n +1) − ψ T ,n ∆ t ii + N X n =1∆ t hh q ( u T ,n ) , ( ∇ ψ ) T ,n ii − N X n =1∆ t nn k (cid:0) ∇ T w T ,n (cid:1) ∇ T e A θ ( w T ,n ) , ∇ T ψ T ,n oo + hh η ( u T , ) , ψ T , ii + N X n =1 ∆ t hh P T ( S ∆ t [ f ]) n , θ ( u T ,n ) ψ T ,n ii ≥ d − d N X n =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ θ ( u K ) ( w n K − w n K ∗ )( ψ n K − ψ n K ∗ ) size ( T )+ R θ [ u M ,n , ψ n ] + R ∗ θ [ u M ∗ ,n , ψ n ] , where A θ ( · ) , e A θ ( · ) and η ( · ) , q ( · ) , R θ [ · , · ] , R ∗ θ [ · , · ] are introduced in Definition andin Proposition , respectively.Moreover, with the specific choice θ ≡ and ψ ∈ D ([0 , T ) × Ω) , there holds (61) N − X n =1 ∆ t hh u T ,n , ψ T , ( n +1) − ψ T ,n ∆ t ii + hh u T , , ψ T , ii + N X n =1 ∆ t hh f ( u T ,n ) , ( ∇ ψ ) T ,n ii − N X n =1 ∆ t nn k (cid:0) ∇ T w T ,n (cid:1) ∇ T w T ,n , ∇ T ψ T ,n oo + N X n =1 ∆ t hh P T ( S ∆ t [ S ]) n , ψ T ,n ii = d − d N X n =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ ( w n K − w n K ∗ )( ψ n K − ψ n K ∗ ) size ( T )+ R [ u M ,n , ψ n ] + R ∗ [ u M ∗ ,n , ψ n ] . Finally, with the specific choices θ ≡ A and ψ ≡ ζ ( t ) , where ζ ∈ D ([0 , T )) is a nonnegative, nonincreasing function with ζ ( t ) ≡ for small t , we have with B ( z ) = R z A ( s ) ds N − X n =1 ∆ t hh B ( u T ,n ) , ζ T , ( n +1) − ζ T ,n ∆ t ii + hh B ( u T , ) , T ii + N X n =1 ∆ t hh P T ( S ∆ t [ S ]) n , w T ,n ζ T ,n ii ≥ N X n =1 ∆ t nn k (cid:0) ∇ T w T ,n (cid:1) ∇ T w T ,n , ∇ T w T ,n ζ T ,n oo . (62) Proof.
Inequality (60) follows by an application of Lemma 4.2, Proposition 4.3,Proposition 4.2 and Lemma 4.1. Note that in (60), we have neglected the positiveterms I θ [ u M ,n , ψ n ], I ∗ θ [ u M ∗ ,n , ψ n ]. In (61) the corresponding terms are zero because θ ≡
1, and we use the equality of Proposition 4.1 instead of the inequality ofProposition 4.2. Also notice that the term with ψ T ,N in Lemma 4.2 disappearsbecause ∆ t is small and ψ vanishes in a neighborhood of t = T . Finally, in (62) wehave treated A ( u T , ∆ t ) ζ T , ∆ t as a mere test function by applying Proposition 4.1 onthe right-hand side, but we have used Lemma 4.2, Proposition 4.3 and the choiceof the constant in x function ψ T , ∆ t to deal with the remaining terms. (cid:3) Control of the remainder terms in Proposition 6.3.
For all ψ ∈ D ( Q ),the terms on the right-hand side of (60),(61) coming from the penalization operatorvanish as h →
0. Indeed, using the estimates of Proposition 6.1(i),(ii), the Cauchy-Schwarz inequality, Proposition 5.1(iv), and the boundedness of θ on [ − M, M ], weobtain (cid:12)(cid:12)(cid:12) N X n =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ θ ( u K ) ( w n K − w n K ∗ )( ψ n K − ψ n K ∗ )size( T ) (cid:12)(cid:12)(cid:12) ≤ C N X n =1 ∆ t hh P T [ w T ,n ] , w T ,n ii! / N X n =1 ∆ t X K ∈ M , K ∗ ∈ M ∗ m K ∩ K ∗ | ψ n K − ψ n K ∗ | size( T ) / ≤ C k ∇ ψ k L ∞ × size( T ) . Let us show that the terms R θ [ u M , ψ ] , R ∗ θ [ u M ∗ , ψ ] in (60),(61) (which are definedin Proposition 4.3) vanish as h →
0. This holds true thanks to their upper boundsin terms the quantities I Id [ u M , I ∗ Id [ u M ∗ , Proposition 6.4.
Let g K , L ∈ C ( R ) be a function with properties (28) (a),(d). For a, b ∈ R , consider I K | L Id ( a, b ) = Z ba ( g K , L ( s, s ) − g K , L ( a, b )) ds,R K | LK ( a, b ) = | g K , L ( a, a ) − g K , L ( a, b ) | , R K | LL ( a, b ) = | g K , L ( b, b ) − g K , L ( a, b ) | . There exists a continuous strictly increasing convex function Π M : R + → R + that only depends on C and ω M ( · ) in (28) (d) such that Π M (0) = 0 , Π ′ M (0) = 0 andthe following bounds hold: (63) ( R K | LK ( a, b ) ≤ Π − M ( I K | L Id ( a, b )) ,R K | LL ( a, b ) ≤ Π − M ( I K | L Id ( a, b )) , for all a, b ∈ [ − M, M ] . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 39
The proof is based upon the following generalization of [48, Lemma 4.5].
Lemma 6.1.
Let g ∈ C ([ a, b ]) be a nondecreasing function equipped with a modulusof continuity ω . Then Z ba ( g ( s ) − g ( a )) ds ≥ Z g ( b ) − g ( a )0 ω − ( r ) dr. Proof.
Set δ = ω − ( g ( b ) − g ( a )). Since | g ( b ) − g ( s ) | ≤ ω ( b − s ) and g is nondecreasing,we have g ( s ) ≥ (cid:26) g ( b ) − ω ( b − s ) , b − δ ≤ s ≤ bg ( a ) , a ≤ s ≤ b − δ. Hence setting z = b − s , integrating by parts, and setting r = ω ( z ), we deduce Z ba ( g ( s ) − g ( a )) ds ≥ Z bb − δ (cid:0) g ( b ) − g ( a ) − ω ( b − s ) (cid:1) ds = δω ( δ ) − Z δ ω ( z ) dz = Z δ z dω ( z ) = Z ω ( δ )0 ω − ( r ) dr. (cid:3) Proof of Proposition . Consider the case a ≤ b . By (28)(a), we have I K | L Id ( a, b ) = Z ba ( g K , L ( s, s ) − g K , L ( a, b )) ds ≥ Z ba ( g K , L ( s, b ) − g K , L ( a, b )) ds ;applying Lemma 6.1 to g ( · ) = g K , L ( · , b ) and recalling (28)(d), we deduce I K | L Id ( a, b ) ≥ Z g K,L ( b,b ) − g K,L ( a,b )0 ( Cω M ) − ( r ) dr = Z R K | LL ( a,b )0 ( Cω M ) − ( r ) dr. Thus in order to estimate R K | LL ( a, b ) as in (63), it is sufficient to take the functionΠ M : R ∈ R + Z R ( Cω M ) − ( r ) dr . Clearly, Π M is continuous, strictly increasing,convex, Π M (0) = 0, and Π ′ M (0) = 0.The other estimate in (63) is obtained in the same way, and the case a > b isobtained by symmetry. (cid:3) Corollary 6.1. (i) Consider I Id [ u M , defined as in (45) , (46) with θ = Id , and ψ ≡ . For general nondecreasing θ ( · ) and general ψ ∈ D (Ω) , consider R θ [ u M , ψ ] defined in (47) , (48) . Assume k u M k ∞ ≤ M . Let Π M be the function given inProposition . Let Π ∗ M be the conjugate convex function of Π M . Then (cid:12)(cid:12) R θ [ u M , ψ ] (cid:12)(cid:12) ≤ k θ k C ([ − M,M ]) inf α> (cid:18) size ( T ) α I Id [ u M ,
1] + Cα Π ∗ M (cid:16) α max K ∈ M , L ∈ N ( K ) | ψ K − ψ K | L | d K , K | L (cid:17)(cid:19) , (64) where C depends on reg( T ) , d and Ω .(ii) Assume we are given a sequence of meshes T with size ( T ) → and time steps ∆ t → . Let u T , ∆ t be the corresponding discrete functions such that k u M , ∆ t k ∞ ≤ M and P Nn =1 ∆ t I Id [ u M ,n , ≤ C uniformly in T , ∆ t . Choose ψ ∈ D ( Q ) and take ψ n = ( S ∆ t [ ψ ]) n . Then P Nn =1 ∆ t R ∗ θ [ u T ,n , ψ n ] → as size ( T ) → .Analogous statements that involve P Nn =1 ∆ t I ∗ Id [ u M ∗ , and ψ K ∗ , ψ K ∗| L ∗ with K ∗ ∈ M ∗ , L ∗ ∈ N ∗ ( K ∗ ) hold for P Nn =1 ∆ t R ∗ θ [ u M ∗ , ψ ] . Proof. (i) By (47) and Proposition 6.4, for all α > (cid:12)(cid:12) R θ [ u M , ψ ] (cid:12)(cid:12) ≤ k θ k C ([ − M,M ]) X K ∈ M , L ∈ N ( K ) (cid:18) α m K | L d K , K | L (cid:19) × Π − M ( I K | L Id ) × (cid:18) α | ψ K − ψ K | L | d K , K | L (cid:19) . Note that d KL ≤ size( T ). Further, even in the case the diamonds are not necessarilyconvex, the definition of reg( T ) permits to control the multiplicity of the coveringof Ω by the convex envelopes of K and K | L , K ∈ M , L ∈ N ( K ). Thus one can upperbound P K ∈ M , L ∈ N ( K ) m K | L d K , K | L by C (reg( T ) , d ) m Ω . Applying the inequality r s ≤ Π M ( r ) + Π ∗ M ( s ) on the right-hand side above, we deduce (64).(ii) First notice that for all ψ ∈ D ( Q ), there exists C > n =1 ,...,N, K ∈ M , L ∈ N ( K ) | ψ n K − ψ n K | L | d K , K | L ≤ C, for all h > . Applying (i) for each n and summing over n = 1 , . . . , N , we get N X n =1 ∆ t R ∗ θ [ u T ,n , ψ n ] ≤ C inf α> size( T ) α N X n =1 ∆ t I Id [ u M ,
1] + T α Π ∗ M (cid:0) Cα ) ! ≤ C inf α> (cid:18) size( T ) α + 1 α Π ∗ M (cid:0) Cα ) (cid:19) , (65)where C stands for a generic constant independent of h .We have (Π M ) ′ (0) = 0. Therefore(Π ∗ M ) ′ (0) = lim b → inf a (cid:18) a − Π M ( a ) b (cid:19) ≤ lim b → (cid:18) b − Π M ( b ) b (cid:19) = 0 . Hence for all
C >
0, lim α → α Π ∗ M ( Cα ) = 0. We deduce that the right-hand side of(65) tends to zero as size( T ) → (cid:3) Remark 6.1.
Notice that if f is locally Lipschitz continuous, both Π M and Π ∗ M are quadratic; thus we can bound (cid:12)(cid:12) R θ [ u T , ψ ] (cid:12)(cid:12) by Const h β for all β < /
2. Usingthe H¨older inequality instead of the Young inequality, one recovers the result of[48] with β = 1 /
2. Whenever f is locally H¨older continuous of order γ ≤
1, we findΠ ∗ M ( s ) = Const s γ . It follows that (cid:12)(cid:12) R θ [ u T , ψ ] (cid:12)(cid:12) ≤ Const h β with β = γγ +1 , underthe assumptions of Corollary 6.1(ii).6.5. Approximate continuous entropy inequalities.
Relying on Proposition6.3, we now deduce the limiting (as h →
0) entropy inequalities and the limitingweak formulation; one should notice that they continue to hold if we replace ( η ± c , q ± c )by regular “boundary” entropy-entropy flux pairs ( η ± c,ε , q ± c,ε ). Proposition 6.5.
Consider a family of double meshes T of Ω and associated timesteps ∆ t > , parametrized by h = max { size ( T ) , ∆ t } , h → . Assume that reg( T ) isuniformly bounded. Denote the corresponding discrete solutions of (33) , (34) , (35) by u T , ∆ t . Fix ψ ∈ D ([0 , T ) × Ω) , ψ ≥ , and set ψ T , ∆ t = P T ◦ S ∆ t [ ψ ] . Fix θ asone of the functions η ± c , c ∈ R . Assume either ( c, ψ ) ∈ R ± × D ([0 , T ) × Ω) , or V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 41 ( c, ψ ) ∈ R × D ([0 , T ) × Ω) . Then lim inf h → (cid:18) Z Q d (cid:16) η ± c ( u M , ∆ t ) + ( d − η ± c ( u M ∗ , ∆ t ) (cid:17) ∂ t ψ + Z Q d (cid:16) q ± c ( u M , ∆ t ) + ( d − q ± c ( u M ∗ , ∆ t ) (cid:17) · ∇ ψ − Z Q k ( ∇ T w T , ∆ t ) ∇ T e A ( η ± c ) ′ ( w T , ∆ t ) · ∇ ψ + Z Ω d (cid:16) η ± c ( u M , ) + ( d − η ± c ( u M ∗ , ) (cid:17) ψ (0 , · )+ Z Q d (cid:16) ( η ± c ) ′ ( u M , ∆ t ) + ( d −
1) ( η ± c ) ′ ( u M ∗ , ∆ t ) (cid:17) S ψ (cid:19) ≥ . (66) Furthermore, if ψ ∈ D ([0 , T ) × Ω) , we have lim h → (cid:18) Z Q d (cid:16) u M , ∆ t + ( d − u M ∗ , ∆ t (cid:17) ∂ t ψ + Z Q (cid:18) d (cid:16) f ( u M , ∆ t ) + ( d − f ( u M ∗ , ∆ t ) (cid:17) − k ( ∇ T w T , ∆ t ) ∇ T w T , ∆ t (cid:19) · ∇ ψ + Z Ω d (cid:16) u M , + ( d − u M ∗ , (cid:17) ψ (0 , · ) + Z Q S ψ (cid:19) = 0 . (67) Proof.
By the choice of ( c, ψ ), (38) holds. Thus, by Proposition 6.3, (60) and (61)hold; it suffices to develop these formulas using the definitions of hh · , · ii , nn · , · oo .The second term in (60) rewrites exactly as the corresponding term in (66).Regarding the other terms on the left-hand side, we also use the uniform boundon u T ,n in L ∞ , the uniform bound on k ( ∇ T w T , ∆ t ) ∇ T w T , ∆ t in ( L p ′ ( Q )) d , and theconvergences N X n =1 X K ∈ M ψ ( n +1) K − ψ n K ∆ t Q nK → ∂ t ψ, N X n =1 X K ∈ M ∗ ψ ( n +1) K ∗ − ψ n K ∗ ∆ t Q nK ∗ → ∂ t ψ in L ( Q ) , N X n =1 X K ∈ M ( P T ( S ∆ t [ S ]) n ) K Q nK → S , N X n =1 X K ∗ ∈ M ∗ ( P T ( S ∆ t [ S ]) n ) K ∗ Q nK ∗ → S in L ( Q ) , N X n =1 X K ∈ M ψ n K Q nK → ψ, N X n =1 X K ∗ ∈ M ∗ ψ n K ∗ Q nK ∗ → ψ in L ∞ ( Q ) , ∇ T ψ T , ∆ t → ∇ ψ in L p ( Q ) and ψ M , ( · ) → ψ (0 , · ) , ψ M ∗ , ( · ) → ψ (0 , · ) in L (Ω) , as h → h →
0, thanks to the initial remarks made inSubsection 6.4 and Corollary 6.1(ii). In the same way, (67) follows from (61). (cid:3) Convergence and statement of main result
We are now in a position to state and prove the main result of this paper.
Theorem 7.1.
Consider a family of double meshes T of Ω and associated time steps ∆ t > , parametrized by h = max { size ( T ) , ∆ t } , h → . Assume that reg( T ) is uni-formly bounded. Then the corresponding discrete solutions u T , ∆ t of (33) , (34) , (35) exist, are uniformly bounded, and converge to the unique entropy solution u of (1) in the following strong sense: u M , ∆ t → u , u M ∗ , ∆ t → u in L s ( Q ) for any s < ∞ , ∇ T w T , ∆ t → ∇ w in L p ( Q ) , where w = A ( u ) . Proof.
We follow step by step the proof of Theorem 2.1. (i)
Discrete solutions u T , ∆ t exist by Proposition 6.2. Besides, they verify theasymptotic entropy inequalities (66) (where we can replace η ± c by η ± c,ε ) and theasymptotic weak formulation (67), both of Proposition 6.5. (ii) Proposition 6.1 yields uniform estimates on both u M , ∆ t and u M ∗ , ∆ t in L ∞ ( Q ); on the time translates of both w M , ∆ t and w M ∗ , ∆ t in L ( Q ); on the pe-nalization term P Nn =1 ∆ t hh P T [ w T ,n ] , w T ,n ii ; and on ∇ T w T , ∆ t in L p ( Q ). The latterestimate implies further uniform estimates: namely, an estimate of the space trans-lates of both w M , ∆ t and w M ∗ , ∆ t in L ( Q ), by Proposition 5.2 (ii); an estimate of ∇ T e A ( η ± c,ε ) ′ ( w T , ∆ t ) in L p ( Q ), because e A ( η ± c,ε ) ′ ( · ) is Lipschitz and by construction of ∇ T [ · ]; and finally an estimate of a ( ∇ T w T , ∆ t ) in L p ′ ( Q ), because of the growthassumption on a . (iii) Thanks to the estimates of (ii) , there exists a (not labelled) sequence of T , ∆ t with h → • by the Fr´echet-Kolmogorov theorem, each of the sequences w M , ∆ t and w M ∗ , ∆ t converges strongly in L ( Q ) and pointwise a.e. in Q ; • by Proposition 5.3, the limits of w M , ∆ t and w M ∗ , ∆ t coincide (we denote thelimit of w M , ∆ t , w M ∗ , ∆ t by w ), and ∇ T w T , ∆ t converges weakly in L p ( Q ) to ∇ w ; • a ( ∇ T w T , ∆ t ) converges weakly in L p ′ ( Q ) to a limit field χ ; • the sequences u M , ∆ t , u M ∗ , ∆ t converge to µ, µ ∗ : Q × (0 , → R , respectively,in the sense of nonlinear L ∞ weak- ⋆ convergence (5). Also by (5), the functions w M , ∆ t = A ( u M , ∆ t ) converge to A ( µ ) in the L ∞ weak- ⋆ sense; since the functions w M , ∆ t also converge strongly, A ( µ ) is independent of α and coincides with w . Inthe same way, we deduce that A ( µ ∗ ) is independent of α and coincides with w .Also observe that u M , , u M ∗ , both converge to u a.e. in Ω and in L (Ω). (iv) As in the proof of Theorem 2.1, we can use the chain rule and the Green-Gauss formula to deduce Z Q Z d ( f ( µ ) + ( d − f ( µ ∗ )) · ∇ A ( u )= 1 d Z Z Q ( f ( µ ) · ∇ A ( µ ) + ( d − f ( µ ∗ ) · ∇ A ( µ ∗ )) = Z T Z ∂ Ω e A f ( w ) · n = 0 , (68)where e A f is defined in (15). (v) Next we pass to the limit in (67). Indeed, by (iii) ,(69) ∂ t e u + div Z d ( f ( µ ) + ( d − f ( µ ∗ )) dα = div χ + S in L p ′ (0 , T ; W − ,p ′ (Ω))+ L ( Q ) , e u | t =0 = u . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 43 where e u ( t, x ) = Z e µ ( t, x, α ) dα, e µ = 1 d ( µ + ( d − µ ∗ ) . Let us identify χ (the weak limit of a ( ∇ T w T , ∆ t )) with a ( ∇ w ), and consequentlyobtain that the weak convergence is in fact strong in L p ( Q ). To this end, we willuse (iv) and (62) to establish the inequality(70) Z Q χ · ∇ w ≥ lim inf h → N X n =1 ∆ t nn a ( ∇ T w T ,n ) , ∇ T w T ,n oo . Indeed, using (69), we can represent the left-hand side of (70) as Z Q χ · ∇ w ζ = − Z T h ∂ t e u, w ζ i + Z Z Q d (cid:16) f ( µ ) + ( d − f ( µ ∗ ) (cid:17) · ∇ w ζ + Z Q S w ζ, (71)where ζ ∈ D ([0 , T )) is nonincreasing with ζ ( t ) ≡ t small.Note that since A is nondecreasing, since e u ( t, x ) is a convex combination of thevalues µ ( t, x, · ) and µ ∗ ( t, x, · ), and because A ( µ ) = A ( µ ∗ ) = w , we conclude that w = A ( e u ) . To control R T h ∂ t e u, w ζ i , we argue along the lines of the proof of Theorem 2.1.The duality product h ∂ t e u, A ( e u ) i is treated via the weak chain rule (cf. [4]). Hence,exploiting also the convexity of B ( z ) = R z A ( s ) ds , Z T h ∂ t e u, A ( e u ) ζ i = − Z Q B ( e u ) ζ ′ − Z Ω B ( u )= Z Q B (cid:18)Z e µ ( t, x, α ) dα (cid:19) ( − ζ ′ ) − Z Ω B ( u ) ≤ − Z Q ζ ′ Z B ( e µ ( t, x, α )) dα − Z Ω B ( u ) . (72)Using (68) and (72), we deduce from (71) that(73) Z Q χ · ∇ w ζ ≥ Z Q ζ ′ Z B ( e µ ( t, x, α )) dα + Z Ω B ( u ) + Z Q S w. On the other hand, Proposition 6.3 permits to evaluate the right-hand side of(70) as follows:lim inf h → N X n =1 ∆ t nn(cid:0) a ( ∇ T w T ,n ) , ∇ T w T ,n ζ T ,n oo! ≤ lim inf h → N − X n =1 ∆ t hh B ( u T ,n ) , ζ T , ( n +1) − ζ T ,n ∆ t ii + hh B ( u T , ) , T ii + N X n =1 ∆ t hh P T ( S ∆ t [ S ]) n , w T ,n ζ T ,n ii(cid:19) (74) By the previously established convergences (see also the proof of Proposition 6.5),the right-hand side of (74) is equal to the right-hand side of (73). Once we let ζ tend to 1l [0 ,T ) , this establishes (70).Starting from (70), we apply the Minty-Browder argument that we employed forthe continuous problem in the proof of Theorem 2.1.Take v ∈ L p (0 , T ; W ,p (Ω)) ∩ L ∞ ( Q ), and set v T , ∆ t = P T ◦ S ∆ t [ v ]. In viewof (70), taking into account the strong convergence of ∇ T v T , ∆ t to ∇ v in L p ( Q ),cf. Proposition 5.1, and the monotonicity of a ( · ) we obtain Z Q χ · ∇ ( w − v ) ≥ lim inf h → N X n =1 ∆ t nn a ( ∇ T w T ,n ) , ∇ T w T ,n − ∇ T v T ,n oo ≥ lim inf h → N X n =1 ∆ t nn a ( ∇ T v T ,n ) , ∇ T w T ,n − ∇ T v T ,n oo . (75)As is a well-known property of Leray-Lions operators, the strong convergence of ∇ T v T , ∆ t to ∇ v in L p ( Q ) implies the strong convergence of a ( ∇ T v T , ∆ t ) to a ( ∇ v )in L p ′ ( Q ). Therefore (75) yields Z Q χ · ∇ ( w − v ) ≥ Z Q a ( ∇ v ) · ∇ ( w − v ) . Choosing v = w ± λψ with λ ↓ ψ ∈ L p (0 , T ; W ,p (Ω)), we conclude χ = a ( ∇ w ) . Moreover, as in the proof of Theorem 2.1 and [11, Theorem 5.1], relying on thestrict monotonicity of a and utilizing an argument of [24, 21], we also deduce thestrong convergence of ∇ T w T , ∆ t to ∇ w in L p ( Q ). (vi) Now we can pass to the limit in the weak and entropy formulations listedin Proposition 6.5. The passage from (67) to ( D’ .2) is straightforward. In (66),we first work with regularized boundary entropies. Taking the limit, all the termsconverge to the corresponding terms in ( D’ .3) in a straightforward way, except forthe third one. Let us show that ∇ T e A ( η ± c,ε ) ′ ( w T , ∆ t ) converges weakly to ∇ e A ( η ± c,ε ) ′ ( w ) in L p ( Q ).Indeed, both e A ( η ± c,ε ) ′ ( w M , ∆ t ) and e A ( η ± c,ε ) ′ ( w M ∗ , ∆ t ) converge to e A ( η ± c,ε ) ′ ( w ) by thea.e. convergence of w M , ∆ t , w M ∗ , ∆ t to w and the continuity of e A ( η ± c,ε ) ′ . Using theboundedness in L p ( Q ) of ∇ T e A ( η ± c,ε ) ′ ( w T , ∆ t )) and the compactness property ofProposition 5.3, we conclude that our claim holds. The subsequent argumentsare the same as in the proof of Theorem 2.1. (vii) We conclude that ( µ, µ ∗ , w ) is an entropy double-process solution of (1).In view of Theorem 2.2, this brings to an end the proof of Theorem 7.1; indeed,we obtain the convergence to u for each sequence of discrete solutions with h → µ and µ ∗ turn out to be independent of α means that theconvergence of u M , ∆ t , u M , ∆ t to u is strong in L s ( Q ) for all finite s . (cid:3) On the choice of FV scheme and various generalizations
In this section we discuss other possible choices of finite volume schemes for (1).
V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 45 • The use of DDFV schemes is motivated by their convenience when it comes tothe discretization of nonlinear diffusion operators. Other possibilities exist; amongthem, let us mention the schemes studied in [56] (see also [3, 12, 8]), in [10], in [42](see also [47]), and in [52] (see also [50, 51, 53]). All these schemes possess somevariant of the “integration-by-parts” property of Proposition 4.1.The 2D schemes of [10] are restricted to Cartesian meshes, so they do not allowfor domains much more general than rectangles. Notice that their generalization to3D appears to be straightforward. The techniques used in the present paper andin the references we cite, such as [48, 49], combined with those of [10], allow todesign rather simple FV schemes on Cartesian meshes for problem (1) and to provetheir convergence. In this case, the notion of entropy double-process solution is notneeded, and the theoretical results in [49] can be adapted directly.This is also the case of the “complementary volumes” schemes as described in[56]. In 2D, ideas quite similar to that of [56] were used to construct the schemesof [3, 12, 8]. All these schemes work on meshes dual to conformal triangular 2Dmeshes, and the discrete gradient is reconstituted by affine per triangle interpola-tion. “Complementary volumes” schemes are simpler than our DDFV scheme fromthe practical point of view, since one discretizes the problem on the same mesh M ∗ using, roughly speaking, half of the unknowns. The discrete duality propertiesfor the 2D “complementary volumes” scheme are shown in the same way as forour DDFV schemes; the proof is based upon Lemma 9.6 (see Appendix B and also[7, 8]). Unfortunately, the straightforward generalization of these “complementaryvolumes” schemes to 3D fails to satisfy the discrete duality property, except forvery constrained geometries of the meshes (see Remark 9.2).The key feature of the 2D schemes of [3, 56, 12, 10, 8] (see also [40]) lies in the factthat the fluxes across interfaces are reconstructed “manually”. The approaches ofDroniou and Eymard [47, 42] and those of the HVF, SUCCES and SUSHI schemes ofEymard, Gallou¨et and Herbin [50, 51, 53, 52] are different; they rely on introducingadditional unknowns (either for the fluxes, or for the values on some of the edges)and on careful penalization of the finite differences.The schemes HVF, SUCCES and SUSHI (among many others) were designed forhandling linear anisotropic, heterogeneous diffusion problems with possibly discon-tinuous coefficients; in this framework, their convergence is justified. These schemesavoid usage of double meshes and thus may have less unknowns; they work both in2D and 3D. We refer to Eymard, Gallou¨et and Herbin [52] for the description andcomparison of these and related (e.g., mimetic finite difference) schemes. Finally,let us also mention the schemes of Aavatsmark et al. (see, e.g., [1, 2]), that arein a sense intermediate. The gradient reconstruction used in [1, 2] also involvesadditional edge unknowns, which are eliminated by solving, locally, an algebraicsystem of equations.The scheme of [42] designed for nonlinear Leray-Lions kind problems can bedirectly compared to the DDFV schemes of [11] and of the present paper. Thescheme of [42] is very interesting because of the extreme generality of the geometriesallowed for the mesh (and it works in any space dimension). For this same reason,theoretical justification of its convergence in the hyperbolic-parabolic framework(1) seems problematic. Indeed, the conformity (orthogonality) condition was usedin an essential way in the derivation of the discrete entropy inequalities (see Remark • Our assumption that M consists of simplexes is a practical one simplifyingthe presentation of the scheme. In 2D, it can be replaced by the more generalassumption that any element of M admits a circumscribed circle. In 3D, we canassume that each K ∈ M admits a circumscribed ball, and each interface K | L is atriangle satisfying (19).Notice that Remark 9.3 (see also [7]) makes it possible to define a consistentdiscrete duality scheme even when the interfaces K | L are not necessarily triangles.Unfortunately, the discrete Poincar´e inequality may fail in this generality; thisundermines the subsequent convergence analysis. Yet one interesting case is thatof a Cartesian mesh M ; the corresponding DDFV schemes are alternatives to thescheme of [10] discussed above. More generally, one can start with a mesh M madeof rectangles (e.g., inside Ω) and triangles (e.g., near the boundary ∂ Ω) in 2D. • As pointed out in Remark 3.5, a different kind of reconstruction formula isneeded for problems in 4D and higher dimensions. It would be interesting to con-ceive discrete gradients consistent with affine functions, following the principle for-mulated in Remark 3.3. One natural way is indicated in [38]. • The choice of penalization in our double scheme can be changed (see Remark3.6). One could also penalize the differences ( u K − u K ∗ ) instead of the differences( w K − w K ∗ ) = ( A ( u K ) − A ( u K ∗ )); this would permit to avoid the use of double-process solutions. But this choice would introduce additional coupling between thesets of variables ( u K ) K ∈ M and ( u K ∗ ) K ∗ ∈ M ∗ in the “hyperbolic” regions. Indeed,if, e.g., A ( u ) ≡
0, there is no coupling at all between the variables sitting on M and those sitting on M ∗ . Therefore our choice seems more convenient in terms ofpractical implementation. • Convection-diffusion problems with anisotropic linear and nonlinear diffusionwere considered in [35, 34] and in [17, 18]. General DDFV schemes do not seemeasy to adapt to the nonlinear anisotropic framework, because of the presence of“privileged” directions of diffusion. In this case, the schemes of [10] on Cartesianmeshes constitute a natural choice, and the geometry of ∂ Ω should be rather takeninto account via the approximation of the domain Ω by domains with piecewiseaxes-aligned boundaries. Notice that for the anisotropic p − Laplace kind diffusions ∂ x (cid:0) | ∂ x A ( u ) | p − ∂ x A ( u ) (cid:1) + ∂ x (cid:0) | ∂ x A ( u ) | p − ∂ x A ( u ) (cid:1) considered by Bendahmane, Karlsen in [17, 18], the discrete entropy inequalitieson Cartesian meshes are as easy to obtain as for the isotropic case a ( ξ ) = k ( ξ ) ξ considered in the present paper. • Taking into account sufficiently smooth dependencies on ( t, x ) of the convectionand diffusion operators is possible, although quite technical; see [48, 11] for some
V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 47 results in that direction, and also [33, 60] for well-posedness results for degener-ate equations with ( t, x ) dependent convection-diffusion operators. Discontinuouscoefficients are important for the modeling of fractured media. DDFV schemesfor Leray-Lions operators div a ( x, ∇ w ) with discontinuous (piecewise smooth) in x nonlinearity a are studied in the recent work [23]. The case of x -discontinuous fluxfunctions f ( x, u ) has received much attention in the last fifteen years (see, e.g., [26]and the references cited therein), both from a theoretical and numerical perspective.Let us mention here that the problem of the choice of the appropriate entropy con-ditions strongly depends on the underlying physical interpretation; different modelslead to qualitatively different admissible solutions. • Inhomogeneous Dirichlet boundary conditions can be taken into account, com-bining the techniques of [67] with those of [11]; both are rather involved, whichexplains our choice of the homogeneous boundary data for the presentation of thescheme and the convergence arguments.
Appendix A: Proof of uniqueness
This appendix is devoted to a proof of Theorem 2.2. The proof is an adaptationof the ones in Carrillo [29] (for entropy solutions) and that in Eymard, Gallou¨et,Herbin, Michel [49] (for entropy process solutions, which can be viewed as entropydouble-process solutions with µ ≡ µ ∗ ). The proof is mainly divided into severallemmas (Lemmas 9.1, 9.3, 9.5 below). For simplicity, let us only consider the casewhere the source term S is zero (see also Remark 9.1).We begin by introducing the set E = { r ∈ R : A − ( r ) is neither empty nor a singleton } , and proving Lemma 9.1.
Let ( ν, ν ⋆ , v ) be an entropy double-process solution of (1) with initialdata v . Then for all W ∈ R d , for any φ ∈ D ([0 , T ) × Ω) , c ∈ R such that A ( c )
6∈ E and also for any φ ∈ D ([0 , T ) × Ω) , c ∈ R ± such that A ( c )
6∈ E , we have with thenotation of Section the following equality: Z Z Q (cid:20) d (cid:0) η ± c ( ν ) + ( d − η ± c ( ν ) (cid:1) ∂ t φ + 1 d (cid:0) q ± c ( ν ) + ( d − q ± c ( ν ⋆ ) (cid:1) · ∇ φ − sign ± ( v − A ( c )) (cid:0) a ( ∇ v ) − W (cid:1)(cid:1) · ∇ φ (cid:21) dx dt dα + Z Ω η ± c ( v ) φ | t =0 dx = lim ε ↓ Z Q (sign ± ε ) ′ ( v − A ( c )) (cid:0) a ( ∇ v ) − W (cid:1) · ∇ v φ dx dt. (76) Proof.
We refer to [29, Lemma 1] and to [49] for details on the proof. The idea is touse ψ := (sign ± ε ( v − A ( c )) φ ) as a test function in ( D’ .2). It is admissible; indeed,we can approximate it by functions in D ([0 , T ) × Ω) and pass to the limit in allterms of ( D’ .2), because ψ ∈ L ∞ ( Q ) ∩ L p (0 , T ; W ,p (Ω)) for any of the two possiblechoices of ( φ, c ) (in particular, notice that sign ± ε ( v − A ( c )) ∈ L p (0 , T ; W ,p (Ω)) incase c ∈ R ± ). We havesign ± ( ν − c ) = sign ± ( v − A ( c )) = sign ± ( ν ∗ − c ) , thanks to the relation ( D’ .1) (which reads A ( ν ) ≡ v ≡ A ( ν ∗ ) in our notation) andto the choice of A ( c ) / ∈ E ; then we use the weak chain rule to deal with the time derivative. We also insert into ( D’ .2) the term Z Z Q sign ± ε ( v − A ( c )) ∇ φ · W − Z Z Q (sign ± ε ) ′ ( v − A ( c )) ∇ w · W φ, which is equal to 0 = Z Q div ( W sign ± ε ( v − A ( c )) φ ) for any of the two possiblechoices of ( φ, c ), by the Gauss-Green formula. As ε ↓
0, the term containing(sign ± ε ) ′ ( v − A ( c )) (cid:0) f ( ν ) − f ( c ) (cid:1) · ∇ v vanishes, as shown in [29, Lemma 1]. (cid:3) We are now interested in comparing two entropy double-process solutions of(1), denoted by ( ν, ν ∗ , θ ) and ( µ, µ ∗ , w )), of which the first one is chosen to satisfy ν ≡ ν ∗ . Consider the distribution I on D ( Q ) defined by I [ φ ] := Z Z Z Q (cid:20) d (cid:0) ( ν − µ ) + + ( d − ν − µ ⋆ ) + (cid:1) ∂ t φ + 1 d (cid:0) sign + ( ν − µ ) ( f ( ν ) − f ( µ ))+ ( d − + ( ν − µ ⋆ ) ( f ( ν ) − f ( µ ⋆ )) (cid:1) · ∇ φ − sign + ( v − w ) (cid:0) a ( ∇ v ) − a ( ∇ w ) (cid:1) · ∇ φ (cid:21) dx dt dα dβ + Z Ω ( v − u ) + φ (0 , x ) dx. (77)Let us prove that we can write I as(78) I = IP + IN where IP [ φ ] is defined by the analogue of (77) with each of ν, v, v , µ, µ ∗ , w, u replaced by its positive part; and IN [ φ ] is defined by the analogue of (77) witheach of ν, v, v , µ, µ ∗ , w, u replaced by − ν − , − v − , − v − , − µ − , − ( µ ∗ ) − , − w − , − u − .To emphasize, whenever necessary, the dependency of I , IP , IN on the involvedsolutions, we will write I ν,ν,vµ,µ ∗ ,w , , IP ν,ν,vµ,µ ∗ ,w , IN ν,ν,vµ,µ ∗ ,w , respectively.To justify (78), we use the identity (79) from the following easy lemma. Lemma 9.2.
For all F : R → R such that F (0) = 0 , for all a, b ∈ R there holds sign + ( a − b ) ( F ( a ) − F ( b )) = sign + ( a + − b + ) ( F ( a + ) − F ( b + ))+ sign + (( − a − ) − ( − b − )) ( F ( − a − ) − F ( − b − )) , (79) and (80) (cid:12)(cid:12)(cid:12)(cid:12) ( i ) sign − ( b − a + ) F ( b ) = − sign + ( a + − b + ) F ( b + ) + sign − ( b ) F ( b ) , ( ii ) sign − ( b − a + ) F ( a + ) = − sign + ( a + − b + ) F ( a + ) . We apply (79) to a = ν , b = µ (or b = µ ∗ ) with F = Id and with F = f i , i =1 , . . . , d . Futhermore, observe that the analogue of (79) still holds for a.e. ( t, x ) ∈ Q if we take a = v ( t, x ), b = w ( t, x ) and replace F ( a ) , F ( b ) and F ( ± a ± ) , F ( ± b ± )by a ( ∇ v ), a ( ∇ w ) and by a ( ±∇ v ± ), a ( ±∇ w ± ), respectively. Indeed, we have, e.g., a ( ∇ v ) = a ( ∇ v + ) + a ( ∇ v − ) a.e. on Q , because v ∈ L p (0 , T ; W ,p (Ω)). Using allaforementioned identities, we split each term in the definition (77) of I into thesum of the corresponding terms in the definitions of IP and IN .Now we estimate I “inside” the domain”. V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 49
Lemma 9.3.
Let ( ν, ν, v ) and ( µ, µ ⋆ , w ) be entropy double-process solutions of (1) with data v , u ∈ L ∞ (Ω) , respectively. Then I [ φ ] ≥ , ∀ φ ∈ D ([0 , T ) × Ω) , φ ≥ .Proof. The proof is an application of the doubling of variables method of Kruzhkov[63]; it follows [29, 49, 17]. We let ν depend on variables ( t, x, α ) ∈ Q × (0 , µ depend on another set of variables ( s, y, β ) ∈ Q × (0 , ∇ v means ∇ x v and ∇ w means ∇ y w . As to the test function φ , it will dependon the variables ( t, s, x, y ), thus we will use the notations ∂ t , ∂ s and ∇ x , ∇ y forthe corresponding derivatives of φ . We will work with nonnegative test functions φ ∈ D (cid:0) ([0 , T ) × Ω) (cid:1) . Let us introduce the sets on which the diffusion term for thefirst, respectively, for the second solution degenerates: E ν = { ( t, x ) ∈ Q | v ( t, x ) ∈ E} , E µ = { ( s, y ) ∈ Q | w ( s, y ) ∈ E} . Denote by E cν , E cµ the complementary sets in Q of E ν , E µ , respectively. Observe that ∇ v = 0 a.e. in E ν and ∇ w = 0 a.e. in E µ (recall ( D’ .1)). (i) First we apply Lemma 9.1 with the solution ( ν, ν, v ). For all ( s, y, β ) ∈E µ × (0 , W = a ( ∇ w ( s, y )) and take the entropy η + c ( · ) = ( · − c ) + with c = µ ( s, y, β ), then with c = µ ⋆ ( s, y, β ) in (76). We multiply the two resultingequations by d and by ( d − d , respectively, and add them together. Then we inte-grate in ( s, y, β ) ∈ E µ × (0 , s, y, β ) ∈ E cµ × (0 , d and ( d − d , respectively, the entropy inequalities ( D’ .3) for ( ν, ν, v )corresponding to η + c ( · ) with c = µ ( s, y, β ) and with c = µ ⋆ ( s, y, β ). We integratethe resulting inequality in ( s, y, β ) ∈ E cµ × (0 , (ii) Next, we exchange the roles of ( ν, ν, v ) and ( µ, µ ∗ , w ). This time we usethe entropy η − c ( · ) = ( · − c ) − ; we use W = a ( ∇ v ( t, x )); and we only use one value c = ν ( t, x, α ) in the analogue of (76) (for all ( t, x, α ) ∈ E ν × (0 , D’ .3) (for all ( t, x, α ) ∈ E cν × (0 , (iii) Adding the inequalities obtained in (i),(ii) , by the symmetry of the expres-sions involved (such as ( ν − µ ) + = ( µ − ν ) − , etc.), we get, keeping in mind Remark2.1, the following inequality: Z Z Z Z Q × Q (cid:20) d (cid:0) ( ν − µ ) + + ( d − ν − µ ⋆ ) + (cid:1) ( ∂ t + ∂ s ) φ + 1 d (cid:0) sign + ( ν − µ ) ( f ( ν ) − f ( µ ))+ ( d − + ( ν − µ ⋆ ) ( f ( ν ) − f ( µ ⋆ )) (cid:1) ( ∇ x + ∇ y ) φ − sign + ( v − w ) (cid:0) a ( ∇ v ) − a ( ∇ w ) (cid:1) · (cid:0) ∇ x + ∇ y (cid:1) φ (cid:21) dx dt dy ds dα dβ + Z Z Z Ω × Q d (cid:0) ( v − µ ) + + ( d − v − µ ⋆ ) + (cid:1) φ dx ( dy ds ) dβ + Z Z Z Q × Ω ( ν − u ) + φ ( dx dt ) dy dα ≥ lim ε ↓ Z Z E cν ×E cµ (cid:0) sign + ε (cid:1) ′ ( v − w ) (cid:0) a ( ∇ v ) − a ( ∇ w ) (cid:1) · (cid:0) ∇ v −∇ w (cid:1) φ dx dt dy ds. (81)The last term in (81) is nonnegative, because a is monotone and φ ≥ (iv) Let us now specify the test function. For l, n ∈ N , let ω n : R d → R , ω l : R → R be standard symmetric mollifiers with supports in { x ∈ R d | k x k ≤ n } and in { t ∈ R | | t | ≤ l } , respectively. We take the test function in (81) to be φ n,l ( t, x, s, y ) = φ ( x, t ) ω n ( x − y ) ω l ( t − s ) ≡ φω n ω l , where φ ∈ D ([0 , T ) × Ω), φ ≥
0. With this choice, we have(82) ( ∂ t + ∂ s ) φ n,l = (cid:0) ∂ t φ (cid:1) ω n ω l , ( ∇ x + ∇ y ) φ n,l = (cid:0) ∇ x φ (cid:1) ω n ω l . Then we let n, l → ∞ . The first term in (81) converges to the first term in theright-hand side of (77). This argument is standard; one can use, e.g., the propertiesof the Lebesgue points of L functions and the upper-semicontinuity of the L “+-bracket” (cid:2) u, f (cid:3) + := Z Ω sign + ( u ) f + Z { u =0 } f + . The two latter terms in the left-hand side of (81) are treated with the help ofthe triangular inequality and of the strong initial trace property (83) proved inLemma 9.4 below. The limit, as n, l → ∞ , of each of these terms is majorated byone half of the last term in (77) (this is because R Q ω l ( t ) dt = = R Q ω l ( − s ) ds ).This concludes the proof of the lemma. (cid:3) Lemma 9.4.
Let ( µ, µ ⋆ , w ) be an entropy double-process solution of (1) with initialdatum u ∈ L ∞ (Ω) . Then the initial datum is also taken in the following strongsense: (83) lim h ↓ h Z h Z Ω Z (cid:16) d | µ − u | + d − d | µ ⋆ − u | (cid:17) dt dx dα = 0 . Notice that another way to formulate (83) is to say thatess lim t ↓ Z Ω Z (cid:16) d | µ − u | + d − d | µ ⋆ − u | (cid:17) dx dα = 0 , in the spirit of the original definition of Kruzhkov [63]. Proof.
The proof follows the one of Panov in [71, Proposition 1]. For c ∈ R and h >
0, consider the functions p h ( · ; c ) : x ∈ Ω h Z h Z (cid:16) d | µ ( t, x ; α ) − c | + d − d | µ ⋆ ( t, x ; α ) − c | (cid:17) dt dα. Because µ, µ ∗ are bounded, the set (cid:0) p h ( · ; c ) (cid:1) h> is bounded in L ∞ (Ω). Thereforefor any sequence h n →
0, there exists a subsequence (not relabeled) such that forall c ∈ Q , (cid:0) p h ( · ; c ) (cid:1) h> converges in L ∞ (Ω) weak- ⋆ to some limit denoted by p ( · ; c ).Fix ξ ∈ D (Ω), ξ ≥
0. From Definition 2.3, taking in ( D’ .2) test functionsapproaching ψ ( t, x ) := (cid:0) − th n (cid:1) + ξ ( x ) we readily infer the inequalities(84) ∀ c ∈ Q Z Ω p ( x ; c ) ξ ( x ) dx ≤ Z Ω | u ( x ) − c | ξ ( x ) dx. By the density argument, we extend (84) to all ξ ∈ L (Ω), ξ ≥ δ >
0, there exists a number N ( δ ) ∈ N , a collection ( c δi ) N ( δ ) i =1 ⊂ Q and a partition of Ω into disjoint union of measurable sets Ω δ , . . . , Ω δN ( δ ) such that k u − u δ k L ≤ δ , where u δ := X N ( δ ) i =1 c δi Ω δi . V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 51
Because 1l Ω = P N ( δ ) i =1 Ω δi , applying (84) with c = c δi and ξ = 1l Ω δi we deducelim n →∞ h n Z h n Z Ω Z (cid:16) d | µ − u δ | + d − d | µ ⋆ − u δ | (cid:17) dt dx dα = lim n →∞ Z Ω X N ( δ ) i =1 p h n ( x ; c δi )1l Ω δi ( x ) dx = Z Ω X N ( δ ) i =1 p ( x ; c δi )1l Ω δi ( x ) dx ≤ Z Ω X N ( δ ) i =1 | u ( x ) − c δi | Ω δi ( x ) dx = (cid:13)(cid:13) u − u δ (cid:13)(cid:13) L ≤ δ. Using once more the bound k u − u δ k L ≤ δ (in the first term of the previouscalculation), we can send δ to zero and infer the analogue of (83), with a limittaken on some subsequence of ( h n ) n> . Because ( h n ) n> was an arbitrary sequenceconvergent to zero, (83) is justified. (cid:3) Lemma 9.3 tells us that I [ · ], which is a distribution on [0 , T ) × Ω, is nonnegativewhen restricted to D ([0 , T ) × Ω) and thus it is a locally finite measure on [0 , T ) × Ω.Now we show that I [ φ ] is nonnegative also for nonnegative test functions φ thatdo not necessarily vanish on the boundary [0 , T ) × ∂ Ω. Lemma 9.5.
Let ( ν, ν, v ) , ( µ, µ ⋆ , w ) be entropy double-process solutions of (1) withinitial data v , u ∈ L ∞ (Ω) , respectively. Then I [ φ ] ≥ , ∀ φ ∈ D ([0 , T ) × Ω) , φ ≥ .Proof. We begin by modifying steps (i)-(iv) of the proof of the previous lemma;we refer to this proof for the notation and a part of the calculations. (i)
We use (76) and ( D’ .3) in the same way as in the proof of Lemma 9.3; but wechoose the values c = µ + ( s, y, β ), c = ( µ ⋆ ) + ( s, y, β ) and W = a ( ∇ w + ( s, y )) insteadof the values c = µ ( s, y, β ), c = µ ⋆ ( s, y, β ) and W = a ( ∇ w ( s, y )), respectively.Notice that for all a, b ∈ R , ε ≥
0, we have sign + ε ( a − b + ) = sign + ε ( a + − b + ) and,moreover, this expression is zero whenever a ≤
0. Thus we can replace ν , v , ∇ v by ν + , v + , ∇ v + everywhere in this calculation and obtain Z Z Z Z Q × Q (cid:20) d (cid:0) ( ν + − µ + ) + + ( d − ν + − ( µ ⋆ ) + ) + (cid:1) ∂ t φ + 1 d (cid:0) sign + (cid:0) ν + − µ + (cid:1) ( f ( ν + ) − f ( µ + ))+ ( d − + (cid:0) ν + − ( µ ⋆ ) + (cid:1) ( f ( ν + ) − f (( µ ⋆ ) + )) (cid:1) ·∇ x φ − sign + (cid:0) v + − w + (cid:1) (cid:0) a ( ∇ v + ) − a ( ∇ w + ) (cid:1) · ∇ x φ (cid:21) dx dt dy ds dα dβ + Z Z Z Ω × Q d (cid:0) ( v +0 − µ + ) + + ( d − v +0 − ( µ ⋆ ) + ) + (cid:1) φ dx ( dyds ) dβ ≥ lim ε ↓ Z Z E cν ×E cµ (cid:0) sign + ε (cid:1) ′ (cid:0) v + − w + (cid:1) (cid:0) a ( ∇ v + ) − a ( ∇ w + ) (cid:1) · ∇ v + φ dx dt dy ds. (85) (ii) We follow the proof of Lemma 9.3 but choose c = ν + ( t, x, α ), W = a ( ∇ v + ( t, x ))instead of c = ν ( t, x, α ), W = a ( ∇ w ( t, x )).Let us apply identities (80) to a = ν , b = µ (or b = µ ∗ ) with F = Id and with F = f i , i = 1 , . . . , d . Moreover, as in the proof of (78), we also have the analogueof (80)( i ) with a = v , b = w with F ( w ) replaced by a ( ∇ w ). In the same way, wealso have the analogue of (80)( ii ) with a = v , b = w , and F ( v ) replaced by a ( ∇ v ). Furthermore, sign ± ( · ) can be replaced by (sign ± ε ) ′ ( · ) in the above properties. Inconclusion, we obtain Z Z Z Z Q × Q (cid:20) d (cid:0) ( ν + − µ + ) + + ( d − ν + − ( µ ⋆ ) + ) + (cid:1) ∂ s φ + 1 d (cid:0) sign + (cid:0) ν + − µ + (cid:1) ( f ( ν + ) − f ( µ + ))+ ( d − + (cid:0) ν + − ( µ ⋆ ) + (cid:1) ( f ( ν + ) − f (( µ ⋆ ) + )) (cid:1) ·∇ y φ − sign + (cid:0) v + − w + (cid:1) (cid:0) a ( ∇ v + ) − a ( ∇ w + ) (cid:1) · ∇ y φ (cid:21) dx dt dy ds dα dβ + Z Z Q × Ω ( ν + − u +0 ) + φ ( dx dt ) dy dα ≥ lim ε ↓ Z Z E cν ×E cµ (cid:0) sign + ε (cid:1) ′ (cid:0) v + − w + (cid:1) (cid:0) a ( ∇ w + ) − a ( ∇ v + ) (cid:1) · ∇ w + φ dx dt dy ds + lim ε ↓ Z Z E cν ×E cµ (cid:0) sign + ε (cid:1) ′ ( − w ) a ( ∇ w ) · ∇ w φ dx dt dy ds − Z Z Z Z Q × Q d (cid:20) sign − ( µ ) (cid:8) µ ∂ s φ + (cid:0) f ( µ ) − a ( ∇ w ) (cid:1) · ∇ y φ (cid:9) + ( d − − ( µ ∗ ) (cid:8) µ ∗ ∂ s φ + (cid:0) f ( µ ∗ ) − a ( ∇ w ) (cid:1) · ∇ y φ (cid:9)(cid:21) dx dt dy ds dα dβ − Z Q × Ω ( u ) − φ ( dx dt ) dy. (86)Notice that the sum of the last two terms in (86) can be rewritten under the form −L µ,µ ∗ ( χ ), where(87) χ ( s, y ) = Z Q φ ( t, s, x, y ) dt dx ∈ D ([0 , T ) × Ω) , and the distribution L µ,µ ∗ is defined on D ([0 , T ) × Ω) by L µ,µ ∗ ( χ ) := Z Z Q (cid:20) d (cid:0) η − ( µ )+( d − η − ( µ ∗ ) (cid:1) ∂ s χ + 1 d (cid:0) q − ( µ )+( d − q − ( µ ∗ ) (cid:1) · ∇ y χ − k ( ∇ w ) ∇ e A ( η − ) ′ ( w ) · ∇ y χ (cid:21) dy ds dβ + Z Ω η − ( u ) χ dy. (88) V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 53 (iii)
Adding (85) and (86), we obtain, for any 0 ≤ φ ∈ D (cid:0) ([0 , T ) × Ω) (cid:1) withcorresponding χ defined in (87), the following inequality: Z Z Z Z Q × Q (cid:20) d (cid:0) ( ν + − µ + ) + + ( d − ν + − ( µ ⋆ ) + ) + (cid:1) ( ∂ t + ∂ s ) φ + 1 d (cid:0) sign + (cid:0) ν + − µ + (cid:1) ( f ( ν + ) − f ( µ + ))+ ( d − + (cid:0) ν + − ( µ ⋆ ) + (cid:1) ( f ( ν + ) − f (( µ ⋆ ) + )) (cid:1) · ( ∇ x + ∇ y ) φ − sign + (cid:0) v + − w + (cid:1) (cid:0) a ( ∇ v + ) − a ( ∇ w + ) (cid:1) · ( ∇ x + ∇ y ) φ (cid:21) dx dt dy ds dα dβ + Z Z Z Ω × Q d (cid:0) ( v +0 − µ + ) + + ( d − v +0 − ( µ ⋆ ) + ) + (cid:1) φ dx ( dy ds ) dβ + Z Z Q × Ω ( ν + − u +0 ) + φ ( dx dt ) dy dα ≥ lim ε ↓ Z Z E cν ×E cµ (cid:0) sign + ε (cid:1) ′ (cid:0) v + − w + (cid:1) (cid:0) a ( ∇ v + ) − a ( ∇ w + ) (cid:1) · ( ∇ v + −∇ w + ) φ dx dt dy ds + lim ε ↓ Z Z E cν ×E cµ (cid:0) sign + ε (cid:1) ′ ( − w ) a ( ∇ w ) · ∇ w φ dx dt dy ds − L µ,µ ∗ ( χ ) ≥ −L µ,µ ∗ ( χ ) , (89)where the last inequality is due to the monotonicity of a ( · ). (iv) Now fix x ∈ ∂ Ω. Since ∂ Ω is supposed sufficiently regular, there exists avector r x and a positive number R x such that the segment ( x, x + r x ] lies withinΩ for all x ∈ ∂ Ω ∩ B ( x , R x ), where B ( x, R ) stands for the ball of R d with centre x and radius R . Choose in (89) the sequence of test functions φ n,l ( t, x, s, y ) = φ ( y, s ) ω n (cid:18) x − y + 2 n r x k r x k (cid:19) ω l ( t − s ) ≡ φω n ω l , for which (82) still holds. Notice that with this choice, the associated function χ n,l ( y, s ) in (87) writes as φ ( y, s ) θ n ( y ) θ l ( s ), where θ n ( y ) := Z Ω ω n (cid:18) x − y + 2 n r x k r x k (cid:19) dx, θ l ( s ) := Z T ω l ( t − s ) dt ;moreover, for all sufficiently large n ∈ N we have(90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φθ n ∈ D (Ω) for all φ ∈ D (cid:16) [0 , T ) × (cid:0) Ω ∩ B ( x , R x ) (cid:1)(cid:17) ; θ n ( y ) = 1 for all y ∈ B ( x , R x ) such that dist ( y, ∂ Ω) ≥ n . As in the proof of Lemma 9.3, passing to the limit as l, n → ∞ and taking intoaccount the definition of the distribution IP , cf (78), from (89) we deduce IP [ φ ] ≥ − lim inf l,n →∞ L µ,µ ∗ [ φθ n θ l ] . (91)Now we remark that according to ( D’ .3), L µ,µ ∗ defined by (88) is a nonnegativedistribution on [0 , T ) × Ω. Notice that the values of θ n are contained in the interval[0 , χ ∈ D ([0 , T ) × Ω), φ ≥
0, one has L µ,µ ∗ [ φθ n ] = L µ,µ ∗ [ φ ] − L µ,µ ∗ [ φ (1 − θ n )] ≤ L µ,µ ∗ [ φ ] . It follows that(92) L µ,µ ∗ : χ ∈ D ([0 , T ) × Ω) lim inf n →∞ L µ,µ ∗ ( χθ n ) is a nonnegative distribution on [0 , T ) × Ω; thus, it is a measure on [0 , T ) × Ω. Since φ ≥ θ l ≤
1, inequality (91) yields(93) IP [ φ ] ≥ − lim inf l,n →∞ L µ,µ ∗ [ χθ n θ l ] ≥ − lim inf n →∞ L µ,µ ∗ [ φθ n ] = −L µ,µ ∗ [ φ ] . It follows that IP is a measure on [0 , T ) × Ω.The remaining steps of the proof are aimed at showing, in an indirect way, thatthe positive part of the measure IP does not charge the boundary [0 , T ) × ∂ Ω (intwo particular cases, a direct proof of this fact is given in [73, 13]). Notice thatthis property is actually equivalent to the claim of the lemma; it accounts for thedissipative nature of the boundary condition imposed for entropy solutions. (v)
Take φ ∈ D (cid:16) [0 , T ) × (cid:0) Ω ∩ B ( x , R x ) (cid:1)(cid:17) . Fix m ∈ N . It is easily checkedfrom (90) that for all sufficiently large n ∈ N , for all ( t, x ) ∈ Q , φ ( s, y )(1 − θ m ( y )) θ n ( y ) = φ ( s, y ) θ n ( y ) − φ ( s, y ) θ m ( y ) . Therefore by (92),lim inf m →∞ L µ,µ ∗ [ φ (1 − θ m )]= lim inf m →∞ lim inf n →∞ L µ,µ ∗ [ φθ n ] − lim inf m →∞ lim inf n →∞ L µ,µ ∗ [ φθ m ] = 0 . Applying (93) to the test function φ (1 − θ m ), we deduce IP ν,ν,vµ,µ ∗ ,w [ φ ] = IP ν,ν,vµ,µ ∗ ,w [ φθ m ] + IP ν,ν,vµ,µ ∗ ,w [ φ (1 − θ m )] ≥ lim sup m →∞ IP ν,ν,vµ,µ ∗ ,w [ φθ m ] . (94) (vi) Definition 2.3 of entropy double-process solution is invariant under thechange of ( µ, µ ∗ , w ), f , f , u into ( − µ, − µ ∗ , − w ), − f , − f , − u . Moreover, one checkseasily from the definition, cf. (78), that IN ν,ν,vµ,µ ∗ ,w = IP − µ, − µ ∗ , − w − ν, − ν, − v . Therefore from(94) we deduce that for all φ ∈ D (cid:16) [0 , T ) × (cid:0) Ω ∩ B ( x , R x ) (cid:1)(cid:17) I ν,ν,vµ,µ ∗ ,w [ φ ] = IP ν,ν,vµ,µ ∗ ,w [ φ ] + IN ν,ν,vµ,µ ∗ ,w [ φ ] = IP ν,ν,vµ,µ ∗ ,w [ φ ] + IP − µ, − µ ∗ , − w − ν, − ν, − v [ φ ] ≥ lim sup m →∞ h IP ν,ν,vµ,µ ∗ ,w [ φθ m ] + IP − µ, − µ ∗ , − w − ν, − ν, − v [ φθ m ] i = lim sup m →∞ I ν,ν,vµ,µ ∗ ,w [ φθ m ] ≥ , where the last inequality is due to (90) and Lemma 9.3. (vii) Not let φ be an arbitrary nonnegative function in D ([0 , T ) × Ω). Choosea covering S Ni =1 B ( x i , R x i ), N ∈ N , of the compact set ∂ Ω. Introduce a partitionof unity ( ξ i ) Ni =0 on Ω associated with the covering Ω S(cid:0)S Ni =1 B ( x i , R x i ) (cid:1) of Ω, andapply Lemma 9.3 and the result of (vi) to the functions φξ ∈ D ([0 , T ) × Ω) andto φξ i ∈ D (cid:16) [0 , T ) × (cid:0) Ω ∩ B ( x i , R x i ) (cid:1)(cid:17) , i = 1 , . . . , N , respectively. The claim of thelemma follows. (cid:3) Now we conclude the proof of Theorem 2.2. We have u = v . By a standardargument, choosing in Lemma 9.5 φ = φ ( t ) ∈ D ([0 , T )), we get for a.e. t ∈ (0 , T ),(95) Z Z Z Ω d (cid:0) ( ν ( t, x, α ) − µ ( t, x, β )) + + ( d − ν ( t, x, α ) − µ ⋆ ( t, x, β )) + (cid:1) ≤ . Now, (95) means that for a.e. ( x, α, β ) ∈ Ω × (0 , × (0 , µ ( t, x, β ) = ν ( t, x, α ) = µ ∗ ( t, x, β ) , V FOR DOUBLY NONLINEAR DEGENERATE EQUATIONS 55 which means that µ ≡ µ ∗ ≡ ν and each of them is independent of α, β . This drawsto a close the proof of Theorem 2.2. Remark 9.1.
The proof of the L contraction and comparison principle for entropysolutions of (1) (with S = 0) is essentially contained in the above proof. For nonzerosource terms S , a more general version of inequalities (80) can be used; see [29] forthe accurate treatment of this term. Appendix B: The reconstruction property
Here we restate the result of [12, Lemma 8] and discuss its possible generaliza-tions.
Lemma 9.6.
Consider a triangle TT with vertices t , t , t and let t be the centreof its circumscribed circle. Denote by | TT | its area. For l ∈ N / N , denote by E l the affine subspace < −−−−−→ t l − t l +1 > ; denote by TT l the triangle formed by t , t l − , t l +1 and by | TT l | its area, with the convention that the area is negative if t and t l lay onopposite sides from the line passing by t l − , t l +1 . Then (96) 2 | TT | X l =1 | TT l | Proj E l ( −→ r ) = −→ r , for all −→ r ∈ R . Remark 9.2.
For a multi-D generalization of the property (96), one could try toreplace the projections on lines < E l > by projections on hyperplanes that containthe faces of the d -dimensional simplex TT . In this case one should replace the factor | TT | by dd − | TT | , since | TT | = P d +1 l =1 | TT l | and because the dimension of Proj E l ( −→ r ) is( d − −→ r is d . The proof of Lemma 9.6 given belowshows that this generalization fails, except for very particular simplexes TT (this isclear from the multi-dimensional analogue of the identity (97) below). Remark 9.3.
Using the “sine theorem”, another proof of Lemma 9.6 can be given,which also works for any 2D polygon that admits a circumscribed circle.
Proof.
Proof of Lemma 9.6 For l ∈ N / N , denote by d l the orthogonal projection ifthe point t l on the affine subspace E l ; set −→ p l = −−→ t d l and −→ a l = −→ t t l . For l, i ∈ N / N ,set −→ b l,i = −→ a i − −→ a l . Denote by −→ n l the exterior to TT unit normal vector to E l .Notice that we have for all l ∈ N / N , −→ d l = ( −→ d l · −→ n l ) −→ n l , and also, for all i ∈ N / N such that i = l , | TT l || TT | = −→ p l · −→ n l −→ b l,i · −→ n l , taking into account the sign of | TT l | . Since Proj E l + Proj < −→ n l > is the identity opera-tor, (96) is equivalent to the statement that | TT | P l =1 | TT l | Proj < −→ n l > is the identityoperator. All vector −→ r ∈ R can be uniquely represented under the form −→ r = X l =1 k l −→ a l with X l =1 k l = 0 , and thus, for all l ∈ N / N , −→ r = P i = l,i =1 k i −→ ( a i − −→ a l ) = P i = l,i =1 k i −→ b l,i . Hence2 | TT | X l =1 | TT l | Proj < −→ n l > ( −→ r ) = 2 X l =1 | TT l || TT | −→ n l ( −→ r · −→ n l ) = 2 X i = l ; i,l =1 −→ n l | TT l || TT | k i ( −→ b l,i · −→ n l )= 2 X l =1 −→ n l X i = l,i =1 −→ p l · −→ n l −→ b l,i · −→ n l k i ( −→ b l,i · −→ n l ) = 2 X l =1 −→ p l X i = l,i =1 k i = − X l =1 k l −→ p l . We conclude that (96) is equivalent to the identity(97) X l =1 k l −→ a l = − X l =1 k l −→ p l for all k , . . . , k ∈ R such that X l =1 k l = 0 . Since t is the centre of the circumscribed circle of TT , the points d l are the centresof the corresponding segments [ t i − , t i +1 ]. Thus for all i, j ∈ N / N , by the Thalestheorem we have −→ p i − −→ p j = − ( −→ a i − −→ a j ). Hence (97) holds with k i ∈ { , , − } , i = 1 , . . . ,
3. Hence it holds for all choice of k i . (cid:3) We refer to [7, 8] for a different kind of generalization of [12, Lemma 8] and adifferent proof of Lemma 9.6.
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Laboratoire de Math´ematiquesUniversit´e de Franche-Comt´e16 route de Gray25 030 Besan ¸c on Cedex, France E-mail address : [email protected] (Mostafa Bendahmane) Departamento de Ingenier´ıa Matem´aticaFacultad de Ciencias F´ısicas y Matem´aticasUniversidad de Concepci´onCasilla 160-C Concepci´on, Chile
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