September  2017, 10(3): 799-822. doi: 10.3934/krm.2017032

On a linear runs and tumbles equation

1. 

Paris-Dauphine, Institut Universitaire de France (IUF), PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassignys, 75775 Paris Cedex 16, France

2. 

Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

* Corresponding author: Stéphane Mischler

Received  February 2016 Revised  September 2016 Published  December 2016

We consider a linear runs and tumbles equation in dimension $d ≥ 1$ for which we establish the existence of a unique positive and normalized steady state as well as its asymptotic stability, improving similar results obtained by Calvez et al. [8] in dimension $d=1$. Our analysis is based on the Krein-Rutman theory revisited in [23] together with some new moment estimates for proving confinement mechanism as well as dispersion, multiplicator and averaging lemma arguments for proving some regularity property of suitable iterated averaging quantities.

Citation: Stéphane Mischler, Qilong Weng. On a linear runs and tumbles equation. Kinetic and Related Models, 2017, 10 (3) : 799-822. doi: 10.3934/krm.2017032
References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[2]

W. Arendt, Kato's inequality: A characterisation of generators of positive semigroups, Proc. Roy. Irish Acad. Sect. A, 84 (1984), 155-174. 

[3]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. 

[4]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 2 (1985), 101–118.

[5]

F. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 19-36.  doi: 10.1017/S030821050002744X.

[6]

N. Bournaveas and V. Calvez, A review of recent existence and blow-up results for kinetic models of chemotaxis, Can. Appl. Math. Q., 18 (2010), 253-265. 

[7]

V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429.

[8]

V. CalvezG. Raoul and C. Schmeiser, Confinement by biased velocity jumps: Aggregation of Escherichia coli, Kinet. Relat. Models, 8 (2015), 651-666.  doi: 10.3934/krm.2015.8.651.

[9]

L. Desvillettes and S. Mischler, About the splitting algorithm for Boltzmann and BGK equations, Math. Models Methods Appl. Sci., 6 (1996), 1079-1101.  doi: 10.1142/S0218202596000444.

[10]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.

[11]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[12]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math. , 65 (2004/05), 361–391 (electronic). doi: 10.1137/S0036139903433232.

[13]

M. EscobedoS. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.  doi: 10.1016/j.anihpc.2004.06.001.

[14]

F. GolseP.-L. LionsB. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.  doi: 10.1016/0022-1236(88)90051-1.

[15]

F. GolseB. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 341-344. 

[16]

M. Gualdani, S. Mischler and C. Mouhot, Factorization for Non-Symmetric Operators and Exponential H-theorem, preprint, arXiv: 1006.5523.

[17]

F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.  doi: 10.1007/s00030-012-0155-4.

[18]

P.-L. Lions and B. Perthame, Lemmes de moments, de moyenne et de dispersion, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 801-806. 

[19]

S. Mischler, Erratum: Spectral analysis of semigroups and growth-fragmentation equations, Submitted.

[20]

S. Mischler, Semigroups in Banach Spaces -Factorization Approach for Spectral Analysis and Asymptotic Estimates, in preparation.

[21]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.

[22]

S. MischlerC. Quiñinao and J. Touboul, On a kinetic fitzhugh-nagumo model of neuronal network, Comm. Math. Phys., 342 (2016), 1001-1042.  doi: 10.1007/s00220-015-2556-9.

[23]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[24]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[25]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.

[26]

B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.

[27]

B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686.  doi: 10.1080/03605309608821201.

[28]

A. R. Schep, Weak Kato-inequalities and positive semigroups, Math. Z., 190 (1985), 305-314.  doi: 10.1007/BF01215132.

[29]

D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 28 (1973/74), 303-315. 

[30]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141.  doi: 10.1090/S0065-9266-09-00567-5.

show all references

References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[2]

W. Arendt, Kato's inequality: A characterisation of generators of positive semigroups, Proc. Roy. Irish Acad. Sect. A, 84 (1984), 155-174. 

[3]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. 

[4]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 2 (1985), 101–118.

[5]

F. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 19-36.  doi: 10.1017/S030821050002744X.

[6]

N. Bournaveas and V. Calvez, A review of recent existence and blow-up results for kinetic models of chemotaxis, Can. Appl. Math. Q., 18 (2010), 253-265. 

[7]

V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429.

[8]

V. CalvezG. Raoul and C. Schmeiser, Confinement by biased velocity jumps: Aggregation of Escherichia coli, Kinet. Relat. Models, 8 (2015), 651-666.  doi: 10.3934/krm.2015.8.651.

[9]

L. Desvillettes and S. Mischler, About the splitting algorithm for Boltzmann and BGK equations, Math. Models Methods Appl. Sci., 6 (1996), 1079-1101.  doi: 10.1142/S0218202596000444.

[10]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.

[11]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[12]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math. , 65 (2004/05), 361–391 (electronic). doi: 10.1137/S0036139903433232.

[13]

M. EscobedoS. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.  doi: 10.1016/j.anihpc.2004.06.001.

[14]

F. GolseP.-L. LionsB. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.  doi: 10.1016/0022-1236(88)90051-1.

[15]

F. GolseB. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 341-344. 

[16]

M. Gualdani, S. Mischler and C. Mouhot, Factorization for Non-Symmetric Operators and Exponential H-theorem, preprint, arXiv: 1006.5523.

[17]

F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.  doi: 10.1007/s00030-012-0155-4.

[18]

P.-L. Lions and B. Perthame, Lemmes de moments, de moyenne et de dispersion, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 801-806. 

[19]

S. Mischler, Erratum: Spectral analysis of semigroups and growth-fragmentation equations, Submitted.

[20]

S. Mischler, Semigroups in Banach Spaces -Factorization Approach for Spectral Analysis and Asymptotic Estimates, in preparation.

[21]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.

[22]

S. MischlerC. Quiñinao and J. Touboul, On a kinetic fitzhugh-nagumo model of neuronal network, Comm. Math. Phys., 342 (2016), 1001-1042.  doi: 10.1007/s00220-015-2556-9.

[23]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[24]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[25]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.

[26]

B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.

[27]

B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686.  doi: 10.1080/03605309608821201.

[28]

A. R. Schep, Weak Kato-inequalities and positive semigroups, Math. Z., 190 (1985), 305-314.  doi: 10.1007/BF01215132.

[29]

D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 28 (1973/74), 303-315. 

[30]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141.  doi: 10.1090/S0065-9266-09-00567-5.

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