Reproductive and time periodic solutions for incompressible fluids
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REPRODUCTIVE AND TIME PERIODIC SOLUTIONS FOR INCOMPRESSIBLE FLUIDS Blanca Climent Ezquerra Francisco Guill´ en Gonz´ alez Marko Rojas Medar Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 1/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 2/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 2/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 2/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 2/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 2/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 3/35
The title Time-conditions Initial condition: u(0) = u0 Condition of reproductivity: u(0) = u(T) Condition of reproductivity (or time-periodic condition) ⇒Reproductive solutions Moreover if u(t+T) = u(t)∀t∈(0,+∞) ⇒Periodic solutions Incompressibility condition ∇ · u=0 Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 4/35
The title Time-conditions Initial condition: u(0) = u0 Condition of reproductivity: u(0) = u(T) Condition of reproductivity (or time-periodic condition) ⇒Reproductive solutions Moreover if u(t+T) = u(t)∀t∈(0,+∞) ⇒Periodic solutions Incompressibility condition ∇ · u=0 Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 4/35
Main classical results for the initial-boundary problem Weak solution For any u0∈Hand f∈L2(0,T;H−1(Ω)), initial-boundary problem has (at least) a weak solution. Regularity If u0∈Vand f∈L∞(0,∞;L2(Ω)): Unique strong solution (u,p)local If (u0,f)are small enough the strong solution is global. Weak/strong uniqueness property If a solution has the strong regularity, it coincides with any weak solution associated with the same data. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 9/35
Main classical results for the initial-boundary problem Weak solution For any u0∈Hand f∈L2(0,T;H−1(Ω)), initial-boundary problem has (at least) a weak solution. Regularity If u0∈Vand f∈L∞(0,∞;L2(Ω)): Unique strong solution (u,p)local If (u0,f)are small enough the strong solution is global. Weak/strong uniqueness property If a solution has the strong regularity, it coincides with any weak solution associated with the same data. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 9/35
Main classical results for the initial-boundary problem Weak solution For any u0∈Hand f∈L2(0,T;H−1(Ω)), initial-boundary problem has (at least) a weak solution. Regularity If u0∈Vand f∈L∞(0,∞;L2(Ω)): Unique strong solution (u,p)local If (u0,f)are small enough the strong solution is global. Weak/strong uniqueness property If a solution has the strong regularity, it coincides with any weak solution associated with the same data. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 9/35
Main classical results for the initial-boundary problem N=2 If u0∈Hand f∈L2(0,T;H−1(Ω)) the weak solutions is unique. If u0∈Vand f∈L∞(0,∞;L2(Ω)) the strong solution is global. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 10/35
On the time-periodic weak solutions Theorem For any f∈L2(0,T;H−1(Ω)), there exists a weak solution of reproductive problem. Time periodic extension, e u, of any weak reproductive solution u to the whole time interval (0,+∞)is a periodic weak solution corresponding to the data,ef, defined as the time periodic extension of f. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 11/35
On the time-periodic weak solutions Theorem For any f∈L2(0,T;H−1(Ω)), there exists a weak solution of reproductive problem. Time periodic extension, e u, of any weak reproductive solution u to the whole time interval (0,+∞)is a periodic weak solution corresponding to the data,ef, defined as the time periodic extension of f. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 11/35
Main ideas proof existence reproductive solutions Let ukthe unique approximate solution of the Galerkin initial boundary problem of Navier-Stokes in the finite-dimensional subspace Vk, spanned by the first kelements of the “spectral” basis of V(orthogonal in Vand orthonormal in H), associated to a initial discrete data uk 0∈Vk. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 12/35
Main ideas proof existence reproductive solutions Energy inequality + Poincar´ e inequality + integrating [0,T]⇒ ec1Tkuk(T)k2 L2≤ kuk(0)k2 L2+CZT 0ec1tkf(t)k2 H−1dt.(1) We define the operator Lk: [0,T]→Rk, Lk(t) = (ck 1(t), ..., ck k(t)) where ck i(t),i=1, . . . , k,are the coefficients of the expansion of uk(t)in Vk. Note that kLk(t)kRk=kuk(t)kL2,(2) Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 13/35
Main ideas proof existence reproductive solutions We define the operator Φk:Rk→Rkas follows: Given Lk 0∈Rk,Φk(Lk 0) = Lk(T), where Lk(t)are the coefficients of the Galerkin solution with initial value with coefficients Lk 0. Leray-Schauder Theorem For all λ∈[0,1], the possible solutions of the equation Lk 0(λ) = λΦk(Lk 0(λ)),are bounded independently of λ? Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 14/35
Main ideas proof existence reproductive solutions We define the operator Φk:Rk→Rkas follows: Given Lk 0∈Rk,Φk(Lk 0) = Lk(T), where Lk(t)are the coefficients of the Galerkin solution with initial value with coefficients Lk 0. Leray-Schauder Theorem For all λ∈[0,1], the possible solutions of the equation Lk 0(λ) = λΦk(Lk 0(λ)),are bounded independently of λ? Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 14/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 21/35
Reproductivity and maximum principle Given u:Q→R3such that ∇ · u=0 in Qand u·n=0 on ∂Ω, we consider the (reproductive) convection-diffusion problem for the unknown c:Q→R(a concentration): ∂tc−∆c+u· ∇c=0,c|Σ=cΣ,c(0) = c(T), where 0 <c≤cΣ≤con Σ, for some constants cand c. Any reproductive solution satisfies the maximum principle. In particular, ∂t(c−c)−∆(c−c)+(u· ∇)(c−c) = 0 in Q. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 22/35
Reproductivity and maximum principle Given u:Q→R3such that ∇ · u=0 in Qand u·n=0 on ∂Ω, we consider the (reproductive) convection-diffusion problem for the unknown c:Q→R(a concentration): ∂tc−∆c+u· ∇c=0,c|Σ=cΣ,c(0) = c(T), where 0 <c≤cΣ≤con Σ, for some constants cand c. Any reproductive solution satisfies the maximum principle. In particular, ∂t(c−c)−∆(c−c)+(u· ∇)(c−c) = 0 in Q. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 22/35
Reproductivity and maximum principle Multiplying by (c−c)+and integrating in Ω: d dt ZΩ |(c−c)+|2+ZΩ |∇(c−c)+|2≤0. Integrating in t∈(0,T)and using the periodic condition c(0) = c(T):ZT 0k∇(c−c)+k2 L2=0. Hence c≤cin Qhold. Similarly c≥cin Qhold. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 23/35
Reproductivity and maximum principle Generalized Boussinesq system ∂tu− ∇ · (ν(θ)∇u)+(u· ∇)u+∇p=αθg+f, ∇ · u=0, ∂tθ− ∇ · (k(θ)∇θ)+(u· ∇)θ=0, u=0, θ =θ∂Ωon ∂Ω×[0,T), u(0) = u(T), θ(0) = θ(T)in Ω. ν:IR →IR+and k:IR →IR+are continuous functions. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 24/35
Reproductivity and maximum principle θm´ın =m´ınθ∂Ωθm´ ax =m´ axθ∂Ω Maximum principle =⇒θm´ın ≤θ≤θm´ ax. Then ∃νm´ın >0,km´ın >0, νm´ ax >0,km´ ax >0 such that νm´ın ≤ν(s)≤νm´ ax,km´ın ≤k(s)≤km´ ax,∀s∈[θm´ın, θm´ ax]. Changing νby eνand kby e k, where eνand e kare bounded functions, the same way that in the Navier-Stokes case. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 25/35
Reproductivity and maximum principle θm´ın =m´ınθ∂Ωθm´ ax =m´ axθ∂Ω Maximum principle =⇒θm´ın ≤θ≤θm´ ax. Then ∃νm´ın >0,km´ın >0, νm´ ax >0,km´ ax >0 such that νm´ın ≤ν(s)≤νm´ ax,km´ın ≤k(s)≤km´ ax,∀s∈[θm´ın, θm´ ax]. Changing νby eνand kby e k, where eνand e kare bounded functions, the same way that in the Navier-Stokes case. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 25/35
Reproductivity and maximum principle Penalized Nematic liquid crystal model ∂tu+ (u· ∇)u−µ∆u+∇p=−λ∇ · (∇dt∇d), ∇ · u=0, ∂td+ (u· ∇)d=γ(∆d−fε(d)). u=0,d=hon ∂Ω×(0,T) u(0) = u(T),d(0) = d(T)in Ω. fε(d) = ε−2(|d|2−1)d Assuming |h| ≤ 1, we can apply the maximum principle argument obtaining |d| ≤ 1. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 26/35
Reproductivity and maximum principle We consider a equivalent problem changing fεbyefε, the auxiliary function efε(d) = (fε(d)if |d| ≤ 1, 0 if |d|>1. The key is that |efε(d)| ≤ 1 ε2∀d∈R3. Then, existence of weak reproductive solution of this model can be proved. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 27/35
Table of contents 1Introduction 2Navier-Stokes equations Main classical results for the initial-boundary problem On the time-periodic weak solutions Relation between weak periodic solutions and global solutions 3Some variants of Navier-Stokes equations Boussinesq equations Micropolar equations 4Reproductivity and maximum principle Generalized Boussinesq system, with diffusion depending on temperature Penalized Nematic liquid crystal model 5Regularity of periodic solutions via regularity of reproductive solutions Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 28/35
Regularity of periodic solutions Generalized Boussinesq model The regularity of a time-periodic solution for the generalized Boussinesq model is an open problem. Generalized Boussinesq model with Neumann boundary conditions for temperature Assuming fsmall enough, reproductive solution has H2(Ω)-velocity and H3(Ω)-temperature regularity. When Dirichlet boundary conditions for uand θare assumed, it is not clear how to obtain appropriate differential inequalities in H2for velocity and H3for temperature. Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 34/35
Generalized Boussinesq model Periodic solutions Uniqueness of regular time periodic solutions remains open: H3regularity for the velocity ↔Dirichlet condition for velocity !! Argument of regular time periodic solution small data ⇒ ku(t?)k2 H1+kθ(t?)k2 H1is small but kθ(t?)k2 H2? Blanca Climent Ezquerra. Universidad de Sevilla. Reprod. and time periodic solutions for incompressible fluids 35/35