On an iterative method for the approximate solution of an initial and boundary-value problem for a generalized Boussinesq model

On an iterative method for the approximate solution of an initial and boundary-value problem for a generalized Boussinesq model Jos´ e Luiz Boldrini Blanca Climent Ezquerra M. Drina Rojas Medar Marko A. Rojas Medar Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 1/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 2/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 2/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 2/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 2/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 2/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 3/33 The model Influence of temperature on hydrodynamics problem Boussinesq approximation The fluid is incompressible except insofar as the thermal expansion produces a buoyancy, represented by a term αgϕ, where gis the acceleration of gravity, αis the coefficient of thermal expansion, ϕis the perturbation temperature. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 4/33 Classical model Continuity equation (Incompressibility) div u=0 in (0,T)×Ω Movement equations (NS) ∂u ∂t+u· ∇u− ∇ · (ν∇u) + ∇p=α ϕg+hin (0,T)×Ω Temperature equation ∂ϕ ∂t+u· ∇ϕ− ∇ · (k∇ϕ) = fin (0,T)×Ω, Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 5/33 Main results Iterative process Let u0(t) = u0and ϕ0(t) = ϕ0for all t∈[0,T]. Step n ≥1:First, given un−1and ϕn−1to find ϕnsuch that ϕn t−div (k(ϕn−1)∇ϕn) + un−1· ∇ϕn=f, ϕn(0) = ϕ0in Ω, ϕn=0 on ∂Ω. Afterwards, given un−1,ϕn−1and ϕn, to find un,pnsuch that un t−div (ν(ϕn−1)∇un) + un−1· ∇un+∇pn=h+αϕng, div un=0, un(0) = u0in Ω, un=0 on ∂Ω. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 12/33 Main results: approximate problems solution Theorem g∈L∞(L6); f,h∈L∞(L2);gt∈L4(L3), ft,ht∈L2(L2). f,h, α, u0, ϕ0small enough ⇒ ∃! (un, ϕn)solution such that un∈L∞(D(A)), ϕn∈L∞(H2), un t∈L∞(H)∩L2(V), ϕn t∈L∞(L2)∩L2(H1 0), sup t{|un t(t)|2+|ϕn t(t)|2} ≤ M, Zt 0|∇un t(τ)|2dτ+Zt 0|∇ϕn t(τ)|2dτ≤M, sup t{|Aun(t)|2+|∆ϕn(t)|2} ≤ M, for all t ∈[0,T]. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 13/33 Main results: solution of the original problem Theorem Under the conditions of the previous theorem (un, ϕn)→(u, ϕ)in L2(D(A)) ×L2(H2(Ω)). (u, ϕ)is the unique (strong) solution of the original problem. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 14/33 Main results: rate of convergence Theorem Rates of convergence: sup 0≤τ≤t{|un(τ)−u(τ)|2+|ϕn(τ)−ϕ(τ)|2} ≤ C(Dt)n n!, sup 0≤τ≤t{Zτ 0|∇un− ∇u|2+Zτ 0|∇ϕn− ∇ϕ|2} ≤ C(Dt)n n!, Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 15/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 16/33 Proof of Theorem 1: Weak estimates (eq. ϕn, ϕn)+(eq. un,un)+ Gronwall’s lemma ⇒ unis bounded in L∞(H)∩L2(V) ϕnis bounded in L∞(L2)∩L2(H1 0). Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 17/33 Proof of Theorem 1: Differential inequalities (eq. ϕn,−∆ϕn)+(eq. un,Aun) +(eq. ϕn, ϕn t)+(eq. un,un t) +((eq. ϕn)t, ϕn t) + ((eq. un)t,un t), ⇒differential inequality Denoting a=a(t) = C2α4|g|4 6∈L∞(0,T) bn=bn(t) = C2(α2|g|2 3|∇ϕn|2+α2|gt|2 3|∇ϕn|2 +|f|2+|h|2+|ft|2+|ht|2)∈L1(0,T). Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 18/33 Proof of Theorem 1 Integrating in [0,t] |∇un|2+|∇ϕn|2+|un t|2+|ϕn t|2 +Zt 0(C−C1(|∆ϕn−1|+|∇un−1|+|ϕn−1 t|)(|∆ϕn|2+|Aun|2)) +CZt 0(|ϕn t|2+|un t|2+|∇ϕn t|2+|∇un t|2) ≤ |∇un(0)|2+|∇ϕn(0)|2+|un t(0)|2+|ϕn t(0)|2 +C1(Zt 0|∇un−1 t|2+|∇ϕn−1 t|2)(|∆ϕn|2+|Aun|2) +C1Zt 0|∇un−1|(|un t|2+|ϕn t|2) + Zt 0(a|ϕn t|2+bn) (1) Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 19/33 Proof of Theorem 1: Differential inequalities (eq. ϕn,−∆ϕn)+(eq. un,Aun)⇒ (k0 2−Cϕ|∆ϕn−1|2−Cϕ|∇un−1|2)|∆ϕn|2 +(ν0 2−Cu|∆ϕn−1|2−Cu|∇un−1|2)|Aun|2 ≤C(|un t|2+|ϕn t|2+|g|2 4|∇ϕ|2) + C(|h|2+|f|2) (2) Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 20/33 Proof of Theorem 1: Induction hypothesis (smallness) sup 0≤t≤T|∆ϕn−1|2< δ2sup 0≤t≤T|Aun−1|2< δ2 sup 0≤t≤T|∇ϕn−1|2< δ2sup 0≤t≤T|∇un−1|2< δ2 sup 0≤t≤T|ϕn−1 t|2< δ2sup 0≤t≤T|un−1 t|2< δ2 sup 0≤t≤TZt 0|∇ϕn−1 t|2< δ2sup 0≤t≤TZt 0|∇un−1 t|2< δ2 δ << Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 21/33 Proof of Theorem 2 d dt(|∇un,s|2+|∇ϕn,s|2) | {z } an(t) +µ(|Aun,s|2+|∆ϕn,s|2) | {z } bn(t) ≤C |{z} dn(t) (|∇un−1,s|2+|∇ϕn−1,s|2) + C |{z} cn(t) (|∇un,s|2+|∇ϕn,s|2) +δ|∆ϕn−1,s|2. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 28/33 Proof of Theorem 2 an(t) = |∇un,s(t)|2+|∇ϕn,s(t)|2,kank∞→0 bn(t) = µ(|Aun,s(t)|2+|∆ϕn,s(t)|2)kbnkL1→0 un→ustrongly in L∞(V)∩L2(D(A)), un t→utstrongly in L2(H), ϕn→ϕstrongly in L∞(H1 0)∩L2(H2), ϕn t→ϕtstrongly in L2(L2). Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 29/33 Proof of Theorem 2 an(t) = |∇un,s(t)|2+|∇ϕn,s(t)|2,kank∞→0 bn(t) = µ(|Aun,s(t)|2+|∆ϕn,s(t)|2)kbnkL1→0 un→ustrongly in L∞(V)∩L2(D(A)), un t→utstrongly in L2(H), ϕn→ϕstrongly in L∞(H1 0)∩L2(H2), ϕn t→ϕtstrongly in L2(L2). Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 29/33 Proof of Theorem 2: Convergence-rate bounds vn=un−u zn=ϕn−ϕ qn=pn−p Subtracting the corresponding equations. an(t) = |vn|2(t) + C|zn|2(t) bn(t) = ν0|∇vn|2(t) + C|∇zn|2(t) an(0) = 0 Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 30/33 Proof of Theorem 2: Convergence-rate bounds Lemma an, bn∈L1(0,T)positive, an(0)≤A0∈R a0 n(t) + bn(t)≤cn(t)an(t) + dn(t)an−1(t),a.e. t ∈(0,T), where cn,dnpositive, bounded in L1(0,T)and L∞(0,T), respectively. Then, an(t) + Zt 0bn(s)ds ≤DA0eDt +|a0|∞ (Dt)n n!. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 31/33 Table of contents 1Introduction 2Main results 3Proof of Theorem 1 4Proof of Theorem 2 5Open problems Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 32/33 Open problems Strong convergence-rate bounds. Nonlinear iterative process. Blanca Climent Ezquerra. Universidad de Sevilla. An iterative method for the generalized Boussinesq model 33/33