## Existence of global solutions to chemotaxis fluid system with logistic source.(English)Zbl 07444210

Summary: We establish the existence of global solutions and $$L^q$$ time-decay of a three dimensional chemotaxis system with chemoattractant and repellent. We show the existence of global solutions by the energy method. We also study $$L^q$$ time-decay for the linear homogeneous system by using Fourier transform and finding Green’s matrix. Then, we find $$L^q$$ time-decay for the nonlinear system using solution representation by Duhamel’s principle and time-weighted estimate.

### MSC:

 35Q31 Euler equations 35Q35 PDEs in connection with fluid mechanics 35Q92 PDEs in connection with biology, chemistry and other natural sciences 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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