Homogenization of scalar and Stokes equations with drift M. Briane, - PowerPoint PPT Presentation

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Homogenization of scalar and Stokes equations with drift M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Colloque EDP-Normandie Universit´ e de Rouen Octobre 25–26 2011 M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives 1 Introduction 2 A scalar drift problem (avec P. G´ erard) Refinement with an equi-integrability condition Sharpness of the equi-integrability condition 3 A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition Sharpness of the equi-integrability condition 4 The periodic case without equi-integrability condition The scalar equation The Stokes equation 5 Homogenization with large drifts A compactness result in dimension two Nonlocal effects in dimension three 6 Perspectives M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A Tartar hydrodynamic problem Ω a bounded domain of R 3 , v ε bounded in L 3 (Ω) 3 ,  − ∆ u ε + curl ( v ε ) × u ε + ∇ p ε = f in Ω  div ( u ε ) = 0 in Ω u ε = 0 on Ω .  curl ( v ε ) × u ε represents an oscillating Coriolis force. Tartar considered for any λ ∈ R 3 the oscillating test function w λ ε solution of  − ∆ w λ ε + curl ( v ε ) × λ + ∇ q λ = 0 in Ω ε  w λ � � div = 0 in Ω ε w λ = 0 on ∂ Ω .  ε M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A Brinkman homogenized problem Theorem (Tartar 77) 0 (Ω) 3 to the homogenized If v ε ⇀ v in L 3 (Ω) 3 , then u ε ⇀ u in H 1 Brinkman problem  − ∆ u + curl ( v ) × u + ∇ p + Mu = f in Ω  div ( u ) = 0 in Ω = 0 on ∂ Ω ,  u where M is the positive definite symmetric matrix-valued function given by: 3 ( Dw λ ε ) T v ε ⇀ M λ 2 (Ω) 3 , for λ ∈ R 3 . weakly in L The boundedness of v ε in L 3 (Ω) 3 is not sharp. Ω | Du ε | 2 dx , since curl ( v ε ) × u ε ⊥ u ε . � The energy is M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition The periodic case without equi-integrability condition Sharpness of the equi-integrability condition Homogenization with large drifts Perspectives A scalar drift problem Let Ω be a regular bounded domain of R N , N ≥ 2. A scalar equivalent to the Stokes drift problem studied by Tartar is: � − ∆ u ε + b ε · ∇ u ε + div ( b ε u ε ) = f in Ω u ε = 0 on ∂ Ω , where b ε ∈ L ∞ (Ω) N for a fixed ε > 0. Ω |∇ u ε | 2 dx , since for � Indeed, as before the associated energy is any v ∈ H 1 0 (Ω) we have � � � b ε · ∇ v + div ( b ε v ) v dx = 0 . Ω M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition The periodic case without equi-integrability condition Sharpness of the equi-integrability condition Homogenization with large drifts Perspectives The strategy Mimicking the Tartar approach leads us to consider for λ ∈ R , � − ∆ w ε + b ε · ∇ λ + div ( b ε λ ) = 0 in Ω = 0 on ∂ Ω , w ε i.e., λ = 1 the test function is defined by w ε ∈ H 1 0 (Ω) , ∆ w ε = div ( b ε ) in Ω. Strategy for the homogenization of this drift problem: Use of the Tartar approach with a condition on div ( b ε ) . Refinement with a sharp condition of equi-integrability on b ε . A new approach based on a parametrix for ∆ . Proof of the sharpness of the condition thanks to an example. M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition The periodic case without equi-integrability condition Sharpness of the equi-integrability condition Homogenization with large drifts Perspectives A Tartar type result Assumption on b ε : in L 2 (Ω) N , � b ε ⇀ b in W 1 , q (Ω) . ∃ q > N , w ε bounded Theorem u ε ⇀ u ∈ H 1 0 (Ω) , where u is a solution of the drift equation in D ′ (Ω) , − ∆ u + b · ∇ u + div ( b u ) + µ u = f |∇ w ε − ∇ w | 2 ⇀ µ q 2 (Ω) . where in L The uniqueness for the limit problem is not obvious. M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition The periodic case without equi-integrability condition Sharpness of the equi-integrability condition Homogenization with large drifts Perspectives An equi-integrability condition ξ ε ∈ L 2 (Ω) N with div ( ξ ε ) = 0. w ε ∈ H 1 0 (Ω) N , b ε = ∇ w ε + ξ ε , New assumption on b ε : b ε bounded in L 2 (Ω) N ∇ w ε equi-integrable in L 2 (Ω) N . and b ε equi-integrable L 2 (Ω) N ∇ w ε , ξ ε equi-integrable L 2 loc (Ω) N . ⇒ The present condition is thus weaker. Theorem Assume that |∇ w ε − ∇ w | 2 converges weakly to µ in L 1 (Ω) . Then, u ε converges weakly in H 1 0 (Ω) to the solution u of − ∆ u + b · ∇ u + div ( b u ) + µ u = f in Ω . But the proof is more delicate since we cannot pass to the limit in the products ( b ε · ∇ u ε ) w ε and ( b ε · ∇ w ε ) u ε . M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition The periodic case without equi-integrability condition Sharpness of the equi-integrability condition Homogenization with large drifts Perspectives A parametrix for the Laplace operator From a partition of the unity one can build a continuous operator P : D ′ (Ω) − → D ′ (Ω) which is a quasi-inverse of ∆ satisfying ∆ ◦ P = I d − K ′ , P ◦ ∆ = I d − K and where K , K ′ are C ∞ -kernel operators properly supported in Ω . By the classical regularity results we have ∀ p > 1 , ∀ s ∈ [0 , 2] , s + 1 P : W − s , p → W 2 − s , p (Ω) − (Ω) p / ∈ N . loc loc M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) Refinement with an equi-integrability condition The periodic case without equi-integrability condition Sharpness of the equi-integrability condition Homogenization with large drifts Perspectives The parametrix method The idea is to write u ε as a solution of a fixed point problem: u ε = P (∆ u ε ) + Ku ε � � �� � � �� = P ∆ u ( w ε − w ) − P div ( w ε − w ) ∇ u � � �� + P ( u ε − u ) ∇ w ε div � � + P b ε · ∇ u ε + ξ ε · ∇ u ε + div ( u ∇ w ) − f + Ku ε , ( w ε − w ) ∇ u → 0 and ( u ε − u ) ∇ w ε → 0 in L p (Ω) N , p ∈ (1 , N ′ ). Hence, we get u ε = u ( w ε − w ) + o W 1 , p loc (Ω) (1) � � + P b ε · ∇ u ε + ξ ε · ∇ u ε + div ( u ∇ w ) − f + Ku ε . M. Briane, IRMA Rennes & P. G´ erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift