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On an $N$-Component Camassa-Holm equation with peakons
Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan |
We prove the existence of solutions of degenerate parabolic-parabolic Keller-Segel system with no-flux and Neumann boundary conditions for each variable respectively, under the assumption that the total mass of the first variable is below a certain constant. The proof relies on the interpretation of the system as a gradient flow in the product space of the Wasserstein space and the standard $L^2$-space. More precisely, we apply the ''minimizing movement'' scheme and show a certain critical mass appears in the application of this scheme to our problem.
References:
| [1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. |
| [2] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics, Birkhäuser, 2005. |
| [3] |
L. Ambrosio and G. Savaré, Gradient flows of probability measures, Handbook of differential
equations: Evolutionary equations, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅲ (2007), 1–136. |
| [4] |
J. Bedrossian, N. Rodríguez and A. Bertozzi,
Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.
|
| [5] |
P. Biler,
Local and global solvability of some parabolic systems modelling chemotaxis, Adv.
Math. Sci. Appl., 8 (1998), 715-743.
|
| [6] |
A. Blanchet, J. A. Carrillo and Ph. Laurençot,
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.
|
| [7] |
A. Blanchet, J. Dolbeault and B. Perthame,
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,
Electron. J. Differential Equations 44 (2006), 32 pp. (electronic). |
| [8] |
A. Blanchet and Ph. Laurençcot,
The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in $\mathbb{R}^d, d ≥q 3$, Comm. Partial Differential Equations, 38 (2013), 658-686.
|
| [9] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in $\mathbb{R}^2$, Commun. Math.
Sci., 6 (2008), 417-447.
|
| [10] |
H. Gajewski and K. Zacharias,
Global behavior of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 195 (1998), 77-114.
|
| [11] |
M. A. Herrero and J. J. L. Velázquez,
Chemotaxis collapse for Keller-Segel model, J. Math.
Biol., 35 (1996), 177-194.
|
| [12] |
M. A. Herrero and J. J. L. Velázquez,
Singularity patterns in a chemotaxis model, Math.
Ann., 306 (1996), 583-623.
|
| [13] |
S. Ishida and T. Yokota,
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.
|
| [14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J.
Theor. Biol., 26 (1970), 399-415.
|
| [15] |
D. Matthes, R. J. McCann and G. Savarß,
A family of nonlinear fourth order equations of gradient flow type, Comm. Part. Diff. Eqs., 34 (2009), 1352-1397.
|
| [16] |
N. Mizoguchi,
Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505.
|
| [17] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
| [18] |
F. Otto, Doubly degenerate diffusion equations as steepest descent, Manuscript, 1996. |
| [19] |
T. Suzuki and R. Takahashi,
Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅱ. Blowup threshold, Differential Integral Equations, 22 (2009), 1153-1172.
|
| [20] |
M. Taylor,
Partial Differential Equations I Springer New York, 1996. |
| [21] |
C. Villani,
Topics in Optimal Transportation Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003. |
show all references
References:
| [1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. |
| [2] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics, Birkhäuser, 2005. |
| [3] |
L. Ambrosio and G. Savaré, Gradient flows of probability measures, Handbook of differential
equations: Evolutionary equations, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅲ (2007), 1–136. |
| [4] |
J. Bedrossian, N. Rodríguez and A. Bertozzi,
Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.
|
| [5] |
P. Biler,
Local and global solvability of some parabolic systems modelling chemotaxis, Adv.
Math. Sci. Appl., 8 (1998), 715-743.
|
| [6] |
A. Blanchet, J. A. Carrillo and Ph. Laurençot,
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.
|
| [7] |
A. Blanchet, J. Dolbeault and B. Perthame,
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,
Electron. J. Differential Equations 44 (2006), 32 pp. (electronic). |
| [8] |
A. Blanchet and Ph. Laurençcot,
The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in $\mathbb{R}^d, d ≥q 3$, Comm. Partial Differential Equations, 38 (2013), 658-686.
|
| [9] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in $\mathbb{R}^2$, Commun. Math.
Sci., 6 (2008), 417-447.
|
| [10] |
H. Gajewski and K. Zacharias,
Global behavior of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 195 (1998), 77-114.
|
| [11] |
M. A. Herrero and J. J. L. Velázquez,
Chemotaxis collapse for Keller-Segel model, J. Math.
Biol., 35 (1996), 177-194.
|
| [12] |
M. A. Herrero and J. J. L. Velázquez,
Singularity patterns in a chemotaxis model, Math.
Ann., 306 (1996), 583-623.
|
| [13] |
S. Ishida and T. Yokota,
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.
|
| [14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J.
Theor. Biol., 26 (1970), 399-415.
|
| [15] |
D. Matthes, R. J. McCann and G. Savarß,
A family of nonlinear fourth order equations of gradient flow type, Comm. Part. Diff. Eqs., 34 (2009), 1352-1397.
|
| [16] |
N. Mizoguchi,
Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505.
|
| [17] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
| [18] |
F. Otto, Doubly degenerate diffusion equations as steepest descent, Manuscript, 1996. |
| [19] |
T. Suzuki and R. Takahashi,
Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅱ. Blowup threshold, Differential Integral Equations, 22 (2009), 1153-1172.
|
| [20] |
M. Taylor,
Partial Differential Equations I Springer New York, 1996. |
| [21] |
C. Villani,
Topics in Optimal Transportation Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003. |
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