On the well-posedness for Keller-Segel system with fractional diffusion
Communicated by M. Costabel In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u0,v0) in critical Fourier‐Herz spaces B˙q2−2αRn×B˙q2−2αRn with q ∈ [2, ∞], where 1 < α ≤ 2. Making...
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| Published in: | Mathematical methods in the applied sciences Vol. 34; no. 14; pp. 1739 - 1750 |
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| Language: | English |
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Chichester, UK
John Wiley & Sons, Ltd
30.09.2011
Wiley |
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| ISSN: | 0170-4214, 1099-1476 |
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| Abstract | Communicated by M. Costabel
In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u0,v0) in critical Fourier‐Herz spaces B˙q2−2αRn×B˙q2−2αRn with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd. |
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| AbstractList | Communicated by M. Costabel
In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u0,v0) in critical Fourier‐Herz spaces B˙q2−2αRn×B˙q2−2αRn with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd. |
| Author | Wu, Gang Zheng, Xiaoxin |
| Author_xml | – sequence: 1 givenname: Gang surname: Wu fullname: Wu, Gang email: wugangmaths@yahoo.com.cn, Gang Wu, Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China., wugangmaths@yahoo.com.cn organization: Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China – sequence: 2 givenname: Xiaoxin surname: Zheng fullname: Zheng, Xiaoxin organization: The Graduate School of China Academy of Engineering Physics, P.O. Box 2101, Beijing, China 100088 |
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| Cites_doi | 10.1016/j.jfa.2008.07.008 10.1016/j.jfa.2005.08.004 10.1137/S0036139996313447 10.4064/bc74-0-9 10.4310/CMS.2008.v6.n2.a8 10.1006/jdeq.2001.4146 10.3233/ASY-2008-0907 10.1016/j.jmaa.2011.02.010 10.1016/j.jmaa.2007.09.060 10.1002/mana.200410472 10.4064/bc74-0-2 10.1016/j.jfa.2010.02.005 |
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| Keywords | Keller-Segel model Numerical linear algebra Fourier analysis Evolution equation Well posed problem Littlewood Paley theory Mathematical method Cauchy problem Non linear equation Regularization method Parabolic equation Numerical analysis fractional diffusion Linear equation III-posedness Applied mathematics nonlinear parabolic equations Ill posed problem Localization nonlinear evolution equations Well-posedness |
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| References_xml | – reference: Biler P, Woyczyński WA. Global and exploding solutions for nonlocal quadratic evolution problems. SIAM Journal on Applied Mathematics 1998; 59:845-869. – reference: Miao C, Yuan B. Solutions to some nonlinear parabolic equations in pseudomeasure spaces. Mathematische Nachrichten 2007; 280:171-186. – reference: Iwabuchi T. Global well-posedness for Keller-Segel system in Besov type spaces. Journal of Mathematical Analysis and Applications 2011; 379(2):930-948. – reference: Bejenaru I, Tao T. Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrodinger equation. Journal of Functional Analysis 2006; 233(1):228-259. – reference: Wu G, Yuan J. Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces. Journal of Mathematical Analysis and Applications 2008; 340:1326-1335. – reference: Biler P. Local and global solvability of parabolic systems modelling chemotaxis. Advances in Mathematical Sciences and Applications 1998; 8:715-743. – reference: Stein EM. Singular Integrals and Differentiability Properties of Functions. Princeton University Press: Princeton, NJ, 1970. – reference: Naito Y, Suzuki T, Yoshida K. Self-similar solutions to a parabolic system modeling chemotaxis. Journal of Differential Equations 2002; 184:386-421. – reference: Calvez V, Corrias L. The parabolic-parabolic Keller-Segel model in ℝ2. Communications in Mathematical Sciences 2008; 6(2):417-447. – reference: Bourgain J, Pavlovic N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. Journal of Functional Analysis 2008; 255(9):2233-2247. – reference: Raczyński A. Stability property of the two-dimensional Keller-Segel model. Asymptotic Analysis 2009; 61:35-59. – reference: Yoneda T. Ill-posedness of the 3D-Navier-Stokes equations in a generalized Besov space near BMO−1. 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volume: 340 start-page: 1326 year: 2008 end-page: 1335 article-title: Well‐posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces publication-title: Journal of Mathematical Analysis and Applications – volume: 258 start-page: 3376 issue: 10 year: 2010 end-page: 3387 article-title: Ill‐posedness of the 3D‐Navier–Stokes equations in a generalized Besov space near publication-title: Journal of Functional Analysis – volume-title: Singular Integrals and Differentiability Properties of Functions year: 1970 ident: 10.1002/mma.1480-BIB0011|mma1480-cit-0011 – volume: 255 start-page: 2233 issue: 9 year: 2008 ident: 10.1002/mma.1480-BIB0008|mma1480-cit-0008 article-title: Ill-posedness of the Navier-Stokes equations in a critical space in 3D publication-title: Journal of Functional Analysis doi: 10.1016/j.jfa.2008.07.008 – volume: 233 start-page: 228 issue: 1 year: 2006 ident: 10.1002/mma.1480-BIB0015|mma1480-cit-0015 article-title: Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrodinger equation publication-title: Journal of Functional Analysis doi: 10.1016/j.jfa.2005.08.004 – volume: 59 start-page: 845 year: 1998 ident: 10.1002/mma.1480-BIB0002|mma1480-cit-0002 article-title: Global and exploding solutions for nonlocal quadratic evolution problems publication-title: SIAM Journal on Applied Mathematics doi: 10.1137/S0036139996313447 – volume: 74 start-page: 149 volume-title: Self-similar Solutions in Nonlinear PDE year: 2006 ident: 10.1002/mma.1480-BIB0006|mma1480-cit-0006 doi: 10.4064/bc74-0-9 – volume: 6 start-page: 417 issue: 2 year: 2008 ident: 10.1002/mma.1480-BIB0003|mma1480-cit-0003 article-title: The parabolic-parabolic Keller-Segel model in ℝ2 publication-title: Communications in Mathematical Sciences doi: 10.4310/CMS.2008.v6.n2.a8 – volume: 184 start-page: 386 year: 2002 ident: 10.1002/mma.1480-BIB0005|mma1480-cit-0005 article-title: Self-similar solutions to a parabolic system modeling chemotaxis publication-title: Journal of Differential Equations doi: 10.1006/jdeq.2001.4146 – volume: 61 start-page: 35 year: 2009 ident: 10.1002/mma.1480-BIB0014|mma1480-cit-0014 article-title: Stability property of the two-dimensional Keller-Segel model publication-title: Asymptotic Analysis doi: 10.3233/ASY-2008-0907 – start-page: 161 volume-title: Handbook of Mathematical Fluid Dynamics year: 2004 ident: 10.1002/mma.1480-BIB0012|mma1480-cit-0012 – volume: 379 start-page: 930 issue: 2 year: 2011 ident: 10.1002/mma.1480-BIB0001|mma1480-cit-0001 article-title: Global well-posedness for Keller-Segel system in Besov type spaces publication-title: Journal of Mathematical Analysis and Applications doi: 10.1016/j.jmaa.2011.02.010 – volume: 340 start-page: 1326 year: 2008 ident: 10.1002/mma.1480-BIB0010|mma1480-cit-0010 article-title: Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces publication-title: Journal of Mathematical Analysis and Applications doi: 10.1016/j.jmaa.2007.09.060 – volume: 280 start-page: 171 year: 2007 ident: 10.1002/mma.1480-BIB0013|mma1480-cit-0013 article-title: Solutions to some nonlinear parabolic equations in pseudomeasure spaces publication-title: Mathematische Nachrichten doi: 10.1002/mana.200410472 – volume: 74 start-page: 33 volume-title: Self-similar Solutions in Nonlinear PDE year: 2006 ident: 10.1002/mma.1480-BIB0007|mma1480-cit-0007 doi: 10.4064/bc74-0-2 – volume: 258 start-page: 3376 issue: 10 year: 2010 ident: 10.1002/mma.1480-BIB0009|mma1480-cit-0009 article-title: Ill-posedness of the 3D-Navier-Stokes equations in a generalized Besov space near BMO−1 publication-title: Journal of Functional Analysis doi: 10.1016/j.jfa.2010.02.005 – volume: 8 start-page: 715 year: 1998 ident: 10.1002/mma.1480-BIB0004|mma1480-cit-0004 article-title: Local and global solvability of parabolic systems modelling chemotaxis publication-title: Advances in Mathematical Sciences and 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In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel... |
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| SubjectTerms | Exact sciences and technology fractional diffusion Functional analysis Global analysis, analysis on manifolds Ill-posedness Keller-Segel model Mathematical analysis Mathematics nonlinear evolution equations nonlinear parabolic equations Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Partial differential equations Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Well-posedness |
| Title | On the well-posedness for Keller-Segel system with fractional diffusion |
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