December  2016, 9(4): 715-748. doi: 10.3934/krm.2016013

Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

2. 

Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708

Received  July 2015 Revised  February 2016 Published  September 2016

This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term $-\nu(-\Delta)^{\frac{\alpha}{2}}\rho~(1<\alpha<2)$. Firstly, the global existence of weak solutions is proved for the initial density $\rho_0\in L^1\cap L^{\frac{d}{\alpha}}(\mathbb{R}^d)~(d\geq2)$ with $\|\rho_0\|_{\frac {d}{\alpha}} < K$, where $K$ is a universal constant only depending on $d,\alpha,\nu$. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in $L^r$ for any $1< r<\infty$. Secondly, for the more general initial data $\rho_0\in L^1\cap L^2(\mathbb{R}^d)$$~(d=2,3)$, the local existence is obtained. Thirdly, for $\rho_0\in L^1\big(\mathbb{R}^d,(1+|x|)dx\big)\cap L^\infty(\mathbb{R}^d)(~d\geq2)$ with $\|\rho_0\|_{\frac{d}{\alpha}} < K$, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant $\alpha$-stable Lévy process $L_{\alpha}(t)$. Also, we prove the weak solution is $L^\infty$ bounded uniformly in time. Lastly, we consider the $N$-particle interacting system with the Lévy process $L_{\alpha}(t)$ and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment $\int_{\mathbb{R}^d}|x|^\gamma\rho_0dx$ for some $1<\gamma<\alpha$ is below a universal constant $K_\gamma$ and $\nu$ is also below a universal constant. Meanwhile, we prove the propagation of chaos as $N\rightarrow\infty$ for the interacting particle system with a cut-off parameter $\varepsilon\sim(\ln N)^{-\frac{1}{d}}$, and show that the mean field limit equation is exactly the generalized KS equation.
Citation: Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013
References:
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D. Applebaum, Lévy Processes and Stochastic Calculus,, $2^{nd}$ edition, (2009). doi: 10.1017/CBO9780511809781. Google Scholar

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S. Bian and J.-G. Liu, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$,, Kinet. Relat. Models, 7 (2014), 9. doi: 10.3934/krm.2014.7.9. Google Scholar

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P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup in two-dimensional chemotaxis models, preprint,, , (). Google Scholar

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P. Biler, T. Funaki and W. A. Woyczyński, Interacting particle approximation for nonlocal quadratic evolution problems,, Probability and Mathematical Statistics-Wroclaw University, 19 (1999), 267. Google Scholar

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P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model,, J. Evol. Equ., 10 (2010), 247. doi: 10.1007/s00028-009-0048-0. Google Scholar

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P. Biler and W. A. Woyczyński, Nonlocal quadratic evolution problems,, Banach Center Publications, 52 (2000), 11. Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

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L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Ration. Mech. Anal., 195 (2010), 1. doi: 10.1007/s00205-008-0181-x. Google Scholar

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J. A. Carrillo, S. Lisini and E. Mainini, Uniqueness for Keller-Segel-type chemotaxis models,, Discrete Contin. Dyn. Syst., 34 (2014), 1319. doi: 10.3934/dcds.2014.34.1319. Google Scholar

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X. Chen, A. Jüngel and J.-G. Liu, A Note on Aubin-Lions-Dubinskiĭ Lemmas,, Acta Appl. Math., 133 (2014), 33. doi: 10.1007/s10440-013-9858-8. Google Scholar

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F. G. Egana and S. Mischler, Uniqueness and long time asymptotic for the Keller-Segel equation: The parabolic-elliptic case,, Arch. Ration. Mech. Anal., 220 (2016), 1159. doi: 10.1007/s00205-015-0951-1. Google Scholar

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C. Escudero, Chemotactic collapse and mesenchymal morphogenesis,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.022903. Google Scholar

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C. Escudero, The fractional Keller-Segel model,, Nonlinearity, 19 (2006), 2909. doi: 10.1088/0951-7715/19/12/010. Google Scholar

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V. Feller, An Introduction to Probability Theory and Its Applications: Volume 2,, $2^{nd}$ edition, (1971). Google Scholar

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N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous vortex model,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1423. doi: 10.4171/JEMS/465. Google Scholar

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M. Kac, Foundations of kinetic theory,, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171. Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[25]

J. Klafter, B. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk,, Biological Motion, 89 (1990), 281. doi: 10.1007/978-3-642-51664-1_20. Google Scholar

[26]

M. Levandowsky, B. White and F. Schuster, Random movements of soil amebas,, Acta Protozoologica, 36 (1997), 237. Google Scholar

[27]

D. Li and J. L. Rodrigo, Finite-time singularities of an aggregation equation in $\mathbbR^n$ with fractional dissipation,, Comm. Math. Phys., 287 (2009), 687. doi: 10.1007/s00220-008-0669-0. Google Scholar

[28]

D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation,, Adv. in Math., 220 (2009), 1717. doi: 10.1016/j.aim.2008.10.016. Google Scholar

[29]

D. Li, J. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem,, Rev. Mat. Iberoamericana, 26 (2010), 295. doi: 10.4171/RMI/602. Google Scholar

[30]

J.-G. Liu and J. Wang, A note on $L^\infty$ bound and uniqueness to a degenerate Keller-Segel model,, Acta Appl. Math., 142 (2016), 173. doi: 10.1007/s10440-015-0022-5. Google Scholar

[31]

J.-G. Liu and R. Yang, Propagation of chaos for keller-segel equation with a logarithmic cut-off,, preprint., (). Google Scholar

[32]

F. Matthäus, M. Jagodič and J. Dobnikar, E. coli superdiffusion and chemotaxis-search strategy, precision, and motility,, Biophys. J., 97 (2009), 946. doi: 10.1016/j.bpj.2009.04.065. Google Scholar

[33]

P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ edition, (2004). doi: 10.1007/978-3-662-10061-5. Google Scholar

[34]

K. Sato, Lévy Processes and Infinitely Divisible Distributions,, Cambridge University Press, (2013). Google Scholar

[35]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

[36]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems,, SIAM J. Appl. Math., 61 (2000), 183. doi: 10.1137/S0036139998342065. Google Scholar

[37]

A.-S. Sznitman, A propagation of chaos result for Burgers' equation,, Probab. Theory Relat. Fields, 71 (1986), 581. doi: 10.1007/BF00699042. Google Scholar

[38]

E. Valdinoci, From the long jump random walk to the fractional Laplacian,, Boletín de la Sociedad Española de Matemática Aplicada, 49 (2009), 33. Google Scholar

[39]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators,, Nonlinear Partial Differential Equations, 7 (2012), 271. doi: 10.1007/978-3-642-25361-4_15. Google Scholar

[40]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857. doi: 10.3934/dcdss.2014.7.857. Google Scholar

[41]

C. Villani, Optimal Transport: Old and New,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-71050-9. Google Scholar

[42]

V. Yudovich, Non-stationary flow of an ideal incompressible liquid,, U.S.S.R. Comput. Math. and Math. Phys., 3 (1963), 1407. doi: 10.1016/0041-5553(63)90247-7. Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus,, $2^{nd}$ edition, (2009). doi: 10.1017/CBO9780511809781. Google Scholar

[2]

F. Bartumeus, F. Peters, S. Pueyo, C. Marraśe and J. Catalan, Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton,, Proceedings of the National Academy of Sciences, 100 (2003), 12771. Google Scholar

[3]

J. Bertoin, Lévy Processes,, Cambridge University Press, (1996). Google Scholar

[4]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$,, Comm. Math. Phys., 323 (2013), 1017. doi: 10.1007/s00220-013-1777-z. Google Scholar

[5]

S. Bian and J.-G. Liu, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$,, Kinet. Relat. Models, 7 (2014), 9. doi: 10.3934/krm.2014.7.9. Google Scholar

[6]

P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup in two-dimensional chemotaxis models, preprint,, , (). Google Scholar

[7]

P. Biler, T. Funaki and W. A. Woyczyński, Interacting particle approximation for nonlocal quadratic evolution problems,, Probability and Mathematical Statistics-Wroclaw University, 19 (1999), 267. Google Scholar

[8]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model,, J. Evol. Equ., 10 (2010), 247. doi: 10.1007/s00028-009-0048-0. Google Scholar

[9]

P. Biler and W. A. Woyczyński, Nonlocal quadratic evolution problems,, Banach Center Publications, 52 (2000), 11. Google Scholar

[10]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1999), 845. doi: 10.1137/S0036139996313447. Google Scholar

[11]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model,, Appl. Math. Lett., 25 (2012), 339. doi: 10.1016/j.aml.2011.09.011. Google Scholar

[12]

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations,, Adv. in Math., 250 (2014), 242. doi: 10.1016/j.aim.2013.09.018. Google Scholar

[13]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[14]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Ration. Mech. Anal., 195 (2010), 1. doi: 10.1007/s00205-008-0181-x. Google Scholar

[15]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math., 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. Google Scholar

[16]

J. A. Carrillo, S. Lisini and E. Mainini, Uniqueness for Keller-Segel-type chemotaxis models,, Discrete Contin. Dyn. Syst., 34 (2014), 1319. doi: 10.3934/dcds.2014.34.1319. Google Scholar

[17]

X. Chen, A. Jüngel and J.-G. Liu, A Note on Aubin-Lions-Dubinskiĭ Lemmas,, Acta Appl. Math., 133 (2014), 33. doi: 10.1007/s10440-013-9858-8. Google Scholar

[18]

F. G. Egana and S. Mischler, Uniqueness and long time asymptotic for the Keller-Segel equation: The parabolic-elliptic case,, Arch. Ration. Mech. Anal., 220 (2016), 1159. doi: 10.1007/s00205-015-0951-1. Google Scholar

[19]

C. Escudero, Chemotactic collapse and mesenchymal morphogenesis,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.022903. Google Scholar

[20]

C. Escudero, The fractional Keller-Segel model,, Nonlinearity, 19 (2006), 2909. doi: 10.1088/0951-7715/19/12/010. Google Scholar

[21]

V. Feller, An Introduction to Probability Theory and Its Applications: Volume 2,, $2^{nd}$ edition, (1971). Google Scholar

[22]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous vortex model,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1423. doi: 10.4171/JEMS/465. Google Scholar

[23]

M. Kac, Foundations of kinetic theory,, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171. Google Scholar

[24]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[25]

J. Klafter, B. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk,, Biological Motion, 89 (1990), 281. doi: 10.1007/978-3-642-51664-1_20. Google Scholar

[26]

M. Levandowsky, B. White and F. Schuster, Random movements of soil amebas,, Acta Protozoologica, 36 (1997), 237. Google Scholar

[27]

D. Li and J. L. Rodrigo, Finite-time singularities of an aggregation equation in $\mathbbR^n$ with fractional dissipation,, Comm. Math. Phys., 287 (2009), 687. doi: 10.1007/s00220-008-0669-0. Google Scholar

[28]

D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation,, Adv. in Math., 220 (2009), 1717. doi: 10.1016/j.aim.2008.10.016. Google Scholar

[29]

D. Li, J. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem,, Rev. Mat. Iberoamericana, 26 (2010), 295. doi: 10.4171/RMI/602. Google Scholar

[30]

J.-G. Liu and J. Wang, A note on $L^\infty$ bound and uniqueness to a degenerate Keller-Segel model,, Acta Appl. Math., 142 (2016), 173. doi: 10.1007/s10440-015-0022-5. Google Scholar

[31]

J.-G. Liu and R. Yang, Propagation of chaos for keller-segel equation with a logarithmic cut-off,, preprint., (). Google Scholar

[32]

F. Matthäus, M. Jagodič and J. Dobnikar, E. coli superdiffusion and chemotaxis-search strategy, precision, and motility,, Biophys. J., 97 (2009), 946. doi: 10.1016/j.bpj.2009.04.065. Google Scholar

[33]

P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ edition, (2004). doi: 10.1007/978-3-662-10061-5. Google Scholar

[34]

K. Sato, Lévy Processes and Infinitely Divisible Distributions,, Cambridge University Press, (2013). Google Scholar

[35]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

[36]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems,, SIAM J. Appl. Math., 61 (2000), 183. doi: 10.1137/S0036139998342065. Google Scholar

[37]

A.-S. Sznitman, A propagation of chaos result for Burgers' equation,, Probab. Theory Relat. Fields, 71 (1986), 581. doi: 10.1007/BF00699042. Google Scholar

[38]

E. Valdinoci, From the long jump random walk to the fractional Laplacian,, Boletín de la Sociedad Española de Matemática Aplicada, 49 (2009), 33. Google Scholar

[39]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators,, Nonlinear Partial Differential Equations, 7 (2012), 271. doi: 10.1007/978-3-642-25361-4_15. Google Scholar

[40]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857. doi: 10.3934/dcdss.2014.7.857. Google Scholar

[41]

C. Villani, Optimal Transport: Old and New,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-71050-9. Google Scholar

[42]

V. Yudovich, Non-stationary flow of an ideal incompressible liquid,, U.S.S.R. Comput. Math. and Math. Phys., 3 (1963), 1407. doi: 10.1016/0041-5553(63)90247-7. Google Scholar

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