Non-Local Porous Media Equations with Fractional Time Derivative
| Autor: | Xinyu Zhang, Maria Pia Gualdani, Jingjing Xu, Nicola Zamponi, Esther S. Daus |
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| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Discretization Applied Mathematics 010102 general mathematics Derivative 01 natural sciences Backward Euler method Convexity 010101 applied mathematics Mathematics - Analysis of PDEs Compact space Time derivative Variational inequality FOS: Mathematics Constant function 0101 mathematics Analysis Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.2010.16332 |
| Popis: | In this paper we investigate existence of solutions for the system: D t α u = div ( u ∇ p ) , D t α p = − ( − Δ ) s p + u 2 , in T 3 for 0 s ≤ 1 , and 0 α ≤ 1 . The term D t α u denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian ( − Δ ) s represents the Levy 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 strong compactness criteria for piecewise constant functions, in the spirit of Aubin–Lions theorem, based on bounds of the discrete Caputo derivative. |
| Databáze: | OpenAIRE |
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