论文标题
趋化模型的时间离散方案的收敛性
Convergence of a time discrete scheme for a chemotaxis-consumption model
论文作者
论文摘要
在目前的工作中，我们提出并研究了以下趋化性消费模型（对于任何$ s \ ge 1 $），$$ \ partial_t u-Δu= - \ nabla \ cdot（u \ nabla v），\ quad \ quad \ partial_t v -Δv -Δv= - ΔV= - ω$，} $$具有隔离的边界条件和初始条件，其中$（u，v）$模型单元密度和化学信号浓度。提出的方案是通过模型的重新制定的，使用辅助变量$ z = \ sqrt {v +α^2} $与$（u，z）$问题的向后欧拉方案结合使用，而在非线性化合理性和消耗期间，$ u $ $ u $ $ u $。然后，提供了两种不同的方法来检索功能$ V $的近似值。我们证明了对时间离散方案的解决方案的存在，并在时间\ emph {a先验估计中建立了统一，从而产生了趋化性消耗模型的方案向弱解决方案$（u，v）$的收敛性。
In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any $s\ge 1$), $$ \partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v \quad \hbox{in $(0,T)\times Ω$,}$$ endowed with isolated boundary conditions and initial conditions, where $(u,v)$ model cell density and chemical signal concentration. The proposed scheme is defined via a reformulation of the model, using the auxiliary variable $z = \sqrt{v + α^2}$ combined with a Backward Euler scheme for the $(u,z)$-problem and a upper truncation of $u$ in the nonlinear chemotaxis and consumption terms. Then, two different ways of retrieving an approximation for the function $v$ are provided. We prove the existence of solution to the time discrete scheme and establish uniform in time \emph{a priori} estimates, yielding the convergence of the scheme towards a weak solution $(u,v)$ of the chemotaxis-consumption model.
