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 and Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013
References:
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Liu, A Note on Aubin-Lions-Dubinskiĭ Lemmas, Acta Appl. Math., 133 (2014), 33-43. doi: 10.1007/s10440-013-9858-8. [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), 1159-1194. doi: 10.1007/s00205-015-0951-1. [19] C. Escudero, Chemotactic collapse and mesenchymal morphogenesis, Phys. Rev. E, 72 (2005), 022903. doi: 10.1103/PhysRevE.72.022903. [20] C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918. doi: 10.1088/0951-7715/19/12/010. [21] V. Feller, An Introduction to Probability Theory and Its Applications: Volume 2, $2^{nd}$ edition, J. Wiley & sons, New York-London-Sydney, 1971. [22] N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous vortex model, J. Eur. Math. Soc. (JEMS), 16 (2014), 1423-1466. doi: 10.4171/JEMS/465. [23] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197. [24] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [25] J. Klafter, B. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk, Biological Motion, 89 (1990), 281-296. doi: 10.1007/978-3-642-51664-1_20. [26] M. Levandowsky, B. White and F. Schuster, Random movements of soil amebas, Acta Protozoologica, 36 (1997), 237-248. [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), 687-703. doi: 10.1007/s00220-008-0669-0. [28] D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. in Math., 220 (2009), 1717-1738. doi: 10.1016/j.aim.2008.10.016. [29] D. Li, J. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoamericana, 26 (2010), 295-332. doi: 10.4171/RMI/602. [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), 173-188. doi: 10.1007/s10440-015-0022-5. [31] J.-G. Liu and R. Yang, Propagation of chaos for keller-segel equation with a logarithmic cut-off,, preprint., (). [32] F. Matthäus, M. Jagodič and J. Dobnikar, E. coli superdiffusion and chemotaxis-search strategy, precision, and motility, Biophys. J., 97 (2009), 946-957. doi: 10.1016/j.bpj.2009.04.065. [33] P. E. Protter, Stochastic Integration and Differential Equations, $2^{nd}$ edition, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10061-5. [34] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 2013. [35] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [36] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212. doi: 10.1137/S0036139998342065. [37] A.-S. Sznitman, A propagation of chaos result for Burgers' equation, Probab. Theory Relat. Fields, 71 (1986), 581-613. doi: 10.1007/BF00699042. [38] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Boletín de la Sociedad Española de Matemática Aplicada, 49 (2009), 33-44. [39] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4_15. [40] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857. [41] C. Villani, Optimal Transport: Old and New, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-71050-9. [42] V. Yudovich, Non-stationary flow of an ideal incompressible liquid, U.S.S.R. Comput. Math. and Math. Phys., 3 (1963), 1407-1456. doi: 10.1016/0041-5553(63)90247-7.

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References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, $2^{nd}$ edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [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), 12771-12775. [3] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996. [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), 1017-1070. doi: 10.1007/s00220-013-1777-z. [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), 9-28. doi: 10.3934/krm.2014.7.9. [6] P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup in two-dimensional chemotaxis models, preprint,, , (). [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), 267-286. [8] P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0. [9] P. Biler and W. A. Woyczyński, Nonlocal quadratic evolution problems, Banach Center Publications, 52 (2000), 11-24. [10] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. doi: 10.1137/S0036139996313447. [11] F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011. [12] M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. in Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018. [13] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [14] L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23. doi: 10.1007/s00205-008-0181-x. [15] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [16] J. A. Carrillo, S. Lisini and E. Mainini, Uniqueness for Keller-Segel-type chemotaxis models, Discrete Contin. Dyn. Syst., 34 (2014), 1319-1338. doi: 10.3934/dcds.2014.34.1319. [17] X. Chen, A. Jüngel and J.-G. Liu, A Note on Aubin-Lions-Dubinskiĭ Lemmas, Acta Appl. Math., 133 (2014), 33-43. doi: 10.1007/s10440-013-9858-8. [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), 1159-1194. doi: 10.1007/s00205-015-0951-1. [19] C. Escudero, Chemotactic collapse and mesenchymal morphogenesis, Phys. Rev. E, 72 (2005), 022903. doi: 10.1103/PhysRevE.72.022903. [20] C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918. doi: 10.1088/0951-7715/19/12/010. [21] V. Feller, An Introduction to Probability Theory and Its Applications: Volume 2, $2^{nd}$ edition, J. Wiley & sons, New York-London-Sydney, 1971. [22] N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous vortex model, J. Eur. Math. Soc. (JEMS), 16 (2014), 1423-1466. doi: 10.4171/JEMS/465. [23] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197. [24] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [25] J. Klafter, B. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk, Biological Motion, 89 (1990), 281-296. doi: 10.1007/978-3-642-51664-1_20. [26] M. Levandowsky, B. White and F. Schuster, Random movements of soil amebas, Acta Protozoologica, 36 (1997), 237-248. [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), 687-703. doi: 10.1007/s00220-008-0669-0. [28] D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. in Math., 220 (2009), 1717-1738. doi: 10.1016/j.aim.2008.10.016. [29] D. Li, J. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoamericana, 26 (2010), 295-332. doi: 10.4171/RMI/602. [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), 173-188. doi: 10.1007/s10440-015-0022-5. [31] J.-G. Liu and R. Yang, Propagation of chaos for keller-segel equation with a logarithmic cut-off,, preprint., (). [32] F. Matthäus, M. Jagodič and J. Dobnikar, E. coli superdiffusion and chemotaxis-search strategy, precision, and motility, Biophys. J., 97 (2009), 946-957. doi: 10.1016/j.bpj.2009.04.065. [33] P. E. Protter, Stochastic Integration and Differential Equations, $2^{nd}$ edition, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10061-5. [34] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 2013. [35] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [36] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212. doi: 10.1137/S0036139998342065. [37] A.-S. Sznitman, A propagation of chaos result for Burgers' equation, Probab. Theory Relat. Fields, 71 (1986), 581-613. doi: 10.1007/BF00699042. [38] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Boletín de la Sociedad Española de Matemática Aplicada, 49 (2009), 33-44. [39] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4_15. [40] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857. [41] C. Villani, Optimal Transport: Old and New, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-71050-9. [42] V. Yudovich, Non-stationary flow of an ideal incompressible liquid, U.S.S.R. Comput. Math. and Math. Phys., 3 (1963), 1407-1456. doi: 10.1016/0041-5553(63)90247-7.
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