Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis - Claes, Leander; Winkler, Michael (Nonlinear Analysis: Real World Applications)

Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid - Claes, Leander; Winkler, Michael (Nonlinear Differential Equations and Applications NoDEA)

Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid - Black, Tobias; Winkler, Michael (Evolution Equations and Control Theory)

Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters - Winkler, Michael (Applied Mathematics & Optimization)

Can diffusion degeneracies enhance complexity in chemotactic aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system - Winkler, Michael (Journal of the European Mathematical Society)
The Cauchy problem in \mathbb{R}^{n} for the cross-diffusion system \begin{cases}u_{t} = \nabla \cdot (D(u)\nabla u) - \nabla\cdot (u\nabla v), \\ 0 = \Delta v +u,\end{cases} is considered for n\ge 2 and under assumptions ensuring that D suitably generalizes the prototype given by D(\xi)=(\xi+1)^{-\alpha}, \quad \xi\ge 0. Under the assumption that \alpha>1 , it is shown that for any r_{\star}>0 and \delta\in (0,1) one can find radially symmetric initial data from C_{0}^{\infty}(\mathbb{R}^{n}) such that the corresponding solution blows up within some finite time, and that this explosion occurs throughout certain spheres in an appropriate sense, with any such sphere being located in the annulus \overline{B}_{r_\star+\delta}(0)\setminus B_{(1-\delta)r_\star}(0) .This is complemented by a result revealing that when \alpha<1 , any finite-mass unbounded radial solution must blow up exclusively at the spatial origin.

A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production - Tao, Youshan; Winkler, Michael (Journal of Differential Equations)

A unified approach to existence theories for singular chemotaxis systems with nonlinear signal production - Tao, Youshan; Winkler, Michael (Science China Mathematics)

Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters - Winkler, Michael (Applied Mathematics & Optimization)
Abstract A Neumann-type initial-boundary value problem for $$\begin{aligned} \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma (\Theta ) \nabla u_t) + a \nabla \cdot (\gamma (\Theta ) \nabla u) + \nabla \cdot f(\Theta ), \\ \Theta _t = D\Delta \Theta + \Gamma (\Theta ) |\nabla u_t|^2 + F(\Theta )\cdot \nabla u_t, \end{array} \right. \end{aligned}$$ u tt = ∇ · ( γ ( Θ ) ∇ u t ) + a ∇ · ( γ ( Θ ) ∇ u ) + ∇ · f ( Θ ) , Θ t = D Δ Θ + Γ ( Θ ) | ∇ u t | 2 + F ( Θ ) · ∇ u t , is considered in a smoothly bounded domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n\ge 1$$ n ≥ 1 . In the case when $$n=1$$ n = 1 , $$\gamma \equiv \Gamma $$ γ ≡ Γ and $$f\equiv F$$ f ≡ F , this system coincides with the standard model for heat generation in a viscoelastic material of Kelvin-Voigt type, well-understood in situations in which $$\gamma =const$$ γ = c o n s t . Covering scenarios in which all key ingredients $$\gamma ,\Gamma ,f$$ γ , Γ , f and F may depend on the temperature $$\Theta $$ Θ here, for initial data which merely satisfy $$u_0\in W^{1,p+2}(\Omega )$$ u 0 ∈ W 1 , p + 2 ( Ω ) , $$u_{0t}\in W^{1,p}(\Omega )$$ u 0 t ∈ W 1 , p ( Ω ) and $$\Theta _0\in W^{1,p}(\Omega )$$ Θ 0 ∈ W 1 , p ( Ω ) with some $$p\ge 2$$ p ≥ 2 such that $$p>n$$ p > n , a result on local-in-time existence and uniqueness is derived in a natural framework of weak solvability.

Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation - Hanfland, Celina; Winkler, Michael (Journal of Elliptic and Parabolic Equations)
Abstract For $$p>2$$ p > 2 , the equation $$\begin{aligned} u_t = u^p u_{xx}, \qquad x\in \mathbb {R}, \ t\in \mathbb {R}, \end{aligned}$$ u t = u p u xx , x ∈ R , t ∈ R , is shown to admit positive and spatially increasing smooth solutions on all of $$\mathbb {R}× \mathbb {R}$$ R × R which are precisely of the form of an accelerating wave for $$t<0$$ t < 0 , and of a wave slowing down for $$t>0$$ t > 0 . These solutions satisfy $$u(· ,t)→ 0$$ u ( · , t ) → 0 in $$L^∞ _{loc}(\mathbb {R})$$ L loc ∞ ( R ) as $$t→ + ∞ $$ t → + ∞ and as $$t→ -∞ $$ t → - ∞ , and exhibit a yet apparently undiscovered phenomenon of transient rapid spatial growth, in the sense that $$\begin{aligned} \lim _{x→ +∞ } x^{-1} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t<0, \end{aligned}$$ lim x → + ∞ x - 1 u ( x , t ) exists for all t < 0 , that $$\begin{aligned} \lim _{x→ +∞ } x^{-\frac{2}{p}} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t>0, \end{aligned}$$ lim x → + ∞ x - 2 p u ( x , t ) exists for all t > 0 , but that $$\begin{aligned} u(x,0)=K e^{α x} \qquad \text{ for } \text{ all } x\in \mathbb {R}\end{aligned}$$ u ( x , 0 ) = K e α x for all x ∈ R with some $$K>0$$ K > 0 and $$α >0$$ α > 0 .

Effects of degeneracies in taxis-driven evolution - Winkler, Michael (Mathematical Models and Methods in Applied Sciences)
Refined investigation of chemotaxis processes has revealed a significant role of degeneracies in corresponding motilities in a number of application contexts. A rapidly growing literature concerned with the analysis of resulting mathematical models has been capable of solving fundamental issues, but various problems have remained open, or even newly arisen. The goal of the paper consists in a summary of some developments in this area, and particularly in the discussion of the question how far the introduction of degeneracies may influence the behavior of solutions to chemotaxis systems.

A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions - Claes, Leander; Lankeit, Johannes; Winkler, Michael (Mathematical Models and Methods in Applied Sciences)

Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities - Winkler, Michael (Zeitschrift für angewandte Mathematik und Physik)
Abstract An initial-boundary value problem for $$\begin{aligned} \left\{ \begin{array}{ll}u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + au_{xx} - \big (f(\Theta )\big )_x, \qquad & x\in \Omega , \ t>0, \\[1mm] \Theta _t = \Theta _{xx} + \gamma (\Theta ) u_{xt}^2 - f(\Theta ) u_{xt}, \qquad & x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$ u tt = ( γ ( Θ ) u xt ) x + a u xx - ( f ( Θ ) ) x , x ∈ Ω , t > 0 , [ 1 m m ] Θ t = Θ xx + γ ( Θ ) u xt 2 - f ( Θ ) u xt , x ∈ Ω , t > 0 , is considered in an open bounded real interval $$\Omega $$ Ω . Under the assumption that $$\gamma \in C^0([0,\infty ))$$ γ ∈ C 0 ( [ 0 , ∞ ) ) and $$f\in C^0([0,\infty ))$$ f ∈ C 0 ( [ 0 , ∞ ) ) are such that $$f(0)=0$$ f ( 0 ) = 0 , and $$k_\gamma \le \gamma \le K_\gamma $$ k γ ≤ γ ≤ K γ as well as $$\begin{aligned} |f(\xi )| \le K_f \cdot (\xi +1)^\alpha \qquad \hbox {for all } \xi \ge 0 \end{aligned}$$ | f ( ξ ) | ≤ K f · ( ξ + 1 ) α for all ξ ≥ 0 with some $$k_\gamma>0, K_\gamma>0, K_f>0$$ k γ > 0 , K γ > 0 , K f > 0 and $$\alpha <\frac{3}{2}$$ α < 3 2 , for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived. By particularly covering the thermodynamically consistent choice $$f\equiv id$$ f ≡ i d of predominant physical relevance, this appears to go beyond previous related literature which seems to either rely on independence of $$\gamma $$ γ on $$\Theta $$ Θ , or to operate on finite time intervals.

Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities - Winkler, Michael (Journal of Evolution Equations)
Abstract The model $$\begin{aligned} \left\{ \begin{array}{l}u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + au_{xx} - \big (f(\Theta )\big )_x, \\[1mm] \Theta _t = \Theta _{xx} + \gamma (\Theta ) u_{xt}^2 - f(\Theta ) u_{xt}, \end{array} \right. \end{aligned}$$ u tt = ( γ ( Θ ) u xt ) x + a u xx - ( f ( Θ ) ) x , [ 1 m m ] Θ t = Θ xx + γ ( Θ ) u xt 2 - f ( Θ ) u xt , for thermoviscoelastic evolution in one-dimensional Kelvin–Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if $$\gamma _0>0$$ γ 0 > 0 is fixed, then there exists $$\delta =\delta (\gamma _0)>0$$ δ = δ ( γ 0 ) > 0 with the property that for suitably regular initial data of arbitrary size an associated initial boundary value problem posed in an open bounded interval admits a global classical solution whenever $$\gamma \in C^2([0,\infty ))$$ γ ∈ C 2 ( [ 0 , ∞ ) ) and $$f\in C^2([0,\infty ))$$ f ∈ C 2 ( [ 0 , ∞ ) ) are such that $$f(0)=0$$ f ( 0 ) = 0 and $$|f(\xi )| \le K_f \cdot (\xi +1)^\alpha $$ | f ( ξ ) | ≤ K f · ( ξ + 1 ) α for all $$\xi \ge 0$$ ξ ≥ 0 and some $$K_f>0$$ K f > 0 and $$\alpha <\frac{3}{2}$$ α < 3 2 , and that $$\begin{aligned} \gamma _0 \le \gamma (\xi ) \le \gamma _0 + \delta \qquad \hbox {for all } \xi \ge 0. \end{aligned}$$ γ 0 ≤ γ ( ξ ) ≤ γ 0 + δ for all ξ ≥ 0 . This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.

Rough solutions in one-dimensional nonlinear thermoelasticity - Winkler, Michael (Calculus of Variations and Partial Differential Equations)
Abstract The hyperbolic-parabolic model $$\begin{aligned} \left\{ \begin{array}{ll} u_{tt} = u_{xx} - \big (f(\Theta )\big )_x, \qquad & x\in \Omega , \ t>0, \\ \Theta _t = \Theta _{xx} - f(\Theta ) u_{xt}, \qquad & x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$ u tt = u xx - ( f ( Θ ) ) x , x ∈ Ω , t > 0 , Θ t = Θ xx - f ( Θ ) u xt , x ∈ Ω , t > 0 , for the evolution of the displacement variable u and the temperature $$\Theta \ge 0$$ Θ ≥ 0 during thermoelastic interaction in a one-dimensional bounded interval $$\Omega $$ Ω is considered. Whereas the literature has provided comprehensive results on global solutions for sufficiently regular initial data $$(u_0,u_{0t},\Theta _0)=(u,u_t,\Theta )|_{t=0}$$ ( u 0 , u 0 t , Θ 0 ) = ( u , u t , Θ ) | t = 0 when $$f\equiv id$$ f ≡ i d , it seems to have remained open so far how far a solution theory can be built solely on the two fundamental physical principles of energy conservation and entropy nondecrease. The present manuscript addresses this by asserting global existence of weak solutions under assumptions which are energy- and entropy-minimal in the sense of allowing for any initial data $$u_0\in W_0^{1,2}(\Omega )$$ u 0 ∈ W 0 1 , 2 ( Ω ) , $$u_{0t} \in L^2(\Omega )$$ u 0 t ∈ L 2 ( Ω ) and $$0\le \Theta _0\in L^1(\Omega )$$ 0 ≤ Θ 0 ∈ L 1 ( Ω ) , and which apply to arbitrary $$f\in C^1([0,\infty ))$$ f ∈ C 1 ( [ 0 , ∞ ) ) with $$f(0)=0$$ f ( 0 ) = 0 and $$f'>0$$ f ′ > 0 on $$[0,\infty )$$ [ 0 , ∞ ) .

A degenerate migration-consumption model in domains of arbitrary dimension - Winkler, Michael (Advanced Nonlinear Studies)
Abstract In a smoothly bounded convex domain Ω ⊂ R n ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ ( v ) , v t = Δ v − u v , $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by ϕ ( ξ ) = ξ α , ξ ∈ [ 0 , ξ 0 ] . $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy C ( T ) ≔ ess sup t ∈ ( 0 , T ) ∫ Ω u ( ⋅ , t ) ln ⁡ u ( ⋅ , t ) < ∞ for all T > 0 , $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){< }\infty \qquad \text{for\,all\,}T{ >}0,$$ with sup T >0 C ( T ) < ∞ if α ∈ [1, 2].

Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system - Winkler, Michael (Mathematische Annalen)
Abstract The Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{l} u_t + (u\cdot \nabla ) u =\Delta u+\nabla P + f(x,t), \\ \nabla \cdot u=0, \end{array} \right. \end{aligned}$$ u t + ( u · ∇ ) u = Δ u + ∇ P + f ( x , t ) , ∇ · u = 0 , is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain $$\Omega $$ Ω . It is firstly, inter alia, observed that if $$T>0$$ T > 0 and $$\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot \ln ^\frac{1}{2} \big (|f(x,t)|+1\big ) dx \bigg \}^2 dt <\infty , \end{aligned}$$ ∫ 0 T { ∫ Ω | f ( x , t ) | · ln 1 2 ( | f ( x , t ) | + 1 ) d x } 2 d t < ∞ , then for all divergence-free $$u_0\in L^2(\Omega ;{\mathbb {R}}^2)$$ u 0 ∈ L 2 ( Ω ; R 2 ) , a corresponding initial-boundary value problem admits a weak solution u with $$u|_{t=0}=u_0$$ u | t = 0 = u 0 . For any positive and nondecreasing $$L\in C^0([0,\infty ))$$ L ∈ C 0 ( [ 0 , ∞ ) ) such that $$\begin{aligned} \frac{L(\xi )}{\ln ^\frac{1}{2} \xi } \rightarrow 0 \qquad \text{ as } \xi \rightarrow \infty , \end{aligned}$$ L ( ξ ) ln 1 2 ξ → 0 as ξ → ∞ , this is complemented by a statement on nonexistence of such a solution in the presence of smooth initial data and a suitably constructed $$f:\Omega \times (0,T)\rightarrow {\mathbb {R}}^2$$ f : Ω × ( 0 , T ) → R 2 fulfilling $$\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot L\big (|f(x,t)|\big ) dx \bigg \}^2 dt < \infty . \end{aligned}$$ ∫ 0 T { ∫ Ω | f ( x , t ) | · L ( | f ( x , t ) | ) d x } 2 d t < ∞ . This resolves a fine structure in the borderline case $$p=1$$ p = 1 and $$q=2$$ q = 2 appearing in results on existence of weak solutions for sources in $$L^q((0,T);L^p(\Omega ;{\mathbb {R}}^2))$$ L q ( ( 0 , T ) ; L p ( Ω ; R 2 ) ) when $$p\in (1,\infty ]$$ p ∈ ( 1 , ∞ ] and $$q\in [1,\infty ]$$ q ∈ [ 1 , ∞ ] satisfy $$\frac{1}{p}+\frac{1}{q}\le \frac{3}{2}$$ 1 p + 1 q ≤ 3 2 , and on nonexistence if here $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) and $$q\in [1,\infty )$$ q ∈ [ 1 , ∞ ) are such that $$\frac{1}{p}+\frac{1}{q}>\frac{3}{2}$$ 1 p + 1 q > 3 2 .

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system - Winkler, Michael (Annales de l'Institut Henri Poincaré C, Analyse non linéaire)
A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of α>0 the Keller–Segel-type migration–consumption system u_{t} = Δ (uv^{-α}) , v_{t} = Δ v-uv , admits a global classical solution.

A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects - Stinner, Christian; Winkler, Michael (Journal of Evolution Equations)
Abstract The quasilinear Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$ ut=∇·(D(u)∇u)-∇·(S(u)∇v),vt=Δv-v+u, endowed with homogeneous Neumann boundary conditions is considered in a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω⊂Rn , $$n \ge 3$$ n≥3 , with smooth boundary for sufficiently regular functions D and S satisfying $$D>0$$ D>0 on $$[0,\infty )$$ [0,∞) , $$S>0$$ S>0 on $$(0,\infty )$$ (0,∞) and $$S(0)=0$$ S(0)=0 . On the one hand, it is shown that if $$\frac{S}{D}$$ SD satisfies the subcritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$ S(s)D(s)≤Csαforalls≥1withsomeα<2n and $$C>0$$ C>0 , then for any sufficiently regular initial data there exists a global weak energy solution such that $${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$ esssupt>0‖u(t)‖Lp(Ω)<∞ for some $$p > \frac{2n}{n+2}$$ p>2nn+2 . On the other hand, if $$\frac{S}{D}$$ SD satisfies the supercritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$ S(s)D(s)≥csαforalls≥1withsomeα>2n and $$c>0$$ c>0 , then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value $$\alpha = \frac{2}{n}$$ α=2n for $$n \ge 3$$ n≥3 , without any additional assumption on the behavior of D ( s ) as $$s \rightarrow \infty $$ s→∞ , in particular without requiring any algebraic lower bound for D . When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type $$Q(s) = \exp (-s^\beta )$$ Q(s)=exp(-sβ) , $$s \ge 0$$ s≥0 , for global solvability the exponent $$\beta = \frac{n-2}{n}$$ β=n-2n is seen to be critical.

Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture - Ding, Mengyao; Winkler, Michael (Nonlinearity)
Abstract The Neumann problem for the Keller-Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = − ∫ Ω u d x , is considered in n -dimensional balls Ω with n ⩾ 2 , with suitably regular and radially symmetric, radially nonincreasing initial data u 0 . The functions D and S are only assumed to belong to C 2 ( [ 0 , ∞ ) ) and to satisfy D > 0 and S ⩾ 0 on [ 0 , ∞ ) as well as S ( 0 ) = 0 ; in particular, diffusivities with arbitrarily fast decay are included. In this general context, it is shown that it is merely the asymptotic behavior as ξ → ∞ of the expression I ( ξ ) := S ( ξ ) ξ 2 n D ( ξ ) , ξ > 0 , which decides about the occurrence of blow-up: Namely, it is seen that • if lim ξ → ∞ I ( ξ ) = 0 , then any such solution is global and bounded, that • if lim sup ξ → ∞ I ( ξ ) < ∞ and ∫ Ω u 0 is suitably small, then the corresponding solution is global and bounded, and that • if lim inf ξ → ∞ I ( ξ ) > 0 , then at each appropriately large mass level m , there exist radial initial data u 0 such that ∫ Ω u 0 = m , and that the associated solution blows up either in finite or in infinite time. This especially reveals the presence of critical mass phenomena whenever lim ξ → ∞ I ( ξ ) ∈ ( 0 , ∞ ) exists.

Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing - Fuest, Mario; Winkler, Michael (Journal of Mathematical Fluid Mechanics)
Abstract The chemotaxis-Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{rcl} n_t+u\cdot \nabla n & =& \Delta \big (n c^{-\alpha } \big ), \\ c_t+ u\cdot \nabla c & =& \Delta c -nc,\\ u_t + (u\cdot \nabla ) u & =& \Delta u+\nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \end{aligned}$$ n t + u · ∇ n = Δ ( n c - α ) , c t + u · ∇ c = Δ c - n c , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain $$\Omega \subset \mathbb R^2$$ Ω ⊂ R 2 . For all $$\alpha > 0$$ α > 0 and all sufficiently regular $$\Phi $$ Φ , we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for u at a point in the proof when knowledge about n is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time $$L^p$$ L p estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time $$L^2$$ L 2 norm of the force term raised to an arbitrary small power.