Sovability of the mixed Formulation for Darcy-Forchheimer Flow in Porous Media
aa r X i v : . [ m a t h . NA ] A ug Mathematical Modelling and Numerical Analysis
Will be set by the publisherMod´elisation Math´ematique et Analyse Num´erique
SOLVABILITY OF THE MIXED FORMULATION FORDARCY–FORCHHEIMER FLOW IN POROUS MEDIA
Peter Knabner and Gerhard Summ Abstract . We consider the mixed formulation of the equations governing Darcy–Forchheimer flow inporous media. We prove existence and uniqueness of a solution for the stationary problem and theexistence of a solution for the transient problem.
R´esum´e . Nous ´etudions la formulation mixte des ´equations pour l’´ecoulement d’un gaz `a traversd’un milieu poreux, qu’on suppose r´egi par la loi de Darcy–Forchheimer. Nous ´etablions des r´esultatsd’existence et d’unicit´e pour le probl`eme stationnaire et un r´esultat d’existence pour le probl`emetransitoire.
AMS Subject Classification.
September 1, 2018.
Introduction
The flow of a gas through a porous medium is governed by the doubly nonlinear parabolic equation φ ( x ) ∂ t ρ ( S ( x , t ) , x , t ) − div ( F ( ∇ S ( x , t ) , x , t )) = f ( x , t ) , ( x , t ) ∈ Ω × [0 , T ] , (1)where the unknown function S (= | p | p ) represents the pressure squared and the nonlinearities ρ and F aredefined by ρ ( S ( x , t ) , x , t ) := γ ( x , t ) S ( x , t ) p | S ( x , t ) | , (2) F ( ∇ S ( x , t ) , x , t ) := p α ( x , t ) + 4 β ( x , t ) |∇ S ( x , t ) | − α ( x , t )2 β ( x , t ) |∇ S ( x , t ) | ∇ S ( x , t ) . (3)This equation together with appropriate initial and Neumann boundary conditions has been studied by Amirat[2]. He restricts his considerations to the case, where γ is constant, and shows the existence of a solution usingthe technique of semi-discretization in time. Under additional regularity conditions on the solution he provesthe uniqueness and positivity of this solution. The technique of semi-discretization in time has been used severaltimes to study similar doubly nonlinear parabolic equations. We mention only the articles of Raviart, e.g. [7]. Keywords and phrases: solvability, Darcy–Forchheimer flow, regularization, monotone operators, semi-discretization Institute for Applied Mathematics, Martensstraße 3, D-91058 Erlangen, Germany; e-mail: [email protected] [email protected] c (cid:13) EDP Sciences, SMAI 1999
PETER KNABNER AND GERHARD SUMM
Equation (1) can be derived from the following system of equations consisting of the Darcy–Forchheimerequation (see e.g. [10]) µ ( x , t ) k ( x ) u ( x , t ) + β Fo ( x ) ρ ( x , t ) | u ( x , t ) | u ( x , t ) + ∇ p ( x , t ) = 0 , ( x , t ) ∈ Ω × [0 , T ] , the continuity equation φ ( x ) ∂ρ ( x , t ) ∂t + div( ρ ( x , t ) u ( x , t )) = f ( x , t ) , ( x , t ) ∈ Ω × [0 , T ]and the ideal gas law as equation of state ρ ( x , t ) = p ( x , t ) W ( x , t ) R Θ( x , t ) =: p ( x , t ) γ ( x , t ) , ( x , t ) ∈ Ω × [0 , T ] . The unknowns here are the pressure p , the density ρ and the volumetric flow rate u of the gas. Porosity φ ,permeability k and Forchheimer coefficient β Fo of the porous medium, viscosity µ , molecular weight W andtemperature Θ of the gas, and the universal gas constant R are given as well as the source term f . Assuming ρ > S = | p | p and m = | ρ | u these equations can be transformed into( α ( x , t ) + β ( x , t ) | m ( x , t ) | ) m ( x , t ) + ∇ S ( x , t ) = 0 , ( x , t ) ∈ Ω × [0 , T ] , (4) φ ( x ) ∂ t ρ ( S ( x , t ) , x , t ) + div ( m ( x , t )) = f ( x , t ) , ( x , t ) ∈ Ω × [0 , T ] , (5)where γ ( x , t ) := W ( x , t ) R Θ( x , t ) , α ( x , t ) := 2 µ ( x , t ) γ ( x , t ) k ( x ) , β ( x , t ) := 2 β Fo ( x ) γ ( x , t ) , and the equation of state ρ = ρ ( S ) is defined in (2). Evidently, the Darcy–Forchheimer equation (4) can beresolved to give m = F ( ∇ S ) with the nonlinear mapping F defined in (3). Substituting m = F ( ∇ S ) into (5)finally yields the parabolic equation (1).The corresponding stationary problem of system (4–5) together with Neumann boundary conditions has beenstudied by Fabrie [4] for constant physical parameters. He obtains the existence and uniqueness of a solution( m , S ) ∈ (cid:0) L (Ω) (cid:1) n × W , / (Ω) and shows additional regularity properties.Throughout this article, for s ∈ [0 , ∞ ] we denote the Lebesgue spaces of s -integrable functions by L s (Ω),for m ≥ , s ∈ [0 , ∞ ] the Sobolev spaces by W m,s (Ω) and the norm of W m,s (Ω) by k · k m,s, Ω ( cf. [1]). Weassume that the domain Ω is bounded and fulfills the uniform C -regularity property. Then the trace operator γ : W m,s (Ω) → W m − /s,s ( ∂ Ω) is onto [1, Thm. 7.53]. We denote by W m,s (Ω) the kernel of γ and its dualspace by W − m,r (Ω) := (W m,s (Ω)) ′ , where 1 /s + 1 /r = 1. Note that for every s ∈ [0 , ∞ ] , m ≥ D (Ω) := C ∞ (Ω) is a dense subset of L s (Ω) and of W m,s (Ω), and that D ( ¯Ω) := (cid:8) Ψ | Ω (cid:12)(cid:12) Ψ ∈ D ( R n ) (cid:9) is a densesubset of W m,s (Ω). In addition, let us introduce the generalization W s (div; Ω) of H(div; Ω), which is defined inAppendix A. For s = 3, this space will turn out to be appropriate for the nonlinearity of the Darcy–Forchheimerlaw (see Proposition 1.2). Finally, we employ the spaces C ([0 , T ]; X ) and L s (0 , T ; X ) of vector-valued functions,where X is one of the above introduced spaces.Note that the restriction of [4] prevents from generalization to relevant situations from applications, wherethe parameters, e.g. the porosity φ , vary discontinuously due to composite media or where some of them, e.g. thetemperature Θ, are unknowns of a more complex model. Both situations appear in the modelling of combustionin porous media, see [ ? ], which is our final goal. Therefore we study the system (4–5) for Dirichlet boundaryconditions and general coefficient functions, imposing only minimal regularity assumptions. We consider thestationary problem in Section 1 and prove the existence and uniqueness of a solution. To this end, we usea regularization of the equations and exploit the monotonicity of the nonlinear mapping F . In Section 2we investigate the semi-discrete problem after discretization of the time-derivative in (5) and show again the IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW The stationary problem
We consider the stationary problem governed by the Darcy–Forchheimer equation and the stationary conti-nuity equation together with Dirichlet boundary conditions:( α ( x ) + β ( x ) | m ( x ) | ) m ( x ) + ∇ S ( x ) = 0 , x ∈ Ω , div ( m ( x )) = f ( x ) , x ∈ Ω , (6) S ( x ) = S b ( x ) , x ∈ ∂ Ω . We require f ∈ L (Ω), S b ∈ W / , / ( ∂ Ω), α, β ∈ L ∞ (Ω) and additionally0 < α ≤ α ( x ) ≤ α < ∞ , < β ≤ β ( x ) ≤ β < ∞ (cid:27) for almost every x ∈ Ω . Mixed formulation of the stationary problem
The mixed formulation of (6) reads as follows: Find ( m , S ) ∈ W (div; Ω) × L / (Ω) such that Z Ω ( α + β | m | ) ( m · v ) d x − Z Ω div( v ) S d x = − Z ∂ Ω S b ( v · n ) dσ for all v ∈ W (div; Ω) , Z Ω div( m ) q d x = Z Ω f q d x for all q ∈ L / (Ω) . (7)Next, we introduce continuous linear forms g : W (div; Ω) → R and f : L / (Ω) → R by means of g ( v ) := − Z ∂ Ω S b ( v · n ) dσ for v ∈ W (div; Ω) , f ( q ) := Z Ω f q d x for q ∈ L / (Ω) , a bilinear form b : W (div; Ω) × L / (Ω) → R and a nonlinear form a : (cid:0) L (Ω) (cid:1) n × (cid:0) L (Ω) (cid:1) n → R by means of b ( v , q ) := Z Ω div( v ) q d x for v ∈ W (div; Ω) , q ∈ L / (Ω) , a ( u , v ) := Z Ω ( α + β | u | ) ( u · v ) d x for u , v ∈ (cid:0) L (Ω) (cid:1) n . The form b obviously is continuous, the continuity of the form a will be shown in Proposition 1.2. Then we canwrite the mixed formulation (7) of (6) in the following way: Find ( m , S ) ∈ W (div; Ω) × L / (Ω) such that a ( m , v ) − b ( v , S ) = g ( v ) for all v ∈ W (div; Ω) ,b ( m , q ) = f ( q ) for all q ∈ L / (Ω) . (8)In the following let V be one of the spaces W (div; Ω) or (cid:0) L (Ω) (cid:1) n . Since a is linear with respect to its secondvariable, we can define a mapping A : V → V ′ by h A u , v i V ′ × V = a ( u , v ), where h· , ·i V ′ × V denotes the dualpairing between V ′ and V . Using H¨older’s inequality we obtain the following bound on k A u k V : k A u k V ′ ≤ C ( α ) k u k V + C ( β ) k u k V , (9)where C ( α ) and C ( β ) are constants depending only on α , β and the domain Ω. Thus A u is a continuous linearform on V for every u ∈ V . The proof of (9) is contained in the proof of Proposition 1.2.The proof of the continuity and monotonicity of the mapping A is based on the following lemma: PETER KNABNER AND GERHARD SUMM
Lemma 1.1.
The following inequalities hold for every x , y ∈ R n : || x | x − | y | y | ≤ ( | x | + | y | ) | x − y | , (10)( | x | x − | y | y ) · ( x − y ) ≥ | x − y | . (11)The proof of (10) and (11) is elementary. In the two-dimensional case, similar inequalities for more generalnonlinearities are derived in [6, Section 5].In the following we use the notions of [11, Def. 25.2]. Proposition 1.2.
The operator A : V → V ′ is continuous and strictly monotone on V = W (div; Ω) , andcontinuous, uniformly monotone and coercive on V = (cid:0) L (Ω) (cid:1) n .Proof. To show the continuity of A we consider u , u ∈ (cid:0) L (Ω) (cid:1) n . Applying H¨older’s inequality we obtain |h A u − A u , v i V × V ′ | ≤ (cid:16) C ( α ) k u − u k , , Ω + β k| u | u − | u | u k , / , Ω (cid:17) k v k , , Ω for all v ∈ (cid:0) L (Ω) (cid:1) n , which also proves (9) for V = (cid:0) L (Ω) (cid:1) n and therefore also for V = W (div; Ω). Applying inequality (10) andagain the H¨older’s inequality yields k A u − A u k V ′ ≤ (cid:0) C ( α ) + C ( β ) ( k u k , , Ω + k u k , , Ω ) (cid:1) k u − u k , , Ω , where C ( α ) and C ( β ) are exactly the same constants as in (9). Using inequality (11) we obtain h A u − A v , u − v i V ′ × V ≥ C ( β )2 k u − v k , , Ω for u , v ∈ (cid:0) L (Ω) (cid:1) n , (12)where C ( β ) depends only on β and Ω, too. Therefore A is strictly monotone for V = W (div; Ω), and uniformlymonotone (and thus coercive) for V = (cid:0) L (Ω) (cid:1) n . Remark 1.3.
Since A is uniformly monotone for V = (cid:0) L (Ω) (cid:1) n we can conclude easily that A is uniformlymonotone (and thus coercive) for V = W (div; Ω) := (cid:8) v ∈ W (div; Ω) (cid:12)(cid:12) div( v ) = 0 (cid:9) . Obviously, the solution m of the homogeneous problem (i.e., f ≡
0) satisfies m ∈ W (div; Ω). Therefore, in the homogeneous case, wecan extend directly the proof of the unique solvability for the linear problem ( cf. [3, Prop. I.1.1 and Thm. I.1.1])to the nonlinear problem (8). Owing to the uniform monotonicity of A we can use the theorem of Browder andMinty [11, Thm. 26.A] to show that there exists a unique solution m ∈ W (div; Ω) of (8). The existence anduniqueness of a solution S then follows exactly like in the proof of [3, Thm. I.1.1]. Regularization of the stationary problem
For general f the situation is not so simple as depicted in Remark 1.3. We use regularization to show theexistence of a solution ( m , S ) ∈ V × Q to (8), where V := W (div; Ω) and Q := L / (Ω). For ε > d ε : V × V → R and c ε : Q × Q → R by d ε ( u , v ) := ε Z Ω | div( u ) | div( u )div( v ) d x for u , v ∈ W (div; Ω) , c ε ( p, q ) := ε Z Ω p p | p | q d x for p, q ∈ L / (Ω) . Then the regularized problem is: Find ( m ε , S ε ) ∈ V × Q such that a ( m ε , v ) + d ε ( m ε , v ) − b ( v , S ε ) = g ( v ) for all v ∈ V ,c ε ( S ε , q ) + b ( m ε , q ) = f ( q ) for all q ∈ Q . (13)
IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW A we define operators A ε : V → V ′ by h A ε u , v i V ′ × V := a ( u , v ) + d ε ( u , v ) and C ε : Q → Q ′ by h C ε p, q i Q ′ × Q := c ε ( p, q ) . It is evident that A ε u is a linear functional on V for all u ∈ V ,and C ε p is a linear functional on Q for all p ∈ Q . Again, continuity of A ε u and C ε p , resp., follow from theboundedness, which is obtained by means of H¨older’s inequality: a ( u , v ) + d ε ( u , v ) ≤ (cid:0) C ( α ) k u k , , Ω + C ( β ) k u k , , Ω (cid:1) k v k , , Ω + ε k div( u ) k , , Ω k div( v ) k , , Ω , | c ε ( p, q ) | ≤ ε k p k / , / , Ω k q k , / , Ω . To show that (13) has a solution, we need continuity, coercivity and monotonicity of A ε and C ε . For A ε these properties are consequences of (10-11) again, for C ε we need an additional lemma: Lemma 1.4.
The real-valued function f : R → R , x
7→ | x | − / x is strictly monotone and H¨older continuous oforder / on R with H¨older constant √ , i.e. for all x, y ∈ R it holds (cid:12)(cid:12)(cid:12) | x | − / x − | y | − / y (cid:12)(cid:12)(cid:12) ≤ √ | x − y | / . (14) Furthermore | x − y | p | x | + p | y | ≤ x p | x | − y p | y | ! ( x − y ) . (15) Proposition 1.5.
For every ε > : a) the operator A ε : V → V ′ is continuous, coercive and strictly monotone on V , b) the operator C ε : Q → Q ′ is continuous, coercive and strictly monotone on Q .Proof. Ad a): The continuity of A ε follows from Proposition 1.2 and the inequality | d ε ( u , v ) − d ε ( u , v ) | ≤ ε ( k div( u ) k , , Ω + k div( u ) k , , Ω ) k div( u ) − div( u ) k , , Ω k div( v ) k , , Ω for all u , u , v ∈ V , which is obtained by means of H¨older’s inequality and of (10). Furthermore, an applicationof (11) yields d ε ( u , u − v ) − d ε ( v , u − v ) ≥ ε k div( u − v ) k , , Ω for all u , v ∈ V .
Together with (12) this inequality implies the uniform monotonicity of A ε : h A ε u − A ε v , u − v i V × V ′ ≥
12 min (cid:0) C ( β ) , ε (cid:1) k u − v k V for all u , v ∈ V .
The coercivity and strict monotonicity of A ε are direct consequences of the uniform monotonicity.Ad b): Using H¨older’s inequality and (14) we obtain the continuity of C ε : k C ε p − C ε q k Q ′ ≤ ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p p | p | − q p | q | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , , Ω ≤ √ ε k p − q k / , / , Ω for all p, q ∈ Q .
The coercivity follows from the equation h C ε q, q i Q ′ × Q = Z Ω ε q p | q | d x = ε k q k / , / , Ω , and strict monotonicity from the strict monotonicity of x
7→ | x | − / x : h C ε p − C ε q, p − q i Q ′ × Q = ε Z Ω p p | p | − q p | q | ! ( p − q ) d x > p, q ∈ Q with p = q . PETER KNABNER AND GERHARD SUMM
Now we are in a position to prove:
Proposition 1.6.
For every ε > there is a unique solution ( m ε , S ε ) ∈ V × Q of the regularized problem (13) .Proof. Adding the left hand sides of (13) we obtain the following nonlinear form defined on V × Q : a ε (( u , p ) , ( v , q )) := a ( u , v ) + d ε ( u , v ) − b ( v , p ) + c ε ( p, q ) + b ( u , q ) for ( u , p ) , ( v , q ) ∈ V × Q and a nonlinear operator A ε : ( V × Q ) → ( V × Q ) ′ defined by hA ε ( u , p ) , ( v , q ) i ( V × Q ) ′ × ( V × Q ) = a ε (( u , p ) , ( v , q )).To study the properties of A ε we introduce the continuous linear operator B : V → Q ′ and its adjoint operator B ′ : Q → V ′ , defined by h B v , q i Q × Q ′ = b ( v , q ) = h B ′ q, v i V × V ′ . Then we can write hA ε ( u , p ) , ( v , q ) i ( V × Q ) ′ × ( V × Q ) = h A ε u , v i V ′ × V − h B ′ p, v i V ′ × V + h C ε p, q i Q ′ × Q + h B u , q i Q ′ × Q for ( u , p ) , ( v , q ) ∈ V × Q . Since A ε , C ε , B and B ′ are continuous, A ε is continuous, too. Furthermore, A ε iscoercive and strictly monotone. This is a straightforward consequence of the corresponding properties of A ε and C ε , since the terms containing B or B ′ cancel each other. For the strict monotonicity this reads hA ε ( u , p ) , ( u − v , p − q ) i ( V × Q ) ′ × ( V × Q ) − hA ε ( v , q ) , ( u − v , p − q ) i ( V × Q ) ′ × ( V × Q ) = h A ε u − A ε v , u − v i V ′ × V − h B ′ ( p − q ) , u − v i V ′ × V + h C ε p − C ε q, p − q i Q ′ × Q + h B ( u − v ) , p − q i Q ′ × Q = h A ε u − A ε v , u − v i V ′ × V − h B ( u − v ) , p − q i Q ′ × Q + h C ε p − C ε q, p − q i Q ′ × Q + h B ( u − v ) , p − q i Q ′ × Q = h A ε u − A ε v , u − v i V ′ × V + h C ε p − C ε q, p − q i Q ′ × Q > , if u = v or p = q . The proof of the coercivity of A ε is even more simple.Thus we can apply the theorem of Browder and Minty [11, Thm. 26.A] to show that for every f ∈ ( V × Q ) ′ there exists a unique solution ( m ε , S ε ) ∈ V × Q of the operator equation A ε ( m ε , S ε ) = f . In particular, wechoose the linear form f defined by f ( v , q ) := g ( v ) + f ( q ), which arises by adding the right hand sides of (13).Therefore (13) has a unique solution.Next, we show that the solution ( m ε , S ε ) is bounded independently of ε . Proposition 1.7.
There exist constants K m , K S , independent of ε , such that for sufficiently small ε > thesolution ( m ε , S ε ) of (13) satisfies the following estimates: k m ε k V ≤ K m , k S ε k Q ≤ K S . (16) Proof.
We begin with a bound for the norm of div( m ε ). Using the second equation of (13) we obtain k div( m ε ) k , , Ω = k div( m ε ) k Q ′ = sup q ∈ Q | b ( m ε , q ) |k q k Q = sup q ∈ Q | f ( q ) − c ε ( S ε , q ) |k q k Q ≤ k f k , , Ω + ε k S ε k / Q . The estimation of k m ε k , , Ω is based on the first equation in (13): C ( β ) k m ε k , , Ω ≤ Z Ω β | m ε | ( m ε · m ε ) d x ≤ a ( m ε , m ε ) + d ε ( m ε , m ε ) = g ( m ε ) + b ( m ε , S ε ) ≤ k g k V ′ k m ε k , , Ω + ( k g k V ′ + k S ε k Q ) k div( m ε ) k , , Ω . Together with the estimate for k div( m ε ) k , , Ω above this yields k m ε k , , Ω ≤ C ( β ) (cid:16) k g k V ′ k m ε k , , Ω + k g k V ′ k f k , , Ω + ε k g k V ′ k S ε k / Q + k f k , , Ω k S ε k Q + ε k S ε k / Q (cid:17) . (17) IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW S ε we employ the inf-sup condition (38). Together with the first equation in (13) and the aboveestimate for k div( m ε ) k , , Ω we obtain θ k S ε k Q ≤ sup v ∈ V b ( v , S ε ) k v k V = sup v ∈ V a ( m ε , v ) + d ε ( m ε , v ) − g ( v ) k v k V ≤ k A m ε k V ′ + ε k div( m ε ) k , , Ω + k g k V ′ ≤ k A m ε k V ′ + k g k V ′ + ε (cid:16) k f k , , Ω + ε k S ε k / Q (cid:17) for some constant θ >
0. Thus for sufficiently small ε ( ε < θ / is enough) it holds k S ε k Q ≤ θ − ε (cid:16) k A m ε k V ′ + k g k V ′ + ε k f k , , Ω + 2 ε k f k , , Ω k S ε k / Q (cid:17) , such that k S ε k / Q ≤ ε θ − ε k f k , , Ω + (cid:18) θ − ε (cid:16) k A m ε k V ′ + k g k V ′ + ε k f k , , Ω (cid:17)(cid:19) / ! . An application of the estimate k A m ε k V ′ ≤ C ( α ) k m ε k , , Ω + C ( β ) k m ε k , , Ω finally yields k S ε k / Q ≤ κ + κ k m ε k / , , Ω + κ k m ε k , , Ω , (18)where the coefficients κ i are bounded, independently of m ε and S ε , for sufficiently small ε , e.g. ε < θ / / κ := 2 ε θ − ε k f k , , Ω + (cid:18) θ − ε (cid:0) k g k V ′ + ε k f k , , Ω (cid:1)(cid:19) / , κ := (cid:18) C ( α ) θ − ε (cid:19) / , κ := (cid:18) C ( β ) θ − ε (cid:19) / . Inserting (18) into (17) we obtain after some calculations the inequality (cid:18) − ε κ βc ℓ (cid:19) k m ε k , , Ω ≤ X i =0 λ i k m ε k i/ , , Ω , where the λ i ( i = 0 , . . . ,
5) are independent of k m ε k , , Ω and bounded for sufficiently small ε < ε ≤ θ / / K < ∞ , independent of ε , such that k m ε k , , Ω ≤ K for ε ≤ ε . Inserting thisestimate into (18) we obtain the bound for k S ε k Q and using the above estimate for k div( m ε ) k , , Ω finally yieldsthe bound for k m ε k V . Solvability of the stationary problem (8)
Theorem 1.8.
The mixed formulation (8) of the stationary problem (6) possesses a unique solution ( m , S ) ∈ W (div; Ω) × L / (Ω) .Proof. Analogously to the definition of a ε we add the left hand sides of (8) and obtain the nonlinear form a defined by a (( u , p ) , ( v , q )) := a ( u , v ) − b ( v , p ) + b ( u , q ) and the nonlinear operator A : ( V × Q ) → ( V × Q ) ′ defined by hA ( u , p ) , ( v , q ) i ( V × Q ) ′ × ( V × Q ) = a (( u , p ) , ( v , q )). Setting ε = 1 /n let ( m n , S n ) be the unique solutionof the regularized problem (13). Since (( m n , S n )) n ∈ N is a bounded sequence in V × Q , there exists a weakly PETER KNABNER AND GERHARD SUMM convergent subsequence, again denoted by (( m n , S n )) n ∈ N , with (weak) limit ( m , S ) ∈ V × Q . As kA ( m n , S n ) − f k ( V × Q ) ′ = sup =( v ,q ) ∈ V × Q | a (( m n , S n ) , ( v , q )) − f ( v , q ) |k ( v , q ) k V × Q = sup ( v ,q ) (cid:12)(cid:12) a /n (( m n , S n ) , ( v , q )) − d /n ( m n , v ) − c /n ( S n , q ) − f ( v , q ) (cid:12)(cid:12) k ( v , q ) k V × Q ≤ n (cid:16) k div( m n ) k , , Ω + k S n k / Q (cid:17) n →∞ −→ A ( m n , S n )) n ∈ N converges strongly in V ′ to f defined by f ( v , q ) := g ( v ) + f ( q ). Thus we canconclude that A ( m , S ) = f in ( V × Q ) ′ (see e.g. [11, p. 474]), i.e., ( m , S ) is a solution of (8).To show uniqueness, we consider two solutions ( m , S ) and ( m , S ) of (8). Using the test functions v = m − m and q = S − S we obtain a ( m , m − m ) − a ( m , m − m ) − b ( m − m , S ) + b ( m − m , S ) = 0 b ( m , S − S ) − b ( m , S − S ) = 0 . Adding these equations yields0 = a ( m , m − m ) − a ( m , m − m ) = h A m − A m , m − m i ( V × Q ) ′ × ( V × Q ) . Since A is strictly monotone it follows m − m = 0.If m ∈ V is given, S ∈ Q is defined as solution of the variational equation b ( v , S ) = g ( v ) − a ( m , v ) for all v ∈ V . Therefore the uniqueness of S is a direct consequence of the injectivity of the operator B ′ : Q → V ′ , cf. [3, § II, Rem. 1.6]. The semi-discrete problem
We return to the transient problem governed by (4) and (5). We discretize (5) in time using the implicitEuler method. This yields not only a method to solve the transient problem numerically, but also an approachto prove its solvability, the technique of semi-discretization.We define a partition 0 = t < t < . . . < t K = T of the segment (0 , T ) into K intervals of constantlength ∆ t = T /K , i.e., t k = k ∆ t for k = 0 , . . . , K . In the following for k = 0 , . . . , K we use the denotations S k := S ( · , t k ) and m k := m ( · , t k ) for the unknown solutions and, analogously defined, α k , β k and γ k forthe coefficient functions, S kb for the boundary conditions and f k for the source term. The initial condition S ( · , t ) = S ( · ) ∈ W , / (Ω) is given.Using the equation of state ρ = ρ ( S ) defined in (2), the discretization in time of the continuity equation (5)with the implicit Euler method yields for each k ∈ { , . . . , K } (cid:0) α k ( x ) + β k ( x ) | m k ( x ) | (cid:1) m k ( x ) + ∇ S k ( x ) = 0 , x ∈ Ω ,φ ( x )∆ t γ k ( x ) S k ( x ) p | S k ( x ) | − γ k − ( x ) S k − ( x ) p | S k − ( x ) | ! + div (cid:0) m k ( x ) (cid:1) = f k ( x ) , x ∈ Ω ,S ( x ) = S kb ( x ) , x ∈ ∂ Ω . (19)Note that for each k ∈ { , . . . , K } the function S k − is known. IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW k ∈ { , . . . , K } we require f k ∈ L (Ω), S kb ∈ W / , / ( ∂ Ω), φ, α k , β k , γ k ∈ L ∞ (Ω) and additionally0 < φ ≤ φ ( x ) ≤ φ < ∞ , < α ≤ α k ( x ) ≤ α < ∞ , < β ≤ β k ( x ) ≤ β < ∞ , < γ ≤ γ k ( x ) ≤ γ < ∞ for almost every x ∈ Ω . Mixed formulation of the semi-discrete problem
We continue to use the spaces V = W (div; Ω) and Q = L / (Ω). Then the variational formulation reads:Find ( m k , S k ) ∈ V × Q such that Z Ω (cid:0) α k + β k | m k | (cid:1) ( m k · v ) d x − Z Ω div( v ) S k d x = − Z ∂ Ω S kb ( v · n ) dσ for all v ∈ V , Z Ω φ γ k ∆ t S k p | S k | q d x + Z Ω div( m k ) q d x = Z Ω f k q d x + Z Ω φ γ k − ∆ t S k − p | S k − | q d x for all q ∈ Q . (20)We introduce additional nonlinear forms c k on Q × Q defined by c k ( p, q ) := Z Ω φ γ k ∆ t p p | p | q d x and nonlinear operators C k : Q → Q ′ by h C k p, q i Q ′ × Q := c k ( p, q ). Again, it is evident that C k p is a linearmapping on Q for all p ∈ Q . The continuity of C k p is equivalent to its boundedness, which in turn is aconsequence of H¨older’s inequality and the boundedness of φ and γ : (cid:12)(cid:12) c k ( p, q ) (cid:12)(cid:12) ≤ φ γ ∆ t k p k / , / , Ω k q k , / , Ω . In the same manner as in the proof of Proposition 1.5 b) we obtain the continuity, coercivity and monotonicityof C k . Proposition 2.1.
The operators C k : Q → Q ′ are continuous, coercive and strictly monotone on Q . We use a , b and g as defined in Section 1, where a = a k and g = g k depend on k , because α , β and theboundary condition S b may change in time. Then we can write the mixed formulation (20) of (19) in thefollowing way: Find (cid:0) m k , S k (cid:1) ∈ V × Q , such that a k ( m k , v ) − b ( v , S k ) = g k ( v ) for all v ∈ V ,c k ( S k , q ) + b ( m k , q ) = ˜ f k ( q ) for all q ∈ Q . (21)Here ˜ f k ∈ Q ′ for k = 1 , . . . , K is defined by˜ f k ( q ) := Z Ω f k + φ γ k − ∆ t S k − p | S k − | ! q d x . For the remainder of this section we restrict our considerations to a fixed time step k . Thus we can omit thesuperscript k .0 PETER KNABNER AND GERHARD SUMM
Regularization of the semi-discrete problem
We use the technique of regularization again. Thus we consider, instead of (21), the following regularizedproblem for ε >
0: Find ( m ε , S ε ) ∈ V × Q such that a ( m ε , v ) + d ε ( m ε , v ) − b ( v , S ε ) = g ( v ) for all v ∈ V ,c ( S ε , q ) + b ( m ε , q ) = ˜ f ( q ) for all q ∈ Q . (22)Here d ε ( u , v ) := ε R Ω | div( u ) | div( u )div( v ) d x is defined as in Section 1.In the same manner as Proposition 1.6 we obtain: Proposition 2.2.
For every ε > there exists a unique solution ( m ε , S ε ) ∈ V × Q of the regularized semi-discrete problem (22) . Next, we show that the solution ( m ε , S ε ) of (22) is bounded independently of ε , too. Since we added in (22)only one regularizing term d ε , we can use different techniques for the estimation of m ε and S ε . In particularwe obtain estimates that hold for every ε > Proposition 2.3.
There exist constants K m , K S , independent of ε , such that the solution ( m ε , S ε ) of (22) satisfies the following estimates: k m ε k V ≤ K m , k S ε k Q ≤ K S . (23) Proof.
As in the proof of Proposition 1.7 we begin with an estimate for the norm of div( m ε ): k div( m ε ) k , , Ω ≤ k ˜ f k Q ′ + φ γ ∆ t k S ε k / Q . The estimation of k m ε k , , Ω uses the following inequality, established in the proof of Proposition 1.7, C ( β ) k m ε k , , Ω ≤ k g k V ′ k m ε k , , Ω + ( k g k V ′ + k S ε k Q ) k div( m ε ) k , , Ω to derive k m ε k , , Ω ≤ (cid:18) C ( β ) k g k V ′ (cid:19) / + (cid:18) C ( β ) (cid:16) k g k V ′ + k S ε k Q (cid:17) k div( m ε ) k , , Ω (cid:19) / . Together with the estimate for k div( m ε ) k , , Ω we obtain the following bound for k m ε k V : k m ε k V ≤ κ + κ k S ε k / Q , where the constants κ and κ are independent of ε and k S ε k Q . To derive an estimation for k S ε k Q , we use in(22) the test functions v = m ε and q = S ε and add the resulting equations. Since b is a bilinear form, we obtainthe inequality c ( S ε , S ε ) ≤ a ( m ε , m ε ) + d ε ( m ε , m ε ) + c ( S ε , S ε ) = g ( m ε ) + ˜ f ( S ε ) . Using the coercivity of C and the bound for k m ε k V derived above we can therefore conclude φ γ ∆ t k S ε k / Q ≤ c ( S ε , S ε ) ≤ g ( m ε ) + ˜ f ( S ε ) ≤ k g k V ′ k m ε k V + k ˜ f k Q ′ k S ε k Q ≤ k g k V ′ (cid:16) κ + κ k S ε k / Q (cid:17) + k ˜ f k Q ′ k S ε k Q . This yields the existence of a bound K S for k S ε k Q . IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW Solvability of the semi-discrete problem (21)
Again, we consider the limit ε → Theorem 2.4.
The mixed formulation (21) of the semi-discrete problem (19) possesses a unique solution ( m , S ) ∈ W (div; Ω) × L / (Ω) .Proof. Like in the proof of Theorem 1.8 we add both equations in (21) and obtain the nonlinear form a , definedon ( V × Q ) × ( V × Q ), and the linear form f ∈ ( V × Q ) ′ , defined by a (cid:0) ( u , p ) , ( v , q ) (cid:1) := a ( u , v ) − b ( v , p ) + c ( p, q ) + b ( u , q ) , f ( v , q ) := g ( v ) + ˜ f ( q ) . Again, the operator A : V × Q → ( V × Q ) ′ is defined by hA ( u , p ) , ( v , q ) i ( V × Q ) ′ × ( V × Q ) = a (cid:0) ( u , p ) , ( v , q ) (cid:1) .Choosing ε = 1 /n for n ∈ N we obtain a sequence of unique solutions ( m n , S n ) of the regularized problems (22).Owing to Proposition 2.3 the sequence (( m n , S n )) n ∈ N is bounded in V × Q . Thus there is a weakly convergentsubsequence, again denoted by (( m n , S n )) n ∈ N , which converges to ( m , S ) ∈ V × Q . In the same manner asin the proof of Theorem 1.8 we obtain the identity A ( m , S ) = f in ( V × Q ) ′ , i.e., ( m , S ) is a solution of thesemi-discrete mixed formulation (21).Uniqueness of the solution ( m , S ) ∈ V × Q follows from the strict monotonicity of A , which, in turn, is aconsequence of the strict monotonicity of A and C . The transient problem
Finally, we address the continuous transient problem. We restrict our considerations here to the case ofhomogeneous Dirichlet boundary conditions. Due to the lack of regularity of the solution m , it is not possibleto handle more general boundary conditions as in the former sections. Thus we consider the following problem:( α ( x , t ) + β ( x , t ) | m ( x , t ) | ) m ( x , t ) + ∇ S ( x , t ) = 0 , ( x , t ) ∈ Ω × [0 , T ] ,φ ( x ) ∂ρ ( S ( x , t ) , x , t ) ∂t + div ( m ( x , t )) = f ( x , t ) , ( x , t ) ∈ Ω × [0 , T ] ,S ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [0 , T ] ,S ( x ,
0) = S ( x ) , x ∈ Ω . (24)Again, we require that S ∈ W , / (Ω), φ ∈ L ∞ (Ω) with lower and upper bound 0 < φ ≤ φ ( x ) ≤ φ < ∞ foralmost every x ∈ Ω. For every t ∈ [0 , T ] the time-varying coefficient functions have to satisfy the followingassumptions: f ( · , t ) ∈ L (Ω) and α ( · , t ) , β ( · , t ) , γ ( · , t ) ∈ L ∞ (Ω) with upper and lower bounds0 < α ≤ α ( x , t ) ≤ α < ∞ , < β ≤ β ( x , t ) ≤ β < ∞ , < γ ≤ γ ( x , t ) ≤ γ < ∞ for almost every x ∈ Ω and every t ∈ [0 , T ] . Furthermore, we require these coefficient functions to be Lipschitz continuous in time, i.e., there exist constants L ( α ), L ( β ), L ( γ ) and L ( f ) such that for every 0 ≤ t ≤ t ≤ T : k α ( t ) − α ( t ) k , ∞ , Ω ≤ L ( α ) | t − t | , k β ( t ) − β ( t ) k , ∞ , Ω ≤ L ( β ) | t − t | , k γ ( t ) − γ ( t ) k , ∞ , Ω ≤ L ( γ ) | t − t | and k f ( t ) − f ( t ) k , , Ω ≤ L ( f ) | t − t | . PETER KNABNER AND GERHARD SUMM
A priori estimates for the solutions of the semi-discrete problems
As mentioned above we use the technique of semi-discretization in time to show the existence of solutions ofthe transient problem (24). One important step has been done in Section 2: The existence and uniqueness ofthe solutions to the semi-discrete problems has been established. In the next step, we have to consider the limit∆ t → K → ∞ ). Similar to the regularization technique employed in the last two sections, we thereforehave to provide a priori estimates for the solutions of the semi-discrete problems, which are independent of ∆ t .The bounds K m and K S of Proposition 2.3 do not fulfill this requirement. Thus we investigate the semi-discreteproblem (20) for homogeneous Dirichlet boundary conditions. In a slightly different notation this problem reads: a k ( m k , v ) − b ( v , S k ) = 0 for all v ∈ V , Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) q d x + b ( m k , q ) = f k ( q ) for all q ∈ Q , (25)where ρ k ( S k ) := γ k S k / p | S k | . Lemma 3.1.
For sufficiently small ∆ t > there exists a constant C S , independent of ∆ t (and of K ), such that k S k k , / , Ω ≤ C S for all ≤ k ≤ K . (26)
Proof.
Choosing v = m k and q = S k in (25) and adding the resulting equations yields a k ( m k , m k ) + Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) S k d x = f k ( S k ) . (27)Since a k ( m k , m k ) ≥ Z Ω φ (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) S k d x ≤ ∆ tf k ( S k ) = ∆ t Z Ω f k S k d x . Estimating the right hand side using Young’s inequality, we obtain: Z Ω f k S k d x ≤ Z Ω | f k || S k | d x ≤ Z Ω | f k | + 23 | S k | / d x = 13 k f k k , , Ω + 23 k S k k / , / , Ω . In a similar manner we can treat the left hand side. Since (cid:12)(cid:12)(cid:12)(cid:12)Z Ω φ ρ k − ( S k − ) S k d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω φ γ k − | S k − | / d x + 23 Z Ω φ γ k − | S k | / d x , it follows that Z Ω φ (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) S k d x ≥ Z Ω φ γ k | S k | / d x − Z Ω φ γ k − | S k − | / d x − Z Ω φ (cid:0) γ k − − γ k (cid:1) | S k | / d x . Due to the assumptions on γ the integrand in the last term can be bounded by (cid:12)(cid:12) φ γ k − − φ γ k (cid:12)(cid:12) = φ (cid:12)(cid:12) γ k − − γ k (cid:12)(cid:12) ≤ φ L ( γ )∆ t ≤ φ γ k L ( γ ) γ ∆ t . Merging all the above estimates together this results in Z Ω φ γ k | S k | / d x − Z Ω φ γ k − | S k − | / d x ≤ ∆ t k f k k , , Ω + 2 (cid:18) φ γ + L ( γ ) γ (cid:19) ∆ t Z Ω φ γ k | S k | / d x . IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW t is sufficient small such that C ∆ t := 2 (cid:16) φ γ + L ( γ ) γ (cid:17) ∆ t <
1, we can conclude that Z Ω φ γ k | S k | / d x ≤ − C ∆ t (cid:18)Z Ω φ γ k − | S k − | / d x + ∆ t k f k k , , Ω (cid:19) for k = 1 , . . . , K . As 1 / (1 − C ∆ t ) >
1, we obtain by induction for all k = 0 , . . . , K Z Ω φ γ k | S k | / d x ≤ (1 − C ∆ t ) − k Z Ω φ γ | S | / d x + k X i =1 ∆ t k f i k , , Ω ! ≤ (1 − C ∆ t ) − K (cid:18)Z Ω φ γ | S | / d x + T C f (cid:19) , where C f is an upper bound for k f k , , Ω . Note that for K → ∞ (i.e., ∆ t = T /K →
0) the expression(1 − C ∆ t ) − K = (1 − CT /K ) − K tends to e CT . In particular, this expression remains bounded. Lemma 3.2.
For sufficiently small ∆ t there exists a constant C m , independent of ∆ t (and of K ), such that k m k k , , Ω ≤ C m for all ≤ k ≤ K . (28)
Proof.
Choosing q = S k − S k − we obtain from the second equation in (25) Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) (cid:0) S k − S k − (cid:1) d x + b ( m k , S k − S k − ) = f k ( S k − S k − ) . Since the first term is non-negative, this implies b ( m k , S k − S k − ) ≤ f k ( S k − S k − ) . Furthermore, we choose v = m k in the first equation of (25) belonging to time step k and k − a k ( m k , m k ) − a k − ( m k − , m k ) = b ( m k , S k ) − b ( m k , S k − ) ≤ f k ( S k − S k − ) . Again, we apply Young’s inequality to show (cid:12)(cid:12)(cid:12)(cid:12)Z Ω α k − (cid:0) m k − · m k (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω α k | m k | d x + 12 Z Ω α k − | m k − | d x + 12 Z Ω (cid:0) α k − − α k (cid:1) | m k | d x , (cid:12)(cid:12)(cid:12)(cid:12)Z Ω β k − | m k − | (cid:0) m k − · m k (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω β k | m k | d x + 23 Z Ω β k − | m k − | d x + 13 Z Ω (cid:0) β k − − β k (cid:1) | m k | d x . Owing to the definition of a we obtain12 Z Ω α k | m k | d x − Z Ω α k − | m k − | d x − Z Ω (cid:0) α k − − α k (cid:1) | m k | d x + 23 Z Ω β k | m k | d x − Z Ω β k − | m k − | d x − Z Ω (cid:0) β k − − β k (cid:1) | m k | d x ≤ a k ( m k , m k ) − a k − ( m k − , m k ) ≤ f k ( S k − S k − ) . Summing this relation for i = 1 , . . . , k yields12 Z Ω α k | m k | d x + 23 Z Ω β k | m k | d x ≤ Z Ω α | m | d x + 23 Z Ω β | m | d x + k X i =1 (cid:20)Z Ω (cid:0) α i − − α i (cid:1) | m i | + 13 (cid:0) β i − − β i (cid:1) | m i | d x + f i ( S i − S i − ) (cid:21) . PETER KNABNER AND GERHARD SUMM
This inequality holds for k = 0 , . . . , K Due to (26) and the Lipschitz-continuity of f , the last term in this sumin bounded. Indeed, for k = 1 , . . . , K it holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =1 f i ( S i − S i − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = f k S k − f S + k − X i =1 ( f i − f i +1 ) S i ≤ k f k k , , Ω k S k k , / , Ω + k f k , , Ω k S k , / , Ω + k − X i =1 k f i − f i +1 k , , Ω k S i k , / , Ω ≤ C f C S + k − X i =1 L ( f )∆ tC S ≤ (2 C f + T L ( f )) C S . Analogously, the estimation of the other terms is based on the Lipschitz-continuity of α and β : Z Ω (cid:0) α i − − α i (cid:1) | m i | + 13 (cid:0) β i − − β i (cid:1) | m i | d x ≤ L ( α ) α ∆ t Z Ω α i | m i | d x + 13 L ( β ) β ∆ t Z Ω β i | m i | d x ≤ C ( α, β )∆ t Z Ω α i | m i | + β i | m i | d x = C ( α, β )∆ t a i ( m i , m i ) , where C ( α, β ) := max n L ( α ) α , L ( β ) β o . Summing up, we obtain for k = 1 , . . . , K k X i =1 Z Ω (cid:0) α i − − α i (cid:1) | m i | + 13 (cid:0) β i − − β i (cid:1) | m i | d x ≤ C ( α, β ) k X i =1 ∆ t a i ( m i , m i ) ≤ C ( α, β ) K X i =1 ∆ t a i ( m i , m i ) . Using (27) finally yields K X k =1 ∆ t a k ( m k , m k ) = K X k =1 (cid:20) ∆ tf k ( S k ) − Z Ω φ (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) S k d x (cid:21) ≤ K X k =1 ∆ t k f k k , , Ω k S k k , / , Ω − K X k =1 (cid:20) Z Ω φγ k | S k | / d x − Z Ω φγ k − | S k − | / d x (cid:21) + K X k =1 Z Ω φ (cid:0) γ k − − γ k (cid:1) | S k | / d x ≤ K X k =1 ∆ tC f C S + 13 Z Ω φ γ | S | / d x − Z Ω φ γ K | S K | / d x + 23 φ K X k =1 (cid:13)(cid:13) γ k − − γ k (cid:13)(cid:13) , ∞ , Ω k S k k / , / , Ω ≤ T C f C S + 23 φ γC / S + 23 φ K X k =1 L ( γ )∆ t C / S ≤ T C f C S + 23 φ ( γ + T L ( γ )) C / S . Summarizing all the relations above, we obtain the following inequality, which holds for k = 0 , . . . , K Z Ω α k | m k | d x + 23 Z Ω β k | m k | d x ≤ Z Ω α | m | d x + 23 Z Ω β | m | d x + (2 C f + T L ( f ) + C ( α, β ) T C f ) C S + C ( α, β ) 23 φ ( γ + T L ( γ )) C / S . Since β k > β >
0, this yields the assertion.
IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW Lemma 3.3.
For sufficiently small ∆ t there exists a constant C S ′ , independent of ∆ t (and of K ), such that K X k =1 ∆ t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) S k − S k − ∆ t (cid:12)(cid:12)(cid:12)(cid:12) / d x ≤ C S ′ . (29) Proof.
In a similar manner as in the proof of Lemma 3.2 we obtain Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) (cid:0) S k − S k − (cid:1) d x + a k ( m k , m k ) − a k − ( m k − , m k ) = Z Ω f k ( S k − S k − ) d x . Summing up for k = 1 , . . . , K yields K X k =1 φ γ ∆ t Z Ω S k p | S k | − S k − p | S k − | ! ( S k − S k − ) d x ≤ K X k =1 (cid:20)Z Ω f k ( S k − S k − ) d x − a k ( m k , m k ) + a k − ( m k − , m k ) (cid:21) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K X k =1 Z Ω f k ( S k − S k − ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + K X k =1 (cid:0) a k − ( m k − , m k ) − a k ( m k , m k ) (cid:1) . As we have seen in the proof of Lemma 3.2, the first term on the right hand side is bounded by (2 C f + T L ( f )) C S .For the second term the following estimate holds K X k =1 (cid:0) a k − ( m k − , m k ) − a k ( m k , m k ) (cid:1) ≤ K X k =1 (cid:18) Z Ω α k − | m k − | d x − Z Ω α k | m k | d x + 12 Z Ω (cid:0) α k − − α k (cid:1) | m k | d x + 23 Z Ω β k − | m k − | d x − Z Ω β k | m k | d x + 13 Z Ω (cid:0) β k − − β k (cid:1) | m k | d x (cid:19) = 12 Z Ω α | m | d x − Z Ω α K | m K | d x + 23 Z Ω β | m | d x − Z Ω β K | m K | d x + K X k =1 (cid:18) Z Ω (cid:0) α k − − α k (cid:1) | m k | d x + 13 Z Ω (cid:0) β k − − β k (cid:1) | m k | d x (cid:19) ≤ C ( α ) k m k , , Ω + 23 C ( β ) k m k , , Ω + C ( α, β ) (cid:18) T C f C S + 23 φ ( γ + T L ( γ )) C / S (cid:19) . Using (15) we therefore showed that there exists a constant
C >
0, independent of ∆ t , such that K X k =1 ∆ t Z Ω p | S k | + p | S k − | (cid:18) S k − S k − ∆ t (cid:19) d x ≤ K X k =1 t Z Ω S k p | S k | − S k − p | S k − | ! (cid:0) S k − S k − (cid:1) d x ≤ C . PETER KNABNER AND GERHARD SUMM
Applying H¨older’s inequality finally yields the assertion: K X k =1 ∆ t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) S k − S k − ∆ t (cid:12)(cid:12)(cid:12)(cid:12) / d x = K X k =1 ∆ t Z Ω (cid:18)q | S k | + q | S k − | (cid:19) / p | S k | + p | S k − | (cid:18) S k − S k − ∆ t (cid:19) ! / d x ≤ K X k =1 ∆ t Z Ω (cid:18)q | S k | + q | S k − | (cid:19) d x ! / ∆ t Z Ω p | S k | + p | S k − | (cid:18) S k − S k − ∆ t (cid:19) d x ! / ≤ K X k =1 ∆ t Z Ω (cid:18)q | S k | + q | S k − | (cid:19) d x ! / K X k =1 ∆ t Z Ω p | S k | + p | S k − | (cid:18) S k − S k − ∆ t (cid:19) d x ! / ≤ T / / C / S C / =: C S ′ . Next, we show that the mixed formulation (25) is equivalent to a variational formulation of the time-discretized parabolic equation (1). To this end, we recall the nonlinear mapping F of (3). For fixed time t = t k , we define the nonlinear mapping F k : (cid:0) L / (Ω) (cid:1) n → (cid:0) L (Ω) (cid:1) n and its inverse G k defined by G k ( v ) = (cid:0) α k + β k | v | (cid:1) v . Note that R Ω G k ( u ) · v d x = a k ( u , v ) for u , v ∈ (cid:0) L (Ω) (cid:1) n . Proposition 3.4. (a) If S k ∈ W , / (Ω) is a solution of the variational formulation: Find S k ∈ W , / (Ω) such that Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) q d x + Z Ω F k (cid:0) ∇ S k (cid:1) · ∇ q d x = f k ( q ) for all q ∈ W , / (Ω) , (30) then (cid:0) F k ( ∇ S k ) , S k (cid:1) is a solution of the mixed formulation (25) . In particular, F k ( ∇ S k ) ∈ W (div; Ω) . (b) If (cid:0) m k , S k (cid:1) ∈ W (div; Ω) × L / (Ω) is a solution of the mixed formulation (25) , then S k is a solution ofthe variational formulation (30) . In particular, S k ∈ W , / (Ω) .Proof. Ad a) Let S k be a solution of (30). We define m k := F ( ∇ S k ). Then Green’s formula yields Z Ω G k ( m k ) · v d x = Z Ω G k ( F k ( ∇ S k )) · v d x = Z Ω ∇ S k · v d x = − Z Ω div( v ) S k d x for all v ∈ W (div; Ω) . This is the first equation in (25). To derive the second equation in (25), we consider (30) for q ∈ D (Ω) ⊂ W , / (Ω), and apply Green’s formula again: Z Ω ( ρ k ( S k ) − ρ k − ( S k − ) − f k ) q d x = − Z Ω F k ( ∇ S k ) · ∇ q d x = − Z Ω m k · ∇ q d x . Thus the difference ρ k ( S k ) − ρ k − ( S k − ) − f ∈ L (Ω) is the generalized divergence of m k ; consequently m k ∈ W (div; Ω). Because D (Ω) is densely embedded into L / (Ω), the second equation in (25) follows.Ad b) Now, let ( m k , S k ) be a solution of (25). Green’s formula then implies Z Ω G k ( m k ) · v d x = − Z Ω div( v ) S k d x for all v ∈ ( D (Ω)) n . Thus in the sense of distributions it holds ∇ S k = G k ( m k ) ∈ (cid:0) L / (Ω) (cid:1) n . Consequently, S k ∈ W , / (Ω) and m k = F k ( ∇ S k ). To prove that S k fulfills (30), we consider q ∈ W , / (Ω) ⊂ L / (Ω) in the first equation of IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW f k ( q ) = Z Ω ( ρ k ( S k ) − ρ k − ( S k − )) q d x − Z Ω div( F k ( ∇ S k )) q d x = Z Ω ( ρ k ( S k ) − ρ k − ( S k − )) q d x + Z Ω F k ( ∇ S k ) · ∇ q d x . Finally, we consider again the first equation of (25) for v ∈ (cid:0) D ( ¯Ω) (cid:1) n . Applying Green’s formula we obtain0 = Z Ω ∇ S k · v d x + Z Ω div( v ) S k d x = Z ∂ Ω γ S k ( v · n ) dσ . Consequently, γ S k = 0 in W / , / ( ∂ Ω), i.e. S k ∈ W , / (Ω).Using this equivalence, we obtain a bound for S k in the norm of W , / (Ω). Lemma 3.5.
For sufficiently small ∆ t there exist constants C S , C ρ ′ and C m , all independent of ∆ t (and of K ), such that k S k k , / , Ω ≤ C S for all ≤ k ≤ K , (31) (cid:13)(cid:13)(cid:13)(cid:13) ρ k ( S k ) − ρ k − ( S k − )∆ t (cid:13)(cid:13)(cid:13)(cid:13) − , , Ω ≤ C ρ ′ for all ≤ k ≤ K , (32) (cid:13)(cid:13) div( m k ) (cid:13)(cid:13) − , , Ω ≤ C m for all ≤ k ≤ K . (33)
Proof.
Proposition 3.4 gives ∇ S k = − G k ( m k ). Therefore (28) implies (cid:13)(cid:13) ∇ S k (cid:13)(cid:13) , / , Ω = (cid:13)(cid:13) G k ( m k ) (cid:13)(cid:13) , / , Ω ≤ C ( α ) C m + C ( β ) C m =: C G . Together with (26) we obtain (31).Also, this equivalence yields (32), because by means of (30) we have for all q ∈ W , / (Ω) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) q d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f k ( q ) − Z Ω F k (cid:0) ∇ S k (cid:1) · ∇ q d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f k ( q ) + Z Ω m k · ∇ q d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k k , , Ω k q k , / , Ω + k m k k , , Ω k∇ q k , / , Ω ≤ (cid:0) k f k k , , Ω + k m k k , , Ω (cid:1) k q k , / , Ω . Finally, we obtain (33), since the first equation of (25) yields (cid:12)(cid:12)(cid:12)(cid:12)Z Ω div( m k ) q d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f k ( q ) − Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) q d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) k f k k , , Ω + φC ρ ′ (cid:1) k q k , / , Ω . for all q ∈ W , / (Ω) Solvability of the continuous problem
Due to the existence of unique solutions to the semi-discrete mixed formulation (25) we obtain for every K ∈ N a K + 1-tuple of solutions (cid:0) ( m k ∆ t , S k ∆ t ) (cid:1) k =0 ,...,K ∈ (cid:0) W (div; Ω) × L / (Ω) (cid:1) K +1 . Recall that ∆ t = T /K .We denote these K + 1-tuples with m ∆ t := ( m k ∆ t ) k =0 ,...,K ∈ (cid:0) W (div; Ω) (cid:1) K +1 and S ∆ t := ( S k ∆ t ) k =0 ,...,K ∈ PETER KNABNER AND GERHARD SUMM (cid:0) L / (Ω) (cid:1) K +1 . We define step functions, e.g. Π ∆ t S ∆ t ∈ L ∞ (0 , T ; W , / (Ω)), which are piecewise constant intime, by (Π ∆ t S ∆ t ) ( t ) := (cid:26) S t , if t = 0 ,S k ∆ t , if ( k − t < t ≤ k ∆ t , k = 1 , . . . , K , and piecewise linear (in time) functions Λ ∆ t S ∆ t ∈ C([0 , T ]; W , / (Ω)) fulfilling(Λ ∆ t S ∆ t ) ( t k ) = S k ∆ t for k = 0 , . . . , K . The time derivative of Λ ∆ t S ∆ t is a piecewise constant step function with valuesΛ ′ ∆ t S ∆ t ( t ) := ∂∂t (Λ ∆ t S ∆ t ) ( t ) = S k ∆ t − S k − t ∆ t , if ( k − t < t < k ∆ t , k = 1 , . . . , K . In addition, we use piecewise constant approximations γ ∆ t and f ∆ t of the coefficient functions γ and f , andpiecewise constant operators ρ ∆ t , F ∆ t and G ∆ t . Owing to the lemmas above the following bounds hold forsufficiently small time step sizes ∆ t : k Π ∆ t S ∆ t k L ∞ (0 ,T ;W , / (Ω)) ≤ C S , k Λ ′ ∆ t S ∆ t k L / (0 ,T ;L / (Ω)) ≤ C S ′ , k Π ∆ t ρ ∆ t ( S ∆ t ) k L ∞ (0 ,T ;L (Ω)) ≤ γ p C S , k Λ ′ ∆ t ρ ∆ t ( S ∆ t ) k L ∞ (0 ,T ;W − , (Ω)) ≤ C ρ ′ , k Π ∆ t m ∆ t k L ∞ (0 ,T ;(L (Ω)) n ) ≤ C m , k Π ∆ t div( m ∆ t ) k L ∞ (0 ,T ;W − , (Ω)) ≤ C m , k Π ∆ t G ∆ t ( m ∆ t ) k L ∞ (0 ,T ;(L / (Ω)) n ) ≤ C G , (cid:13)(cid:13) ρ K ( S K ) (cid:13)(cid:13) , , Ω ≤ γ p C S . The third (and the last) inequality follow from Z Ω (cid:12)(cid:12) ρ k ( S k ) (cid:12)(cid:12) d x = Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ k S k p | S k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x ≤ γ Z Ω (cid:12)(cid:12) S k (cid:12)(cid:12) / d x ≤ γ C / S . Thus there exist subsequences, again indexed by ∆ t , that converge in the corresponding weak*-topology; indetail Π ∆ t S ∆ t ∗ ⇀ S in L ∞ (cid:0) , T ; W , / (Ω) (cid:1) , Λ ′ ∆ t S ∆ t ⇀ S ′ in L / (cid:0) , T ; L / (Ω) (cid:1) , Π ∆ t ρ ∆ t ( S ∆ t ) ∗ ⇀ R in L ∞ (cid:0) , T ; L (Ω) (cid:1) , Λ ′ ∆ t ρ ∆ t ( S ∆ t ) ∗ ⇀ R ′ in L ∞ (cid:0) , T ; W − , (Ω) (cid:1) , Π ∆ t m ∆ t ∗ ⇀ m in L ∞ (cid:0) , T ; (L (Ω)) n (cid:1) , Π ∆ t div( m ∆ t ) ∗ ⇀ m in L ∞ (cid:0) , T ; W − , (Ω) (cid:1) , Π ∆ t G ∆ t ( m ∆ t ) ∗ ⇀ g in L ∞ (cid:0) , T ; (L / (Ω)) n (cid:1) ρ K ( S K ) ⇀ R T in L (Ω) . (34) Proposition 3.6.
The limits S of Π ∆ t S ∆ t and R of Π ∆ t ρ ∆ t ( S ∆ t ) from (34) satisfy ρ ( S ) = R almost everywherein (0 , T ) × Ω . IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW Proof.
As well as Π ∆ t S ∆ t , also Λ ∆ t S ∆ t is bounded in L ∞ (cid:0) , T ; W , / (Ω) (cid:1) . In particular, Λ ∆ t S ∆ t and itspartial derivatives ( ∂/∂x i )Λ ∆ t S ∆ t are bounded in L / (cid:0) , T ; L / (Ω) (cid:1) . Owing to (29) ( ∂/∂t )Λ ∆ t S ∆ t = Λ ′ ∆ t S ∆ t is bounded, too, such that Λ ∆ t S ∆ t is bounded in W , / ((0 , T ) × Ω). The Rellich–Kondrachov-Theorem yieldsthat W , / is embedded compactly in L / . Thus there exists a subsequence (again denoted by Λ ∆ t ), whichconverges (strongly) in L / ((0 , T ) × Ω) to S . Choosing a further subsequence, we obtain that Λ ∆ t S ∆ t convergesalmost everywhere in (0 , T ) × Ω to S . Applying the mapping ρ ∆ t yields lim ∆ t → Λ ∆ t ρ ∆ t ( S ∆ t ) = ρ ( S ) a.e. in(0 , T ) × Ω. Since Λ ∆ t ρ ∆ t ( S ∆ t ) is bounded in L ((0 , T ) × Ω), we can conclude that Λ ∆ t ρ ∆ t ( S ∆ t ) weakly convergesto ρ ( S ) in L ((0 , T ) × Ω), i.e. for all q ∈ L / ((0 , T ) × Ω) it holdslim ∆ t → Z T Z Ω Λ ∆ t ρ ∆ t ( S ∆ t ) q d x dt = Z T Z Ω ρ ( S ) q d x dt . On the other handlim ∆ t → Z T Z Ω Λ ∆ t ρ ∆ t ( S ∆ t ) q d x dt = lim ∆ t → Z T Z Ω Π ∆ t ρ ∆ t ( S ∆ t ) q d x dt = Z T Z Ω Rq d x dt . Since the limit of a convergent sequence is unique, the assertion follows.For the remainder of this section, we denote by h· , ·i the dual pairing between W − , (Ω) and W , / (Ω). Proposition 3.7. a) The identity S ′ = ( ∂/∂t ) S holds in the sense of distributions from (0 , T ) to L / (Ω) ,i.e., for all ϕ ∈ D ((0 , T )) it holds: Z T S ′ ( t ) ϕ ( t ) dt = − Z T S ( t ) ϕ ′ ( t ) dt in L / (Ω) . b) The identity R ′ = ( ∂/∂t ) R holds in the sense of distributions from (0 , T ) to W − , (Ω) , i.e., for all ϕ ∈ D ((0 , T )) it holds: Z T R ′ ( t ) ϕ ( t ) dt = − Z T R ( t ) ϕ ′ ( t ) dt in W − , (Ω) . c) The identity m = div( m ) in the sense of distributions on Ω holds almost everywhere in (0 , T ) , i.e., for all Ψ ∈ D (Ω) it holds: h m , Ψ i = − Z Ω m · ∇ Ψ d x a.e. in (0 , T ) . d) The identity g = −∇ S holds in L ∞ (cid:0) , T ; (L / (Ω)) n (cid:1) , i.e., for all v ∈ L (cid:0) , T ; (L (Ω)) n (cid:1) it holds: Z T Z Ω g · v d x dt = − Z T Z Ω ∇ S · v d x dt . Proof.
Ad a) From the second equation in (34) we can conclude that for all ϕ ∈ D ((0 , T ))lim ∆ t → Z T Λ ′ ∆ t S ∆ t ( t ) ϕ ( t ) dt = Z T S ′ ( t ) ϕ ( t ) dt in L / (Ω) . PETER KNABNER AND GERHARD SUMM
On the other hand, partial integration yields Z T Λ ′ ∆ t S ∆ t ( t ) ϕ ( t ) dt = K X k =1 Z k ∆ t ( k − t Λ ′ ∆ t S ∆ t ( t ) ϕ ( t ) dt = K X k =1 (cid:16) Λ ∆ t S ∆ t ( k ∆ t ) ϕ ( k ∆ t ) − Λ ∆ t S ∆ t (( k − t ) ϕ (( k − t ) (cid:17) − K X k =1 Z k ∆ t ( k − t Λ ∆ t S ∆ t ( t ) ϕ ′ ( t ) dt = − Z T Λ ∆ t S ∆ t ( t ) ϕ ′ ( t ) dt , such that Z T S ′ ( t ) ϕ ( t ) dt = lim ∆ t → Z T Λ ′ ∆ t S ∆ t ( t ) ϕ ( t ) dt = lim ∆ t → − Z T Λ ∆ t S ∆ t ( t ) ϕ ′ ( t ) dt = lim ∆ t → − Z T Π ∆ t S ∆ t ( t ) ϕ ′ ( t ) dt = − Z T S ( t ) ϕ ′ ( t ) dt . The identity in b) follows in a similar manner as the identity in a).Ad c) Let Ψ ∈ D (Ω) and ϕ ∈ D ((0 , T )) be arbitrarily chosen. Thenlim ∆ t → Z T h Π ∆ t div( m ∆ t ) , Ψ i ϕ ( t ) dt = lim ∆ t → Z T h Π ∆ t div( m ∆ t ) , ϕ ( t )Ψ i dt = Z T h m ( t ) , ϕ ( t )Ψ i dt = Z T h m ( t ) , Ψ i ϕ ( t ) dt , because m is the limit of (Π ∆ t div( m ∆ t )) ∆ t in L ∞ (cid:0) , T ; W − , (Ω) (cid:1) . On the other hand Z T h m ( t ) , Ψ i ϕ ( t ) dt = lim ∆ t → Z T h Π ∆ t div( m ∆ t ) , Ψ i ϕ ( t ) dt = lim ∆ t → − Z T Z Ω Π ∆ t m ∆ t · ∇ Ψ d x ϕ ( t ) dt = lim ∆ t → − Z T Z Ω Π ∆ t m ∆ t · ∇ Ψ ϕ ( t ) d x dt = − Z T Z Ω m · ∇ Ψ ϕ ( t ) d x dt = − Z T Z Ω m · ∇ Ψ d x ϕ ( t ) dt . Since ϕ is arbitrarily chosen, the assertion follows.Ad d) We have seen in the proof of Proposition 3.4 that Z Ω G k ( m k ) · v d x = − Z Ω ∇ S k · v d x for all v ∈ (L (Ω)) n . Consequently, for v ∈ L (cid:0) , T ; (L (Ω)) n (cid:1) it holds Z T Z Ω g · v d x dt = lim ∆ t → Z T Z Ω Π ∆ t G ∆ t ( m ∆ t ) · v d x dt = lim ∆ t → − Z T Z Ω ∇ (cid:0) Π ∆ t S ∆ t (cid:1) · v d x dt = − Z T Z Ω ∇ S · v d x dt . IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW , T ] we have that S ∈ C (cid:0) [0 , T ]; L / (Ω) (cid:1) and R ∈ C (cid:0) [0 , T ]; W − , (Ω) (cid:1) . Thus for every t ∈ [0 , T ] the value S ( t ) ∈ L / (Ω) is well defined. Proposition 3.8.
The following identity holds in L ∞ (cid:0) , T ; W − , (Ω) (cid:1) : φ ∂ρ ( S ) ∂t + div( m ) = f . Furthermore ρ ( S (0)) = ρ ( S ) and ρ ( S ( T )) = R T .Proof. For ϕ ∈ D ((0 , T )) we define a step function ϕ ∆ t by ϕ ∆ t ( t ) := (cid:26) ϕ (( k − t ) , if ( k − t ≤ t < k ∆ t , k = 1 , . . . , K ,ϕ ( T ) , if t = T .
Using the test function q = Ψ ∈ D (Ω) in (25), multiplying by ∆ t ϕ (( k − t ) and summing up for k = 1 , . . . , K ,we obtain K X k =1 ∆ t Z Ω φ ρ k ( S k ) − ρ k − ( S k − )∆ t Ψ d x ϕ (( k − t )+ K X k =1 ∆ t Z Ω div( m k ) Ψ d x ϕ (( k − t ) = K X k =1 ∆ t Z Ω f k Ψ d x ϕ (( k − t ) (35)Employing the piecewise constant functions Π ∆ t and Λ ′ ∆ t this reads K X k =1 Z k ∆ t ( k − t Z Ω φ Λ ′ ∆ t ρ ∆ t ( S ∆ t )Ψ d x ϕ ∆ t dt + K X k =1 Z k ∆ t ( k − t Z Ω Π ∆ t div( m ∆ t ) Ψ d x ϕ ∆ t dt = K X k =1 Z k ∆ t ( k − t Z Ω Π ∆ t f ∆ t Ψ d x ϕ ∆ t dt , and after joining the integrals over t Z T Z Ω φ Λ ′ ∆ t ρ ∆ t ( S ∆ t ) Ψ ϕ ∆ t d x dt + Z T Z Ω Π ∆ t div( m ∆ t ) Ψ ϕ ∆ t d x dt = Z T Z Ω Π ∆ t f ∆ t Ψ ϕ ∆ t d x dt . Since (Ψ ϕ ∆ t ) ∆ t strongly converges to Ψ ϕ in L (cid:16) , T ; W , / (Ω) (cid:17) , we can pass to the limit ∆ t → Z T (cid:28) φ ∂ρ ( S ) ∂t , Ψ ϕ (cid:29) dt + Z T h ( m ) , Ψ ϕ i dt = Z T Z Ω f Ψ ϕ d x dt . (36)But the set n Ψ( x ) ϕ ( t ) (cid:12)(cid:12) Ψ ∈ D (Ω) , ϕ ∈ D ((0 , T )) o is a dense subset of L (cid:16) , T ; W , / (Ω) (cid:17) . Therefore the firstidentity in Proposition (3.8) is established.2 PETER KNABNER AND GERHARD SUMM
To prove the remaining two identities we first conclude from (35) that K X k =1 (cid:18)Z Ω φρ k ( S k )Ψ d x ϕ (( k − t ) − Z Ω φρ k − ( S k − )Ψ d x ϕ (( k − t ) (cid:19) + K X k =1 ∆ t Z Ω div( m k ) Ψ d x ϕ (( k − t ) = K X k =1 ∆ t Z Ω f k Ψ d x ϕ (( k − t ) . Rearranging the terms in the first line yields K X k =1 (cid:18)Z Ω φρ k ( S k )Ψ d x ϕ (( k − t ) − Z Ω φρ k − ( S k − )Ψ d x ϕ (( k − t ) (cid:19) = − K X k =1 Z Ω φρ k ( S k )Ψ d x (cid:16) ϕ ( k ∆ t ) − ϕ (( k − t ) (cid:17) + Z Ω φρ K ( S K )Ψ d x ϕ ( K ∆ t ) − Z Ω φρ ( S )Ψ d x ϕ (0) . Like above this leads to − Z T Z Ω φ Π ∆ t ρ ∆ t ( S ∆ t ) Ψ d x ϕ ( k ∆ t ) − ϕ (( k − t )∆ t dt + Z T Z Ω Π ∆ t div( m ∆ t ) Ψ d x ϕ ∆ t dt = Z T Z Ω Π ∆ t f ∆ t Ψ d x ϕ ∆ t dt + Z Ω φρ ( S )Ψ d x ϕ (0) − Z Ω φρ K ( S K )Ψ d x ϕ ( T ) . Passing to the limit ∆ t → − Z T Z Ω φρ ( S ) Ψ d x ∂ϕ∂t dt + Z T h ( m ) , Ψ i ϕ dt = Z T Z Ω f Ψ d x ϕ dt + Z Ω φρ ( S )Ψ d x ϕ (0) − Z Ω φR T Ψ d x ϕ ( T ) . In the other hand, partial integration of (36) yields − Z T Z Ω φρ ( S ) Ψ d x ∂ϕ∂t dt + Z T h div( m ) , Ψ i ϕ dt = Z T Z Ω f Ψ d x ϕ dt + Z Ω φρ ( S (0))Ψ d x ϕ (0) − Z Ω φρ ( S ( T ))Ψ d x ϕ ( T ) . Subtracting the last two equations we can conclude (cid:18)Z Ω φρ ( S )Ψ d x − Z Ω φρ ( S (0))Ψ d x (cid:19) ϕ (0) − (cid:18)Z Ω φR T Ψ d x − Z Ω φρ ( S ( T ))Ψ d x (cid:19) ϕ ( T ) = 0 . Since ϕ (0) and ϕ ( T ) are arbitrary, this implies Z Ω φρ ( S )Ψ d x = Z Ω φρ ( S (0))Ψ d x and Z Ω φR T Ψ d x = Z Ω φρ ( S ( T ))Ψ d x and finally ρ ( S (0)) = ρ ( S ) and ρ ( S ( T )) = R T .Only the identity g = G ( m ) is missing yet. To show this, we need an auxiliary result, a generalization ofLemma 1.2 from [7]. IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW Lemma 3.9.
The limit S of (Π ∆ t S ∆ t ) ∆ t satisfies: Z T (cid:28) φ ∂ρ ( S ) ∂t , S (cid:29) dt = 13 (cid:18)Z Ω φ | ρ ( S ( T )) | | S ( T ) | d x − Z Ω φ | ρ ( S (0)) | | S (0) | d x (cid:19) + 23 Z T Z Ω φ ∂γ∂t | S | / d x dt . Proof.
We prolongate S to a function ˜ S , defined on [ − T, T ], by˜ S ( t ) := S ( − t ) , if − T ≤ t ≤ ,S ( t ) , if 0 ≤ t ≤ T ,S (2 T − t ) , if T ≤ t ≤ T ,
Owing to the corresponding properties of S , we can conclude that ˜ S ∈ C (cid:0) [ − T, T ]; L / (Ω) (cid:1) and ( ∂/∂t ) ρ ( ˜ S ) ∈ L ∞ (cid:0) − T, T ; W − , (Ω) (cid:1) . For ∆ t > X ∆ t := 1∆ t Z T Z Ω φ (cid:16) ρ ( ˜ S ( t )) − ρ ( ˜ S ( t − ∆ t )) (cid:17) ˜ S ( t ) d x . In the limit ∆ t → cf. [7, proof of Lemma 1.2])lim ∆ t → X ∆ t = Z T (cid:28) φ ∂ρ ( S ) ∂t , S (cid:29) dt . Like in the proof of Lemma 3.1 an application of Young’s inequality yields X ∆ t ≥ t Z T Z Ω φ | ρ ( ˜ S ( t )) | | ˜ S ( t ) | d x − Z Ω φ | ρ ( ˜ S ( t − ∆ t )) | | ˜ S ( t − ∆ t ) | d x dt + 1∆ t Z T Z Ω ( φ γ ( t ) − φ γ ( t − ∆ t )) | ˜ S ( t ) | / d x dt = 13 φ ∆ t Z TT − ∆ t Z Ω | ρ ( ˜ S ( t )) || ˜ S ( t ) | d x dt − Z − ∆ t Z Ω | ρ ( ˜ S ( t )) || ˜ S ( t ) | d x dt ! + 23 Z T Z Ω φ ∆ t ( γ ( t ) − γ ( t − ∆ t )) | ˜ S ( t ) | / d x dt . Therefore we obtain in the limit ∆ t → Z T (cid:28) φ ∂ρ ( S ) ∂t , S (cid:29) dt = lim ∆ t → X ∆ t ≥ (cid:18)Z Ω φ | ρ ( S ( T )) | | S ( T ) | d x − Z Ω φ | ρ ( S (0)) | | S (0) | d x (cid:19) + 23 Z T Z Ω φ ∂γ∂t | S ( t ) | / d x dt . Applying the same transformations and estimations to Y ∆ t := 1∆ t Z T Z Ω φ (cid:16) ρ ( ˜ S ( t + ∆ t )) − ρ ( ˜ S ( t )) (cid:17) ˜ S ( t ) d x , PETER KNABNER AND GERHARD SUMM we find Z T (cid:28) φ ∂ρ ( S ) ∂t , S (cid:29) dt = lim ∆ t → Y ∆ t ≤ (cid:18)Z Ω φ | ρ ( S ( T )) || S ( T ) | d x − Z Ω φ | ρ ( S (0)) || S (0) | d x (cid:19) + 23 Z T Z Ω φ ∂γ∂t | S ( t ) | / d x dt . Together with the estimate from the consideration of X ∆ t above, the assertion follows. Proposition 3.10.
The limits m of (Π ∆ t m ∆ t ) ∆ t and g of (Π ∆ t G ∆ t ( m ∆ t )) ∆ t satisfy g = G ( m ) , i.e., for all v ∈ L (cid:0) , T ; (L (Ω)) n (cid:1) : Z T Z Ω g · v d x dt = Z T Z Ω G ( m ) · v d x dt . Proof.
Again, we employ (27), replacing a k ( m k , m k ) by R Ω G k ( m k ) · m k d x , i.e., Z Ω G k ( m k ) · m k d x + Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) S k d x = Z Ω f k S k d x and the inequality (see the proof of Lemma 3.1)13 Z Ω φ ∆ t | ρ k ( S k ) || S k | d x − Z Ω φ ∆ t | ρ k − ( S k − ) || S k − | d x + 23 Z Ω φ ∆ t (cid:0) γ k − γ k − (cid:1) | S k | / d x ≤ Z Ω φ ∆ t (cid:0) ρ k ( S k ) − ρ k − ( S k − ) (cid:1) S k d x . Multiplying with ∆ t and summing up for k = 1 , . . . , K , we obtain13 Z Ω φ | ρ K ( S K ) || S K | d x − Z Ω φ | ρ ( S ) || S | d x + 23 Z T ∆ t Z Ω φ (Π ∆ t γ ∆ t ( t ) − Π ∆ t γ ∆ t ( t − ∆ t )) | Π ∆ t S ∆ t | / d x dt + Z T Z Ω Π ∆ t G ∆ t ( m ∆ t ) · Π ∆ t m ∆ t d x dt ≤ Z T Z Ω Π ∆ t f ∆ t Π ∆ t S ∆ t d x dt . Taking the limes inferior, we can conclude that13 Z Ω φ | ρ ( S ( T )) || S ( T ) | d x − Z Ω φ | ρ ( S (0)) || S (0) | d x + 23 Z T Z Ω φ ∂γ∂t | S | / d x dt + lim inf ∆ t → Z T Z Ω Π ∆ t G ∆ t ( m ∆ t ) · Π ∆ t m ∆ t d x dt ≤ Z T Z Ω f S d x dt . Thus Lemma 3.9 yields the following inequality Z T (cid:28) φ ∂ρ ( S ) ∂t , S (cid:29) dt + lim inf ∆ t → Z T Z Ω Π ∆ t G ∆ t ( m ∆ t ) · Π ∆ t m ∆ t d x dt ≤ Z T Z Ω f S d x dt . On the other hand, Proposition 3.8 implies Z T (cid:28) φ ∂ρ ( S ) ∂t , S (cid:29) dt + Z T Z Ω g · m d x dt = Z T Z Ω f S d x dt , IXED FORMULATION FOR DARCY–FORCHHEIMER FLOW Z T h div( m ) , S i dt = − Z T Z Ω m · ∇ S d x dt = Z T Z Ω m · g d x dt . Consequently, lim inf ∆ t → Z T Z Ω Π ∆ t G ∆ t ( m ∆ t ) · Π ∆ t m ∆ t d x dt ≤ Z T Z Ω g · m d x dt . Thus we have shown that for arbitrary v ∈ L ∞ (cid:0) , T ; (L (Ω)) n (cid:1)Z T Z Ω ( g − G ( v )) · ( m − v ) d x dt ≥ lim inf ∆ t → Z T Z Ω (Π ∆ t G ∆ t ( m ∆ t ) − Π ∆ t G ∆ t ( v )) · (Π ∆ t m ∆ t − v ) d x dt ≥ . Now the assertion follows from the fact that G is a maximal monotone operator on L ∞ (cid:0) , T ; (L (Ω)) n (cid:1) ( cf. theproof of Thm. 1.1 in [7]).Now we are in a position to formulate and prove our main result: Theorem 3.11.
For all f ∈ L ∞ (cid:0) , T ; L (Ω) (cid:1) that are Lipschitz continuous in t there exists a pair ( m , S ) ∈ L ∞ (cid:0) , T ; (L (Ω)) n (cid:1) × L ∞ (cid:0) , T ; W , / (Ω)) (cid:1) such that Z T Z Ω G ( m ) · v d x dt − Z T h div( v ) , S i dt = 0 for all v ∈ L (cid:0) , T ; (L (Ω)) n (cid:1) , Z T (cid:28) φ ∂ρ ( S ) ∂t , q (cid:29) dt + Z T h div( m ) , q i dt = Z T Z Ω f q d x dt for all q ∈ L (cid:0) , T ; W , / (Ω)) (cid:1) . Proof.
Let m be the limit of (Π ∆ t m ∆ t ) ∆ t and S be the limit of (Π ∆ t S ∆ t ) ∆ t . Then Proposition 3.10 andProposition 3.7 d) imply that Z T Z Ω G ( m ) · v d x dt = Z T Z Ω g · v d x dt = − Z T Z Ω ∇ S · v d x dt = Z T h div( v ) , S i dt for all v ∈ L (cid:0) , T ; (L (Ω)) n (cid:1) . Thus ( m , S ) satisfies the first equation above. In Proposition 3.8 we have seenthat ( m , S ) fulfills the second equation, too. Remark 3.12. a) By means of the definition of the generalized divergence, we can replace the dual pairing h div( v ) , q i for v ∈ (cid:0) L (Ω) (cid:1) n and q ∈ W , / (Ω) with the integral − R Ω v · ∇ q d x . Thus, in the case ofthe continuous transient problem, we have established the existence of a solution of the primal mixedformulation ( cf. [8, Sect. I.3.2]). In contrast, we considered the uniqueness and existence of a solution ofthe dual mixed formulation for the stationary and semi-discrete transient Problem. This lack of regularityof the vector solution m hinders the consideration of more general boundary conditions.b) Assuming additional regularity properties of the solution, Amirat [2] showed that the solution to thecorresponding parabolic Neumann-problem is unique. Furthermore, he proved that the solution is positiveprovided that the initial and boundary conditions satisfy corresponding requirements. Appendix A. Properties of W s (div; Ω) We introduce the generalization W s (div; Ω) of H(div; Ω), defined byW s (div; Ω) := (cid:8) v ∈ (L s (Ω)) n (cid:12)(cid:12) div( v ) ∈ L s (Ω) (cid:9) , PETER KNABNER AND GERHARD SUMM and equip it with the norm k v k W s (div;Ω) := Z Ω n X i =1 | v i ( x ) | s d x + Z Ω | div( v ( x )) | s d x ! /s , where v = ( v , . . . , v n ) T . Since W s (div; Ω) is a closed subspace of ( L s (Ω)) n +1 , it follows that W s (div; Ω) is areflexive Banach space.It is straightforward to extend the proofs of Thm. 2.4 and Thm. 2.5 in [5] to show the next two lemmas: Lemma A.1.
The space D ( ¯Ω) n is dense in W s (div; Ω) . Lemma A.2.
The mapping γ n : v v · n defined on D ( ¯Ω) n can be extended by continuity to a linear andcontinuous mapping, still denoted by γ n , from W s (div; Ω) into (cid:0) W /s,r ( ∂ Ω) (cid:1) ′ . In particular, Green’s formula Z Ω v · ∇ ψ d x + Z Ω div( v ) ψ d x = Z ∂ Ω ψ ( v · n ) dσ (37) holds for every v ∈ W s (div; Ω) and ψ ∈ W ,r (Ω) , where /s + 1 /r = 1 . For s >
1, the well known inf-sup condition (see e.g. [3, § II.1]) can be extended, too. Generalizing thedefinition of the bilinear form b from Section 1 onto W s (div; Ω) × L r (Ω), we define b ( v , q ) := R Ω div( v ) q d x for v ∈ W s (div; Ω), q ∈ L r (Ω). Lemma A.3.
Let s > and /s + 1 /r = 1 . Then there exists a constant θ > such that θ k q k ,r, Ω ≤ sup v ∈ W s (div;Ω) b ( v , q ) k v k W s (div;Ω) for all v ∈ W s (div; Ω) , q ∈ L r (Ω) . (38) Proof.
We define a mapping B : W s (div; Ω) → L s (Ω) = (L r (Ω)) ′ by means of h B v , q i = b ( v , q ). Since s >
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