论文标题
具有分数时间导数的非本地多孔介质方程
Non-Local Porous Media Equations with Fractional Time Derivative
论文作者
论文摘要
在本文中，我们调查了系统的解决方案的存在：\ begin {equination*} \ left \ {\ oken {array} {l} d^α_tu= \ textrm {div}（u \ nabla p） \正确的。 \ end {equation*} in $ \ mathbb {t}^3 $ for $ 0 <s \ leq 1 $，$ 0 <α\ le 1 $。一词$ d^α_tu $表示Caputo衍生物，该衍生物会在时间上对内存效应进行建模。分数拉普拉斯$（ - δ）^{s} $表示Lévy扩散。我们证明了满足各种不平等的非负弱解决方案的全球存在。证明使用了几个近似步骤，包括隐式Euler时间离散化。我们表明，提出的离散Caputo衍生物满足了几种重要特性，包括保留，凸度和与连续Caputo衍生物的严格融合。最重要的是，我们根据离散的caputo衍生物的界限，本着Aubin-Lions定理的精神给出了强烈的紧凑性标准。
In this paper we investigate existence of solutions for the system: \begin{equation*} \left\{ \begin{array}{l} D^α_tu=\textrm{div}(u \nabla p),\\ D^α_tp=-(-Δ)^{s}p+u^{2}, \end{array} \right. \end{equation*} in $\mathbb{T}^3$ for $0< s \leq 1$, and $0< α\le 1$. The term $D^α_t u$ denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian $(-Δ)^{s}$ represents the Lévy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. Most importantly, we give a strong compactness criteria for piecewise constant functions, in the spirit of Aubin-Lions theorem, based on bounds of the discrete Caputo derivative.
