Global bounded weak solutions in a two-dimensional Keller-Segel-Navier-Stokes system with indirect signal production and nonlinear diffusion

This paper considers the coupled quasilinear Keller-Segel-Navier-Stokes system with indirect signal production and nonlinear diffusion ⎧ ⎨ ⎪⎪⎪⎪ ⎪ ⎪⎪⎪⎪⎪⎩ nt & PLUSMN; u & BULL; Vn = & UDelta;nm - V & BULL; (nVv), xE S2, t > 0, vt & PLUSMN;u & BULL; Vv= & UDelta;v - v & PLUSMN;w, xE S2, t > 0, wt & PLUSMN;u & BULL; Vw = & UDelta;w - w & PLUSMN; n, xE S2, t > 0, ut & PLUSMN; (u & BULL; V)u & PLUSMN; VP = & UDelta;u & PLUSMN;nV & phi;, xE S2, t > 0, V & BULL;u= 0, xE S2, t > 0 (*) under no-flux boundary conditions for n, v and w and no-slip boundary condition for u in a bounded domain S2 C R2 with smooth boundary, where & phi; E W2,& INFIN;(S2) and m > 0. If m > 1, then for any sufficiently smooth initial data, there exists at least one globally defined weak solution which is bounded for the corresponding initial-boundary value problem of system (*).& COPY; 2023 Elsevier Inc. All rights reserved.