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Regularity and nonexistence of solutions for a system involving the fractional Laplacian
Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces
1. | Department of Mathematics, Yonsei University, 120-749 SeoDaeMun-gu, Seoul, South Korea |
2. | Mathematics Department,Yonsei University, Seoul 120-749, South Korea |
3. | School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, South Korea |
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics. Vol. III, North-Holland, Amsterdam, 2004, 161-244. |
[3] |
M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[4] |
R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[5] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. |
[6] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[7] |
A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[8] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. |
[9] |
Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100.
doi: 10.1016/j.nonrwa.2013.10.008. |
show all references
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics. Vol. III, North-Holland, Amsterdam, 2004, 161-244. |
[3] |
M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[4] |
R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[5] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. |
[6] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[7] |
A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[8] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. |
[9] |
Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100.
doi: 10.1016/j.nonrwa.2013.10.008. |
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