Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates

Transcription

1 Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates Daniel Balagué joint work with José A. Cañizo and Pierre GABRIEL (in preparation) Universitat Autònoma de Barcelona SIAM Conference on Analysis of PDE s November 15th, San Diego

2 Introduction We consider the Fragmentation-Drift equation: t g t (x) + x (τ(x)g t (x)) + λg t (x) = L[g t ](x) g t () = (t ) g (x) = g in (x) (x > ). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 2/16

3 Introduction We consider the Fragmentation-Drift equation: t g t (x) + x (τ(x)g t (x)) + λg t (x) = L[g t ](x) g t () = (t ) g (x) = g in (x) (x > ). g t (x) represents the density of the objects of study (cells or polymers) of size x at a given time t. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 2/16

4 Introduction We consider the Fragmentation-Drift equation: t g t (x) + x (τ(x)g t (x)) + λg t (x) = L[g t ](x) g t () = (t ) g (x) = g in (x) (x > ). g t (x) represents the density of the objects of study (cells or polymers) of size x at a given time t. The function τ(x) is the growth rate of cells of size x. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 2/16

5 Introduction We consider the Fragmentation-Drift equation: t g t (x) + x (τ(x)g t (x)) + λg t (x) = L[g t ](x) g t () = (t ) g (x) = g in (x) (x > ). g t (x) represents the density of the objects of study (cells or polymers) of size x at a given time t. The function τ(x) is the growth rate of cells of size x. λ is the largest eigenvalue of the operator g x (τg) + Lg acting on a function g depending only on x. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 2/16

6 Introduction The fragmentation operator L acts on a function g as Lg(x) := L + g(x) B(x)g(x), where the positive part L + is given by L + g(x) := x b(y, x)g(y) dy. The coefficient b(y, x), defined for y > x >, is the fragmentation coefficient, and B(x) is the total fragmentation rate of cells of size x >. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 3/16

7 Introduction The fragmentation operator L acts on a function g as Lg(x) := L + g(x) B(x)g(x), where the positive part L + is given by L + g(x) := x b(y, x)g(y) dy. The coefficient b(y, x), defined for y > x >, is the fragmentation coefficient, and B(x) is the total fragmentation rate of cells of size x >. It is obtained from b through B(x) := x y b(x, y) dy (x > ). x Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 3/16

8 Hypotheses 1. Suppose that b(x, y) = 2x γ 1 (x > y > ). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 4/16

9 Hypotheses 1. Suppose that b(x, y) = 2x γ 1 (x > y > ). In consequence, for the total fragmentation rate we have: B(x) = x γ (x > ). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 4/16

10 Hypotheses 1. Suppose that b(x, y) = 2x γ 1 (x > y > ). In consequence, for the total fragmentation rate we have: B(x) = x γ (x > ). 2. We consider the growth rate τ as a power: τ(x) = x α (x > ). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 4/16

11 Hypotheses 1. Suppose that b(x, y) = 2x γ 1 (x > y > ). In consequence, for the total fragmentation rate we have: B(x) = x γ (x > ). 2. We consider the growth rate τ as a power: τ(x) = x α (x > ). 3. The exponents α and γ satisfy γ α + 1 >. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 4/16

12 The dual eigenproblem φ, where τ(x) x φ + (B(x) + λ) φ(x) = L +φ(x), L +φ(x) := x G(x)φ(x) dx = 1, b(x, y)φ(y) dy, and we have chosen the normalization Gφ = 1. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 5/16

13 The dual eigenproblem where τ(x) x φ + (B(x) + λ) φ(x) = L +φ(x), φ, L +φ(x) := x G(x)φ(x) dx = 1, b(x, y)φ(y) dy, and we have chosen the normalization Gφ = 1.This dual eigenproblem is interesting because φ gives a conservation law for the starting problem: φ(x) g t (x) dx = φ(x) g in (x) dx = Cst (t ). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 5/16

14 Asymptotic behavior of the profile G Theorem There exists a constant C > such that There exists C > such that G(x) Cx 1 α (x ). G(x) Ce Λ(x) x 2 α (x + ) where Λ(x) = x λ+b(y) τ(y) dy. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 6/16

15 Asymptotic behavior of the profile G Idea of the proof: we prove first that, for any m > 3 and any x > it holds that 1 G(x) e Λ(x) x α m dx < +, where Λ(x) = x λ+b(y) τ(y) dy. I.e, the exponential moments are bounded. We exploit this fact to prove the result. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 7/16

16 The truncated problem To prove the estimates on the dual eigenfunction φ we use a truncated problem: τ(x) x φ L (x) + (B(x) + λ L )φ L (x) = L +(φ L )(x), φ L (L) = or φ L (L) = δ > or φ L (L) = δl, φ L, L G(x)φ L (x) dx = 1. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 8/16

17 The truncated problem To prove the estimates on the dual eigenfunction φ we use a truncated problem: τ(x) x φ L (x) + (B(x) + λ L )φ L (x) = L +(φ L )(x), φ L (L) = or φ L (L) = δ > or φ L (L) = δl, φ L, L G(x)φ L (x) dx = 1. Lemma There exists L > such that for each L L the previous truncated problem has a unique solution (λ L, φ L ) with λ L > and φ L W 1, loc (R +). Moreover we have λ L L + λ A >, φ L L + φ uniformly on [, A). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 8/16

18 Supersolutions The function φ L (x) satisfies the following equation Sφ L (x) = (x (, L)), where S is the operator given by Sφ L (x) := τ(x)φ L (x) + (B(x) + λ L)φ L (x) x b(x, y)φ L (y)dy, defined for all functions φ W 1, (, L) and for x (, L). Definition We say that w W 1, (, L) is a supersolution of S on the interval I (, L) when Sw(x) (x I ). Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 9/16

19 Bounds for φ Theorem If γ >, there are two positive constants C 1 and C 2 such that C 1 x φ(x) C 2 x, x > 1. For B(x) B > we have φ(x) 1. For γ <, there exist two positive constants C 1 and C 2 such that C 1 x γ 1 φ(x) C 2 x γ 1, x > 1. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 1/16

20 Bounds for φ Idea of the proof: We distinguish between two cases. Case γ >. - v(x) := Cx + 1 x k is a supersolution on [A, L]. - ṽ(x) := x + x k 1 is a subsolution on [A, L]. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 11/16

21 Bounds for φ Idea of the proof: We distinguish between two cases. Case γ >. - v(x) := Cx + 1 x k is a supersolution on [A, L]. - ṽ(x) := x + x k 1 is a subsolution on [A, L]. Case γ <. - For any η > C, v(x) = (η + x) γ 1 is a supersolution. { for < x < ɛ, - ṽ(x) := (x ɛ)x γ 2 is a subsolution on for x > ɛ. [A, L] for ɛ < C. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 11/16

22 Bounds for φ Idea of the proof: We distinguish between two cases. Case γ >. - v(x) := Cx + 1 x k is a supersolution on [A, L]. - ṽ(x) := x + x k 1 is a subsolution on [A, L]. Case γ <. - For any η > C, v(x) = (η + x) γ 1 is a supersolution. { for < x < ɛ, - ṽ(x) := (x ɛ)x γ 2 is a subsolution on for x > ɛ. [A, L] for ɛ < C. In both cases, we conclude using a maximum principle. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 11/16

23 Maximum principle Lemma (Maximum principle for S) Take A > 1 λ L. If ω is a supersolution of S on (A, L), ω on [, A] and ω (L) then ω on [A, L]. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 12/16

24 Entropy The general relative entropy applies. where u = g(x) G(x). H 2 [g G] := φg(u 1) 2 dx Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 13/16

25 Entropy The general relative entropy applies. H 2 [g G] := φg(u 1) 2 dx where u = g(x) G(x). In this particular case, we have where we define D b [g G] := d dt H 2[g G] = D b [g G], x φ(x)g(y)b(y, x)(u(x) u(y)) 2 dydx. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 13/16

26 Entropy Theorem The following relation is satisfied H 2 [g G] C D b [g G], for some constant C > and for any nonnegative measurable function g : (, ) R such that φg = 1. In consequence, if g t is a solution for the problem, the speed of convergence to equilibrium is exponentially fast in the L 2 -weighted norm = L 2 (G 1 φdx) sense, i.e., H 2 [g t G] H[g G]e λt. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 14/16

27 Entropy Idea of the proof: One can check that H 2 [g G] = D 2 [g G] where D 2 [g G] := with u(x) = g(x) G(x). x φ(x)g(x)φ(y)g(y)(u(x) u(y)) 2 dydx Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 15/16

28 Entropy Idea of the proof: One can check that H 2 [g G] = D 2 [g G] where D 2 [g G] := with u(x) = g(x) G(x).Consider two cases. x φ(x)g(x)φ(y)g(y)(u(x) u(y)) 2 dydx Case γ <. One can compare pointwise D 2 and D b. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 15/16

29 Entropy Idea of the proof: One can check that H 2 [g G] = D 2 [g G] where D 2 [g G] := with u(x) = g(x) G(x).Consider two cases. x φ(x)g(x)φ(y)g(y)(u(x) u(y)) 2 dydx Case γ <. One can compare pointwise D 2 and D b. Case γ >. Split D 2 in two terms: one for which the pointwise comparison holds and one where it does not. The idea for the second part is to break u(x) u(y) into intermediate reactions in order to obtain a new expression of D 2 [g G] for which the pointwise estimate applies. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 15/16

30 Conclusions We give accurate estimates of the profiles G and φ. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 16/16

31 Conclusions We give accurate estimates of the profiles G and φ. We have described the asymptotic behavior of the solutions when we have τ(x) = x α. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 16/16

32 Conclusions We give accurate estimates of the profiles G and φ. We have described the asymptotic behavior of the solutions when we have τ(x) = x α. We have an entropy - entropy dissipation inequality Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 16/16

33 Conclusions We give accurate estimates of the profiles G and φ. We have described the asymptotic behavior of the solutions when we have τ(x) = x α. We have an entropy - entropy dissipation inequality Thank you for your attention. Daniel Balagué Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates 16/16