Incompressible limit of the linearized NavierStokes equations. N.A. - PowerPoint PPT Presentation

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of the linearized Navier–Stokes equations. N.A. Gusev 1 1 Moscow Institute of Physics and Technology Padova, June 25, 2012 N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Plan of the talk 1 Motivation: Incompressible Limit of Full NSE; 2 Model problem for linearised equations; 3 Discussion of the results. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Incompressible limit of NSE ∙ Incompressible NSE: }︄ div U = 0 , IE ( U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U ∙ Compressible isentropic NSE: }︄ 𝜛 t + div ( 𝜛 U ) = 0 , F ( 𝜛, P ) = 0 , CE ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜆 ∇ div U . ∙ Formally IE can be viewed as CE with F ( 𝜛, P ) := 𝜛 − 1 . Can we pass to the limit from CE to IE ? N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Incompressible limit of NSE ∙ Incompressible NSE: }︄ div U = 0 , IE ( U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U ∙ Compressible isentropic NSE: }︄ 𝜛 t + div ( 𝜛 U ) = 0 , F ( 𝜛, P ) = 0 , CE ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜆 ∇ div U . ∙ Formally IE can be viewed as CE with F ( 𝜛, P ) := 𝜛 − 1 . Can we pass to the limit from CE to IE ? N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Incompressible limit of NSE ∙ Incompressible NSE: }︄ div U = 0 , IE ( U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U ∙ Compressible isentropic NSE: }︄ 𝜛 t + div ( 𝜛 U ) = 0 , F ( 𝜛, P ) = 0 , CE ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜆 ∇ div U . ∙ Formally IE can be viewed as CE with F ( 𝜛, P ) := 𝜛 − 1 . Can we pass to the limit from CE to IE ? N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Incompressible limit of NSE ∙ Incompressible NSE: }︄ div U = 0 , IE ( U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U ∙ Compressible isentropic NSE: }︄ 𝜛 t + div ( 𝜛 U ) = 0 , F ( 𝜛, P ) = 0 , CE ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜆 ∇ div U . ∙ Formally IE can be viewed as CE with F ( 𝜛, P ) := 𝜛 − 1 . Can we pass to the limit from CE to IE ? N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Introduction of compressibility When F ε ( 𝜛, P ) = 𝜛 − 1 − 𝜁 P the compressible equations take form }︄ 𝜛 t + div ( 𝜛 U ) = 0 , 𝜛 = 1 + 𝜁 P , CE ε ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜇 ∇ div U When 𝜁 = 0 we formally obtain the incompressible NSE. √︁ dP / d 𝜛 = 1 / √ 𝜁 . The speed of sound is c = Deҥnition 1 The parameter 𝜁 > 0 in the equations CE ε ( 𝜛, U , P ) = 0 is called the compressibility . N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Introduction of compressibility When F ε ( 𝜛, P ) = 𝜛 − 1 − 𝜁 P the compressible equations take form }︄ 𝜛 t + div ( 𝜛 U ) = 0 , 𝜛 = 1 + 𝜁 P , CE ε ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜇 ∇ div U When 𝜁 = 0 we formally obtain the incompressible NSE. √︁ dP / d 𝜛 = 1 / √ 𝜁 . The speed of sound is c = Deҥnition 1 The parameter 𝜁 > 0 in the equations CE ε ( 𝜛, U , P ) = 0 is called the compressibility . N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Introduction of compressibility When F ε ( 𝜛, P ) = 𝜛 − 1 − 𝜁 P the compressible equations take form }︄ 𝜛 t + div ( 𝜛 U ) = 0 , 𝜛 = 1 + 𝜁 P , CE ε ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜇 ∇ div U When 𝜁 = 0 we formally obtain the incompressible NSE. √︁ dP / d 𝜛 = 1 / √ 𝜁 . The speed of sound is c = Deҥnition 1 The parameter 𝜁 > 0 in the equations CE ε ( 𝜛, U , P ) = 0 is called the compressibility . N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Introduction of compressibility When F ε ( 𝜛, P ) = 𝜛 − 1 − 𝜁 P the compressible equations take form }︄ 𝜛 t + div ( 𝜛 U ) = 0 , 𝜛 = 1 + 𝜁 P , CE ε ( 𝜛, U , P ) = 0 ( 𝜛 U ) t + div ( 𝜛 U ⊗ U ) + ∇ P = 𝜈 ∆ U + 𝜇 ∇ div U When 𝜁 = 0 we formally obtain the incompressible NSE. √︁ dP / d 𝜛 = 1 / √ 𝜁 . The speed of sound is c = Deҥnition 1 The parameter 𝜁 > 0 in the equations CE ε ( 𝜛, U , P ) = 0 is called the compressibility . N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Linearization of the original problem Problem 1 (Original) Do the solutions { 𝜛 ε , U ε , P ε } of CE ε ( 𝜛, U , P ) = 0 converge towards the solution { U , P } of IE ( U , P ) = 0 as 𝜁 → +0 ? ⃒ ⃒ d ⃒ d 𝜐 CE ε ( 𝜛 + 𝜐𝜍, U + 𝜐 u , P + 𝜐 p ) = 0 → LCE ε ( 𝜍, u , p ) = 0 , ⃒ τ =0 ⃒ ⃒ d ⃒ d 𝜐 IE ( U + 𝜐 u , P + 𝜐 p ) = 0 → LIE ( u , p ) = 0 . ⃒ τ =0 Problem 2 (Simpliҥed) Do the solutions { 𝜍 ε , u ε , p ε } of LCE ε ( 𝜍, u , p ) = 0 converge towards the solution { u , p } of LIE ( u , p ) = 0 as 𝜁 → +0 ? N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Linearization of the original problem Problem 1 (Original) Do the solutions { 𝜛 ε , U ε , P ε } of CE ε ( 𝜛, U , P ) = 0 converge towards the solution { U , P } of IE ( U , P ) = 0 as 𝜁 → +0 ? ⃒ ⃒ d ⃒ d 𝜐 CE ε ( 𝜛 + 𝜐𝜍, U + 𝜐 u , P + 𝜐 p ) = 0 → LCE ε ( 𝜍, u , p ) = 0 , ⃒ τ =0 ⃒ ⃒ d ⃒ d 𝜐 IE ( U + 𝜐 u , P + 𝜐 p ) = 0 → LIE ( u , p ) = 0 . ⃒ τ =0 Problem 2 (Simpliҥed) Do the solutions { 𝜍 ε , u ε , p ε } of LCE ε ( 𝜍, u , p ) = 0 converge towards the solution { u , p } of LIE ( u , p ) = 0 as 𝜁 → +0 ? N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Incompressible limit of NSE Motivation Introduction of compressibility Results for the linearized equations Linearization of the original problem Discussion Linearized equations | Cauchy problem on torus Linearization of the original problem Problem 1 (Original) Do the solutions { 𝜛 ε , U ε , P ε } of CE ε ( 𝜛, U , P ) = 0 converge towards the solution { U , P } of IE ( U , P ) = 0 as 𝜁 → +0 ? ⃒ ⃒ d ⃒ d 𝜐 CE ε ( 𝜛 + 𝜐𝜍, U + 𝜐 u , P + 𝜐 p ) = 0 → LCE ε ( 𝜍, u , p ) = 0 , ⃒ τ =0 ⃒ ⃒ d ⃒ d 𝜐 IE ( U + 𝜐 u , P + 𝜐 p ) = 0 → LIE ( u , p ) = 0 . ⃒ τ =0 Problem 2 (Simpliҥed) Do the solutions { 𝜍 ε , u ε , p ε } of LCE ε ( 𝜍, u , p ) = 0 converge towards the solution { u , p } of LIE ( u , p ) = 0 as 𝜁 → +0 ? N.A. Gusev Incompressible limit of the linearised NSE