Theoretical and numerical results for a chemo-repulsion model with quadratic production

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1 Theoretical and numerical results for a chemo-repulsion model with quadratic production F. Guillén-Gonzalez, M. A. Rodríguez-Bellido & and D. A. Rueda-Gómez Dpto. Ecuaciones Diferenciales y Análisis Numérico Facultad de Matemáticas Universidad de Sevilla, Spain Escuela de Matemáticas, Universidad Industrial de Santander, Colombia Waves in Flows Workshop, August 29, 2018 Prague, Czech Republic

2 Powered by TCPDF ( Powered by TCPDF ( The model Chemotaxis Chemorepulsion - Production Some properties Chemotaxis is the biological process of the movement of living organisms in response to a chemical stimulus which can be given towards a higher (attractive) or lower (repulsive) concentration of a chemical substance. At the same time, the presence of living organisms can produce or consume chemical substance.

3 Chemotaxis Chemorepulsion - Production Some properties Chemotaxis system [Keller-Segel, ] t u D u u + χ (u v) = 0 in Ω, t > 0, t v D v v + βv = αu in Ω, t > 0, v n = u n = 0 on Ω, t > 0, v(x, 0) = v 0 (x) > 0, u(x, 0) = u 0 (x) > 0 in Ω, (1) v is the chemical concentration, u denotes the cell density, The term χu v models the transport of cells towards the higher concentrations of chemical signal if χ > 0, and towards the lower concentrations of chemical signal if χ < 0.

4 Chemotaxis Chemorepulsion - Production Some properties Chemorepulsion production system with quadratic production term t u D u u (u v) = 0 in Ω, t > 0, t v D v v + v = u 2 in Ω, t > 0, v n = u (2) n = 0 on Ω, t > 0, v(x, 0) = v 0 (x) > 0, u(x, 0) = u 0 (x) > 0 in Ω, In the case of linear production, [Cieslak et all, 2008], prove that model (1) is well-posed: there exists global in time weak solution (based on an energy inequality) and, for 2D domains, there exists a unique global in time strong solution. The quadratic production term allows to control an energy in L 2 (Ω)-norm for u which is very useful for performing numerical analysis.

5 Chemotaxis Chemorepulsion - Production Some properties Weak solution u L (0, + ; L 2 (Ω)) L 2 (0, T ; H 1 (Ω)), v L (0, + ; H 2 (Ω)) L 2 (0, T ; H 2 (Ω)) ( t u, t v) L q (0, T ; H 1 (Ω) L 2 (Ω)), (3) where q = 2 in the (2D) case and q = 4/3 in the (3D) case It is possible to prove that model (2) is well-posed: there exists global in time weak solution (based on an energy inequality) and, for 2D domains, there exists a unique global in time strong solution satisfying u L (0, + ; H 1 (Ω)) L 2 (0, T ; H 2 (Ω)), v L (0, + ; H 2 (Ω)) L 2 (0, T ; H 3 (Ω)) t u L (0, + ; L 2 (Ω)) L 2 (0, + ; H 1 (Ω)), t v L (0, + ; H 1 (Ω)) L 2 (0, + ; H 2 (Ω)). (4)

6 Chemotaxis Chemorepulsion - Production Some properties Some Properties The problem is conservative in u, as we can check integrating (2) 1 in Ω, ( ) d u = 0, i.e. dt Ω Ω u(t) = u 0, t > 0. Ω Also, integrating (2) 2 in Ω, we deduce the following behavior of v, u 0 and v 0. ( ) d v = u 2 v. dt Ω Ω Ω The following energy inequality holds for a.e. t 0, t 1 with t 1 t 0 0: t1 E(u(t 1 ), v(t 1 )) E(u(t 0 ), v(t 0 ))+ ( u(s) t 0 2 v(s) v(s) 2 0 ) ds 0, for E(u(t), v(t)) = u(t) v(t) 2 0 (5)

7 Chemotaxis Chemorepulsion - Production Some properties Chemorepulsion production system with quadratic production term tu D u u (u v) = 0 in Ω, t > 0, tv D v v v = u 2 in Ω, t > 0, v n = u = 0 on Ω, t > 0, n v(x, 0) = v 0 (x) > 0, u(x, 0) = u 0 (x) > 0 in Ω, (6)

8 Chemotaxis Chemorepulsion - Production Some properties Chemorepulsion production system with quadratic production term tu D u u (u v) = 0 in Ω, t > 0, tv D v v v = u 2 in Ω, t > 0, v n = u = 0 on Ω, t > 0, n v(x, 0) = v 0 (x) > 0, u(x, 0) = u 0 (x) > 0 in Ω, (6) An equivalent problem (σ = v), [Zhang & Zhu, 2016] tu ( u) = (uσ) in Ω, t > 0, tσ ( σ) + rot(rot σ) + σ = (u 2 ) in Ω, t > 0, u = 0 on Ω, t > 0, n σ n = 0, [rot σ n] tang = 0 on Ω, t > 0, u(x, 0) = u 0 (x) 0, σ(x, 0) = v 0 (x) in Ω, Once solved (7), we can recover v from u 2 solving tv v + v = u 2 in Ω, t > 0, v = 0 on Ω, t > 0, n v(x, 0) = v 0 (x) 0 in Ω. (7) (8) Problems (2) and (7)-(8) are equivalents!

9 Chemotaxis Chemorepulsion - Production Some properties Some Properties of (7) The problem is conservative in u, as we can check integrating (7) 1 in Ω, ( ) d u = 0, i.e. dt Ω Ω u(t) = u 0, t > 0. Ω Also, integrating (8) in Ω, we deduce the following behavior of v, u 0 ( ) d v = u 2 v. dt Ω Ω Ω The following energy inequality holds for a.e. t 0, t 1 with t 1 t 0 0: E(u(t 1 ), v(t 1 )) E(u(t 0 ), v(t 0 )) t1 ( + u(s) t 0 2 σ ) 2 rot σ σ 2 0 ds 0, (9) for E(u(t), v(t)) = u(t) σ(t) 2 0

10 Chemotaxis Chemorepulsion - Production Some properties For the variable σ we will use the following space H 1 σ(ω) := {u H 1 (Ω) : u n = 0 on Ω} endowed with the equivalent norm in H 1 (Ω) ([Amrouche and Seloula, 2013]): σ 2 1 = σ rot σ σ 2 0, σ H 1 σ(ω).

11 Time-Discrete Scheme Fully Discrete Scheme TIME-DISCRETE SCHEME Scheme US Time step n: Given (u n 1, σ n 1 ), compute (u n, σ n) solving { (δtun, ū) + ( u n, ū) + (u nσ n, ū) = 0, ū, (δ tσ n, σ) + B 2 σ n, σ 2(u n u n, σ) = 0, σ. (10) where B 2 σ, σ = (σ, σ) + ( σ, σ) + (rot σ, rot σ). Moreover, we recover v n solving δ tv n + B 1 v n = u 2 n, a.e. x Ω, (11) Main results 1. Scheme US is conservative in u, that is, Ω un = Ω u n 1 = = Ω u u n 0, v n 0 3. It can be proved the existence of nonnegative solution (u n, σ n, v n) of scheme US, and under a smallness condition on k, this solution is unique.

12 TIME-DISCRETE SCHEME Scheme US Time-Discrete Scheme Fully Discrete Scheme Time step n: Given (u n 1, σ n 1 ), compute (u n, σ n) solving { (δtun, ū) + ( u n, ū) + (u nσ n, ū) = 0, ū, (δ tσ n, σ) + B 2 σ n, σ 2(u n u n, σ) = 0, σ. (12) where B 2 σ, σ = (σ, σ) + ( σ, σ) + (rot σ, rot σ). Moreover, we recover v n solving δ tv n + B 1 v n = u 2 n, a.e. x Ω, (13) Main results 4. Scheme US is unconditionally energy-stable: δ te(u n, σ n) + k 2 δtun k 4 δtσn un σn 2 1 = 0, (14) holds, where E(u n, σ n) = 1 2 un σn 2 0.

13 TIME-DISCRETE SCHEME Scheme US Time-Discrete Scheme Fully Discrete Scheme Time step n: Given (u n 1, σ n 1 ), compute (u n, σ n) solving { (δtun, ū) + ( u n, ū) + (u nσ n, ū) = 0, ū, (δ tσ n, σ) + B 2 σ n, σ 2(u n u n, σ) = 0, σ. (15) where B 2 σ, σ = (σ, σ) + ( σ, σ) + (rot σ, rot σ). Moreover, we recover v n solving δ tv n + B 1 v n = u 2 n, a.e. x Ω, (16) Main results 5. As consequence, we have estimates (u n, σ n) L 2 l 2 H Moreover, we obtain the estimate Ω vn K 0, and therefore v n l H 1 l 2 H Convergence towards weak solutions. [Marion-Temam] 8. Error estimates (O(k)) for eu n = u(tn) un, en σ = σ(tn) σn, en v = v(tn) vn 9. Uniform strong estimates in 2D domains.

14 Time-Discrete Scheme Fully Discrete Scheme Lemma (Convergence) There exists a subsequence (k ), with k 0, and a weak solution (u, σ) of (7) in (0, T ), such that the sequence (u k, σ k ) of solutions of discrete scheme US corresponding to k converges to (u, σ) weakly-* in L (0, T ; L 2 (Ω)), weakly in L 2 (0, T ; H 1 (Ω)) and strongly in L 2 (0, T ; L 2 (Ω)).

15 Time-Discrete Scheme Fully Discrete Scheme Lemma (Error estimates in weak norms for US) Let (u n, σ n ) be a solution of scheme US, v n solution of (26) and assume the following regularity for (u, σ) exact solution of (7) and v exact solution of (8): If k < u L (0, T ; H 1 (Ω)), u tt L 2 (0, T ; (H 1 (Ω)) ), (17) σ L (0, T ; H 1 (Ω)), σ tt L 2 (0, T ; (H 1 (Ω)) ), (18) v tt L 2 (0, T ; (H 1 (Ω)) ). (19) 1 4C u, σ 4 L L 2, then the a priori error estimates hold, where C(T ) = K 1 exp(k 2 T ). (eu,, n eσ) n l L 2 l 2 H 1 C(T ) k ev n l L 2 l 2 H 1 C(T ) k

16 Time-Discrete Scheme Fully Discrete Scheme A LINEAR TIME-DISCRETE SCHEME Scheme LC Time step n: Given (u n 1, σ n 1 ), compute (u n, σ n) solving { (δtu n, ū) + Au n, ū = (u n 1 σ n, ū), ū, (δ tσ n, σ) + B 2 σ n, σ = 2(u n 1 u n, σ), σ (20) Moreover, we recover v n solving δ tv n + B 1 v n = u 2 n, a.e. x Ω, (21) Main results 1. The same results hold; but: It is not clear u n We do not have the relation σ n = v n.

17 FULLY DISCRETE SCHEME Scheme US Time-Discrete Scheme Fully Discrete Scheme Time step n: Given (u n 1, σ n 1 ), compute (u n, σ n) solving { (δtun, ū) + ( u n, ū) + (u nσ n, ū) = 0, ū, (δ tσ n, σ) + B 2 σ n, σ 2(u n u n, σ) = 0, σ. (22) where B 2 σ, σ = (σ, σ) + ( σ, σ) + (rot σ, rot σ). Moreover, we recover v n solving δ tv n + B 1 v n = u 2 n, a.e. x Ω, (23) Main results 1. Total mass conservation. 2. Existence of solution (u n, σ n, v n), and under a smallness condition on k, this solution is unique. 3. Unconditional energy-stability: δ te(u n, σ n) + k 2 δtun k 4 δtσn un σn 2 1 = 0. (24)

18 FULLY DISCRETE SCHEME Scheme US Time-Discrete Scheme Fully Discrete Scheme Time step n: Given (u n 1, σ n 1 ), compute (u n, σ n) solving { (δtun, ū) + ( u n, ū) + (u nσ n, ū) = 0, ū, (δ tσ n, σ) + B 2 σ n, σ 2(u n u n, σ) = 0, σ. (25) where B 2 σ, σ = (σ, σ) + ( σ, σ) + (rot σ, rot σ). Moreover, we recover v n solving δ tv n + B 1 v n = u 2 n, a.e. x Ω, (26) Main results 4. Weak estimates. 5. Convergence towards weak solutions. 6. Optimal error estimates for e n u = u(tn) un h, Ru h : H1 (Ω) U h, e n u = (I R u h )u(tn) + Ru h u(tn) un h = en u,i + en u,h, (e n u,h, en σ,h ) l L 2 l 2 H 1, en v,h l H 1 l 2 W 1,6 C(T )(k + hm+1 ). 7. Uniform strong estimates in 2D domains.

19 ASYMPTOTIC ANALYSIS Convergence at infinite time Main results Convergence for the numerical schemes Let (u, v) be any weak-strong solution of problem (2) obtained by Galerkin approximations. Then, the following estimates hold (u(t) m 0, v(t)) 2 0 C 0e 2t, a.e. t 0. (27) v(t) (m 0 ) C 0e t, t 0, (28) where m 0 := 1 u 0 and C 0 is a positive constant depending on the data (u 0, v 0 ), Ω Ω but independent of t. Convergence in stronger norms Let ε > 0. There exists a constant C 1 > 0 such that if ε 2 1 2C 1 it holds (u(t) m 0, v(t)) 2 1 2εe 1 2 (t t 2), a.e. t t 2 (ε), (29) with t 2 := t 2 (ε) 0 a large enough time that will be obtained in the proof.

20 Main results Convergence for the numerical schemes Main results 1. Total mass conservation. 2. Unconditional existence, 3. Unconditional energy-stability: 4. Weak estimates. 5. Convergence towards weak solutions.

21 Main results Convergence for the numerical schemes Large-time behavior of the scheme UV Let (uh n, v h n ) be a solution of the scheme UV associated to an initial data (uh 0, v h 0) U h V h which is a suitable approximation of (u 0, v 0 ) L 2 (Ω) H 1 (Ω), as h 0, with 1 Ω Ω u0 h = 1 Ω Ω u 0 = m 0. Then, (u n h m 0, v n h ) 2 0 C 0 e 2 1+2k kn, n 0, (30) vh n (m 0) C 0 e 1 1+k kn, n 0, (31) k ( ũh m (A h I )vh n ) 2 v h n 2 0 C 0 e 2 1+2k kn, n 0, m>n (32) where C 0 is a positive constant depending on the data (u 0, v 0 ), but independent of (k, h) and n.

22 Main results Convergence for the numerical schemes Large-time behavior of scheme US Let (uh n, σn h ) be a solution of the scheme US associated to an initial data (uh 0, σ0 h ) U h Σ h which is a suitable approximation of (u 0, σ 0 ) L 2 (Ω) L 2 (Ω), as h 0, with 1 Ω Ω u 0 h = 1 Ω k m>n Ω u 0 = m 0. Then, (u n h m 0, σ n h ) 2 0 C 0 e 2 1+2k kn, n 0, (33) ( ũh m ) 2 σm h 2 1 C 0 e 2 1+2k kn, n 0, (34) where C 0 is a positive constant depending on the data (u 0, σ 0 ), but independent of (k, h) and n.

23 Positivity Energy-stability Numeric error orders Numerical Simulations We are considering a finite element discretization in space associated to the variational formulation of schemes BE and LC, where the P 1 -continuous approximation is taken for u h, σ h and v h. We have chosen the domain Ω = [0, 2] 2 using a structured mesh. All the simulations are carried out using FreeFem++ software. The linear iterative method used to approach the nonlinear scheme US is the Newton Method, and in all the cases, the iterative method stops when the relative error in L 2 -norm is less than ε = 10 6.

24 Positivity The model Positivity Energy-stability Numeric error orders Positivity of u h, v h in fully discrete schemes??? Remember that In scheme US, we prove u n 0 and v n 0. In scheme LC, we prove v n 0, but u n 0??? it is not clear. Then, we compare the positivity of the variables u h and v h in both schemes, taking meshes in space increasingly thinner (h = 2 70, h = and h = ). In all the cases, we choose k = 10 5 and the initial conditions are u 0 = 10xy(2 x)(2 y)exp( 10(y 1) 2 10(x 1) 2 ) and v 0 =200xy(2 x)(2 y)exp( 30(y 1) 2 30(x 1) 2 )

25 Positivity Energy-stability Numeric error orders

26 US LC US LC The model Positivity Energy-stability Numeric error orders Minimum of u h Time Figure : Minimum values of u h, with h = Minimum of u h Time 10-3 Figure : Minimum values of u h, with h = 1 75

27 Positivity Energy-stability Numeric error orders Minimum of u h US LC Time 10-3 Figure : Minimum values of u h, with h =

28 Positivity Energy-stability Numeric error orders Minimum of v h US LC Time

29 Positivity Energy-stability Numeric error orders Energy-Stability: If (u n, σ n ) is any solution of the fully discrete schemes corresponding to schemes US (or. LC), the following relation holds RE(u n, σ n ) := δ t E(u n, σ n ) + u n σ n 2 1 0, n, (35) where E(u n, σ n ) was defined as: E(u n, σ n ) = 1 2 u n σ n 2 0. (36)

30 Positivity Energy-stability Numeric error orders Taking k = 10 6, h = 1 25 and the initial conditions u 0 = 10xy(2 x)(2 y)exp( 10(y 1) 2 10(x 1) 2 ) and v 0 =20xy(2 x)(2 y)exp( 30(y 1) 2 30(x 1) 2 ) , we obtain US UV E(u n h,vn h ) Time Figure : Energy E(uh n, v h n ) of schemes UV and US.

31 Positivity Energy-stability Numeric error orders RE(u n h,vn h ) US UV Time Figure : Residue RE(uh n, v h n ) of schemes UV and US.

32 Positivity Energy-stability Numeric error orders Numeric error orders: We consider an exact solution for problem (2) but in the variables (u, σ, v), k = 10 5 and obtain the following spatial numerical error orders: Error - h 1/40-1/50 1/50-1/60 1/60-1/70 1/70-1/80 eu n l 2 L eu n l L eu n l 2 H eu n l H ev n l 2 L ev n l L ev n l 2 H ev n l H

33 Extensions Design unconditional energy stables and mass-conservative numerical schemes for the model t u D u u (u v) = 0 in Ω, t > 0, t v D v v + v = u p in Ω, t > 0, v n = u (37) n = 0 on Ω, t > 0, v(x, 0) = v 0 (x) > 0, u(x, 0) = u 0 (x) > 0 in Ω, for p = 1 and p (1, 2). The attractive case: Pattern formation.

34 C. Amrouche and N. E. H. Seloula, Lp-theory for vector potentials and Sobolev?s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23 (2013), no. 1, T. Cieslak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system. Parabolic and Navier-Stokes equations. Part 1, , Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, F. Guillén-González, M. A. Roríguez-Bellido, D. A. Rueda-Gómez, Study of a chemo-repulsion model with quadratic production. Part I: Analysis of the continuous problem and time-discrete numerical schemes. Submitted. F. Guillén-González, M. A. Roríguez-Bellido, D. A. Rueda-Gómez, Study of a chemo-repulsion model with quadratic production. Part II: Analysis of an unconditional energy-stable fully discrete scheme. Submitted. F. Guillén-González, M. A. Roríguez-Bellido, D. A. Rueda-Gómez, Asymptotic behaviour for a chemo-repulsion system with quadratic production: The continuous problem and two fully discrete numerical schemes. Submitted. M. Marion, R. Temam, Navier-Stokes equations: theory and approximation. Handbook of numerical analysis, Vol. VI, , Handb. Numer. Anal., VI, North-Holland, Amsterdam (1998). J. Zhang, J. Zhu and R. Zhang, Characteristic splitting mixed finite element analysis of Keller-Segel chemotaxis models. Appl. Math. Comput. 278 (2016),

35 Acknowledgments This research was partially supported by MINECO grant MTM P with the participation of FEDER. D. A. Rueda-Gómez has also been partially supported by the VIE of the Universidad Industrial de Santander (Colombia).

36 Thank you very much for your attention!