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 n t + u ⋅ ∇ n = ∇ ⋅ ( D ( n ) ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , c t + u ⋅ ∇ c = Δ c − n c , u t = Δ u + ∇ 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 ⊂ R 3 \Omega \subset {{\mathbb{R}}}^{3} , with given suitably regular functions D : [ 0 , ∞ ) → [ 0 , ∞ ) D:{[}0,\infty )\to {[}0,\infty ) , S : Ω ¯ × [ 0 , ∞ ) × ( 0 , ∞ ) → R 3 × 3 S:\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 > 0 D\gt 0 on ( 0 , ∞ ) \left(0,\infty ) . It is shown that if with some nondecreasing S 0 : ( 0 , ∞ ) → ( 0 , ∞ ) {S}_{0}:\left(0,\infty )\to \left(0,\infty ) we have ∣ S ( x , n , c ) ∣ ≤ S 0 ( c ) c 1 2 for 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 > 0 M\gt 0 there exists L ( M ) > 0 L\left(M)\gt 0 such that whenever liminf n → ∞ D ( n ) > L ( M ) and liminf n ↘ 0 D ( 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 ( n 0 , c 0 , u 0 ) \left({n}_{0},{c}_{0},{u}_{0}) fulfilling ‖ c 0 ‖ 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 (

A chemotaxis‐Stokes system with nonhomogeneous boundary condition: Global large‐datum solution and asymptotic behavior

“…However, the study for 3‐D case is rather tortuous, through the efforts of many mathematicians [7, 32, 35, 36], the existence of global bounded weak solutions are finally proved for $$m>1$$ by Refs. [17, 37] using different methods.…”