Global existence and boundedness in a chemotaxis‐Stokes system with arbitrary porous medium diffusion

Yu, Peifeng

doi:10.1002/mma.5920

Cited by 6 publications

(2 citation statements)

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“…33 Furthermore, the global weak solutions have been constructed for any m > 1 for this chemotaxis-Stokes system. 34 Besides, if 𝜅 = 1 and m > 1, Wang proved that the initial-boundary value problem of the fully chemotaxis-Navier-Stokes system possesses at least one global weak solution. 35 To the best of our knowledge, the well-posedness for the initial-boundary value problem of (1.4) is much less than that of (1.2) and its variant (1.3).…”

Section: Introductionmentioning

confidence: 99%

“…$$ in a smooth bounded domain

\Omega \subset {\mathbb{R}}&amp;amp;amp;#x0005E;3

, they obtained the global existence of weak solutions to the Neumann–Neumann–Dirichlet boundary problem for the system () with

\kappa &amp;amp;amp;#x0003D;0

m&amp;amp;gt;\frac{4}{3}

33 . Furthermore, the global weak solutions have been constructed for any

m&amp;amp;gt;1

for this chemotaxis–Stokes system 34 . Besides, if

\kappa &amp;amp;amp;#x0003D;1

and

m&amp;amp;gt;1

, Wang proved that the initial‐boundary value problem of the fully chemotaxis–Navier–Stokes system possesses at least one global weak solution 35 …”

Section: Introductionmentioning

confidence: 99%

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Boundedness in a two‐dimensional self‐consistent chemotaxis–Navier–Stokes system with nonlinear diffusion

Zhou

Dong

2022

Math Methods in App Sciences

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This paper deals with the self‐consistent chemotaxis–Navier–Stokes system with porous medium diffusion {left left leftarrayarraynt+u·∇n=Δnm−∇·(n∇c)+∇·(n∇ϕ),arrayx∈Ω,t>0,arrayarrayct+u·∇c=Δc−nc,arrayx∈Ω,t>0,arrayarrayut+(u·∇)u+∇P=Δu−n∇ϕ+n∇c,arrayx∈Ω,t>0,arrayarray∇·u=0,arrayx∈Ω,t>0$$ \left\{\begin{array}{lll}&amp; {n}_t&amp;#x0002B;u\cdotp \nabla n&amp;#x0003D;\Delta {n}&amp;#x0005E;m-\nabla \cdotp \left(n\nabla c\right)&amp;#x0002B;\nabla \cdotp \left(n\nabla \phi \right),\kern2em &amp; x\in \Omega, t&gt;0,\\ {}&amp; {c}_t&amp;#x0002B;u\cdotp \nabla c&amp;#x0003D;\Delta c- nc,\kern2em &amp; x\in \Omega, t&gt;0,\\ {}&amp; {u}_t&amp;#x0002B;\left(u\cdotp \nabla \right)u&amp;#x0002B;\nabla P&amp;#x0003D;\Delta u-n\nabla \phi &amp;#x0002B;n\nabla c,\kern2em &amp; x\in \Omega, t&gt;0,\\ {}&amp; \nabla \cdotp u&amp;#x0003D;0,\kern2em &amp; x\in \Omega, t&gt;0\end{array}\right. $$ in the two‐dimensional smoothly bounded domain, where the gravitational potential ϕ∈W1,∞false(normalΩfalse)$$ \phi \in {W}&amp;#x0005E;{1,\infty}\left(\Omega \right) $$ is given function. Systems of this type involve the effect of gravity on cells ( ∇·false(n∇ϕfalse)$$ \nabla \cdotp \left(n\nabla \phi \right) $$) and the effect of the chemotactic force on fluid ( n∇c$$ n\nabla c $$), which leads to a stronger coupling than usual chemotaxis‐fluid system that proposed by Tuval et al. (Proc. Nat. Acad. Sci. 102 (2005), 2277‐2282) and studied widely in the most existing literatures. Under no‐flux boundary conditions for n$$ n $$ and c$$ c $$ and no‐slip boundary condition for u$$ u $$, it is proved that the system admits at least one global weak solution which is uniformly bounded if m>1$$ m&gt;1 $$. In particular, our result improves that established by Wang (Discr. Cont. Dyn. Syst. S 13 (2020), 329‐349), which asserts the global existence of weak solutions under the constraint m>1$$ m&gt;1 $$.

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Section: Introductionmentioning

confidence: 99%

“…$$ in a smooth bounded domain