# American Institute of Mathematical Sciences

December  2020, 25(12): 4703-4719. doi: 10.3934/dcdsb.2020120

## Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation

Received  October 2019 Revised  January 2020 Published  March 2020

We study the chemotaxis–Navier–Stokes system
 \left\{\;\; \begin{aligned} n_t + u \cdot \nabla n &\;\; = \;\; \Delta n - \nabla \cdot (nS(x,n,c) \nabla c), \\ c_t + u\cdot \nabla c &\;\; = \;\; \Delta c - n f(c), \\ u_t + (u\cdot \nabla) u &\;\; = \;\; \Delta u + \nabla P + n \nabla \phi, \;\;\;\;\;\; \nabla \cdot u = 0, \;\;\;\;\;\; \end{aligned} \right. \tag{\star$}$
with no-flux boundary conditions for
 $n$
,
 $c$
and Dirichlet boundary conditions for
 $u$
in a bounded, convex, smooth domain
 $\Omega \subseteq \mathbb{R}^2$
, which is motivated by recent modeling approaches from biology for aerobic bacteria suspended in a sessile water drop. We further do not assume the chemotactic sensitivity
 $S$
to be scalar as is common, but to be able to attain values in
 $\mathbb{R}^{2\times2}$
, which allows for more complex modeling of bacterial behavior.
While there have been various results for scalar
 $S$
and some for the non-scalar case with only a Stokes fluid equation simplifying the analysis of the third equation in (
 $\star$
), we consider the fully combined case giving us very little to go on in terms of a priori estimates. We nonetheless manage to still achieve sufficient estimates using Trudinger–Moser type inequalities to extend the existence results seen in a recent work by Winkler for the Stokes case with non-scalar
 $S$
to the full Navier–Stokes case. Namely, we construct a similar global mass-preserving solution for (
 $\star$
) in planar convex domains under fairly weak assumptions on the parameter functions.
Citation: Frederic Heihoff. Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4703-4719. doi: 10.3934/dcdsb.2020120
##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar [2] X. Cao, Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term, J. Differential Equations, 261 (2016), 6883-6914.  doi: 10.1016/j.jde.2016.09.007.  Google Scholar [3] X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.  Google Scholar [4] M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar [5] S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.  doi: 10.4310/jdg/1214441783.  Google Scholar [6] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North Holland Mathematics Studies, Vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar [7] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103. doi: 10.1103/PhysRevLett.93.098103.  Google Scholar [8] 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.  Google Scholar [9] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [10] T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.  doi: 10.1142/S0218202515500177.  Google Scholar [11] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 2011,643–652. doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar [12] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar [13] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar [14] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar [15] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. U.S.A., 102, (2005) 2277–2282. doi: 10.1073/pnas.0406724102.  Google Scholar [16] Y. Wang, M. Winkler and and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 421-466.   Google Scholar [17] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027.  Google Scholar [18] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.  Google Scholar [19] M. Winkler, Can fluid interaction influence the critical mass for taxis-driven blow-up in bounded planar domains?, Acta Appl. Math., published online, (2020). doi: 10.1007/s10440-020-00312-2.  Google Scholar [20] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar [21] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar [22] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity., Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar [23] M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.  Google Scholar [24] M. Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar [25] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar [26] M. Winkler, Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components, J. Evol. Equ., 18 (2018), 1267-1289.  doi: 10.1007/s00028-018-0440-8.  Google Scholar [27] M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?, in International Mathematics Research Notices, (2019), rnz056. doi: 10.1093/imrn/rnz056.  Google Scholar [28] C. Xue, Macroscopic equations for bacterial chemotaxis: Integration of detailed biochemistry of cell signaling, J. Math. Biol., 70 (2015), 1-44.  doi: 10.1007/s00285-013-0748-5.  Google Scholar [29] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.  Google Scholar [30] Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.  Google Scholar

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##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar [2] X. Cao, Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term, J. Differential Equations, 261 (2016), 6883-6914.  doi: 10.1016/j.jde.2016.09.007.  Google Scholar [3] X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.  Google Scholar [4] M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar [5] S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.  doi: 10.4310/jdg/1214441783.  Google Scholar [6] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North Holland Mathematics Studies, Vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar [7] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103. doi: 10.1103/PhysRevLett.93.098103.  Google Scholar [8] 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.  Google Scholar [9] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [10] T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.  doi: 10.1142/S0218202515500177.  Google Scholar [11] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 2011,643–652. doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar [12] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar [13] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar [14] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar [15] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. U.S.A., 102, (2005) 2277–2282. doi: 10.1073/pnas.0406724102.  Google Scholar [16] Y. Wang, M. Winkler and and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 421-466.   Google Scholar [17] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027.  Google Scholar [18] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.  Google Scholar [19] M. Winkler, Can fluid interaction influence the critical mass for taxis-driven blow-up in bounded planar domains?, Acta Appl. Math., published online, (2020). doi: 10.1007/s10440-020-00312-2.  Google Scholar [20] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar [21] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar [22] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity., Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar [23] M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.  Google Scholar [24] M. Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar [25] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar [26] M. Winkler, Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components, J. Evol. Equ., 18 (2018), 1267-1289.  doi: 10.1007/s00028-018-0440-8.  Google Scholar [27] M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?, in International Mathematics Research Notices, (2019), rnz056. doi: 10.1093/imrn/rnz056.  Google Scholar [28] C. Xue, Macroscopic equations for bacterial chemotaxis: Integration of detailed biochemistry of cell signaling, J. Math. Biol., 70 (2015), 1-44.  doi: 10.1007/s00285-013-0748-5.  Google Scholar [29] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.  Google Scholar [30] Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.  Google Scholar
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