On stochastic 2D Navier Stokes equations and hydrodynamical models Annie Millet (amillet@univ-paris1.fr ∗ ) SAMM, Universit´ e Paris 1 Workshop Stochastic control and finance Roscoff, March 19, 2010 A. Millet On stochastic 2D NS equations
Outline 1 Introduction The evolution equations Random perturbation 2 Well posedeness and apriori estimates The well posedeness results Proof of the well posedeness and apriori estimates 3 Further results Some control of time increments Large Deviations Support characterization Stochastic 2D Euler equation Stochastic 3D Navier Stokes equations A. Millet On stochastic 2D NS equations
Outline 1 Introduction The evolution equations Random perturbation 2 Well posedeness and apriori estimates The well posedeness results Proof of the well posedeness and apriori estimates 3 Further results Some control of time increments Large Deviations Support characterization Stochastic 2D Euler equation Stochastic 3D Navier Stokes equations A. Millet On stochastic 2D NS equations
Outline 1 Introduction The evolution equations Random perturbation 2 Well posedeness and apriori estimates The well posedeness results Proof of the well posedeness and apriori estimates 3 Further results Some control of time increments Large Deviations Support characterization Stochastic 2D Euler equation Stochastic 3D Navier Stokes equations A. Millet On stochastic 2D NS equations
1 Introduction The evolution equations Random perturbation 2 Well posedeness and apriori estimates The well posedeness results Proof of the well posedeness and apriori estimates 3 Further results Some control of time increments Large Deviations Support characterization Stochastic 2D Euler equation Stochastic 3D Navier Stokes equations A. Millet On stochastic 2D NS equations
Introduction The Navier Stokes equations D bounded domain of R 2 , x = ( x 1 , x 2 ), u ( x , t ) = ( u 1 ( x , t ) , u 2 ( x , t )) fluid velocity, p ( x , t ) pressure divergence: div u = � i =1 , 2 ∂ i u i i =1 , 2 ∂ 2 i u k , k = 1 , 2) (Stokes operator if one Laplace operator: ∆ u = ( � adds the incompressibility condition div u = 0 on D ) Find a pair ( u , p ) ( u velocity, p pressure) such that ∂ t u − ν ∆ u + u . ∇ u + ∇ p = f , div u = 0 in D , u = 0 on ∂ D ν > 0 viscosity, n outwards normal to ∂ D and f ( x , t ) external force � 2 : div f = 0 in D and f . n = 0 on ∂ D } L 2 ( D ) � H = { f ∈ � 2 ∩ H , H 1 � V ≡ 0 ( D ) V ⊂ H , ( H , | | ) and ( V , � � ) Hilbert spaces Project on divergence free fields (integration by parts: if, div u = 0 then ( ∇ p , u ) = 0 if div u = 0 A. Millet On stochastic 2D NS equations
Introduction The operators A and B A : V → V ′ et B : V × V → V ′ defined by: � ∇ u j · ∇ v j dx � � Au , v � = ν D j =1 , 2 � � u j ∂ j v i w i dx , ∀ u , v , w ∈ V � � B ( u , v ) , w � = [ u ·∇ v ] w dx ≡ D D i , j =1 , 2 Properties of A and B A = − ν ∆ non-negative, unbounded, self-adjoint linear operator on H , B : V × V → V ′ bilinear continuous ∀ u 1 , u 2 , u 3 ∈ V � B ( u 1 , u 2 ) , u 3 � = −� B ( u 1 , u 3 ) , u 2 � D u j ∂ j v i w i dx = − � w i ∂ j u j + u j ∂ j w i � D v i � (Proof: � � � dx i , j i , j j ∂ j u j = div u = 0 ) for fixed i , � A. Millet On stochastic 2D NS equations
Introduction Interpolation space and B V ⊂ L 4 ( D ) ⊂ H and for u ∈ V , � u � 2 L 4 ( D ) ≤ | u | � u � (Proof: for real-valued functions f , g � fg � 1 ≤ 1 4 � ∂ 1 f � 1 ∂ 2 g � 1 with f 2 and g 2 , � f 2 g 2 � 1 ≤ � f ∂ 1 f � 1 � g ∂ 2 g � 1 Schwarz’s inequality : � f 2 g 2 � 1 ≤ � f � 2 �∇ f � 2 � g � 2 �∇ g � 2 ; then f = g ) For η > 0 there exists C η > 0 such that |� B ( u 1 , u 1 ) , u 3 �| ≤ η � u 1 � 2 + C η | u 1 | 2 � u 3 � 4 L 4 ( D ) (Proof: H¨ older’s inequality 3 1 2 | u 1 | 2 � u 3 � L 4 ( D ) � | D u 1 ∇ u 1 u 3 | ≤ � u 1 � L 4 ( D ) |∇ u 1 |� u 3 � L 4 ( D ) ≤ � u 1 � Then Young’s inequality with exponents 4 / 3 and 4 yields 3 � u 1 � 2 + 1 4 α | u 1 | 2 � u 3 � 4 |� B ( u 1 , u 1 ) , u 3 �| ≤ 4 α L 4 ( D ) ) If B ( u ) := B ( u , u ) then |� B ( u 1 ) − B ( u 2 ) , u 1 − u 2 �| ≤ η � u 1 − u 2 � 2 + C η | u 1 − u 2 | 2 � u 2 � 4 L 4 ( D ) A. Millet On stochastic 2D NS equations
Introduction Interpolation space and B V ⊂ L 4 ( D ) ⊂ H and for u ∈ V , � u � 2 L 4 ( D ) ≤ | u | � u � (Proof: for real-valued functions f , g � fg � 1 ≤ 1 4 � ∂ 1 f � 1 ∂ 2 g � 1 with f 2 and g 2 , � f 2 g 2 � 1 ≤ � f ∂ 1 f � 1 � g ∂ 2 g � 1 Schwarz’s inequality : � f 2 g 2 � 1 ≤ � f � 2 �∇ f � 2 � g � 2 �∇ g � 2 ; then f = g ) For η > 0 there exists C η > 0 such that |� B ( u 1 , u 1 ) , u 3 �| ≤ η � u 1 � 2 + C η | u 1 | 2 � u 3 � 4 L 4 ( D ) (Proof: H¨ older’s inequality 3 1 2 | u 1 | 2 � u 3 � L 4 ( D ) � | D u 1 ∇ u 1 u 3 | ≤ � u 1 � L 4 ( D ) |∇ u 1 |� u 3 � L 4 ( D ) ≤ � u 1 � Then Young’s inequality with exponents 4 / 3 and 4 yields 3 � u 1 � 2 + 1 4 α | u 1 | 2 � u 3 � 4 |� B ( u 1 , u 1 ) , u 3 �| ≤ 4 α L 4 ( D ) ) If B ( u ) := B ( u , u ) then |� B ( u 1 ) − B ( u 2 ) , u 1 − u 2 �| ≤ η � u 1 − u 2 � 2 + C η | u 1 − u 2 | 2 � u 2 � 4 L 4 ( D ) A. Millet On stochastic 2D NS equations
Introduction Interpolation space and B V ⊂ L 4 ( D ) ⊂ H and for u ∈ V , � u � 2 L 4 ( D ) ≤ | u | � u � (Proof: for real-valued functions f , g � fg � 1 ≤ 1 4 � ∂ 1 f � 1 ∂ 2 g � 1 with f 2 and g 2 , � f 2 g 2 � 1 ≤ � f ∂ 1 f � 1 � g ∂ 2 g � 1 Schwarz’s inequality : � f 2 g 2 � 1 ≤ � f � 2 �∇ f � 2 � g � 2 �∇ g � 2 ; then f = g ) For η > 0 there exists C η > 0 such that |� B ( u 1 , u 1 ) , u 3 �| ≤ η � u 1 � 2 + C η | u 1 | 2 � u 3 � 4 L 4 ( D ) (Proof: H¨ older’s inequality 3 1 2 | u 1 | 2 � u 3 � L 4 ( D ) � | D u 1 ∇ u 1 u 3 | ≤ � u 1 � L 4 ( D ) |∇ u 1 |� u 3 � L 4 ( D ) ≤ � u 1 � Then Young’s inequality with exponents 4 / 3 and 4 yields 3 � u 1 � 2 + 1 4 α | u 1 | 2 � u 3 � 4 |� B ( u 1 , u 1 ) , u 3 �| ≤ 4 α L 4 ( D ) ) If B ( u ) := B ( u , u ) then |� B ( u 1 ) − B ( u 2 ) , u 1 − u 2 �| ≤ η � u 1 − u 2 � 2 + C η | u 1 − u 2 | 2 � u 2 � 4 L 4 ( D ) A. Millet On stochastic 2D NS equations
Introduction General framework Project on divergence free functions, suppress the pressure div ∇ p = 0 add a Coriolis term (replace the forcing term f by f − Ru where R ( u 1 , u 2 ) = c 0 ( − u 2 , u 1 ), c 0 constant. ∂ t u − ν ∆ u + u . ∇ u + Ru = f Abstract setting ( H , | . | ) Hilbert, R linear continuous operator on H A non negative, self-adjoint operator (unbounded) operator on H 1 1 2 ); for v ∈ V set � v � = | A 2 v | V = Dom ( A H Banach space such that V ⊂ H ⊂ H and � v � 2 H ≤ K 0 | v | � v � B : V × V → V ′ bilinear continuous such that ∀ u 1 , u 2 , u 3 ∈ V � B ( u 1 , u 2 ) , u 3 � = −� B ( u 1 , u 3 ) , u 2 � for η > 0 there exists C η > 0 such that |� B ( u 1 , u 1 ) , u 3 �|≤ η � u 1 � 2 + C η | u 1 | 2 � u 3 � 4 H � � � � d t u ( t ) + Au ( t ) + B u ( t ) + Ru ( t ) dt = σ ( u ( t )) dW t , u (0) ∈ H A. Millet On stochastic 2D NS equations
Introduction Other examples of evolution equations D = (0 , l ) × (0 , 1), x = ( x 1 , x 2 ) spatial variable p pressure, φ = ( u , θ, β ) satisfy the coupled non-linear equations u ∈ R 2 velocity field, θ ∈ R temperature field, β ∈ R 2 magnetic field ν, κ, η and S physical constants, ∂ ∂ t u + u ·∇ u − ν ∆ u + ∇ p � 1 2 | β | 2 ) − S β ·∇ β − θ e 2 = σ 1 ( t , φ ) dW 1 ( t ) , + ∇ ∂ ∂ t θ + u ·∇ θ − κ ∆ θ − u 2 = σ 2 ( t , φ ) dW 2 ( t ) , ∂ ∂ t β − η ∆ β + u ·∇ β − β ·∇ u = σ 3 ( t , φ ) dW 3 ( t ) , where ∆ is the Laplace operator (Stokes operator after Leray projection) A. Millet On stochastic 2D NS equations
Introduction Examples of evolution equations - Conditions div ( u ) = div ( β ) = 0 ∂ u = θ = β 2 = β 1 = 0 on x 2 ∈ { 0 , 1 } ∂ x 2 u , p , θ, β, u x 1 , θ x 1 , β x 1 period l in x 1 H = L 2 ( D ) 5 with divergence, periodicity and boundary conditions V = H 1 ( D ) 5 with the same conditions → H = H ′ ֒ H = L 4 ( D ) 5 ∩ H and � u � 2 → V ′ V ֒ H ≤ K 0 | u | � u � B ( φ ) = ( B 1 ( u , u ) − SB 1 ( β, β ) , B 2 ( u , θ ) , B 1 ( u , β ) − B 1 ( β, u )) � � � � B 1 ( u , v ) , w � = [ u ·∇ v ] wdx := u i ∂ i v j w j dx , D D i , j =1 , 2 � � � � B 2 ( u , θ ) , η � = [ u ·∇ θ ] η dx := u i ∂ i θ η dx . D D i =1 , 2 A. Millet On stochastic 2D NS equations
Recommend
More recommend
