In this paper we prove the local well-posedness of strong solutions to a chemotaxis-shallow water system with initial vacuum in a bounded domain $\Omega\subset\mathbb{R}^2$ without the standard compatibility condition for the initial data. This improves some results obtained in [J. Differential Equations 261(2016), 6758-6789].
Jishan FAN
,
Fucai LI
,
Gen NAKAMURA
. THE LOCAL WELL-POSEDNESS OF A CHEMOTAXIS-SHALLOW WATER SYSTEM WITH VACUUM[J]. Acta mathematica scientia, Series B, 2021
, 41(1)
: 231
-240
.
DOI: 10.1007/s10473-021-0113-5
[1] Che J, Chen L, Duan B, Luo Z. On the existence of local strong solutions to chemotaxis-shallow water system with large data and vacuum. J Differential Equations, 2016, 261:6758-6789
[2] Choe H J, Kim H. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J Differential Equations, 2003, 190:504-523
[3] Huang X. On local strong and classical solutions to the three-dimensional barotropic compressible NavierStokes equations with vacuum. Sci China Math, 2020, DOI:10.1007/s11425-019-9755-3
[4] Huang X, Li J. Existence and blowup behavior of global strong solutions to the two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data. J Math Pures Appl, 2016, 106:123-154
[5] Li J, Liang Z. On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum. J Math Pures Appl, 2014, 102:640-671
[6] Li J, Xin Z-P. Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum. Ann PDE, 2019, 5:7, 37pp
[7] Metivier G, Schochet S. The incompressible limit of the non-isentropic Euler equations. Arch Ration Mech Anal, 2001, 158:61-90
[8] Salvi R, Straskraba I. Global existence for viscous compressible fluids and their behavior as t → ∞. J Fac Sci Univ Tokyo Sect IA Math, 1993, 40:17-51
[9] Tao Q, Yao Z. Global existence and large time behavior for a two-dimensional chemotaxis-shallow water system. J Differential Equations, 2018, 265:3092-3129
[10] Tuval I, Cisneros L, Dombrowski C, Wolgemuth C, Kessler J, Glodstein R. Bacterial swimming and oxygen transport near contact lines. Proc Natl Acad Sci USA, 2005, 102(7):2277-2282
[11] Wang W, Wang Y. The Lp decay estimates for the chemotaxis-shallow water system. J Math Anal Appl, 2019, 474:640-665
[12] Wang Z. Existence results for chemotaxis-shallow water system with degenerate viscosity coefficients and initial vacuum. Nonlinear Anal, 2019, 188:265-293
[13] Vol'pert A I, Hudjaev S I. The Cauchy problem for composite systems of nonlinear differential equations. (Russian) Mat Sb, 1972, 87(129):504-528
