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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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