# American Institute of Mathematical Sciences

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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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. 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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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