On distributional solutions of local and nonlocal problems of porous medium type
| Autor: | Félix del Teso, Jørgen Endal, Espen R. Jakobsen |
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| Rok vydání: | 2017 |
| Předmět: |
Generator (category theory)
010102 general mathematics Mathematics::Analysis of PDEs General Medicine Type (model theory) 01 natural sciences Lévy process 010101 applied mathematics Elliptic curve Mathematics - Analysis of PDEs Numerical approximation Bounded function FOS: Mathematics Uniqueness 0101 mathematics Porous medium Mathematical physics Mathematics Analysis of PDEs (math.AP) |
| Zdroj: | Comptes rendus. Mathematique |
| DOI: | 10.48550/arxiv.1706.05306 |
| Popis: | We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\varphi$ is merely continuous and nondecreasing and $\mathfrak{L}^{\sigma,\mu}$ is the generator of a general symmetric L\'evy process. This means that $\mathfrak{L}^{\sigma,\mu}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{\sigma,\mu}=\Delta-(-\Delta)^{\frac12}$. New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for $\mathfrak{L}^{\sigma,\mu}$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained. Comment: 6 pages. Minor revision. Added details to Step 2 of the proof of Theorem 3.1 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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