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. Gusev1

1Moscow 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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Incompressible limit of NSE

∙ Incompressible NSE: div U = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U }︄ IE(U, P) = 0 ∙ Compressible isentropic NSE: 𝜛t + div(𝜛U) = 0, F(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜆∇ div U. }︄ CE(𝜛, U, P) = 0 ∙ 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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Incompressible limit of NSE

∙ Incompressible NSE: div U = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U }︄ IE(U, P) = 0 ∙ Compressible isentropic NSE: 𝜛t + div(𝜛U) = 0, F(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜆∇ div U. }︄ CE(𝜛, U, P) = 0 ∙ 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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Incompressible limit of NSE

∙ Incompressible NSE: div U = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U }︄ IE(U, P) = 0 ∙ Compressible isentropic NSE: 𝜛t + div(𝜛U) = 0, F(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜆∇ div U. }︄ CE(𝜛, U, P) = 0 ∙ 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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Incompressible limit of NSE

∙ Incompressible NSE: div U = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U }︄ IE(U, P) = 0 ∙ Compressible isentropic NSE: 𝜛t + div(𝜛U) = 0, F(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜆∇ div U. }︄ CE(𝜛, U, P) = 0 ∙ 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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Introduction of compressibility

When Fε(𝜛, P) = 𝜛 − 1 − 𝜁P the compressible equations take form 𝜛t + div(𝜛U) = 0, 𝜛 = 1 + 𝜁P, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U }︄ CEε(𝜛, U, P) = 0 When 𝜁 = 0 we formally obtain the incompressible NSE. The speed of sound is c = √︁ dP/d𝜛 = 1/√𝜁. Deҥnition 1 The parameter 𝜁 > 0 in the equations CEε(𝜛, U, P) = 0 is called the compressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Introduction of compressibility

When Fε(𝜛, P) = 𝜛 − 1 − 𝜁P the compressible equations take form 𝜛t + div(𝜛U) = 0, 𝜛 = 1 + 𝜁P, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U }︄ CEε(𝜛, U, P) = 0 When 𝜁 = 0 we formally obtain the incompressible NSE. The speed of sound is c = √︁ dP/d𝜛 = 1/√𝜁. Deҥnition 1 The parameter 𝜁 > 0 in the equations CEε(𝜛, U, P) = 0 is called the compressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Introduction of compressibility

When Fε(𝜛, P) = 𝜛 − 1 − 𝜁P the compressible equations take form 𝜛t + div(𝜛U) = 0, 𝜛 = 1 + 𝜁P, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U }︄ CEε(𝜛, U, P) = 0 When 𝜁 = 0 we formally obtain the incompressible NSE. The speed of sound is c = √︁ dP/d𝜛 = 1/√𝜁. Deҥnition 1 The parameter 𝜁 > 0 in the equations CEε(𝜛, U, P) = 0 is called the compressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Introduction of compressibility

When Fε(𝜛, P) = 𝜛 − 1 − 𝜁P the compressible equations take form 𝜛t + div(𝜛U) = 0, 𝜛 = 1 + 𝜁P, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U }︄ CEε(𝜛, U, P) = 0 When 𝜁 = 0 we formally obtain the incompressible NSE. The speed of sound is c = √︁ dP/d𝜛 = 1/√𝜁. Deҥnition 1 The parameter 𝜁 > 0 in the equations CEε(𝜛, U, P) = 0 is called the compressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearization of the original problem

Problem 1 (Original) Do the solutions {𝜛ε, Uε, Pε} of CEε(𝜛, U, P) = 0 converge towards the solution {U, P} of IE(U, P) = 0 as 𝜁 → +0? d d𝜐 CEε(𝜛 + 𝜐𝜍, U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LCEε(𝜍, u, p) = 0, d d𝜐 IE(U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LIE(u, p) = 0. Problem 2 (Simpliҥed) Do the solutions {𝜍ε, uε, pε} of LCEε(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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearization of the original problem

Problem 1 (Original) Do the solutions {𝜛ε, Uε, Pε} of CEε(𝜛, U, P) = 0 converge towards the solution {U, P} of IE(U, P) = 0 as 𝜁 → +0? d d𝜐 CEε(𝜛 + 𝜐𝜍, U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LCEε(𝜍, u, p) = 0, d d𝜐 IE(U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LIE(u, p) = 0. Problem 2 (Simpliҥed) Do the solutions {𝜍ε, uε, pε} of LCEε(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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearization of the original problem

Problem 1 (Original) Do the solutions {𝜛ε, Uε, Pε} of CEε(𝜛, U, P) = 0 converge towards the solution {U, P} of IE(U, P) = 0 as 𝜁 → +0? d d𝜐 CEε(𝜛 + 𝜐𝜍, U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LCEε(𝜍, u, p) = 0, d d𝜐 IE(U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LIE(u, p) = 0. Problem 2 (Simpliҥed) Do the solutions {𝜍ε, uε, pε} of LCEε(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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearization of the original problem

Problem 1 (Original) Do the solutions {𝜛ε, Uε, Pε} of CEε(𝜛, U, P) = 0 converge towards the solution {U, P} of IE(U, P) = 0 as 𝜁 → +0? d d𝜐 CEε(𝜛 + 𝜐𝜍, U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LCEε(𝜍, u, p) = 0, d d𝜐 IE(U + 𝜐u, P + 𝜐p) ⃒ ⃒ ⃒ ⃒

τ=0

= 0 → LIE(u, p) = 0. Problem 2 (Simpliҥed) Do the solutions {𝜍ε, uε, pε} of LCEε(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 Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearized equations

For simplicity assume that U is smooth and divergence-free and 𝜛 = 1, 𝜈 = 1, 𝜇 = 0. Then the linearized equations have the form 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u. }︄ LCEε(𝜍, u, p) = 0 div u = 0, ut + (U, ∇)u + ∇p = ∆u. }︄ LIE(u, p) = 0 Let us study Cauchy problems for these equations on torus.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearized equations

For simplicity assume that U is smooth and divergence-free and 𝜛 = 1, 𝜈 = 1, 𝜇 = 0. Then the linearized equations have the form 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u. }︄ LCEε(𝜍, u, p) = 0 div u = 0, ut + (U, ∇)u + ∇p = ∆u. }︄ LIE(u, p) = 0 Let us study Cauchy problems for these equations on torus.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearized equations

For simplicity assume that U is smooth and divergence-free and 𝜛 = 1, 𝜈 = 1, 𝜇 = 0. Then the linearized equations have the form 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u. }︄ LCEε(𝜍, u, p) = 0 div u = 0, ut + (U, ∇)u + ∇p = ∆u. }︄ LIE(u, p) = 0 Let us study Cauchy problems for these equations on torus.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Linearized equations

For simplicity assume that U is smooth and divergence-free and 𝜛 = 1, 𝜈 = 1, 𝜇 = 0. Then the linearized equations have the form 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u. }︄ LCEε(𝜍, u, p) = 0 div u = 0, ut + (U, ∇)u + ∇p = ∆u. }︄ LIE(u, p) = 0 Let us study Cauchy problems for these equations on torus.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Cauchy problems on torus

On (0, T) × Td, where T = R/Z and T > 0 we consider:

• 1. Compressible problem:

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1)

• 2. Incompressible problem:

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2)

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Cauchy problems on torus

On (0, T) × Td, where T = R/Z and T > 0 we consider:

• 1. Compressible problem:

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1)

• 2. Incompressible problem:

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2)

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Incompressible limit of NSE Introduction of compressibility Linearization of the original problem Linearized equations | Cauchy problem on torus

Cauchy problems on torus

On (0, T) × Td, where T = R/Z and T > 0 we consider:

• 1. Compressible problem:

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1)

• 2. Incompressible problem:

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2)

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Solvability of the compressible problem

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) Theorem 1 ∀ 𝜁 > 0, u∘ ∈ L2 and p∘ ∈ L2 there exists a unique weak solution {u, p} = {uε, pε} ∈ L2(0, T; H1) × L∞(0, T; L2) to the problem (1).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Solvability of the incompressible problem

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 2 ∀ v∘ ∈ H there exists a weak solution {v, q} ∈ L2(0, T; V ) × L∞(0, T; L2) to the problem (2).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Weak convergence of the velocity

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 3 If PHu∘ = v∘ then uε ⇁ v in L2(0, T; H1)–weak and L∞(0, T; L2)–weak*, 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Strong convergence of the velocity

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 4 If v∘ is smooth and u∘ = v∘ then uε → v in L2(0, T; H1) and L∞(0, T; L2) as 𝜁 → 0. Remark: in this case ‖uε − v‖ = O(√𝜁) as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Strong convergence of the velocity

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 4 If v∘ is smooth and u∘ = v∘ then uε → v in L2(0, T; H1) and L∞(0, T; L2) as 𝜁 → 0. Remark: in this case ‖uε − v‖ = O(√𝜁) as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Convergence of the pressure

What about the convergence of the pressure?

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Mass conservation property

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) implies that d dt ∫︂

Td p dx = 0. However in

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) generally d dt ∫︂

Td q dx ̸= 0, since ∀Q = Q(t)

q(t, x) + Q(t) is also a pressure for (2).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Mass conservation property

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) implies that d dt ∫︂

Td p dx = 0. However in

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) generally d dt ∫︂

Td q dx ̸= 0, since ∀Q = Q(t)

q(t, x) + Q(t) is also a pressure for (2).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Mass conservation property

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) implies that d dt ∫︂

Td p dx = 0. However in

div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) generally d dt ∫︂

Td q dx ̸= 0, since ∀Q = Q(t)

q(t, x) + Q(t) is also a pressure for (2).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Correction of pressure

If q ∈ W 1,2(0, T; L2), then there exists unique Q = Q(t) such that ̂︁ q = q − Q ∈ W 1,2(0, T; L2) satisҥes d dt ∫︂

Td ̂︁

q dx = 0, ∫︂

Td ̂︁

q dx ⃒ ⃒ ⃒ ⃒

t=0

= ∫︂

Td p∘ dx.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Weak convergence of the pressure

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 5 If v∘ is smooth and u∘ = v∘ then pε → ̂︁ q in L∞(0, T; L2)–weak* as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Strong convergence of the pressure

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 6 If v∘ is smooth and u∘ = v∘ and p∘ = q|t=0 + const then pε → ̂︁ q in L∞(0, T; L2) as 𝜁 → 0. Remark: in this case ‖uε − v‖ = O(𝜁) and ‖pε − ̂︁ q‖ = O(√𝜁) as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Solvability of the compressible & incompressible problems Convergence of velocity Convergence of the pressure

Strong convergence of the pressure

𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) div v = 0, vt + (U, ∇)v + ∇q = ∆v, v|t=0 = v∘. ⎫ ⎪ ⎬ ⎪ ⎭ (2) Theorem 6 If v∘ is smooth and u∘ = v∘ and p∘ = q|t=0 + const then pε → ̂︁ q in L∞(0, T; L2) as 𝜁 → 0. Remark: in this case ‖uε − v‖ = O(𝜁) and ‖pε − ̂︁ q‖ = O(√𝜁) as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Summary

Assumption ↓ Result → uε ⇁ v uε → v pε

*

⇁ ̂︁ q pε → ̂︁ q PHu∘ = v∘ + − − − u∘ = v∘ + √𝜁 + − u∘ = v∘, p∘ = q|t=0 + const + 𝜁 + √𝜁 Some of the results above are similar to well-known results for NSE: P.L. Lions, N. Masmoudi (’98); E. Feireisl, A. Novotny (’07) Artiҥcial compressibility method (see e.g. R. Temam).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Summary

Assumption ↓ Result → uε ⇁ v uε → v pε

*

⇁ ̂︁ q pε → ̂︁ q PHu∘ = v∘ + − − − u∘ = v∘ + √𝜁 + − u∘ = v∘, p∘ = q|t=0 + const + 𝜁 + √𝜁 Some of the results above are similar to well-known results for NSE: P.L. Lions, N. Masmoudi (’98); E. Feireisl, A. Novotny (’07) Artiҥcial compressibility method (see e.g. R. Temam).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Summary

Assumption ↓ Result → uε ⇁ v uε → v pε

*

⇁ ̂︁ q pε → ̂︁ q PHu∘ = v∘ + − − − u∘ = v∘ + √𝜁 + − u∘ = v∘, p∘ = q|t=0 + const + 𝜁 + √𝜁 Some of the results above are similar to well-known results for NSE: P.L. Lions, N. Masmoudi (’98); E. Feireisl, A. Novotny (’07) Artiҥcial compressibility method (see e.g. R. Temam).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Summary

Assumption ↓ Result → uε ⇁ v uε → v pε

*

⇁ ̂︁ q pε → ̂︁ q PHu∘ = v∘ + − − − u∘ = v∘ + √𝜁 + − u∘ = v∘, p∘ = q|t=0 + const + 𝜁 + √𝜁 Some of the results above are similar to well-known results for NSE: P.L. Lions, N. Masmoudi (’98); E. Feireisl, A. Novotny (’07) Artiҥcial compressibility method (see e.g. R. Temam).

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Explicit solution

Suppose that U = 0, d = 2, 𝜔(x) = sin(2𝜌x1) sin(2𝜌x2). Then ∆𝜔 = −𝜇𝜔 where 𝜇 = 8𝜌2. If we take u∘ = 0 and p∘ = B𝜔 then the solution to (1) is given by uε(t, x) = − 𝜁 𝜇 4𝜕2

ε + 𝜇2

4𝜕ε Be− λt

2 sin(𝜕εt)∇𝜔(x),

pε(t, x) = Be− λt

2

(︃ cos(𝜕εt) + 𝜇 2𝜕ε sin(𝜕εt) )︃ 𝜔(x), where 𝜕ε = √︂

λ ε − λ2 4 . The solution to (2) is v = 0, ̂︁

q = 0.

1 ‖uε − v‖ ∼ √𝜁, 2 p∘ = q|t=0 + const ⇔ B = 0, 3 when B ̸= 0 the convergence of pε is weak but not strong. N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 1. Low Mach Number Limit

Dimensionless form of CE: ⇒ ⎧ ⎨ ⎩ Sr 𝜛t + div(𝜛U) = 0, F(𝜛, P) = 0, Sr(𝜛U)t + div(𝜛U ⊗ U) + 1 Ma2 ∇P = 1 Reµ ∆U + 1 Reλ ∇ div U, Compressibility 𝜁 := «Mach Number» Ma → +0. (P.-L. Lions, N. Masmoudi ’98, E. Feireisl ’07, T. Alazard ’06, . . . ) Remark 1 It is not obvious that 𝜁 → 0 implies incompressibility!

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 1. Low Mach Number Limit

Dimensionless form of CE: ⇒ ⎧ ⎨ ⎩ Sr 𝜛t + div(𝜛U) = 0, F(𝜛, P) = 0, Sr(𝜛U)t + div(𝜛U ⊗ U) + 1 Ma2 ∇P = 1 Reµ ∆U + 1 Reλ ∇ div U, Compressibility 𝜁 := «Mach Number» Ma → +0. (P.-L. Lions, N. Masmoudi ’98, E. Feireisl ’07, T. Alazard ’06, . . . ) Remark 1 It is not obvious that 𝜁 → 0 implies incompressibility!

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 2. Special equation of state

Consider {︄ 𝜛t + div(𝜛U) = 0, Fε(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U, with Fε(𝜛, p) = 0 equivalent to one of the following:

1 P = k𝜛1/ε, k = const.

(D. Ebin ’75)

2 P = P(𝜛)/𝜁2.

(A. Majda, S. Klainerman ’81)

3 𝜛 = 𝜛0 + 𝜁R(P).

(E. Shifrin ’99) with compressibility 𝜁 → +0. Remark 2 Formally, 𝜁 = 0 clearly implies incompressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 2. Special equation of state

Consider {︄ 𝜛t + div(𝜛U) = 0, Fε(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U, with Fε(𝜛, p) = 0 equivalent to one of the following:

1 P = k𝜛1/ε, k = const.

(D. Ebin ’75)

2 P = P(𝜛)/𝜁2.

(A. Majda, S. Klainerman ’81)

3 𝜛 = 𝜛0 + 𝜁R(P).

(E. Shifrin ’99) with compressibility 𝜁 → +0. Remark 2 Formally, 𝜁 = 0 clearly implies incompressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 2. Special equation of state

Consider {︄ 𝜛t + div(𝜛U) = 0, Fε(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U, with Fε(𝜛, p) = 0 equivalent to one of the following:

1 P = k𝜛1/ε, k = const.

(D. Ebin ’75)

2 P = P(𝜛)/𝜁2.

(A. Majda, S. Klainerman ’81)

3 𝜛 = 𝜛0 + 𝜁R(P).

(E. Shifrin ’99) with compressibility 𝜁 → +0. Remark 2 Formally, 𝜁 = 0 clearly implies incompressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 2. Special equation of state

Consider {︄ 𝜛t + div(𝜛U) = 0, Fε(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U, with Fε(𝜛, p) = 0 equivalent to one of the following:

1 P = k𝜛1/ε, k = const.

(D. Ebin ’75)

2 P = P(𝜛)/𝜁2.

(A. Majda, S. Klainerman ’81)

3 𝜛 = 𝜛0 + 𝜁R(P).

(E. Shifrin ’99) with compressibility 𝜁 → +0. Remark 2 Formally, 𝜁 = 0 clearly implies incompressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

• 2. Special equation of state

Consider {︄ 𝜛t + div(𝜛U) = 0, Fε(𝜛, P) = 0, (𝜛U)t + div(𝜛U ⊗ U) + ∇P = 𝜈∆U + 𝜇∇ div U, with Fε(𝜛, p) = 0 equivalent to one of the following:

1 P = k𝜛1/ε, k = const.

(D. Ebin ’75)

2 P = P(𝜛)/𝜁2.

(A. Majda, S. Klainerman ’81)

3 𝜛 = 𝜛0 + 𝜁R(P).

(E. Shifrin ’99) with compressibility 𝜁 → +0. Remark 2 Formally, 𝜁 = 0 clearly implies incompressibility.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Comparison with low Mach Number framework

If the compressibility is deҥned using equation of state, we have 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) If the compressibility is deҥned using «Mach» number, we have 𝜍′

t + (U, ∇)𝜍′ + div u′ = 0,

𝜍′ = Ap′, u′

t + (U, ∇)u′ + 1

𝜁∇p′ = ∆u′, u′|t=0 = u′∘, p′|t=0 = p′∘. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (1′) These systems can be identiҥed if u = u′, 𝜍 = 𝜍′ and p = p′/𝜁. Hence p′

ε = 𝜁pε → 0 as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Comparison with low Mach Number framework

If the compressibility is deҥned using equation of state, we have 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) If the compressibility is deҥned using «Mach» number, we have 𝜍′

t + (U, ∇)𝜍′ + div u′ = 0,

𝜍′ = Ap′, u′

t + (U, ∇)u′ + 1

𝜁∇p′ = ∆u′, u′|t=0 = u′∘, p′|t=0 = p′∘. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (1′) These systems can be identiҥed if u = u′, 𝜍 = 𝜍′ and p = p′/𝜁. Hence p′

ε = 𝜁pε → 0 as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Comparison with low Mach Number framework

If the compressibility is deҥned using equation of state, we have 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) If the compressibility is deҥned using «Mach» number, we have 𝜍′

t + (U, ∇)𝜍′ + div u′ = 0,

𝜍′ = Ap′, u′

t + (U, ∇)u′ + 1

𝜁∇p′ = ∆u′, u′|t=0 = u′∘, p′|t=0 = p′∘. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (1′) These systems can be identiҥed if u = u′, 𝜍 = 𝜍′ and p = p′/𝜁. Hence p′

ε = 𝜁pε → 0 as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Comparison with low Mach Number framework

If the compressibility is deҥned using equation of state, we have 𝜍t + (U, ∇)𝜍 + div u = 0, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) If the compressibility is deҥned using «Mach» number, we have 𝜍′

t + (U, ∇)𝜍′ + div u′ = 0,

𝜍′ = Ap′, u′

t + (U, ∇)u′ + 1

𝜁∇p′ = ∆u′, u′|t=0 = u′∘, p′|t=0 = p′∘. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (1′) These systems can be identiҥed if u = u′, 𝜍 = 𝜍′ and p = p′/𝜁. Hence p′

ε = 𝜁pε → 0 as 𝜁 → 0.

N.A. Gusev Incompressible limit of the linearised NSE

Plan of the talk Motivation Results for the linearized equations Discussion Explicit solution Defenitions of compressibility Comparison with low Mach Number framework

Thank you for your attention!

N.A. Gusev Incompressible limit of the linearised NSE

Main estimates Results by other authors

Main estimates

Consider the problem with right-hand side: 𝜍t + (U, ∇)𝜍 + div u = 𝜏, 𝜍 = 𝜁p, ut + (U, ∇)u + ∇p = ∆u + s, u|t=0 = u∘, p|t=0 = p∘. ⎫ ⎪ ⎬ ⎪ ⎭ (1) The energy estimate: ‖u‖L2(0,T;H1) + ‖u‖L∞(0,T;L2) + √𝜁‖p‖L∞(0,T;L2) C (︃ ‖u∘‖L2 + √𝜁‖p∘‖L2 + ‖s‖L2(0,T;H−1) + 1 √𝜁‖𝜏‖L2(0,T;L2) )︃ . A modiҥed estimate: ‖u‖L2(0,T;H1) + ‖u‖L∞(0,T;L2) + √𝜁‖p‖L∞(0,T;L2) C (︁ ‖u∘‖L2 + √𝜁‖p∘‖L2 + ‖s‖L2(0,T;H−1) + ‖𝜏‖W 1,2(0,T;L2/R) )︁ .

N.A. Gusev Incompressible limit of the linearised NSE

Main estimates Results by other authors

Results by other authors

1 Solvability of (1) when U ≡ 0:

• R. Ikehata , T. Koboyashi , T. Matsuyama ’01

P.B. Mucha, W.M. Zajaczkowski ’02 ...

2 Incompressible limit

D.G. Ebin ’75

• S. Klainerman, A. Majda ’81
• R. Temam 80

P.-L. Lions, N. Masmoudi ’98

• E. Shifrin ’99
• V. Puchnachev ’02
• E. Feireisl, A. Novotn´

y ’07 ...

N.A. Gusev Incompressible limit of the linearised NSE