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Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems
1. | Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419 |
2. | Department of Mathematics and Information Technology, The Hong Kong Institute of Education, Rm 19A, 1/F, Block D4, 10, Lo Ping Road, Tai Po, New Territories, Hong Kong, China |
References:
[1] |
L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Mathematical and Computer Modelling, 47 (2008), 755-764.
doi: 10.1016/j.mcm.2007.06.005. |
[2] |
R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Communications in Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[3] |
L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998. |
[4] |
M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.
doi: 10.1137/S0036141001385046. |
[5] |
T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[6] |
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J. Diff. Eqns., 120 (1995), 215-254.
doi: 10.1006/jdeq.1995.1111. |
[7] |
D. Hoff, Compressible Flow in a Half-Space with Navier Boundary Conditions, J. Math. Fluid Mech., 7 (2005), 315-338.
doi: 10.1007/s00021-004-0123-9. |
[8] |
D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760.
doi: 10.1137/040618059. |
[9] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[10] |
T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[11] |
T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Diff. Eqns., 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[12] |
M. Rascle, On a system of non linear strongly coupled partial differential equations arising in biology, Lectures Notes in Math., Springer, Berlin, 846 (1981), 290-298. |
[13] |
A. Suen and D. Hoff, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics, Arch. Rational Mechanics Anal., 205 (2012), 27-58.
doi: 10.1007/s00205-012-0498-3. |
[14] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher- dimensional Keller-Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[15] |
M. Winkler, Global Large-Data Solutions in a Chemotaxis-(Navier)Stokes System Modeling Cellular Swimming in Fluid Drops, Communications in Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
show all references
References:
[1] |
L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Mathematical and Computer Modelling, 47 (2008), 755-764.
doi: 10.1016/j.mcm.2007.06.005. |
[2] |
R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Communications in Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[3] |
L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998. |
[4] |
M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.
doi: 10.1137/S0036141001385046. |
[5] |
T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[6] |
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J. Diff. Eqns., 120 (1995), 215-254.
doi: 10.1006/jdeq.1995.1111. |
[7] |
D. Hoff, Compressible Flow in a Half-Space with Navier Boundary Conditions, J. Math. Fluid Mech., 7 (2005), 315-338.
doi: 10.1007/s00021-004-0123-9. |
[8] |
D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760.
doi: 10.1137/040618059. |
[9] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[10] |
T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[11] |
T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Diff. Eqns., 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[12] |
M. Rascle, On a system of non linear strongly coupled partial differential equations arising in biology, Lectures Notes in Math., Springer, Berlin, 846 (1981), 290-298. |
[13] |
A. Suen and D. Hoff, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics, Arch. Rational Mechanics Anal., 205 (2012), 27-58.
doi: 10.1007/s00205-012-0498-3. |
[14] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher- dimensional Keller-Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[15] |
M. Winkler, Global Large-Data Solutions in a Chemotaxis-(Navier)Stokes System Modeling Cellular Swimming in Fluid Drops, Communications in Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
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