Tobias Black

We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\ &x\inΩ,& t>0,\\ c_{t}+u\cdot\nabla c=Δc-cn,\ &x\inΩ,& t>0,\\ u_{t}+(u\cdot\nabla)u=Δu+\nabla P+n\nablaΦ,\quad \nabla\cdot u=0,\ &x\inΩ,& t>0,\\ \big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c)\cdotν=\nabla c\cdotν=0,\ u=0,\ &x\in\partialΩ,& t>0,\\ n(\cdot,0)=n_0,\ c(\cdot,0)=c_0,\ u(\cdot,0)=u_0,\ &x\inΩ. \end{array}\right. \end{align*} in a smoothly bounded domain $Ω\subset\mathbb{R}^2$. Assuming $S:\overlineΩ\times[0,\infty)\times(0,\infty)\rightarrow \mathbb{R}^{2\times 2}$ to be sufficiently regular and such that with $γ\in[0,\frac56]$ and some non-decreasing $S_0:(0,\infty)\to(0,\infty)$, we have \begin{align*} \big|S(x,n,c)\big|\leq \frac{S_0(c)}{c^γ}\quad\text{for all }(x,n,c)\in\overlineΩ\times[0,\infty)\times(0,\infty), \end{align*} we show that if $D:[0,\infty)\to[0,\infty)$ is suitably regular and positive throughout $(0,\infty)$, then for all $M>0$ one can find $L(M)>0$ such that whenever $$\liminf_{n\to\infty} D(n)>L\quad\text{and}\quad \liminf_{n\searrow0}\frac{D(n)}{n}>0$$ are satisfied and the initial data $(n_0,c_0,u_0)$ are suitably regular and satisfy $\|c_0\|_{L^{\infty}(Ω)}\leq M$ there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of $D(0)>0$ this solution is moreover a classical solution of the same problem.

In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\ &v_t=Δv- uv,\\ \end{array}\right. \end{align*} in a bounded domain $Ω\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^2((0,\infty))$ is assumed to satisfy $D(0)=0$, $D(s)>0$ on $(0,\infty)$, $D'(s)\geq 0$ on $(0,\infty)$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $$s D'(s)\leq C_D D(s)\quad\text{and}\quad C_D s^{p-1}\leq D(s)\quad\text{for }s\in[0,s_0].$$ The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ are supposed to be nonnegative. We show that for all suitably regular initial data $(u_0,v_0)$ satisfying $u_0\geq δ_0>0$ and $v_0\not\equiv 0$ there is a time-local classical solution and - despite the degeneracy at $0$ - the solution satisfies an extensibility criterion of the form $$\text{either}\quad T_{max}=\infty,\quad\text{or}\quad\limsup_{t\nearrow T_{max}}\|u(\cdot,t)\|_{L^\infty(Ω)}=\infty.$$ Moreover, as a by-product of our analysis, we prove that a classical solution on $Ω\times(0,T)$ obeying $\|u(\cdot,t)\|_{L^\infty(Ω)}\leq M_u$ for all $t\in(0,T)$ and emanating from initial data $(u_0,v_0)$ as specified above remains strictly positive throughout $Ω\times(0,T)$, i.e. one can find $δ_u=δ_u(T,δ_0, M_u,\|v_0\|_{W^{1,\infty}(Ω)})>0$ such that $$u(x,t)\geqδ_u\quad\text{for all }(x,t)\inΩ\times(0,T).$$ Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time.

We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form "Equation missing"for parameter choices $$\kappa \ge 0$$κ ≥ 0 and $$\mu >0$$μ > 0 . Herein, we moreover impose a nonnegative and time-constant prescribed concentration $$c_\star \in C^2({\overline{\Omega }})$$c ⋆ ∈ C 2 ( Ω ¯ ) for the signal chemical on the boundary of the domain $$\Omega \subset {\mathbb {R}}^{\mathcal {N}}$$Ω ⊂ R N with $${\mathcal {N}}\in \{2,3\}$$N ∈ { 2 , 3 } . After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of $$c_\star $$c ⋆ being also constant in space. We show that sufficiently strong logistic influence, in the sense that for $$\omega >0$$ω > 0 and $$\mu _0>0$$μ 0 > 0 there is some $$\eta =\eta (\omega ,\mu _0,c_\star )>0$$η = η ( ω , μ 0 , c ⋆ ) > 0 with the property that whenever $$\begin{aligned} \mu _0\le \mu \quad \text {and}\quad \frac{\kappa }{\min \{\mu ,\mu ^{\frac{{\mathcal {N}}+6}{6}+\omega }\}}<\eta \end{aligned}$$μ 0 ≤ μ and κ min { μ , μ N + 6 6 + ω } < η are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on $$\omega ,\mu _0,\eta ,c_\star $$ω , μ 0 , η , c ⋆ and the initial data.

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form "Equation missing"where $$\kappa \ge 0$$κ≥0 , $$\mu >0$$μ>0 and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $$\Omega \subset \mathbb {R}^N$$Ω⊂RN with $$N\in \{2,3\}$$N∈{2,3} is a prescribed time-independent nonnegative function $$c_*\in C^{2}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) $$c∗∈C2Ω¯ . Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.

We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - \frac1{|\Omega|} \int_\Omega u + u \end{cases} \tag{JL} \end{align*} and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - v + u, \end{cases}\tag{PE} \end{align*} where $\lambda$ and $\mu$ are given spatially radial nonnegative functions and $m, \kappa > 0$ are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $m,\kappa$ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $\lambda, \mu > 0$, we find that there are initial data which lead to blow-up in (JL) if \begin{alignat*}{2} 0 \leq \kappa &< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq \kappa&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if } m\in\left(0,\frac{2}{n}\right), \end{alignat*} and in (PE) if $m \in [1, \frac{2n-2}{n})$ and \begin{align*} 0 \leq \kappa < \min\left\{\frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}

We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - \frac{1}{|\Omega |} \int \limits _\Omega u + u \end{array}\right. } \end{aligned}$$u t = ∇ · ( ( u + 1 ) m - 1 ∇ u - u ∇ v ) + λ u - μ u 1 + κ , 0 = Δ v - 1 | Ω | ∫ Ω u + u and $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - v + u, \end{array}\right. } \end{aligned}$$u t = ∇ · ( ( u + 1 ) m - 1 ∇ u - u ∇ v ) + λ u - μ u 1 + κ , 0 = Δ v - v + u , where $$\lambda $$λ and $$\mu $$μ are given spatially radial nonnegative functions and $$m, \kappa > 0$$m , κ > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $$m,\kappa $$m , κ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $$\lambda , \mu > 0$$λ , μ > 0 , we find that there are initial data which lead to blow-up in (JL) if $$\begin{aligned} 0 \le \kappa&< \min \left\{ \frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&\quad \text {if } m\in \left[ \frac{2}{n},\frac{2n-2}{n}\right) \\ \text { or }\quad 0 \le \kappa&<\min \left\{ \frac{1}{2},\frac{n-1}{n}-\frac{m}{2}\right\}&\quad \text {if } m\in \left( 0,\frac{2}{n}\right) , \end{aligned}$$0 ≤ κ < min 1 2 , n - 2 n - ( m - 1 ) + if m ∈ 2 n , 2 n - 2 n or 0 ≤ κ < min 1 2 , n - 1 n - m 2 if m ∈ 0 , 2 n , and in (PE) if $$m \in [1, \frac{2n-2}{n})$$m ∈ [ 1 , 2 n - 2 n ) and $$\begin{aligned} 0 \le \kappa < \min \left\{ \frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\} . \end{aligned}$$0 ≤ κ < min ( m - 1 ) n + 1 2 ( n - 1 ) , n - 2 - ( m - 1 ) n n ( n - 1 ) .