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 & 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

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

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.   Google Scholar

[30]

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

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

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

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.   Google Scholar

[30]

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

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