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FINITE DIMENSIONAL REDUCTION OF NONAUTONOMOUS DISSIPATIVESYSTEMSAlain MiranvilleUniversit´e de Poitiers Collaborators:Long time behavior of equations of the formy′=F(t,y)For autonomous systems:y′=F(y)In many situations,the evolution of the sys-tem is described by a system of ODEs:y=(y1,...,y N)∈R N,F=(F1,...,F N)Assuming that the Cauchy problemy′=F(y),y(0)=y0,is well-posed,we can define the family of solv-ing operators S(t),t≥0,acting on a subset φ⊂R N:S(t):φ→φy0→y(t)This family of operators satisfiesS(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0We say that it forms a semigroup onφQualitative study of such systems:goes back to Poincar´eMuch is known nowadays,at least in low di-mensionsEven relatively simple systems can generate very complicated chaotic behaviorsThese systems are sensitive to perturbations: trajectories with close initial data may diverge exponentially→Temporal evolution unpredictable on ti-me scales larger than some critical value→Show typical stochastic behaviorsExample:Lorenz systemx′=σ(y−x)y′=−xy+rx−yz′=xy−bzObtained by truncature of the Navier-Stokes equationsGives an approximate description of a layer of fluid heated from belowSimilar to what is observed in the atmosphereFor a sufficiently intense heating:sensitive dependence on the initial conditions,repre-sents a very irregular convection→Butterfly effectVery often,the trajectories are localized in some subset of the phase space having a very complicated geometric structure(e.g.,locally homeomorphic to the product of R m and a Cantor set)→Strange attractor(Ruelle and Takens)Main feature of a strange attractor:dimen-sionSensitivity to initial conditions:>2(dimen-sion of the phase space≥3,say,3)Contraction of volumes:its volume is equal to0→noninteger,strictly between2and3→Fractal dimensionExample:Lorenz system:dim F A=2.05...Distributed systems:systems of PDEsφis a subset of an infinite dimensional func-tion space(e.g.,L2(Ω)or L∞(Ω))Solution:y:R+→φt→y(t)x→y(t,x)If the problem is well-posed,we can define the semigroup S(t):S(t):φ→φy0→y(t)The analytic structure of a PDE is much more complicated than that of an ODE:the global well-posedness can be a very difficult problemSuch results are known for a large class of PDEs→it is natural to investigate whether the notion of a strange attractor extends to PDEsSuch chaotic behaviors can be observed in dissipative PDEsChaotic behaviors arise from the interaction of•Energy dissipation in the higher part of the Fourier spectrum•External energy income in the lower part•Energyflux from the lower to the higher modesThe trajectories are localized in a”thin”in-variant region of the phase space having a very complicated geometric structure→the global attractor1.The global attractor.S(t)semigroup acting on E:S(t):E→E,t≥0S(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0 Continuity:x→S(t)x is continuous on E,∀t≥0A set A⊂E is the global attractor for S(t)if(i)it is compact(ii)it is invariant:S(t)A=A,t≥0(iii)∀B⊂A,lim t→+∞dist(S(t)B,A)=0dist(A,B)=supa∈A infb∈Ba−b EEquivalently:∀B⊂φbounded,∀ǫ>0,∃t0= t0(B,ǫ)s.t.t≥t0implies S(t)B⊂UǫThe global attractor is uniqueIt is the smallest closed set enjoying(iii)It is the maximal bounded invariant setTheorem:(Babin-Vishik)We assume that S(t)possesses a compact attracting set K, i.e.,∀B⊂E bounded,lim t→+∞dist(S(t)B,K)=0Then S(t)possesses the global attractor A.The global attractor is oftenfinite dimen-sional:the dynamics,restricted to A isfinite dimensionalFractal dimension:Let X be a compact setdim F X=lim supǫ→0+ln Nǫ(X)ǫNǫ(X):minimum number of balls of radius ǫnecessary to cover XIf Nǫ(X)≤c(1Theorem:(H¨o lder-Ma˜n´e theorem)Let X⊂E compact satisfy dim F X=d and N>2d be an integer.Then almost every bounded linear projector P:E→R N is one-to-one on X and has a H¨o lder continuous inverse.This result is not valid for other dimensions (e.g.,the Hausdorffdimension)If A hasfinite fractal dimension,then,fixing a projector P satisfying the assumptions of the theorem,we obtain a reduced dynamical system(S),S= P(A),which isfinite dimensional(in R N)and H¨o lder continuousDrawbacks:(S)cannot be realized as a system of ODEs which is well-posedReasonable assumptions on A which would ensure that the Ma˜n´e projectors are Lipschitz are not knownComplicated geometric structure of A and AThe lower semicontinuitydist(A0,Aǫ)→0asǫ→0is more difficult to prove and may not hold It may be unobservable:∂y∂x2+y3−y=0,x∈[0,1],ν>0y(0,t)=y(1,t)=−1,t≥0A={−1}There are many metastable”almost station-ary”equilibria which live up to t⋆≡eν−12.Inertial manifolds.A Lipschitzfinite dimensional manifold M⊂E is an inertial manifold for S(t)if(i)S(t)M⊂M,∀t≥0(ii)∀u0∈E,∃v0∈M s.t.S(t)u0−S(t)v0 E≤Q( u0 E)e−αt,α>0,Q monotonicM contains A and attracts the trajectories exponentiallyConfirms in a perfect way thefinite dimen-sional reduction principle:The dynamics reduced to M can be realized as a Lipschitz system of ODEs(inertial form)Perfect equivalence between the initial sys-tem and the inertial formDrawback:all the known constructions are based on a restrictive condition,the spectral gap condition→The existence of an inertial manifold is not known for several important equations, nonexistence results for damped Sine-Gordon equations3.Exponential attractors.A compact set M⊂E is an exponential at-tractor for S(t)if(i)It hasfinite fractal dimension(ii)S(t)M⊂M,∀t≥0(iii)∀B⊂E bounded,dist(S(t)B,M)≤Q( B E)e−αt,α>0,Q monotonicM contains AIt is stillfinite dimensional and one has a uni-form exponential control on the rate of at-traction of trajectoriesIt is no longer smoothDrawback:it is not unique→One looks for a simple algorithm S→M(S)Initial construction:non-constructible and valid in Hilbert spaces onlyConstruction in Banach spaces:Efendiev, Miranville,Zelik→Exponential attractors are as general as global attractorsMain tool:Compact smoothing property on the difference of2solutionsLet S:E→E.We consider the discrete dynamical system generated by the iterations of S:S n=S◦...◦S(n times)Theorem:(Efendiev,Miranville,Zelik)We consider2Banach spaces E and E1s.t.E1⊂E is compact.We assume that•S maps theδ-neighborhood Oδ(B)of a bounded subset B of E into B•∀x1,x2∈Oδ(B),≤K x1−x2 ESx1−Sx2 E1Then the discrete dynamical system gener-ated by the iterations of S possesses an ex-ponential attractor M(S)s.t.(i)M(S)⊂B,is compact in E anddim F M(S)≤c1(ii)S M(S)⊂M(S)(iii)dist(S k B,M(S))≤c2e−c3k,k∈N,c3>0 (iv)The map S→M(S)is H¨o lder continu-ous:∀S1,S2,dist sym(M(S1),M(S2))≤c4 S1−S2 c5,c5>0, wheredist sym(A,B)=max(dist(A,B),dist(B,A))S =supSh Eh∈Oδ(B)Furthermore all the constants only depend on B,E,E1,δand K and can be computed explicitly.Remarks:1)We have a mapping S→M(S)and,due to the H¨o lder continuity,we can construct continuous families of exponential attractors2)Exponential attractors for a continuous semigroup S(t):Prove that∃t⋆>0s.t.S⋆=S(t⋆)satisfies the assumptions of the theorem→M⋆for S⋆If(x,t)→S(t)x is Lipschitz(or H¨o lder)on B×[0,t⋆],setS(t)M⋆M=∪t∈[0,t⋆]We again have a mapping S(t)→M(S)which is H¨o lder continuous3)For damped hyperbolic equations:asymp-totically smoothing property4.Finite dimensional reduction of nonau-tonomous systems.Systems of the form∂yDrawback:the uniform attractor has infinite dimension in general.Example:∂yThe family{A(t),t∈R}is a pullback attrac-tor for U(t,τ)if(i)A(t)is compact in E,∀t∈R(ii)U(t,τ)A(τ)=A(t),∀t≥τ(iii)∀B⊂E bounded,dist(U(t,t−s)B,A(t))=0lims→+∞Remarks:1)The pullback attractor is unique2)If the system is autonomous,we recover the global attractor3)In general,A(t)hasfinite fractal dimen-sion,∀t∈RDrawback:The forward convergence does not hold in generalExample:y′=f(t,y),where f(t,y)=−y if y≤0,(−1+2t)y−ty2 if t∈[0,1],and y−y2if t≥1Then A(t)={0},∀t∈R,but every trajectory starting from a neighborhood of0leaves this neighborhood never to enter it againThe forward convergence does not hold be-cause the rate of attraction is not uniform in t→This can be solved by constructing ex-ponential attractorsWe can construct a family{M(t),t∈R}, called nonautonomous exponential attractor, s.t.(i)dim F M(t)≤c1,∀t∈R,c1independent of t(ii)U(t,τ)M(τ)⊂M(t),∀t≥τ,(iii)∀B⊂E bounded,dist(U(t,τ)B,M(t+τ))≤Q( B E)e−αt,t∈R,t≥τ,α>0,Q monotonic(iii)implies the pullback attraction,but also the forward attraction→(i)and(iii)yield a satisfactoryfinite di-mensional reduction principle for nonautono-mous systemsRemarks:1)The time dependence is arbitrary2)The map U(t,τ)→{M(t),t∈R}is also H¨o lder continuous。