Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems

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February 2016, 36(2): 861-875. doi: 10.3934/dcds.2016.36.861

Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems

Tong Li ^1, and Anthony Suen ^2,

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

Received February 2014 Revised February 2015 Published August 2015

Abstract
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We prove the global-in-time existence of intermediate weak solutions of the equations of chemotaxis system in a bounded domain of $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small in $H^1$. No smallness assumption is imposed on the initial cell density which is in $L^2$. We first show that when the initial chemical concentration $c_0$ is small only in $H^1$ and $(n_0-n_\infty,c_0)$ is smooth, the classical solution exists for all time. Then we construct weak solutions as limits of smooth solutions corresponding to mollified initial data. Finally we determine the asymptotic behavior of the global solutions.

Keywords: chemotaxis, energy estimates, intermediate weak solution, asymptotic behavior., global existence, Keller-Segel model.

Mathematics Subject Classification: 35L45, 35L60, 92B05, 92B99, 92C15, 92C1.

Citation: Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861

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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