Small-amplitude homogenisation of parabolic equations Nenad Antoni - PowerPoint PPT Presentation

Small-amplitude homogenisation of parabolic equations Nenad Antoni´ c Department of Mathematics Faculty of Science University of Zagreb Dubrovnik, 13 th October, 2008 Joint work with Marko Vrdoljak

H-convergence and G-convergence Homogenisation: in the sense of G-convergence (S. Spagnolo) and H-convergence (F. Murat & L. Tartar)

H-convergence and G-convergence Homogenisation: in the sense of G-convergence (S. Spagnolo) and H-convergence (F. Murat & L. Tartar) Recall small-amplitude homogenisation for − div ( A ∇ u ) = f .

Small-amplitude homogenisation Consider − div ( A n γ ∇ u n ) = f , where A n γ is a perturbation of A 0 ∈ C(Ω; M d × d ) , which is bounded from below; for small γ function A n γ is analytic in γ : A n γ ( x ) = A 0 + γ B n ( x ) + γ 2 C n ( x ) + o ( γ 2 ) , ∗ where B n , C n ⇀ 0 in L ∞ ( Q ; M d × d ) ). − −

Small-amplitude homogenisation Consider − div ( A n γ ∇ u n ) = f , where A n γ is a perturbation of A 0 ∈ C(Ω; M d × d ) , which is bounded from below; for small γ function A n γ is analytic in γ : A n γ ( x ) = A 0 + γ B n ( x ) + γ 2 C n ( x ) + o ( γ 2 ) , ∗ where B n , C n ⇀ 0 in L ∞ ( Q ; M d × d ) ). − − Then (after passing to a subsequence, if needed) H ⇀ A ∞ γ = A 0 + γ B 0 + γ 2 C 0 + o ( γ 2 ) ; A n − − − γ the limit being measurable in x , and analytic in γ . A ∞ γ is the effective conductivity.

No first-order term on the limit The effective conductivity matrix A ∞ Theorem. γ admits the expansion A ∞ γ ( x ) = A 0 ( x ) + γ 2 C 0 ( x ) + o ( γ 2 ) .

No first-order term on the limit The effective conductivity matrix A ∞ Theorem. γ admits the expansion A ∞ γ ( x ) = A 0 ( x ) + γ 2 C 0 ( x ) + o ( γ 2 ) . C 0 depends only on a subsquence of B n (and A 0 ), and there is an explicit formula involving the H-measure of the above subsequence: � � � µ , ϕ ⊠ ξ ⊗ ξ − ϕ C 0 = . A 0 ξ · ξ

No first-order term on the limit The effective conductivity matrix A ∞ Theorem. γ admits the expansion A ∞ γ ( x ) = A 0 ( x ) + γ 2 C 0 ( x ) + o ( γ 2 ) . C 0 depends only on a subsquence of B n (and A 0 ), and there is an explicit formula involving the H-measure of the above subsequence: � � � µ , ϕ ⊠ ξ ⊗ ξ − ϕ C 0 = . A 0 ξ · ξ This might provide a precise sense for some formulas in the book by Landau & Lifschitz.

No first-order term on the limit The effective conductivity matrix A ∞ Theorem. γ admits the expansion A ∞ γ ( x ) = A 0 ( x ) + γ 2 C 0 ( x ) + o ( γ 2 ) . C 0 depends only on a subsquence of B n (and A 0 ), and there is an explicit formula involving the H-measure of the above subsequence: � � � µ , ϕ ⊠ ξ ⊗ ξ − ϕ C 0 = . A 0 ξ · ξ This might provide a precise sense for some formulas in the book by Landau & Lifschitz. The method also works on the system of linearised elasticity (see Tartar’s paper in the Proceedings of SIAM conference in Leesburgh, Dec 1988)

Our goal What can be done for parabolic equations? � ∂ t − div ( A ∇ u ) = f u (0 , · ) = u 0 . with some boundary conditions.

Our goal What can be done for parabolic equations? � ∂ t − div ( A ∇ u ) = f u (0 , · ) = u 0 . with some boundary conditions. Things to check: 1. H-convergence and G-convergence (in particular, analytical dependence of the H-limit on a parameter) 2. Parabolic variant od H-measures 3. What result do we get for small-amplitude homogenisation in this case (possible applications)

Known results for elliptic equations Homogenisation of parabolic equations H-convergence and G-convergence H-convergent sequence depending on a parameter A parabolic variant of H-measures What are H-measures and variants ? A brief comparative description Small-amplitude homogenisation Setting of the problem (parabolic case) Variant H-measures in small-amplitude homogenisation

Parabolic problems If A does not depend on t , the problem reduces to the elliptic case.

Parabolic problems If A does not depend on t , the problem reduces to the elliptic case. For A depending on both t and x , only a few papers (a few more than three, in fact): 1977 S. Spagnolo: Convergence of parabolic operators 1981 V.V. ˇ Zikov et al.: O G-shodimosti paraboliˇ ceskih operatorov 1997 A. Dall’Aglio, F. Murat: A corrector result . . .

Parabolic problems If A does not depend on t , the problem reduces to the elliptic case. For A depending on both t and x , only a few papers (a few more than three, in fact): 1977 S. Spagnolo: Convergence of parabolic operators 1981 V.V. ˇ Zikov et al.: O G-shodimosti paraboliˇ ceskih operatorov 1997 A. Dall’Aglio, F. Murat: A corrector result . . . There are some interesting differences in comparison to the elliptic case.

Non-stationary diffusion Consider a domain Q = � 0 , T � × Ω , where Ω ⊆ R d is open: � ∂ t u − div ( A ∇ u ) = f u (0 , · ) = u 0 .

Non-stationary diffusion Consider a domain Q = � 0 , T � × Ω , where Ω ⊆ R d is open: � ∂ t u − div ( A ∇ u ) = f u (0 , · ) = u 0 . 0 (Ω) , V ′ := H − 1 (Ω) and H := L 2 (Ω) , More precisely: V := H 1 → V ′ . the Gel’fand triple: V ֒ → H ֒

Non-stationary diffusion Consider a domain Q = � 0 , T � × Ω , where Ω ⊆ R d is open: � ∂ t u − div ( A ∇ u ) = f u (0 , · ) = u 0 . 0 (Ω) , V ′ := H − 1 (Ω) and H := L 2 (Ω) , More precisely: V := H 1 → V ′ . the Gel’fand triple: V ֒ → H ֒ For time dependent functions: V := L 2 (0 , T ; V ) , V ′ := L 2 (0 , T ; V ′ ) , W = { u ∈ V : ∂ t u ∈ V ′ } and H := L 2 (0 , T ; H ) , again: V ֒ → V ′ . → H ֒

Non-stationary diffusion Consider a domain Q = � 0 , T � × Ω , where Ω ⊆ R d is open: � ∂ t u − div ( A ∇ u ) = f u (0 , · ) = u 0 . 0 (Ω) , V ′ := H − 1 (Ω) and H := L 2 (Ω) , More precisely: V := H 1 → V ′ . the Gel’fand triple: V ֒ → H ֒ For time dependent functions: V := L 2 (0 , T ; V ) , V ′ := L 2 (0 , T ; V ′ ) , W = { u ∈ V : ∂ t u ∈ V ′ } and H := L 2 (0 , T ; H ) , again: V ֒ → V ′ . → H ֒ Additionally assume A ∈ L ∞ ( Q ; M d × d ) satisfies: A ( t, x ) ξ · ξ � α | ξ | 2 A ( t, x ) ξ · ξ � 1 β | A ( t, x ) ξ | 2 , i.e. it belongs to M ( α, β ; Q ) .

Non-stationary diffusion Consider a domain Q = � 0 , T � × Ω , where Ω ⊆ R d is open: � ∂ t u − div ( A ∇ u ) = f u (0 , · ) = u 0 . 0 (Ω) , V ′ := H − 1 (Ω) and H := L 2 (Ω) , More precisely: V := H 1 → V ′ . the Gel’fand triple: V ֒ → H ֒ For time dependent functions: V := L 2 (0 , T ; V ) , V ′ := L 2 (0 , T ; V ′ ) , W = { u ∈ V : ∂ t u ∈ V ′ } and H := L 2 (0 , T ; H ) , again: V ֒ → V ′ . → H ֒ Additionally assume A ∈ L ∞ ( Q ; M d × d ) satisfies: A ( t, x ) ξ · ξ � α | ξ | 2 A ( t, x ) ξ · ξ � 1 β | A ( t, x ) ξ | 2 , i.e. it belongs to M ( α, β ; Q ) . With such coefficients the problem is well posed: � u � W � c 1 � u 0 � H + c 2 � f � V ′ .

Parabolic operators Parabolic operator P ∈ L ( W ; V ′ ) P u := ∂ t u − div ( A ∇ u ) is an isomorphisms of W 0 := { u ∈ W : u (0 , · ) = 0 } onto V ′ .

Parabolic operators Parabolic operator P ∈ L ( W ; V ′ ) P u := ∂ t u − div ( A ∇ u ) is an isomorphisms of W 0 := { u ∈ W : u (0 , · ) = 0 } onto V ′ . Spagnolo introduced G-convergence for more general parabolic operators: → V ′ , P A := ∂ t + A : W − where ( A u )( t ) := A ( t ) u ( t ) , with A ( t ) ∈ L ( V ; V ′ ) such that for ϕ, ψ ∈ V t �→ � A ( t ) ϕ, ψ � is measurable λ 0 � ϕ � 2 V � � A ( t ) ϕ, ϕ � � Λ 0 � ϕ � 2 V � � |� A ( t ) ϕ, ψ �| � M � A ( t ) ϕ, ϕ � � A ( t ) ψ, ψ � , where λ 0 , Λ 0 and M are some positive constants.

Parabolic operators Parabolic operator P ∈ L ( W ; V ′ ) P u := ∂ t u − div ( A ∇ u ) is an isomorphisms of W 0 := { u ∈ W : u (0 , · ) = 0 } onto V ′ . Spagnolo introduced G-convergence for more general parabolic operators: → V ′ , P A := ∂ t + A : W − where ( A u )( t ) := A ( t ) u ( t ) , with A ( t ) ∈ L ( V ; V ′ ) such that for ϕ, ψ ∈ V t �→ � A ( t ) ϕ, ψ � is measurable λ 0 � ϕ � 2 V � � A ( t ) ϕ, ϕ � � Λ 0 � ϕ � 2 V � � |� A ( t ) ϕ, ψ �| � M � A ( t ) ϕ, ϕ � � A ( t ) ψ, ψ � , where λ 0 , Λ 0 and M are some positive constants. The set of all such operators P A we denote by P ( λ 0 , Λ 0 , M ) . For A ( t ) = − div ( A ( t, · ) , · ) we write P A instead of P A .

G-convergence and compactness A sequence P A n ∈ P ( λ 0 , Λ 0 , M ) G-converges to P A (and we write G ⇀ P A ) if for any f ∈ V ′ P A n − − − P − 1 ⇀ P − 1 A n f − A f in W 0 .

G-convergence and compactness A sequence P A n ∈ P ( λ 0 , Λ 0 , M ) G-converges to P A (and we write G ⇀ P A ) if for any f ∈ V ′ P A n − − − P − 1 ⇀ P − 1 A n f − A f in W 0 . → V ′ (continuous inclusions), if they are also compact, If V ֒ → H ֒ Spagnolo proved the compactness of G-convergence: For any P A n ∈ P ( λ 0 , Λ 0 , M ) there is a subsequence P A n ′ and a � G P A ∈ P ( λ 0 , M 2 Λ 0 , Λ 0 /λ 0 M ) , such that P A n ′ − − − ⇀ P A .