Journal of Differential Equations, ISSN 0022-0396, 12/2015, Volume 259, Issue 12, pp. 7578 - 7609

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled Keller–Segel–Stokes system{nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)⋅∇c...

Keller–Segel–Stokes system | Global existence | Tensor-valued sensitivity | Boundedness | Keller-Segel-Stokes system | CHEMOTAXIS-FLUID MODEL | LP | STABILIZATION | EQUATIONS | NONLINEAR DIFFUSION | MATHEMATICS | BLOW-UP | WEAK SOLUTIONS | DOMAINS

Keller–Segel–Stokes system | Global existence | Tensor-valued sensitivity | Boundedness | Keller-Segel-Stokes system | CHEMOTAXIS-FLUID MODEL | LP | STABILIZATION | EQUATIONS | NONLINEAR DIFFUSION | MATHEMATICS | BLOW-UP | WEAK SOLUTIONS | DOMAINS

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 09/2016, Volume 261, Issue 6, pp. 3414 - 3462

... is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis...

Boundary concentration | Keller–Segel system | MATHEMATICS | Keller-Segel system | PERTURBED NEUMANN PROBLEM | STEADY-STATES | CHEMOTAXIS | CURVES

Boundary concentration | Keller–Segel system | MATHEMATICS | Keller-Segel system | PERTURBED NEUMANN PROBLEM | STEADY-STATES | CHEMOTAXIS | CURVES

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2015, Volume 430, Issue 1, pp. 585 - 591

This paper deals with the higher dimension quasilinear parabolic–parabolic Keller–Segel system involving a source term of logistic type ut...

Logistic source | Global existence | Keller–Segel system | Chemotaxis | Boundedness | Keller-Segel system | MATHEMATICS | MATHEMATICS, APPLIED | CHEMOTAXIS SYSTEM | EQUATIONS | MODELING CHEMOTAXIS | DOMAINS | TIME BLOW-UP

Logistic source | Global existence | Keller–Segel system | Chemotaxis | Boundedness | Keller-Segel system | MATHEMATICS | MATHEMATICS, APPLIED | CHEMOTAXIS SYSTEM | EQUATIONS | MODELING CHEMOTAXIS | DOMAINS | TIME BLOW-UP

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 11/2016, Volume 443, Issue 1, pp. 445 - 452

We consider a parabolic–parabolic Keller–Segel system of chemotaxis model with singular sensitivity: ut=Δu−χ∇⋅(uv∇v), vt=kΔv−v...

Keller–Segel system | Chemotaxis | Singular sensitivity | Boundedness | MATHEMATICS | MATHEMATICS, APPLIED | Keller-Segel system

Keller–Segel system | Chemotaxis | Singular sensitivity | Boundedness | MATHEMATICS | MATHEMATICS, APPLIED | Keller-Segel system

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2017, Volume 447, Issue 1, pp. 499 - 528

In this paper, we consider the following Keller–Segel(–Navier)–Stokes system(⋆){nt+u⋅∇n=Δn−∇⋅(nχ(c)∇c),x∈Ω, t>0,ct+u⋅∇c=Δc−c+n,x∈Ω, t>0,ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ,x∈Ω...

Navier–Stokes | Global existence | Stokes | Keller–Segel | Boundedness | MATHEMATICS, APPLIED | STOKES SYSTEM | MODEL | MATHEMATICS | SINGULAR SENSITIVITY | LOGISTIC SOURCE | Navier-Stokes | PARABOLIC CHEMOTAXIS SYSTEM | BLOW-UP | Keller-Segel | AGGREGATION

Navier–Stokes | Global existence | Stokes | Keller–Segel | Boundedness | MATHEMATICS, APPLIED | STOKES SYSTEM | MODEL | MATHEMATICS | SINGULAR SENSITIVITY | LOGISTIC SOURCE | Navier-Stokes | PARABOLIC CHEMOTAXIS SYSTEM | BLOW-UP | Keller-Segel | AGGREGATION

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 07/2017, Volume 263, Issue 2, pp. 1477 - 1521

We prove the time-global existence of solutions of the degenerate Keller–Segel system in higher dimensions, under the assumption that the mass of the first component is below a certain critical value...

Degenerate diffusion | Wasserstein distance | Gradient flows | Chemotaxis | Keller–Segel | Keller-Segel | GLOBAL EXISTENCE | BEHAVIOR | EQUATIONS | MODEL | MATHEMATICS | CRITICAL MASS | HIGHER DIMENSIONS | ASYMPTOTICS | BLOW-UP

Degenerate diffusion | Wasserstein distance | Gradient flows | Chemotaxis | Keller–Segel | Keller-Segel | GLOBAL EXISTENCE | BEHAVIOR | EQUATIONS | MODEL | MATHEMATICS | CRITICAL MASS | HIGHER DIMENSIONS | ASYMPTOTICS | BLOW-UP

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 06/2015, Volume 426, Issue 1, pp. 105 - 124

In this study, we consider an attraction–repulsion chemotaxis system{ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0 with homogeneous Neumann boundary conditions...

Asymptotic stability | Chemotaxis | Boundedness | Attraction–repulsion | Attraction-repulsion | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | SENSITIVITY | MODEL | MATHEMATICS | KELLER-SEGEL SYSTEM | EQUILIBRIA | BLOW-UP | AGGREGATION

Asymptotic stability | Chemotaxis | Boundedness | Attraction–repulsion | Attraction-repulsion | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | SENSITIVITY | MODEL | MATHEMATICS | KELLER-SEGEL SYSTEM | EQUILIBRIA | BLOW-UP | AGGREGATION

Journal Article

8.
Full Text
Vanishing cross-diffusion limit in a Keller–Segel system with additional cross-diffusion

Nonlinear Analysis, ISSN 0362-546X, 03/2020, Volume 192, p. 111698

Keller–Segel systems in two and three space dimensions with an additional cross-diffusion term in the equation for the chemical concentration are analyzed...

Higher-order estimates | Numerical simulations | Asymptotic analysis | Keller–Segel model | Vanishing cross-diffusion limit | Entropy method | MATHEMATICS | Keller-Segel model | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | CHEMOTAXIS | PREVENTING BLOW-UP | MODEL | Diffusion rate | Parameters | Elliptic functions | Convergence

Higher-order estimates | Numerical simulations | Asymptotic analysis | Keller–Segel model | Vanishing cross-diffusion limit | Entropy method | MATHEMATICS | Keller-Segel model | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | CHEMOTAXIS | PREVENTING BLOW-UP | MODEL | Diffusion rate | Parameters | Elliptic functions | Convergence

Journal Article

Zeitschrift für angewandte Mathematik und Physik, ISSN 0044-2275, 6/2019, Volume 70, Issue 3, pp. 1 - 18

...–Segel system with the diffusion exponent $$\frac{2n}{2+n} Engineering | Mathematical Methods in Physics | Supercritical exponent | Degenerate Keller–Segel system | Secondary 35K55 | Primary 35B44 | Theoretical and Applied Mechanics | L^\infty $$ L ∞ estimate | Free energy | CRITICAL DIFFUSION | MATHEMATICS, APPLIED | Degenerate Keller-Segel system | GLOBAL EXISTENCE | L-infinity estimate | BLOW-UP | MODEL | Equality | Public-private sector cooperation

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 02/2015, Volume 258, Issue 4, pp. 1158 - 1191

We prove existence of global weak solutions to the chemotaxis systemut=Δu−∇⋅(u∇v)+κu−μu2vt=Δv−v+u under homogeneous Neumann boundary conditions in a smooth...

Logistic source | Weak solutions | Chemotaxis | Eventual smoothness | Existence | Secondary | Primary | EQUATIONS | BOUNDEDNESS | ATTRACTOR | MODEL | GROWTH SYSTEM | MATHEMATICS | KELLER-SEGEL SYSTEM | DIMENSION | GLOBAL-SOLUTIONS

Logistic source | Weak solutions | Chemotaxis | Eventual smoothness | Existence | Secondary | Primary | EQUATIONS | BOUNDEDNESS | ATTRACTOR | MODEL | GROWTH SYSTEM | MATHEMATICS | KELLER-SEGEL SYSTEM | DIMENSION | GLOBAL-SOLUTIONS

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 09/2017, Volume 263, Issue 5, pp. 2606 - 2629

The coupled quasilinear Keller–Segel–Navier–Stokes system(KSNF){nt+u⋅∇n=Δnm−∇⋅(n∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+κ(u⋅∇)u+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0...

Navier–Stokes system | Global existence | Nonlinear diffusion | Keller–Segel model | EXISTENCE | Keller-Segel model | TENSOR-VALUED SENSITIVITY | STABILIZATION | BOUNDEDNESS | MODEL | FLUID EQUATIONS | Navier-Stokes system | MATHEMATICS | LINEAR CHEMOTAXIS SYSTEM | LOGISTIC SOURCE | CHEMOATTRACTANT | BLOW-UP | Fluid dynamics

Navier–Stokes system | Global existence | Nonlinear diffusion | Keller–Segel model | EXISTENCE | Keller-Segel model | TENSOR-VALUED SENSITIVITY | STABILIZATION | BOUNDEDNESS | MODEL | FLUID EQUATIONS | Navier-Stokes system | MATHEMATICS | LINEAR CHEMOTAXIS SYSTEM | LOGISTIC SOURCE | CHEMOATTRACTANT | BLOW-UP | Fluid dynamics

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 8/2016, Volume 55, Issue 4, pp. 1 - 39

The coupled chemotaxis fluid system $$\begin{aligned} \left\{ \begin{array}{lll} n_t=\Delta n-\nabla \cdot (n S(x,n,c)\cdot \nabla c)-u\cdot \nabla n, &{}\quad...

Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | 35B35 | Analysis | Theoretical, Mathematical and Computational Physics | 35Q35 | Mathematics | 92C17 | 35B40 | 35K55 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | BACTERIA | PARABOLIC EQUATIONS | INITIAL-VALUE PROBLEM | BOUNDED DOMAINS | FLUID MODEL | NONLINEAR DIFFUSION | WEAK SOLUTIONS | KELLER-SEGEL MODELS | ASYMPTOTIC-BEHAVIOR

Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | 35B35 | Analysis | Theoretical, Mathematical and Computational Physics | 35Q35 | Mathematics | 92C17 | 35B40 | 35K55 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | BACTERIA | PARABOLIC EQUATIONS | INITIAL-VALUE PROBLEM | BOUNDED DOMAINS | FLUID MODEL | NONLINEAR DIFFUSION | WEAK SOLUTIONS | KELLER-SEGEL MODELS | ASYMPTOTIC-BEHAVIOR

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 09/2014, Volume 35, Issue 1, pp. 29 - 34

We consider the system ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),τvt=Δv+αu−βv,τwt=Δw+γu−δw, which has been proposed to describe the aggregation of microglia...

Aggregation | Parabolic | Elliptic | Alzheimer | Keller–Segel model | Blow-up | Keller-Segel model | MATHEMATICS, APPLIED | DISEASE SENILE PLAQUES | Mathematical analysis | Agglomeration

Aggregation | Parabolic | Elliptic | Alzheimer | Keller–Segel model | Blow-up | Keller-Segel model | MATHEMATICS, APPLIED | DISEASE SENILE PLAQUES | Mathematical analysis | Agglomeration

Journal Article

Journal of Evolution Equations, ISSN 1424-3199, 6/2018, Volume 18, Issue 2, pp. 561 - 581

...–fluid system $$\begin{aligned} n_t + u\cdot \nabla n&= \Delta n - \chi \nabla \cdot \left( \frac{n}{c}\nabla c\right) \\ c_t + u\cdot \nabla c&= \Delta c - c + n...

Secondary 35Q30 | Analysis | Chemotaxis–fluid | Global existence | Navier–Stokes | Mathematics | 92C17 | Primary 35A01 | Singular sensitivity | Keller–Segel | 35Q92 | 35A09 | EXISTENCE | TENSOR-VALUED SENSITIVITY | MATHEMATICS, APPLIED | BEHAVIOR | STABILIZATION | BOUNDEDNESS | NAVIER-STOKES SYSTEM | NONLINEAR DIFFUSION | MATHEMATICS | Navier-Stokes | PARABOLIC CHEMOTAXIS SYSTEM | Chemotaxis-fluid | GLOBAL WEAK SOLUTIONS | Keller-Segel

Secondary 35Q30 | Analysis | Chemotaxis–fluid | Global existence | Navier–Stokes | Mathematics | 92C17 | Primary 35A01 | Singular sensitivity | Keller–Segel | 35Q92 | 35A09 | EXISTENCE | TENSOR-VALUED SENSITIVITY | MATHEMATICS, APPLIED | BEHAVIOR | STABILIZATION | BOUNDEDNESS | NAVIER-STOKES SYSTEM | NONLINEAR DIFFUSION | MATHEMATICS | Navier-Stokes | PARABOLIC CHEMOTAXIS SYSTEM | Chemotaxis-fluid | GLOBAL WEAK SOLUTIONS | Keller-Segel

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 11/2017, Volume 263, Issue 9, pp. 6115 - 6142

We consider a two dimensional parabolic–elliptic Keller–Segel equation with a fractional diffusion of order α∈(0,2) and a logistic term. In the case of an...

Logistic source | Keller–Segel system | Global-in-time smoothness | Fractional dissipation | Active scalar equations | Nonlocal maximum principle | EXISTENCE | AGGREGATION EQUATION | INSTABILITY | CHEMOTAXIS | MODEL | DIFFUSION EQUATION | GLOBAL WELL-POSEDNESS | MATHEMATICS | Keller-Segel system | REGULARITY | DYNAMICS | QUASI-GEOSTROPHIC EQUATION

Logistic source | Keller–Segel system | Global-in-time smoothness | Fractional dissipation | Active scalar equations | Nonlocal maximum principle | EXISTENCE | AGGREGATION EQUATION | INSTABILITY | CHEMOTAXIS | MODEL | DIFFUSION EQUATION | GLOBAL WELL-POSEDNESS | MATHEMATICS | Keller-Segel system | REGULARITY | DYNAMICS | QUASI-GEOSTROPHIC EQUATION

Journal Article

16.
Full Text
Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

Journal of Differential Equations, ISSN 0022-0396, 12/2016, Volume 261, Issue 12, pp. 6883 - 6914

The coupled chemotaxis fluid system(⋆){nt=Δn−∇⋅(nS(x,n,c)⋅∇c)−u⋅∇n,(x,t)∈Ω×(0,T),ct=Δc−nc−u⋅∇c,(x,t)∈Ω×(0,T),ut=Δu−κ(u⋅∇)u+∇P+n∇ϕ,(x,t)∈Ω×(0,T),∇⋅u=0,(x,t)∈Ω×(0...

Navier–Stokes | Global existence | Chemotaxis | Large time behavior | EXISTENCE | CHEMOTAXIS-STOKES SYSTEM | STABILIZATION | EQUATIONS | BOUNDEDNESS | NONLINEAR DIFFUSION | DEPENDENT SENSITIVITY | MATHEMATICS | KELLER-SEGEL SYSTEM | MODELS | Navier-Stokes | WEAK SOLUTIONS

Navier–Stokes | Global existence | Chemotaxis | Large time behavior | EXISTENCE | CHEMOTAXIS-STOKES SYSTEM | STABILIZATION | EQUATIONS | BOUNDEDNESS | NONLINEAR DIFFUSION | DEPENDENT SENSITIVITY | MATHEMATICS | KELLER-SEGEL SYSTEM | MODELS | Navier-Stokes | WEAK SOLUTIONS

Journal Article

SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2015, Volume 47, Issue 4, pp. 3092 - 3115

The chemotaxis system u(t) = Delta u - del. (uS(x, u, v) . del v); v(t) = Delta v - uf(v) (referred to as (star) in this abstract), for the density u...

Generalized solution | Global existence | Chemotaxis | MATHEMATICS, APPLIED | global existence | KELLER-SEGEL SYSTEM | MODELS | STABILIZATION | EQUATIONS | BOUNDEDNESS | DIFFUSION | BLOW-UP | generalized solution | chemotaxis

Generalized solution | Global existence | Chemotaxis | MATHEMATICS, APPLIED | global existence | KELLER-SEGEL SYSTEM | MODELS | STABILIZATION | EQUATIONS | BOUNDEDNESS | DIFFUSION | BLOW-UP | generalized solution | chemotaxis

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 06/2015, Volume 258, Issue 12, pp. 4275 - 4323

In this paper, we are concerned with a general class of quasilinear parabolic–parabolic chemotaxis systems with/without growth source, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn with n≥2...

Characterizations | Global existence | Chemotaxis systems | Growth source | Boundedness | Secondary | Primary | BEHAVIOR | SENSITIVITY | MODEL | ATTRACTOR | MATHEMATICS | KELLER-SEGEL SYSTEM | LOGISTIC SOURCE | TIME BLOW-UP

Characterizations | Global existence | Chemotaxis systems | Growth source | Boundedness | Secondary | Primary | BEHAVIOR | SENSITIVITY | MODEL | ATTRACTOR | MATHEMATICS | KELLER-SEGEL SYSTEM | LOGISTIC SOURCE | TIME BLOW-UP

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 02/2017, Volume 64, pp. 1 - 7

We consider the parabolic–elliptic chemotaxis-growth system {ut=Δu−χ∇⋅(um∇v)+μu(1−uα),x∈Ω,t>0,−Δv+v=uγ,x∈Ω,t>0, under no-flux boundary conditions in a smoothly bounded domain...

Chemotaxis-growth system | Critical parameter condition | Boundedness | MATHEMATICS, APPLIED | KELLER-SEGEL SYSTEM | LOGISTIC SOURCE | GLOBAL-SOLUTIONS

Chemotaxis-growth system | Critical parameter condition | Boundedness | MATHEMATICS, APPLIED | KELLER-SEGEL SYSTEM | LOGISTIC SOURCE | GLOBAL-SOLUTIONS

Journal Article