Kinetic models for chemotaxis

Transcription

1 Kinetic models for chemotaxis FABIO A. C. C. CHALUB Centro de Matemática e Aplicações Fundamentais Universidade de Lisboa Kinetic models for chemotaxis p.

2 Main Goal To study two different levels of description for chemotaxis and to study how these two levels are related. Kinetic models for chemotaxis p.

3 General Picture Consider a kinetic model M ε with a certain non-dimensional parameter ε > 0. Kinetic models for chemotaxis p.

4 General Picture Consider a kinetic model M ε with a certain non-dimensional parameter ε > 0. Consider the solution Ψ ε := (f ε,s ε ) (microscopic variables), Kinetic models for chemotaxis p.

5 General Picture Consider a kinetic model M ε with a certain non-dimensional parameter ε > 0. Consider the solution Ψ ε := (f ε,s ε ) (microscopic variables), and consider Φ ε := (ρ ε,s ε ) := ( V f εdv,s ε ) (macroscopic variables). Kinetic models for chemotaxis p.

6 General Picture Consider a kinetic model M ε with a certain non-dimensional parameter ε > 0. Consider the solution Ψ ε := (f ε,s ε ) (microscopic variables), and consider Φ ε := (ρ ε,s ε ) := ( V f εdv,s ε ) (macroscopic variables). Let us define the limit Φ := lim ε 0 (ρ ε,s ε ). Kinetic models for chemotaxis p.

7 General Picture Consider a kinetic model M ε with a certain non-dimensional parameter ε > 0. Consider the solution Ψ ε := (f ε,s ε ) (microscopic variables), and consider Φ ε := (ρ ε,s ε ) := ( V f εdv,s ε ) (macroscopic variables). Let us define the limit Question: Φ := lim ε 0 (ρ ε,s ε ). Which is the set of equations that Φ obey? Kinetic models for chemotaxis p.

8 General Picture Model ε > 0 Limit model ε 0 Kinetic models for chemotaxis p.

9 General Picture Model ε > 0 Limit model ε 0 Initial conditions Ψ I ε Φ I := lim ε 0 Φ I ε Kinetic models for chemotaxis p.

10 General Picture Model ε > 0 Limit model ε 0 Initial conditions Ψ I ε Φ I := lim ε 0 Φ I ε Time evolution M ε [Ψ ε ] = 0 M[Φ] = 0 Kinetic models for chemotaxis p.

11 General Picture Model ε > 0 Limit model ε 0 Initial conditions Ψ I ε Φ I := lim ε 0 Φ I ε Time evolution M ε [Ψ ε ] = 0 M[Φ] = 0 Final state Ψ ε (T)? Φ(T) Kinetic models for chemotaxis p.

12 General Picture Model ε > 0 Limit model ε 0 Initial conditions Ψ I ε Φ I := lim ε 0 Φ I ε Time evolution M ε [Ψ ε ] = 0 M[Φ] = 0 If Final state Ψ ε (T)? Φ(T) Φ(t) = lim ε 0 Φ ε (t), t < T (in some sense) then M is the limit model of M ε. Kinetic models for chemotaxis p.

13 Chemotaxis Figure 1: Life cycle of Dictyostelium discoideum. Picture made by Florian Seigert and Kees Wiejer (Zoologisches Institut München Ludwig- Maximilians-Universität München). Kinetic models for chemotaxis p.

14 Velocity jump model Introduced by Alt (1980) and Othmer, Dunbar and Alt (1988). Kinetic models for chemotaxis p.

15 Velocity jump model Introduced by Alt (1980) and Othmer, Dunbar and Alt (1988). The cell goes in straight line for a certain characteristic time and then changes its direction from v to v (in a space-time point (x,t) in the presence of the substance S and cell density ρ) according to a certain turning kernel T[S,ρ](x,v,v,t). Kinetic models for chemotaxis p.

16 Velocity jump model Introduced by Alt (1980) and Othmer, Dunbar and Alt (1988). The cell goes in straight line for a certain characteristic time and then changes its direction from v to v (in a space-time point (x,t) in the presence of the substance S and cell density ρ) according to a certain turning kernel T[S,ρ](x,v,v,t). The set of all possible velocities is given by a compact, spherically symmetric set V. Kinetic models for chemotaxis p.

17 Velocity jump model V t f(x,v,t) + v f(x,v,t) = (T[S,ρ](x,v,v,t)f(x,v,t) T[S,ρ](x,v,v,t)f(x,v,t))dv. Kinetic models for chemotaxis p.

18 Velocity jump model V t f(x,v,t) + v f(x,v,t) = (T[S,ρ](x,v,v,t)f(x,v,t) T[S,ρ](x,v,v,t)f(x,v,t))dv. f(x,v,t) = cell density in space-time point (x,t) with velocity v (phase-space density). Kinetic models for chemotaxis p.

19 Velocity jump model Notation f = f(x,v,t), f = f(x,v,t), T[S,ρ] = T[S,ρ](x,v,v,t), T [S,ρ] = T[S,ρ](x,v,v,t). Kinetic models for chemotaxis p.

20 Velocity jump model Notation f = f(x,v,t), f = f(x,v,t), T[S,ρ] = T[S,ρ](x,v,v,t), T [S,ρ] = T[S,ρ](x,v,v,t). Equation t f + v f = V (T[S,ρ]f T [S,ρ]f)dv. Kinetic models for chemotaxis p.

21 Velocity jump model This is an example of a Boltzmann-type integro-differential equation (kinetic model). Kinetic models for chemotaxis p.

22 Velocity jump model This is an example of a Boltzmann-type integro-differential equation (kinetic model). The macroscopic density ρ is related to the microscopic density f by ρ(x,t) = f(x,v,t)dv. V Kinetic models for chemotaxis p.

23 Velocity jump model This is an example of a Boltzmann-type integro-differential equation (kinetic model). The macroscopic density ρ is related to the microscopic density f by ρ(x,t) = f(x,v,t)dv. We should consider also an equation for S: V t S = D 0 S + ϕ(s,ρ). Kinetic models for chemotaxis p.

24 Keller-Segel model Mathematical model for chemotaxis introduced by Patlak (1953) and Keller and Segel (1970). Kinetic models for chemotaxis p. 1

25 Keller-Segel model Mathematical model for chemotaxis introduced by Patlak (1953) and Keller and Segel (1970). Keller-Segel equations: t ρ = (D ρ χ(s)β(ρ)ρ S), t S = D 0 S + ϕ(s,ρ). ρ = cell density, S = density of chemo-attractant, χ = chemotactic sensitivity, D,D 0 = diffusion coefficients, ϕ = interaction between ρ and S. Kinetic models for chemotaxis p. 1

26 Keller-Segel model Typical example of interaction with α > 0, β 0. ϕ(s,ρ) = αρ βs, Kinetic models for chemotaxis p. 1

27 Keller-Segel model Typical example of interaction ϕ(s,ρ) = αρ βs, with α > 0, β 0. Finite-time-blow-up: lim t T ( ) ρ(,t) L (R n ) + S(,t) L (R n ) =. Ex: ρ(,t) δ a. Kinetic models for chemotaxis p. 1

28 Re-scaling Let us go back to the Othmer-Dunbar-Alt model: t f + v f = (T[S,ρ]f T [S,ρ])dv. V Kinetic models for chemotaxis p. 1

29 Re-scaling Let us go back to the Othmer-Dunbar-Alt model: t f + v f = (T[S,ρ]f T [S,ρ])dv. V Re-scaling x = x/x 0, t = t/t 0, v = v/v 0, T = T/T 0, S = S/S 0, ρ = ρ/ρ 0, f = f/f 0. Kinetic models for chemotaxis p. 1

30 Re-scaling f t + v 0 x 0 /t 0 v f = T 0 v n 0t 0 S t = t 0 x 2 0 ρ = f 0v n 0 ρ 0 V (Tf T f) dv, D 0 S + α 1ρ 0 t 0 ρ α 2 t 0 S, S 0 f dv. V Kinetic models for chemotaxis p. 1

31 Re-scaling f t + v 0 x 0 /t 0 v f = T 0 v n 0t 0 S t = t 0 x 2 0 ρ = f 0v n 0 ρ 0 V (Tf T f) dv, D 0 S + α 1ρ 0 t 0 ρ α 2 t 0 S, S 0 f dv. We impose the diffusive scaling: t 0 x 2 0, normalizations and V ε = x 0/t 0 v 0. Kinetic models for chemotaxis p. 1

32 Re-scaling f t + 1 ε v f = 1 ε 2 The kernel T depends on ε... S = S + ρ S, t ρ = f dv. V V (T ε f T ε f) dv, Kinetic models for chemotaxis p. 1

33 Re-scaling f ε t + 1 ε v f ε = 1 ε 2 The solution depends on ε... V (T ε f ε T ε f ε ) dv, S ε = S ε + ρ ε S ε, t ρ ε = f ε dv. V Kinetic models for chemotaxis p. 1

34 Formal computations Consider the turning kernel such that T ε [S,ρ] = T 0 [S,ρ] + εt 1 [S,ρ] + T 0 [S,ρ] = λ(s,ρ)(x,t)f(v), T 1 [S,ρ] = F(v)a(S,ρ)v S, where F > 0, V Fdv = 1, V vfdv = 0, λ λ min > 0 Kinetic models for chemotaxis p. 1

35 Formal computations Two possible examples are given by T ε [S,ρ] = F(v)λ(S,ρ) + εf(v)a(s,ρ)v S, T ε [S,ρ] = ψ(s(x,t),s(x + εµ(ρ)v,t))f(v), Kinetic models for chemotaxis p. 1

36 Formal computations Two possible examples are given by T ε [S,ρ] = F(v)λ(S,ρ) + εf(v)a(s,ρ)v S, T ε [S,ρ] = ψ(s(x,t),s(x + εµ(ρ)v,t))f(v), We consider the formal expansion of the solutions: f ε = f 0 + εf 1 + ε 2 f 2 +, S ε = S 0 + εs 1 + ε 2 S 2 +. Kinetic models for chemotaxis p. 1

37 Formal computations Two possible examples are given by T ε [S,ρ] = F(v)λ(S,ρ) + εf(v)a(s,ρ)v S, T ε [S,ρ] = ψ(s(x,t),s(x + εµ(ρ)v,t))f(v), We consider the formal expansion of the solutions: f ε = f 0 + εf 1 + ε 2 f 2 +, S ε = S 0 + εs 1 + ε 2 S 2 +. We put these expansions in the model, match terms with the same order of ε and solve the resulting system. Kinetic models for chemotaxis p. 1

38 Formal computations In both cases the formal drift-diffusion limit is given by the zeroth order equations (Othmer, Hillen) t ρ 0 = (D(S 0,ρ 0 ) ρ 0 χ(s 0,ρ 0 )ρ 0 S 0 ), t S 0 = S 0 + ρ 0 S 0, Kinetic models for chemotaxis p. 1

39 Formal computations In both cases the formal drift-diffusion limit is given by the zeroth order equations (Othmer, Hillen) t ρ 0 = (D(S 0,ρ 0 ) ρ 0 χ(s 0,ρ 0 )ρ 0 S 0 ), t S 0 = S 0 + ρ 0 S 0, where D(S 0,ρ 0 ) := χ(s 0,ρ 0 ) := 1 nλ(s 0,ρ 0 ) a(s,ρ) nλ(s 0,ρ 0 ) v 2 FdvI, V ( ) v 2 Fdv V. Kinetic models for chemotaxis p. 1

40 General Picture (again...) Model ε > 0 Limit model ε 0 Initial conditions Ψ I ε Φ I := lim ε 0 Φ I ε Time evolution M ε [Ψ ε ] = 0 M[Φ] = 0 If Final state Ψ ε (T)? Φ(T) Φ(t) = lim ε 0 Φ ε (t), t < T (in some sense) then M is the limit model of M ε. Kinetic models for chemotaxis p. 1

41 Rigorous results Theorem (C., Markowich, Perthame, Schmeiser, Hwang, Kang, Stevens, Rodrigues): Consider turning kernels T ε depending on S ε, S ε and ρ ε under mild assumptions. Then, the solution of the kinetic model (f ε,s ε ) is such that ρ ε ρ 0 in L 2 loc (Rn ), S ε S 0 in L q loc (Rn ), 1 q <, S ε S 0 in L q loc, 1 q <. where (ρ 0,S 0 ) is the solution of the associated Keller-Segel model. Kinetic models for chemotaxis p. 1

42 Rigorous Results By mild assumptions we mean: V where φ S ε [ρ,s] γ(1 ελ( S W 1, ))FF, φ A ε [ρ,s] 2 Fφ S ε [ρ,s] dv ε 2 Λ( S W 1, ), φ S ε φ A ε := T ε[s,ρ]f + Tε [ρ,s]f 2 := T ε[s,ρ]f Tε [ρ,s]f 2 and other more technical ones.,, Kinetic models for chemotaxis p. 1

43 Global Existence Theorem (C., Markowich, Perthame, Schmeiser, Hwang, Kang, Stevens): Consider turning kernels such that: 0 T ε [S ε, S ε ] c 1 + c 2 S(x + εv,t) + c 3 S(x εv,t) +c 4 S(x + εv,t) + c 5 S(x εv,t), T ε [S ε, S ε ] c 2 S(x + εv,t) + c 3 S(x εv,t) +c 4 2 S(x + εv,t) + c 5 2 S(x εv,t). The kinetic model has global existence of solutions. Kinetic models for chemotaxis p. 2

44 Global Existence Let us consider T ε [S](x,v,v,t) = ψ(s(x + εv,t) S(x,t)) with ψ ψ min > 0, increasing and ψ(y) Ay + B. Kinetic models for chemotaxis p. 2

45 Global Existence Let us consider T ε [S](x,v,v,t) = ψ(s(x + εv,t) S(x,t)) with ψ ψ min > 0, increasing and ψ(y) Ay + B. Then the kinetic model has global existence of solution, converges (in the drift diffusion limit) to the classical Keller-Segel model (which presents blow up). Kinetic models for chemotaxis p. 2

46 Global Existence Theorem (C., Rodrigues): For certain classes of turning kernels T ε [S ε, S ε,ρ ε ], such that if ρ ε (x,t) ρ, then T ε [S ε, S ε,ρ ε ](x,v,v,t) = T 0 [S ε, S ε,ρ ε ](x,v,v,t), and for ε small enough, we conclude global existence of solution (f ε,s ε ). Kinetic models for chemotaxis p. 2

47 Global Existence Theorem (C., Rodrigues): For certain classes of turning kernels T ε [S ε, S ε,ρ ε ], such that if ρ ε (x,t) ρ, then T ε [S ε, S ε,ρ ε ](x,v,v,t) = T 0 [S ε, S ε,ρ ε ](x,v,v,t), and for ε small enough, we conclude global existence of solution (f ε,s ε ). Furthermore, ρ ε (,t) L (R n ) max{ ρ, ρ I L (R n )}. Kinetic models for chemotaxis p. 2

48 Global Existence Example: T ε [S,ρ] = λ(s,ρ)f + εa(s,ρ)fv S, T ε [S,ρ] = ψ(s(x + εµ(ρ)v,t) S(x,t))F with a(s,ρ) = 0, µ(ρ) = 0, ρ ρ > 0. Kinetic models for chemotaxis p. 2

49 Global Existence Example: with T ε [S,ρ] = λ(s,ρ)f + εa(s,ρ)fv S, T ε [S,ρ] = ψ(s(x + εµ(ρ)v,t) S(x,t))F a(s,ρ) = 0, µ(ρ) = 0, ρ ρ > 0. Corollary: For the limit Keller-Segel model, we can conclude global existence of solutions. Furthermore, the cell density is bounded. (Hillen, Painter) t ρ = ( ρ β(ρ)ρ S), β(ρ) = 0,ρ ρ > 0. Kinetic models for chemotaxis p. 2

50 Beyond Keller-Segel Q:What is the meaning of blow up? Kinetic models for chemotaxis p. 2

51 Beyond Keller-Segel Q:What is the meaning of blow up? A:Changes of orders of magnitude with respect to the certain variables. Kinetic models for chemotaxis p. 2

52 Beyond Keller-Segel Q:What is the meaning of blow up? A:Changes of orders of magnitude with respect to the certain variables. Q:Is blow up good or bad? Kinetic models for chemotaxis p. 2

53 Beyond Keller-Segel Q:What is the meaning of blow up? A:Changes of orders of magnitude with respect to the certain variables. Q:Is blow up good or bad? A:Bad: Blow up does not exists in nature, we should look for models without blow up. Kinetic models for chemotaxis p. 2

54 Beyond Keller-Segel Q:What is the meaning of blow up? A:Changes of orders of magnitude with respect to the certain variables. Q:Is blow up good or bad? A:Bad: Blow up does not exists in nature, we should look for models without blow up. A:Good: Keller-Segel model cannot be valid after certain time, so a realistic model should indicate that it breaks down. Kinetic models for chemotaxis p. 2

55 Beyond Keller-Segel Q: Can we prevent blow up? Kinetic models for chemotaxis p. 2

56 Beyond Keller-Segel Q: Can we prevent blow up? A: Yes. This was done in the kinetic level. Other possibility is given by Velazquez: t ρ = ( ρ ρ ) 1 + µρ S For any µ > 0, there is global existence of solutions. For µ = 0, solutions blow up in finite time. Kinetic models for chemotaxis p. 2

57 Beyond Keller-Segel With T ε,µ [S,ρ] = ψ ( S ( x + ε ) 1 + µρ v ) S (x,t) the solution exists globally and the drift-diffusion limit (globally in time) is the Velazquez model (C., Kang). Kinetic models for chemotaxis p. 2

58 Beyond Keller-Segel µ Global existence of solutions Velazquez Model global in time convergence local in time convergence Blow up Kinetic Models Classical Keller Segel ε Kinetic models for chemotaxis p. 2

59 Conclusions Model (ε > 0) Limit model (ε 0) Closer to first principles. Phenomenological. Kinetic models for chemotaxis p. 2

60 Conclusions Model (ε > 0) Limit model (ε 0) Closer to first principles. Phenomenological. Detailed description. Simpler picture. Kinetic models for chemotaxis p. 2

61 Conclusions Model (ε > 0) Limit model (ε 0) Closer to first principles. Phenomenological. Detailed description. Simpler picture. Difficult to handle. Easy to find solutions. Kinetic models for chemotaxis p. 2

62 Conclusions Model (ε > 0) Limit model (ε 0) Closer to first principles. Phenomenological. Detailed description. Simpler picture. Difficult to handle. Easy to find solutions. THE END Kinetic models for chemotaxis p. 2