Advanced Nonlinear Studies (Mar 2022)
Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings
Abstract
The chemotaxis–Stokes system nt+u⋅∇n=∇⋅(D(n)∇n)−∇⋅(nS(x,n,c)⋅∇c),ct+u⋅∇c=Δc−nc,ut=Δu+∇P+n∇Φ,∇⋅u=0,\left\{\begin{array}{l}{n}_{t}+u\cdot \nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+\nabla P+n\nabla \Phi ,\hspace{1.0em}\nabla \cdot u=0,\end{array}\right. is considered in a smoothly bounded convex domain Ω⊂R3\Omega \subset {{\mathbb{R}}}^{3}, with given suitably regular functions D:[0,∞)→[0,∞)D:{[}0,\infty )\to {[}0,\infty ), S:Ω¯×[0,∞)×(0,∞)→R3×3S:\overline{\Omega }\times {[}0,\infty )\times \left(0,\infty )\to {{\mathbb{R}}}^{3\times 3} and Φ:Ω¯→R\Phi :\overline{\Omega }\to {\mathbb{R}} such that D>0D\gt 0 on (0,∞)\left(0,\infty ). It is shown that if with some nondecreasing S0:(0,∞)→(0,∞){S}_{0}:\left(0,\infty )\to \left(0,\infty ) we have ∣S(x,n,c)∣≤S0(c)c12for all(x,n,c)∈Ω¯×[0,∞)×(0,∞),| S\left(x,n,c)| \le \frac{{S}_{0}\left(c)}{{c}^{\tfrac{1}{2}}}\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\left(x,n,c)\in \overline{\Omega }\times {[}0,\infty )\times \left(0,\infty ), then for all M>0M\gt 0 there exists L(M)>0L\left(M)\gt 0 such that whenever liminfn→∞D(n)>L(M)andliminfn↘0D(n)n>0,\mathop{\mathrm{liminf}}\limits_{n\to \infty }D\left(n)\gt L\left(M)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{liminf}}\limits_{n\searrow 0}\frac{D\left(n)}{n}\gt 0, for all sufficiently regular initial data (n0,c0,u0)\left({n}_{0},{c}_{0},{u}_{0}) fulfilling ‖c0‖L∞(Ω)≤M\Vert {c}_{0}{\Vert }_{{L}^{\infty }\left(\Omega )}\le M an associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global bounded weak solution, classical if additionally D(0)>0D\left(0)\gt 0. When combined with previously known results, this particularly implies global existence of bounded solutions when D(n)=nm−1D\left(n)={n}^{m-1}, n≥0n\ge 0, with arbitrary m>1m\gt 1, but beyond this asserts global boundedness also in the presence of diffusivities which exhibit arbitrarily slow divergence to +∞+\infty at large densities and of possibly singular chemotactic sensitivities.
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