Abstract
We establish the global well-posedness for the multidimensional Chemotaxis model with some classes of large initial data, especially the case when the rate of variation of ln v0 (v0 is the chemical concentration) contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D. Besides, we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption, which include specially the situations of d = 2, 3 and improve the previous time-decay rates. Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.
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Abidi, H.: Équation de Navier—Stokes avec densitéetviscosité variables dans l’espace critique. Rev. Mat. Iberoamericana, 23, 537–586 (2007)
Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin/Heidelberg, 2011
Biler, P.: Singularities of Solutions to Chemotaxis Systems, De Gruyter, Berlin, Boston, 2019
Biler, P., Karch, G.: Blow-up of solutions to generalized Keller—Segel model. J. Evol. Equ., 10, 247–262 (2010)
Cannone, M.: Ondelettes, paraproduits et Navier—Stokes, Nouveaux essais, Diderot éditeurs, Paris, 1995
Cannone, M.: Harmonic Analysis Tools for Solving the Incompressible Navier—Stokes Equations, Handbook of Mathematical fluid Dynamics, Vol. III, North-Holland, Amsterdam, 2004
Cannone, M.: A generalization of a theorem by Kato on Navier—Stokes equations. Rev. Mat. Iberoamericana, 13, 515–141 (1997)
Cannone, M., Meyer, Y., Planchon, F.: Solutions auto-similaires des équations de Navier—Stokes. Séminaire “Équations aux Dérivées Partielle” de l’École polytechnique, Exposé VIII, 1–10 (1993–1994)
Chen, Q., Hao, X.: Large global solutions to the three dimensional chemotaxis-Navier—Stokes equations slowly varying in one direction. Appl. Math. Lett., 112, 106773 (2021)
Chen, Q., Miao, C., Zhang Z.: Global well-posedness for compressible Navier—Stokes equations with highly oscillating initial velocity. Comm. Pure Appl. Math., 63, 1173–1224 (2010)
Chen, Q., Miao, C., Zhang, Z.: A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Comm. Math. Phys., 271, 821–838 (2007)
Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimemsions. Milan J. Math., 72, 1–28 (2004)
Danchin, R., Xu, J.: Optimal time-decay estimates for the compressible Navier—Stokes equations in the critical Lp framework. Arch. Ration. Mech. Anal., 224, 53–90 (2017)
Fang, D., Zhang, T., Zi, R.: Global solutions to the isentropic compressible Navier—Stokes equations with a class of large initial data. SIAM J. Math. Anal., 50, 4983–5026 (2018)
Fujita, H., Kato, T.: On the Navier—Stokes initial value problem I. Arch. Ration. Mech. Anal., 16, 269–315 (1964)
Hao, C.: Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces. Z. Angew. Math. Phys., 63, 825–834 (2012)
Haspot, B.: Existence of global strong solutions in critical spaces for barotropic viscous fluids. Arch. Ration. Mech. Anal., 202, 427–460 (2011)
He, H., Zhang, Q.: Global existence of weak solutions for the 3D chemotaxis-Navier—Stokes equations. Nonlinear Anal. Real World Appl., 35, 336–349 (2017)
Hoff, D.: Global solutions of the Navier—Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differential Equations, 120, 215–254 (1995)
Horstmann, D.: From 1970 until present: the Keller—Segel model in chemotaxis and its consequences: I. Jahresber. Deutsch. Math.-Verein., 105, 103–165 (2003)
Lemarié-Rieusset, P. G.: Recent Developments in the Navier—Stokes Problem. Res. Notes Math., Vol. 43, Chapman & Hall/CRC, Boca Raton, FL, 2002
Lemarié-Rieusset, P. G.: The Navier—Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016
Li, D., Li, T., Zhao, K.: On a hyperbolic-parabolic system modeling chemotaxis. Math. Models Methods Appl. Sci., 21, 1631–1650 (2011)
Tao, Y., Winkler, M.: A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source. SIAM J. Math. Anal., 43, 685–704 (2011)
Wang, Z., Hillen, T.: Shock formation in a chemotaxis model. Math. Methods Appl. Sci., 31, 45–70 (2008)
Winkler, M.: Global large-data solutions in a chemotaxis-(Navier—)Stokes system modeling cellular swimming in fluid drops. Comm. Partial Differential Equations, 37, 319–352 (2012)
Xin, Z., Xu, J.: Optimal decay for the compressible Navier—Stokes equations without additional smallness assumptions. J. Differential Equations, 274, 543–575 (2020)
Zhang, Q., Zheng, X.: Global well-posedness for the two-dimensional incompressible chemptaxis-Navier—Stokes equations. SIAM J. Math. Anal., 46, 3078–3105 (2014)
Zhang, Q: Local well-posedness for the chemotaxis-Navier—Stokes equations in Besov spaces. Nonlinear Anal. Real World Appl., 17, 89–100 (2014)
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Supported by the National Natural Science Foundation of China (Grant No. 12071043), the National Key Research and Development Program of China (Grant No. 2020YFA0712900)
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Chen, Q.L., Hao, X.N. & Li, J.Y. Global Fujita—Kato’s Type Solutions and Long-time Behavior for the Multidimensional Chemotaxis Model. Acta. Math. Sin.-English Ser. 38, 311–330 (2022). https://doi.org/10.1007/s10114-022-1001-1
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DOI: https://doi.org/10.1007/s10114-022-1001-1