On a linear runs and tumbles equation

Stéphane Mischler; Qilong Weng

doi:10.3934/krm.2017032

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2017, Volume 10, Issue 3: 799-822. Doi: 10.3934/krm.2017032

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On a linear runs and tumbles equation

Stéphane Mischler^1, , and
Qilong Weng^2,

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
* Corresponding author: Stéphane Mischler

Received: February 2016

Revised: September 2016

Published: September 2017

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Abstract

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.

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Mathematics Subject Classification: 35B40, 35Q92, 47D06, 92C17.

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References

	W. Alt , Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980) , 147-177. doi: 10.1007/BF00275919.
	W. Arendt , Kato's inequality: A characterisation of generators of positive semigroups, Proc. Roy. Irish Acad. Sect. A, 84 (1984) , 155-174.
	J. M. Ball , Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977) , 370-373.
	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.
	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.
	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.
	V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429.
	V. Calvez , G. 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.
	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.
	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.
	J. Dolbeault , C. 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.
	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.
	M. Escobedo , S. 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.
	F. Golse , P.-L. Lions , B. 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.
	F. Golse , B. 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.
	M. Gualdani, S. Mischler and C. Mouhot, Factorization for Non-Symmetric Operators and Exponential H-theorem, preprint, arXiv: 1006.5523.
	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.
	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.
	S. Mischler, Erratum: Spectral analysis of semigroups and growth-fragmentation equations, Submitted.
	S. Mischler, Semigroups in Banach Spaces -Factorization Approach for Spectral Analysis and Asymptotic Estimates, in preparation.
	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.
	S. Mischler , C. 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.
	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.
	H. G. Othmer , S. R. Dunbar and W. Alt , Models of dispersal in biological systems, J. Math. Biol., 26 (1988) , 263-298. doi: 10.1007/BF00277392.
	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.
	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.
	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.
	A. R. Schep , Weak Kato-inequalities and positive semigroups, Math. Z., 190 (1985) , 305-314. doi: 10.1007/BF01215132.
	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.
	C. Villani , Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009) , ⅳ+141. doi: 10.1090/S0065-9266-09-00567-5.