Generalized Forchheimer Equations for Porous Media. Part V.

Transcription

1 Generalized Forchheimer Equations for Porous Media. Part V. Luan Hoang,, Akif Ibragimov, Thinh Kieu and Zeev Sobol Department of Mathematics and Statistics, Texas Tech niversity Mathematics Department, Swansea niversity lhoang/ Project on Generalized Forchheimer Flows: also with Eugenio Aulisa (TT), Lidia Bloshanskaya (TT), Tuoc Phan (Tennessee) Applied Mathematics Seminars Texas Tech niversity March 20, 2013 Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

2 Outline 1 Introduction 2 Bounds for the solutions 3 Dependence on the boundary data Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

3 Introduction Fluid flows in porous media with velocity u and pressure p: Darcy s Law: αu = p, the two term law αu + β u u = p, the three term law Au + B u u + C u 2 u = p. the power law au + c n u n 1 u = p, Here α, β, a, c, n, A, B, and C are empirical positive constants. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

4 General Forchheimer equations Generalizing the above equations as follows g( u )u = p. Let G(s) = sg(s). Then G( u ) = p u = G 1 ( p ). Hence where p u = g(g 1 ( p )) u = K( p ) p, K(ξ) = K g (ξ) = 1 g(s) = 1 g(g 1, sg(s) = ξ. (ξ)) We derived a non-linear equation of Darcy type from Forchheimer equations. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

5 IBVP for pressure Let ρ be the density. Continuity equation dρ + (ρu) = 0. dt For slightly compressible fluid: dρ dp = 1 κ ρ, where κ 1. Then dp ( ) dt = κ K( ) p + K( p ) p 2. Since κ 1, we neglect the last terms, after scaling the time variable: dp ( ) dt = K( ) p. Consider the equation on a bounded domain in R 3 with boundary Γ. Dirichlet boundary data: u = ψ(x, t) is known on Γ. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

6 Historical remarks Darcy-Dupuit: 1865 Forchheimer: 1901 Other nonlinear models: 1940s 1960s Incompressible fluids: Payne, Straughan and collaborators since 1990 s, Celebi-Kalantarov-gurlu since 2005 (Brinkman-Forchheimer) Derivation of non-darcy, non-forchheimer flows: Marusic-Paloka and Mikelic 2009 (homogenization for Navier Stokes equations), Balhoff et. al (computational) Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

7 Works on generalized Forchheimer flows A. Single-phase flows s Numerical study L 2 -theory (for slightly compressible flows): Aulisa-Bloshanskaya-Ibragimov-LH (2009), Ibragimov-LH: Dirichlet B.C. (2011), Ibragimov-LH Flux B.C. (2012), Aulisa-Bloshanskaya-Ibragimov total flux, productivity index (2011, 2012). L α -theory : Ibragimiv-Kieu-Sobol-LH (this talk). L -theory : Kieu-Phan-LH (in progress). B. Multi-phase flows. One-dimensional case: Ibragimov-Kieu-LH (2013). Multi-dimensional case: Ibragimov-Kieu-LH (in preparation). Note: there are more works on Forchheimer flows (2-terms or 3 terms). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

8 Class FP(N, α) We introduce a class of Forchheimer polynomials Definition A function g(s) is said to be of class FP(N, α) if g(s) = a 0 s α 0 + a 1 s α 1 + a 2 s α a N s α N = N a j s α j, j=0 where N > 0, 0 = α 0 < α 1 < α 2 <... < α N, and a 0, a N > 0, a 1,..., a N 1 0. Notation: α N = deg(g), a = (a 0, a 1,..., a N ), a = α N a (0, 1), b = α N a = α N (0, 1). α N + 2 Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

9 Degeneracy - The Degreee Condition Lemma Let g(s, a) be in class FP(N, α). One has for any ξ 0 that Degree Condition (DC) C 1 ( a) (1 + ξ) a K(ξ, a) C 2( a) (1 + ξ) a, C 3 ( a)(ξ 2 a 1) K(ξ, a)ξ 2 C 2 ( a)ξ 2 a. deg(g) 4 n 2 2 (2 a) = n(2 a) n (2 a). nder the (DC), the Sobolev space W 1,2 a () is continuously embedded into L 2 (). (NDC) deg(g) > 4 n 2 2 > (2 a) = n(2 a) n (2 a). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

10 Definitions and notations Let p(x, t) = p(x, t) Ψ(x, t) where Ψ(x, t) is an extension of ψ(x, t) from to. [ ] 2 a [ ] 1 G 1 (t)= Ψ(x, t) 2 dx + Ψ t (x, t) r r 0 (1 a) 0 dx + Ψ t (x, t) r r 0 0 dx G 2 (t) = Ψ t (x, t) 2 dx + Ψ t (x, t) 2 dx, G 3 (t) = G 1 (t) + G 2 (t), G 4 (t) = G 3 (t) + Ψ tt 2 dx, where r 0 = n(2 a) (2 a)(n+1) n. For any α > 0, we define α = na 2 a = nα N α N + 2, { max{α, α } in the NDC case α = max{α, 2, α } = max{α, 2} in the DC case. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

11 Estimates of solutions - I: for p For α > 0 and t 0, we define A(α, t) = Ψ α(2 a) 2 dx + B(α, t) = e c 5(α)t Theorem Let α > 0. Then for all t 0, and lim sup Ψ t α dx, t 0, t p(x, 0) α dx + e c5(α)(t τ) A(α, τ)dτ. 0 p(x, t) α dx C(1 + B( α, t)) p α dx C(1 + lim sup A( α, t)). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

12 Ideas of proof α α a. Multiplying equation of p by p α 1 sgn( p), integrating over Let γ 0 =, and many other calculations give d ( p α dx C p 2 a p α 2 dx+ε dt Poincaré-Sobolev inequality: from the Sobolev space W 1,2 a () into L α(2 a) α a gives sing this, d ( p α dx C dt ) 1 p α γ dx 0 +C ε (1+A(α, t)). for α max{2, α }, applying the imbedding α a () to function u 2 a u 2 a u α 2 dx c 2 ( ) 1 p α γ dx 0 Selecting ε sufficiently small, we obtain d ( p α dx C p α dx dt ) 1 u α γ dx 0. ( ) 1 + ε p α γ dx 0 ) 1 γ 0 + C(A(α, t) + 1). + C ε (1 + A(α, t)). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

13 Differential inequality Then we use the following estimates. Lemma Let φ : [0, ) [0, ) be continuous, strictly increasing. Suppose y(t) 0 is a continuous function on [0, ) such that y h(t)φ 1 (y(t)) + f (t), t > 0, where h(t) > 0, f (t) ) 0 for t 0 are continuous functions on [0, ). (i) If M(t) = Env on [0, ) then ( f (t) h(t) (ii) If 0 h(τ)dτ = then y(t) y(0) + φ(m(t)) for all t 0. lim sup y(t) φ ( lim sup ) f (t). h(t) Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

14 Estimates of solutions - II: for p Improving L 2 -theory. Previously, d H( p )dx d 5 dt H( p )dx + C where H( p) K( p ) p 2 p 2 a ± 1. Theorem If t 0 then p(x, t) 2 a dx C + Ce d 5t + C t 0 p 2 dx + CG 3 (t), p(x, 0) 2 + p(x, 0) 2 a dx e d 5(t τ) (A( 2, τ) + G 3 (τ))dτ, where d 5 > 0. Consequently, lim sup p(x, t) 2 a dx C lim sup(1 + A( 2, t) + G 3 (t)). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

15 Estimates of solutions - III: for p t Theorem If t t 0 > 0 then p(x, t) 2 a + p t (x, t) 2 dx C + Ce d 7t { + Ce Ct 0 t0 1 e d 7t p(x, 0) 2 + p(x, 0) 2 a dx + + C t 0 e d 7(t τ) (A( 2, τ) + G 4 (τ))dτ, p(x, 0) 2 dx t0 0 } G 3 (τ)dτ where d 7 > 0. Consequently, lim sup p(x, t) 2 a + p t (x, t) 2 dx C lim sup(1 + A( 2, t) + G 4 (t)). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

16 Estimates of solutions - IV: for mixed terms d ( ) 1 p α dx C p 2 a p α 2 dx+ε p α γ dx 0 +C ε (1+A(α, t)). dt Integrate in time or p 2 a p α 2 dx α p t p α 1 dx +... Corollary If α max{2, α } and t > 0 then t p(x, τ) 2 a p(x, τ) α 2 dx dτ 0 ( t ) C p(x, 0) α dx + (1 + A(α, τ))dτ, 0 p(x, t) 2 a p(x, t) α 2 dx ( ) C 1 + A(α, t) + p(x, t) 2(α 1) dx + p t (x, t) 2 dx. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

17 Dependence on the boundary data Let N 1, the exponent vector α and coefficient vector a be fixed. Let p k = p k (x, t) (for i = 1, 2) be the solution with initial data p k (x, 0) and boundary data ψ k. Let Ψ 1, Ψ 2 be extensions of ψ 1, ψ 2, and Φ = Ψ 1 Ψ 2. Let p k = p k Ψ k for k = 1, 2, and z = p 1 p 2, z = p 1 p 2 = z Φ. Then { z t = (K( p 1 ) p 1 ) (K( p 2 ) p 2 ) + Φ t in (0, ), z(x, t) = 0 on Γ (0, ). Goal. Estimating z(x, t) in terms of z(x, 0) and Φ. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

18 Main differential inequalities for z Define N(λ, t) = 2 k=1 ( p k(x, t) λ dx) 1/λ for λ > 0, and N(0, t) = 1. Lemma Let α 2. (i) In the DC case, for t > 0, d [ z(x, t) α dx c 6 z(x, t) α dx ]M 1 (t) a 2 a + CF (α, t)d(α, t) dt where M 1 (t) = 1 + p 1 (x, t) 2 a + p 2 (x, t) 2 a dx, F (α, t) = 1 + N(α, t) α 1 + N(γ 1, t) α 2 M 1 (t) 1 a 2 a, with γ 1 = γ 1 (α) == def 2(α 2)(2 a), and { Φ(, t) D(α, t) = L 2(2 a) + Φ(, t) 2 L + Φ α t (, t) L α if α > 2, Φ(, t) L 2 a + Φ(, t) 2 L + Φ 2 t (, t) L 2 if α = 2. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

19 Lemma (cont.) (ii) In the NDC case, d [ z(x, t) α dx c 7 dt for all t > 0, where ] θm2 z(x, t) α dx (α, t) 2 θ 1 θ 1 +CF (α, t)d(α, t) M 2 (α, t) = 1 + p 1 (x, t) 2 a + p 2 (x, t) 2 a + p 1 (x, t) θ2α + p 2 (x, t) θ2α dx. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

20 Monotonicity Multiplying the equation of z by z α 1 sgn( z),... 1 d z α dx α dt = (α 1) (K( p 1 ) p 1 K( p 2 ) p 2 ) ( p 1 p 2 ) z α 2 dx +.. Notation. for max. Lemma (Degenerate monotonicity) For any y, y R n, one has ( K( y )y K( y )y ) (y y ) (1 a)k( y y ) y y 2. d z α dx C K( p 1 p 2 ) z 2 z α 2 dx +... dt Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

21 where θ 2 = θ 1(θ 1)(2 a) 2(2 a θ 1 ) > 0. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20, Weighted Poincaré Sobolev inequality Lemma Let ξ(x) 0 on and u(x) = 0 on. Assume α 2. (i) In the DC case, [ ][ ] a u α dx c 3 K(ξ) u 2 u α 2 dx 1 + ξ 2 a 2 a dx. (ii) In the NDC case, given two numbers θ and θ 1 that satisfy then θ > 2 { (2 a) and max 2n } 1, θ 1 < 2 a, nθ + 2 [ ] 1 [ ] 2 θ u α dx c 4 K(ξ) u 2 u α 2 θ dx 1 + ξ 2 a + u θ2α 1 θθ dx 1,

22 Dependence - Ia. DC case. Theorem (i) Assume (DC) and α 2. Then z(x, t) α dx e c t 8 0 M 1(τ) 2 a a dτ + C t e c 8 0 z(x, 0) α dx for all t 0, where c 8 = c 8 (α) > 0. Moreover, if 0 then lim sup t τ M 1(s) a 2 a ds F (α, τ)d(α, τ)dτ M 1 (t) a 2 a dt = z(x, t) α dx C lim sup [ F (α, t) M 1 (t) a 2 a D(α, t) ]. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

23 Dependence - Ib. NDC case. Theorem (cont.) (ii) Assume (NDC) and α α. Then z(x, t) α dx for all t 0. Moreover, if 0 [ )] z(x, 0) α dx+c Env (F (α, t)m 2 ( θ 2 α, t) 2 θ 1 1 θ θ 1 D(α, t) M 2 (t) 2 θ 1 θ 1 dt = then [ ] lim sup z(x, t) α dx C lim sup F (α, t) M 2 ( θ 2 α, t) 2 θ 1 1 θ θ 1 D(α, t). Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

24 Dependence - Ic Corollary Let α max{2, α }. Assume the functions Ψ k (k = 1, 2) satisfy sup { Ψ k (, t) L, (Ψ k ) t (, t) L, (Ψ k ) t (, t) L } <, [0, ) lim Φ(, t) L γ = lim Φ t(, t) L α = 0, where γ = max{α, 2(2 a)}. Then z(x, t) α dx = 0. lim Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

25 Gradient Lemma In the NDC case, we have for all t > 0 that [ ] 2 z 2 a 2 a dx where M 3 (t) = CM 1 (t) 1 2 a Φ(, t) L 2 a [ ] + CM 1 (t) a 2 a M3 (t) z α dx α, ( p 1 ) t (x, t) 2 + ( p 2 ) t (x, t) 2 dx + Φ t (x, t) 2 dx. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

26 Dependence II. Finite time. Theorem Assume (NDC). (i) Then ( ) 2 z(x, t) 2 a 2 a dx for all t 1, where CM 1 (t) 1 2 a Φ(, t) L 2 a + CM 1 (t) a 2 a M3 (t) 1 2 D(t) [ ( D(t) = z(, 0) L α + Env F (α, t)m 2 ( θ )] 2 α, t) 2 θ 1 1 θ 1 D(α, t) θα. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

27 Dependence III. Asymptotics (a) Theorem (ii) Let η 1 = 1 + Ā(µ 1 ) + lim sup G 4 (t). If η 1 < then ( ) 2 lim sup z(x, t) 2 a 2 a dx + Cη µ 3 1 lim sup 1 2 a Cη1 lim sup Φ(, t) L 2 a { Φ(, t) L 2(2 a) + Φ(, t) 2 L α + Φ t (, t) L α } 1 θα. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

28 Case η 1 = Assume (NDC) and η 1 =. Define for t 0, 1 + Ā(µ 1) + t 0 e d7(t τ) G 4 (τ)dτ if Ā(µ 1) <, 1 + β(µ 1 ) µ 1 µ 1 2a + Ã(µ 1, t) if Ā(µ 1 ) =, ω(t) = + t 0 e d7(t τ) (Ã(µ 1, τ) + G 4 (τ))dτ β(µ 1 ) <, 1 + EnvÃ(µ 1, t) + t 0 e d7(t τ) G 4 (τ)dτ if Ā(µ 1 ) =, β(µ 1 ) =. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

29 Dependence III. Asymptotics (b) Theorem Assume (NDC) and η 1 =. If 0 ω(t) µ 4 dt = and then ( lim sup == lim sup[(ω(t) µ 5 ) ] + < η 2 def ) 2 z(x, t) 2 a 2 a dx [ + C lim sup ω(t) µ 6 D(t) ] 1 θα + Cη C lim sup 1 θα 2 [ ) (ω(t) 1 2 a Φ(, t) L 2 a lim sup ω(t) µ 7 D(t) ] 1 θ 2 α. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

30 Dependence on the Forchheimer polynomial Collaborator Thinh Kieu will present this topic next week in this Applied Math Seminar. Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,

31 THANK YO FOR YOR ATTENTION! Luan Hoang - Texas Tech Generalized Forchheimer Equations (V) TT - Mar. 20,