Outline Introduction Attractors Other less standard topologies On the long-time behavior of 2D dissipative Euler equations Luigi C. Berselli Dipartimento di Matematica Applicata “U. Dini” Universit` a di Pisa June 26, 2012 hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies 1 Introduction 2 Attractors 3 Other less standard topologies hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model We consider the following system 2D in a bounded and smooth domain Ω with boundary Γ ∂ t u + ( u · ∇ ) u + χ u + ∇ p = f ∇ · u = 0 (1) ( u · n ) | Γ = 0 with initial condition u (0 , x ) = u 0 ( x ) This are called dissipative Euler equations. There is a damping term, not a smoothing one. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model 1 The term χ u (with χ > 0) may model the bottom friction in some 2D oceanic models (in that case, is called the viscous Charney-Stommel barotropic ocean circulation model of the gulf stream) hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model 1 The term χ u (with χ > 0) may model the bottom friction in some 2D oceanic models (in that case, is called the viscous Charney-Stommel barotropic ocean circulation model of the gulf stream) 2 It is related with Rayleigh friction in the planetary boundary layer (with space-periodic boundary conditions). hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model 1 The term χ u (with χ > 0) may model the bottom friction in some 2D oceanic models (in that case, is called the viscous Charney-Stommel barotropic ocean circulation model of the gulf stream) 2 It is related with Rayleigh friction in the planetary boundary layer (with space-periodic boundary conditions). 3 The constant χ is the Rayleigh friction coefficient (or the Ekman pumping/dissipation constant) or also the sticky viscosity, when the model is used to study motion in presence of rough boundaries. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model The model (1) represents (probably) the “weakest” dissipative modification of the Euler equations; hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model The model (1) represents (probably) the “weakest” dissipative modification of the Euler equations; Results on the long-time behavior of the damped Navier-Stokes do not directly pass to the limit “viscosity goes to zero,” hence a completely different treatment is required to study the problem without dissipation. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies The model The model (1) represents (probably) the “weakest” dissipative modification of the Euler equations; Results on the long-time behavior of the damped Navier-Stokes do not directly pass to the limit “viscosity goes to zero,” hence a completely different treatment is required to study the problem without dissipation. Early studies are in Barcilon Constantin & Titi (SIMA 1988); Hauk (PhD Thesis, Irvine 1997) Gallavotti (Quaderni CNR 1996). hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Long time behavior Since there is a damping term, one has chances of studying the long-time behavior. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Long time behavior Since there is a damping term, one has chances of studying the long-time behavior. The theory of attractors has been studied by Il’in (Math. Sbornik 1991) Bessaih & Flandoli (NODEA 2000). hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Long time behavior Since there is a damping term, one has chances of studying the long-time behavior. The theory of attractors has been studied by Il’in (Math. Sbornik 1991) Bessaih & Flandoli (NODEA 2000). The theory is based on the following energy-type estimates. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies A priori estimates Energy estimate: Testing with u itself one obtains dt � u � 2 + χ � u � 2 ≤ 1 d χ � f � 2 hence the estimate � t � u ( t ) � 2 ≤ � u ( t 0 ) � 2 e − χ ( t − t 0 ) + � f � 2 e − χ ( t − s ) ds , a. e. t ≥ t 0 ≥ 0 , χ t 0 and consequently a UNIFORM bound for the kinetic energy for all positive times. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies A priori estimates Enstrophy estimate: Taking the 2D curl ξ := ∂ 1 u 2 − ∂ 2 u 1 φ := curl f we get ∂ t ξ + χ ξ + ( u · ∇ ) ξ = φ (2) one immediately obtain a uniform bound for ξ � t � ξ ( t ) � 2 ≤ � ξ ( t 0 ) � 2 e − χ ( t − t 0 ) + � φ � 2 e − χ ( t − s ) ds , a. e. t ≥ t 0 ≥ 0 , χ t 0 hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Existence of weak solutions From the above estimates one obtains directly existence of weak solutions, by adapting Yudovich, (USSR Comp. Math. Math. Phys. 1963) and Bardos (JMAA 1972) theorems, based on vanishing viscosity approximation. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Existence of weak solutions From the above estimates one obtains directly existence of weak solutions, by adapting Yudovich, (USSR Comp. Math. Math. Phys. 1963) and Bardos (JMAA 1972) theorems, based on vanishing viscosity approximation. One has then an absorbing set in L 2 and boundedness in H 1 (since if ∇ · u = 0 and u · n = 0, then curl u ∼ ∇ u . hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Existence of weak solutions From the above estimates one obtains directly existence of weak solutions, by adapting Yudovich, (USSR Comp. Math. Math. Phys. 1963) and Bardos (JMAA 1972) theorems, based on vanishing viscosity approximation. One has then an absorbing set in L 2 and boundedness in H 1 (since if ∇ · u = 0 and u · n = 0, then curl u ∼ ∇ u . Seemingly this should be enough to construct an attractor in a standard way by taking the ω -limit closure of an absorbing set. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies On the global attractor This approach does not work, since the map u 0 �→ u ( t ) is not well defined (not a semigroup in the phase space): Lack of uniqueness hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies On the global attractor This approach does not work, since the map u 0 �→ u ( t ) is not well defined (not a semigroup in the phase space): Lack of uniqueness Apart small improvements a requirement to have uniqueness is that curl u 0 , curl f ∈ L ∞ hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies On the global attractor This approach does not work, since the map u 0 �→ u ( t ) is not well defined (not a semigroup in the phase space): Lack of uniqueness Apart small improvements a requirement to have uniqueness is that curl u 0 , curl f ∈ L ∞ The mapping u 0 �→ u ( t ) is well defined, but not continuous in this setting, namely in W 1 , ∞ !! hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies On the global attractor One main point when dealing with the Euler equations is that they are a (very peculiar) semi-linear hyperbolic system hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies On the global attractor One main point when dealing with the Euler equations is that they are a (very peculiar) semi-linear hyperbolic system The usual splitting of the semigroup S ( t ) = S 1 ( t ) + S 2 ( t ) with a compact term, plus a second one decaying at infinity is not simple to be obtained. hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies On the global attractor One main point when dealing with the Euler equations is that they are a (very peculiar) semi-linear hyperbolic system The usual splitting of the semigroup S ( t ) = S 1 ( t ) + S 2 ( t ) with a compact term, plus a second one decaying at infinity is not simple to be obtained. The well-established techniques for damped hyperbolic equations, as summarized in Temam (Springer 1997) seem to be not applicable hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Weak attractor This explains why in Il’in and Bessaih and Flandoli it is studied a weak attractor, that is the attractor is considered in the path space and the semi-group is made with the time-shifts u ( t ) �→ u ( t + h ) hyp2012, Padova, June 25–29, 2012
Outline Introduction Attractors Other less standard topologies Links with 2D NSE In the context of 2D Navier-Stokes equations, the fact that vorticity is conserved can be used to improve known results. hyp2012, Padova, June 25–29, 2012
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