Strong estimates for some coupled Navier-Stokes type systems

Strong estimates for some coupled Navier-Stokes type systems Chill´ an, agosto de 2012 Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 1/21 Table of contents 1Navier-Stokes 3D: Strong estimates for small data or large viscosity 2Some Models Navier-Stokes type 3Third option: large time Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 2/21 Table of contents 1Navier-Stokes 3D: Strong estimates for small data or large viscosity 2Some Models Navier-Stokes type 3Third option: large time Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 3/21 Navier-Stokes Navier-Stokes 3D (NS)       ∂tu−ν∆u+ (u· ∇)u+∇p=f, ∇ · u=0 in Ω×(0,T), u(0) = u0u|∂Ω=0 (NS)m: Approximated Galerkin Pb. ( eigenfunctions Stokes Pb. “special” basis of V) V={u∈H1 0:∇ · u=0 in Ω} Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 4/21 Navier-Stokes Aumtest function (AStokes operator) (∂tum,Aum)−ν(∆um,Aum) + ((um· ∇)um,Aum)=(f,Aum), 1 2 d dt kumk2 1+νkumk2 2=−((um· ∇)um,Aum)+(f,Aum). |(f,Aum)| ≤ ενkumk2 2+C|f|2 2 Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 5/21 Navier-Stokes Aumtest function (AStokes operator) (∂tum,Aum)−ν(∆um,Aum) + ((um· ∇)um,Aum)=(f,Aum), 1 2 d dt kumk2 1+νkumk2 2=−((um· ∇)um,Aum)+(f,Aum). |(f,Aum)| ≤ ενkumk2 2+C|f|2 2 Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 5/21 Navier-Stokes Small data |((um· ∇)um,Aum)|≤|um|6|∇um|3|Aum|2≤Ckumk3/2 2kumk3/2 1 ≤ενkumk2 2+Ckumk6 1 Φm(t) = kumk2 1,Ψm(t) = kumk2 2. Φ0 m+νΨm≤C1Φ3 m+C2|f|2 2, Φ(0)=Φm0 Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 6/21 Navier-Stokes Small data |((um· ∇)um,Aum)|≤|um|6|∇um|3|Aum|2≤Ckumk3/2 2kumk3/2 1 ≤ενkumk2 2+Ckumk6 1 Φm(t) = kumk2 1,Ψm(t) = kumk2 2. Φ0 m+νΨm≤C1Φ3 m+C2|f|2 2, Φ(0)=Φm0 Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 6/21 Navier-Stokes Φm(0)≤δ,kfkL2(L2)< δ/C2⇒Φm(t)<2δ,∀t∈[0,T]. Indeed, (by contradiction) if there exists T∗∈[0,T], Φm(T∗) = 2δand Φm(s)<2δ∀s∈[0,T∗), then (PPoincar´ e constant), Φ0 m+νPΦm≤C1(2δ)2Φm+C2|f|2 2in [0,T∗]. δ <<: Φ0 m+CΦm≤C2|f|2 2in [0,T∗]. Integrating in [0,T∗]with a Gronwall’s technique, Φm(T∗)≤Φm(0)e−CT∗+C2ZT∗ 0 |f|2 2<2δ.!! um∈L∞(H1)∩L2(H2). Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 7/21 Some models A Nematic Liquid Crystal Model                    ∂tu+ (u· ∇)u−ν∆u+∇p= (∇d)tω, ∇ · u=0,in Ω×[0,∞), ∂td+ (u· ∇)d+ω=0 u(x,t) = 0,d(x,t) = h(x,t)on [0,∞)×∂Ω, u(0) = u0,d(0) = d0or u(0) = u(T),d(0) = d(T)in Ω. f(d) = 1 ε2(|d|2−1)d,ω=−∆d+f(d), Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 12/21 Some models A Nematic Liquid Crystal Model                    ∂tu+ (u· ∇)u−ν∆u+∇p= (∇d)tω, ∇ · u=0,in Ω×[0,∞), ∂td+ (u· ∇)d+ω=0 u(x,t) = 0,d(x,t) = h(x,t)on [0,∞)×∂Ω, u(0) = u0,d(0) = d0or u(0) = u(T),d(0) = d(T)in Ω. f(d) = 1 ε2(|d|2−1)d,ω=−∆d+f(d), Φm(t) = kumk2 1+|ωm+∂te d|2 2, Ψ1 m(t) = kumk2 2,Ψ2 m(t) = |∇(ωm+∂te d)|2 2 ν >> ⇒Strong solution of (IVP) in (0,+∞)and regular time-periodic solution. Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 13/21 Some models A Smectic-A Liquid Crystal Model                    ∂tu+ (u· ∇)u−ν∆u−ω∇ϕ+∇p=0 ∇ · u=0,in Ω×[0,∞), ∂tϕ+u· ∇ϕ+ω=0, u|∂Ω=0, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2on [0,∞)×∂Ω, u(0) = u0, ϕ(0) = ϕ0or u(0) = u(T), ϕ(0) = ϕ(T)in Ω. ω= ∆2ϕ−∇·f(∇ϕ), ϕ1=ϕ1(x,t), ϕ2=ϕ2(x,t) Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 14/21 Some models A Smectic-A Liquid Crystal Model                    ∂tu+ (u· ∇)u−ν∆u−ω∇ϕ+∇p=0 ∇ · u=0,in Ω×[0,∞), ∂tϕ+u· ∇ϕ+ω=0, u|∂Ω=0, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2on [0,∞)×∂Ω, u(0) = u0, ϕ(0) = ϕ0or u(0) = u(T), ϕ(0) = ϕ(T)in Ω. ω= ∆2ϕ−∇·f(∇ϕ), ϕ1=ϕ1(x,t), ϕ2=ϕ2(x,t) Φm(t) = kumk2 1+|ωm−∂teϕ|2 2, Ψ1 m(t) = kumk2 2,Ψ2 m(t) = kωm−∂teϕk2 2 ν >> ⇒Strong solution of (IVP) in (0,+∞)and regular time-periodic solution. Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 15/21 Some models A Bioconvective Model              ∂u ∂t−2∇ · (ν(m)D(u)) + u· ∇u+∇q=−mχ+f, ∇ · u=0,in (0,T)×Ω. ∂m ∂t−θ∆m+u· ∇m+U∂m ∂x3 =0,          u=0 on (0,T)×S,u·n=0 on (0,T)×Γ, ν(m)[D(u)n−n·(D(u)n)n] = 0 on (0,T)×Γ, θ∂m ∂n−Umn3=0 on (0,T)×∂Ω. u(0) = u(T),m(0) = m(T),in Ω. Γ S Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 16/21 Some models Φ1 n(t) = ZΩ (ν(mn) + 1)|D(un−uα)|2 Φ2 n(t) = kmn−mαk2 2+|∂tmn|2 2, Ψ1 n(t) = kun−uαk2 2, Ψ2 n=|∂tun|2 2+kmn−mαk2 3+k∂tmnk2 1 νmin >> and u0−uα,m0−mα<< ⇒ Strong time-periodic solution. Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 17/21 Table of contents 1Navier-Stokes 3D: Strong estimates for small data or large viscosity 2Some Models Navier-Stokes type 3Third option: large time Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 18/21 Large time Lemma Let Φ∈L1(0,+∞)and Ψ∈L1 loc(0,+∞)be two positive functions satisfying Φ0(t) + CΨ(t)≤A(Φ(t)) + B(Φ(t))Ψ(t) A(Φ) will be an addition of powers (≥1) and B(Φ) an addition of powers (>0) of Φ. Then, lim t→+∞Φ(t) = 0. In particular, there exists t∗≥0such that Φ∈Cb[t∗,+∞). Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 19/21 Large time A Double Penalized Smectic-A Model              ∂tu+ (u· ∇)u−∆u−(∇n)tω+∇q=0, ∇ · u=0, ∂tn+u· ∇n−γ∆ω=0, Aε2(n) + fε1(n)−ω=0,in Ω×(0,+∞) u|∂Ω=0,n|∂Ω=n∂Ω,ω|∂Ω=0 u(0) = u0,n(0) = n0in Ω fε1(n) = 1 ε2 1 (|n|2−1)n (Aε2(n),n) := (∇n,∇n) + 1 ε2 2 (∇ × n,∇ × n). Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 20/21 Large time Φm(t) = |∇um|2 2+|∇ωm|2 2, Ψm(t) = 1 2|Aum|2 2+Kk∂tnmk2 1 Φ0 m+ Ψm≤C1+ Φ3 m. Strong solution of (PVI) in (t∗,+∞). Blanca Climent Ezquerra. Universidad de Sevilla. WIMA 2012. 21/21