Large time behavior of coagulation-fragmentation equations with - PowerPoint PPT Presentation

Large time behavior of coagulation-fragmentation equations with degenerate diffusion Laurent Desvillettes CMLA, ENS Cachan & IUF In collaboration with Klemens Fellner – p. 1

A strange decay rate Convergence towards equilibrium in || f ( t ) − f eq || ≤ Cst e − Cst (log t ) β , for all β < 2 . Better than any power; Worse than any Cst exp( − Cst t γ ) , with γ > 0 . – p. 2

Ingredients √ Entropy (Entropy dissipation) method √ Almost exponential convergence to equilibrium √ Tracking explicitly the evolution of moments – p. 3

Entropy method for large time behavior Abstract equation : ∂ t f = A f. We suppose that there exists a (bounded below) Lyapounov functional H := H ( f ) (entropy) and a functional D := D ( f ) (entropy dissipation) such that ∂ t H ( f ) = − D ( f ) ≤ 0 , and D ( f ) = 0 ⇐ ⇒ A f = 0 ⇐ ⇒ f = f eq , where f eq is a given function. – p. 4

De La Salle’s Principle Then, one can often prove that 1. the decreasing function t �→ H ( f ( t )) converges toward its minimum H ( f eq ) , 2. t �→ D ( f ( t )) converges toward 0 , 3. t → + ∞ f ( t ) = f eq lim in a convenient topology. – p. 5

Explicit rate of convergence toward equilibrium One looks for functional inequalities like D ( f ) ≥ Cst ( H ( f ) − H ( f eq )) . Then, using the Lyapounov functional, we get the differential inequality ∂ t ( H ( f ) − H ( f eq )) ≤ − Cst ( H ( f ) − H ( f eq )) , and Gronwall’s lemma ensures that H ( f ) − H ( f eq ) ≤ Cst e − Cst t . Finally, if H has good properties of coercivity, we obtain || f − f eq || ≤ Cst e − Cst t . – p. 6

Coagulation-fragmentation Aizenmann-Bak model ∂ t f ( t, y ) = Q AB ( f )( t, y ) . 1. Break-ups of clusters of size y ′ larger than y contribute to create clusters of size y : Z ∞ Q + f ( t, y ′ ) dy ′ . frag ( f )( t, y ) := 2 y 2. Break-up of polymers of size y reduces its concentration: Q − frag ( f )( t, y ) := y f ( t, y ) . – p. 7

1. Coalescence of clusters of size y ′ ≤ y and y − y ′ results into clusters of size y : Z y Q + f ( t, y − y ′ ) f ( t, y ′ ) dy ′ . coag ( f )( t, y ) := 0 2. Polymerization of clusters of size y with other clusters of size y ′ produces a loss in its concentration: Z ∞ Q − f ( t, y ′ ) dy ′ . coag ( f )( t, y ) := 2 f ( t, y ) 0 Aizenmann-Bak kernel : Q AB ( f )( t, y ) = Q + frag ( f )( t, y ) − Q − frag ( f )( t, y )+ Q + coag ( f )( t, y ) − Q − coag ( f )( t, y ) . – p. 8

Entropy structure for Aizenmann-Bak model Entropy: Z ∞ „ « H AB ( f )( t ) = f ( t, y ) log f ( t, y ) − f ( t, y ) dy , 0 Entropy dissipation: Z ∞ Z ∞ „ « f ( t, y + y ′ ) − f ( t, y ) f ( t, y ′ ) D AB ( f )( t ) = 0 0 „ « log[ f ( t, y + y ′ )] − log[ f ( t, y ) f ( t, y ′ )] dy ′ dy ≥ 0 , × Entropy relation: Z ∞ d dtH AB ( f )( t ) = Q AB ( f, f )( t, y ) log f ( t, y ) dy = − D AB ( f )( t ) . 0 – p. 9

Case of equality in the entropy structure for Aizenmann-Bak kernel D AB ( f ) = 0 ⇐ ⇒ ∀ y ≥ 0 , Q AB ( f )( y ) = 0 ∀ y, y ′ ≥ 0 , f ( y + y ′ ) = f ( y ) f ( y ′ ) ⇐ ⇒ ⇐ ⇒ ∃ c > 0 , f ( y ) = exp ( − c y ) Analogous to the second part (case of equality) of Boltzmann H-theorem. – p. 10

Entropy/Entropy Dissipation estimate for AB model Theorem (M. Aizenmann, T. Bak): For any f := f ( y ) ≥ 0 such that N f ≥ N ∗ > 0 , D AB ( f ) ≥ Cst ( N ∗ ) ( H AB ( f ) − H ( M AB ( f ))) , where √ Z ∞ y − Nf , M AB ( f )( y ) = e N f = f ( y ) y dy. 0 Main tool : Elementary convexity inequalities Analogous result for a discrete model : P.-E. Jabin, B. Niethammer – p. 11

Explicit rate of convergence toward equilibrium (AB) Theorem (M. Aizenmann, T. Bak): Let f in := f in ( y ) ≥ 0 be an initial datum such that Z ∞ f in ( y ) (1 + y + | log f in ( y ) | ) dy < + ∞ . 0 Then there exists a unique solution to the Aizenmann-Bak equation ∂ t f ( t, y ) = Q AB ( f )( t, y ) , f (0 , y ) = f in ( y ) , such that Z ∞ Z ∞ N ( f ( t, · )) = f ( t, y ) y dy = f (0 , y ) y dy = N ( f in ) . 0 0 Moreover this solution satisfies (for some explicit C 1 , C 2 > 0 ) ˛ ˛ «˛ ˛ „ y ≤ C 1 e − C 2 t . ˛ ˛ ˛ ˛ ˛ f ( t, y ) − exp − ˛ ˛ ˛ ˛ p N ( f in ) ˛ ˛ ˛ L 1 ( R + ) – p. 12

Inhomogeneous Aizenmann-Bak equation New variables for the unknown: f ( t, y ) → f ( t, x, y ) . Equation (for some a ( y ) ≥ 0 ): ∂ t f ( t, x, y ) − a ( y ) ∆ x f ( t, x, y ) = Q AB ( f )( t, x, y ) . Homogeneous Neumann boundary conditions: ∀ x ∈ ∂ Ω , ∇ x f ( t, x, y ) · n ( x ) = 0 . Initial datum: f (0 , x, y ) = f in ( x, y ) . – p. 13

Entropy structure for the Inhomogeneous AB equation Entropy: Z ∞ „ « Z H IAB ( f )( t ) = f ( t, x, y ) log f ( t, x, y ) − f ( t, x, y ) dydx . Ω 0 Entropy relation: ∂ t H IAB ( f ) = − ( D 1 ( f ) + D 2 ( f )) . Entropy dissipation: Z ∞ a ( y ) |∇ x f ( t, x, y ) | 2 Z D 1 ( f )( t ) = dydx, f ( t, x, y ) Ω 0 Z ∞ Z ∞ „ « Z f ( t, x, y + y ′ ) − f ( t, x, y ) f ( t, x, y ′ ) D 2 ( f )( t ) = Ω 0 0 „ « log[ f ( t, x, y + y ′ )] − log[ f ( t, x, y ) f ( t, x, y ′ )] dy ′ dydx . × – p. 14

Case of equality in the entropy structure of the inhomogeneous AB model D 1 ( f ) + D 2 ( f ) = 0 f ( x, y ) = e − c ( x ) y ⇐ ⇒ ∇ x c ( x ) = 0 and f ( x, y ) = e − c y . ⇐ ⇒ – p. 15

Entropy/entropy dissipation estimate for the inhomogeneous AB equation Proposition (J. Carrillo, LD, K. Fellner): Let f := f ( x, y ) ≥ 0 be such that Z ∞ Z ∞ M ( x ) := f ( x, y ) dy ≥ M ∗ > 0 , N ( x ) := f ( x, y ) y dy ≥ N ∗ > 0 . 0 0 Then Cst ( M ∗ , N ∗ , inf a, sup a ) D 1 ( f ) + D 2 ( f ) ≥ || M || L ∞ (Ω) log( || M || L ∞ (Ω) ) ( H IAB ( f ) − H IAB ( f eq )) , with r | Ω | − y N ( x ) dx . R f eq ( x, y ) = e – p. 16

Large time behavior of the inhomogeneous AB equation Theorem (J. Carrillo, LD, K. Fellner): Let 0 < a ∗ ≤ a ( y ) ≤ a ∗ , and f in := f in ( x, y ) ≥ 0 be an initial datum such that Z 1 Z ∞ f in ( x, y ) (1 + y + | log f in ( x, y ) | ) dydx < + ∞ . 0 0 According to a theorem by Ph. Laurençot, S. Mischler, there exists a unique solution f := f ( t, x, y ) ≥ 0 to the spatially inhomogeneous Aizenmann-Bak equation with Neumann BC and initial datum f in . Moreover, for c given by the conservation of total mass and for all q ∈ N , Z ∞ (1 + y ) q || f ( t, · , y ) − e − c y || L ∞ (]0 , 1[) dy ≤ C 1 e − C 2 t , 0 where C 1 , C 2 > 0 are explictly computable in terms of a ∗ , a ∗ , q , f in . – p. 17

Estimates used in the proof 1. Bounds from above using the entropy dissipation M ∈ ( L 1 + L ∞ )([0 , + ∞ [; L ∞ (]0 , 1[)) , 2. Bounds from below using the heat kernel and the conservation of total mass M ( t, x ) ≥ M ∗ > 0 , N ( t, x ) ≥ N ∗ > 0 , 3. Cziszar-Kullback-Pinsker inequality, 4. Smoothness estimates using the heat kernel : for all q ∈ N , Z ∞ (1 + y ) q || f ( t, · , y ) || H 1 (]0 , 1[) dy ≤ Pol ( t ) . 0 – p. 18

Almost exponential convergence to equilibrium Sometimes, the functional inequality D ( f ) ≥ C ( H ( f ) − H ( f eq )) cannot be proven, and one has instead D ( f ) ≥ C A − 1 ( H ( f ) − H ( f eq )) − C p A − ( p +1) for some or all p > 0 , and all A > 0 . Together with “slowly growing a priori coefficients”: G. Toscani, C. Villani – p. 19

Almost exponential convergence to equilibrium D ( f ) ≥ C A − 1 ( H ( f ) − H ( f eq )) − C p A − ( p +1) for some or all p > 0 , and all A > 0 . Then by interpolation, D ( f ) ≥ C p ( H ( f ) − H ( f eq )) ( p +1) /p , and we get the differential inequality ∂ t ( H ( f ) − H ( f eq )) ≤ − C p ( H ( f ) − H ( f eq )) ( p +1) /p , so that thanks to Gronwall’s lemma, H ( f )( t ) − H ( f eq ) ≤ C p (1 + t ) − p . – p. 20

Strange rate of convergence toward equilibrium If C p = 2 2 p 2 , then 2 − 2 p 2 t p [ H ( f )( t ) − H ( f eq )] ≤ 1 . Taking p = (log t ) 1 − ε for some ε > 0 , we get e (log t ) 2 − ε e 2 log 2 (log t ) 2 − 2 ε ( H ( f )( t ) − H ( f eq )) ≤ 1 , so that H ( f )( t ) − H ( f eq ) ≤ Cst e − Cst (log t ) 2 − ε . – p. 21

Large time behavior of the degenerate inhomogeneous AB equation a ∗ 1+ y ≤ a ( y ) ≤ a ∗ , and f in := f in ( x, y ) ≥ 0 Theorem (LD, K. Fellner): Let 0 < be an initial datum such that Z 1 Z ∞ f in ( x, y ) (1 + y + | log f in ( x, y ) | ) dydx < + ∞ . 0 0 According to a theorem by Ph. Laurençot, S. Mischler, there exists a unique solution f := f ( t, x, y ) ≥ 0 to the spatially inhomogeneous Aizenmann-Bak equation with Neumann BC and initial datum f in . Moreover, for c given by the conservation of total mass and for all β < 2 , || f ( t, · , ∗ ) − e − c ∗ || L 1 (]0 , 1[ × R + ) ≤ C β e − (log t ) β . where C β > 0 is explictly computable in terms of a ∗ , a ∗ , β , f in . – p. 22

Estimates used in the proof 1. Bounds from above using the entropy dissipation M ∈ ( L 1 + L ∞ )([0 , + ∞ [; L ∞ (]0 , 1[) , 2. Bounds from below using the heat kernel and the conservation of total mass M ( t, x ) ≥ M ∗ > 0 , 3. Cziszar-Kullback-Pinsker inequality, R 1 R ∞ y p f ( t, x, y ) dydx . 4. Estimates on the moments M p ( f )( t ) := 0 0 – p. 23