Traveling waves of a kinetic transport model for the KPP-Fisher equation

Transcription

1 Traveling waves of a kinetic transport model for the KPP-Fisher equation Christian Schmeiser Universität Wien and RICAM homepage.univie.ac.at/christian.schmeiser/ Joint work with C. Cuesta (Bilbao), S. Hittmeir (Wien) (to appear in SIAM J. Math. Anal.) C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

2 Traveling waves of the KPP-Fisher equation The KPP-Fisher equation t u = D 2 u + u( ρ u) Theorem: For every s s 0 = 2 D ρ there eists a unique (up to a shift) traveling wave u TW ( st), which satisfies u TW ( ) = ρ, u TW ( ) = 0, and which is monotone. Theorem: For s > s 0 and (u u TW )(t = 0) small enough, [u(ξ + st, t) u TW (ξ)] 2 (1 + ep(sξ/d)) dξ decays eponentially. Proof: weighted L 2 -norm as Lyapunov function, boundedness of weighted H 1 -norm (for L -control). Fisher (1937), Kolmogorov et al. (1937), Canosa (1973), Sattinger (1976),... C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

3 A kinetic transport model for the KPP-Fisher equation Distribution f (t,, v) in phase space with R, v V R t f + v f = Lf + Q(f ) Collisions or reorientation: Lf = (M(v)f (v ) M(v )f (v))dv = Mρ f f V with even equilibrium velocity distribution M, satisfying M dv = 1, v 2 M dv = D. Macrocopic density ρ f := f dv Birth/death: Q(f ) = V [ ρm(v)f (v ) f (v)f (v )]dv = ρ f ( ρm f ) C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

4 The macroscopic limit Scaling assumption: Reorientation much more frequent than birth/death (by a factor ε 2 1) Macroscopic scaling (parabolic, since vm dv = 0): Formal macroscopic limit: ε 2 t f + εv f = Lf + ε 2 Q(f ) lim f (t,, v) = ρ 0(t, )M(v) ε 0 where ρ 0 satisfies the KPP-Fisher equation. C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

5 Traveling waves of kinetic equations Kinetic shock profiles: Caflisch, Nikolaenko (1982): Boltzmann profiles for weak ideal gas shocks Bose, Illner, Ukai (1998), Nishibata (2002): Discrete velocity models Golse (1998): Perthame-Tadmor profiles for scalar conservation laws Liu, Yu (2004): Stability of Caflisch-Nikolaenko profiles Cuesta, CS (2006, 2007): BGK profiles for scalar conservation laws BenAbdallah, Chaker, CS (2007): Fermions in high electric fields Cuesta, Hittmeir, CS (2010): BGK profiles for weak isentropic gas dynamics shocks Cuesta, Hittmeir, CS (2009): Review Reaction-kinetic models: Bouin, Calvez (2012, in preparation): Kinetic KPP-Fisher C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

6 Kinetic shock profiles: fermions in a scattering background and a strong electric field (BenAbdallah, Chaker, CS (2007)) Q(f ) = ε t f + εv f + E v f = Q(f ) [Mf (1 f ) M f (1 f )]dv Lemma: (BenAbdallah, Chaker (2001)) E v f = Q(f ), iff f = ˆM(ρ, E, v), where ˆM is increasing in ρ, 0 < ˆM < 1, v ˆM(ρ, E, v)dv = σ(ρ, E )E. Macroscopic limit: f (t,, v) ˆM(ρ(t, ), E, v) as ε 0, with t ρ + (σ(ρ, E )E) = 0 C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

7 Fermions in a scattering background and a strong electric field Theorem: Convergence to entropy solutions of the macroscopic equation. Traveling wave problem: (v s) ξ f + E v f = Q(f ), f (±, v) = ˆM(ρ ±, E, v) Theorem: Let 2 ρσ 0, (s, ρ +, ρ ) satisfy the Rankine-Hugoniot conditions and the entropy condition. Then there eists a dynamically stable traveling wave. C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

8 Global eistence for the kinetic KPP-Fisher equation Initial value problem: ε 2 t f + εv f = ρ f M f + ε 2 ρ f ( ρm f ) f (t = 0) = f 0 Theorem: Let 0 f 0 (, v) ˆρM(v) hold. Then the initial value problem has a unique mild solution f C([0, ); L (R V )), satisfying 0 f (t,, v) ma{ ρ, ˆρ}M(v). Proof: Local eistence in weighted L -space ( f = sup f /M). Global eistence by maimum principle. C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

9 Eistence of traveling waves of the kinetic KPP-Fisher equation The traveling wave problem: ε(v εs) ξ f = ρ f M f + ε 2 ρ f ( ρm f ) f ( = ) = ρm, f ( = ) = 0 Step 1: Formal (Chapman-Enskog) approimation f as = Mu TW εvmu TW + ε2 (v 2 D)Mu TW, produces a massless O(ε 3 ) residual. BC satisfied. C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

10 Eistence of traveling waves of the kinetic KPP-Fisher equation Step 2: Micro-macro decomposition of the error (ideas from Caflisch, Nikolaenko (1982)) g = f f as ε 2 = z(ξ)φ(v) + εw(ξ, v) with Φ = M + O(ε), (v εs) 2 w dv = 0 This leads to the linearized system Dz + sz + z( ρ 2u TW ) = h z ε(v εs) ξ w Lw = h w C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

11 Eistence of traveling waves of the kinetic KPP-Fisher equation Step 3: Removal of the null space (again motivated by Caflisch, Nikolaenko) Problem: semidefiniteness of L Cure: Modify L by a term K, such that M := L K is definite and Kw = 0 should be zero. Kw := (v εs) 2 M (v εs) 2 w dv Lemma: Solutions of the modified problem satisfy Kw = 0 Proof: ξ (v εs) 2 w dv = 2sD V (v εs)2 w dv Lemma: Eistence of solutions of the modified linearized equation Proof: discrete velocity approimation V C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

12 Eistence of traveling waves of the kinetic KPP-Fisher equation Step 4: Solution of the nonlinear problem by contraction Theorem: Let s s 0. For ε small enough, there eists a TW f TW, which satisfies f TW f as H 2 ξ (L 2 v ) = O(ε2 ) Note: No claims of monotonicity and/or positivity! C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

13 Stability of traveling waves of the kinetic KPP-Fisher equation Theorem: Let V be bounded, s > s 0, and (f f TW )(t = 0) small enough. Then [f (ξ + st, t) f TW (ξ)] 2 (1 + ep(sξ/d)) dv dξ V decays eponentially. Proof: weighted H 1 ξ (L2 v )-norm as Lyapunov function, micro-macro-decomposition of the error, macroscopic part: as for KPP-Fisher, microscopic part: entropy. Corollary: Let V be bounded. Then 0 f TW ρ. C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14

14 Discussion Kinetic model for KPP-Fisher Bouin, Calvez have an eistence result for bounded V without smallness of ε. Numerical results by Bouin, Calvez suggest that stability is not true for unbounded V. Probably, for unbounded V, all traveling waves oscillate as ξ. Valeur de la vitesse data1 data2 data3 data4 data5 data6 data7 data8 data9 data10 data11 data12 data13 data14 data15 data20 y=[(0.085*t) 0,5 +(0.09*t) 0.49 ]/2 y=(0.085*t) 0,5 y=(0.09*t) 0, Temps dt= C. Schmeiser (Univ. Wien) Hρακλɛιo, October 5, / 14