论文标题
均匀的尾巴估计和$ l^p（\ mathbb {r}^n）$ - 非线性扩散方程有限差近似的收敛
Uniform tail estimates and $L^p(\mathbb{R}^N)$-convergence for finite-difference approximations of nonlinear diffusion equations
论文作者
论文摘要
我们获得了新的quitightness和$ c（[[0，t]; l^p（\ mathbb {r}^n））$ - 收敛结果，用于形式$ \ partial_tu- \ mathfrak \ partial_tu- \ mathfrak {l} {l} [l} [l} [l} [φ（u）= g \ q \ q \ q \ quad的有限多孔介质方程$ \ mathbb {r}^n \ times（0，t）$}，$φ：$φ：\ mathbb {r} \ to \ mathbb {r} $是连续且不crea的，而$ \ mathfrak {l} $是本地或非元素扩散型运算师。我们的结果包括缓慢的扩散，强烈退化的Stefan问题以及高于关键指数的快速扩散。这些结果改善了先前的$ c（[[0，t]; l _ {\ text {loc}}^p（\ mathbb {r}^n））$ - 作者在有关该主题的一系列论文中获得的融合。要具有均衡性和全局$ l^p（\ mathbb {r}^n）$ - 收敛，需要对$ \ mathfrak {l} $和$φ$进行一些其他限制。仍然包括最常用的对称操作员$ \ mathfrak {l} $：Laplacian，分数Laplacians和其他具有一定分数时刻的对称Lévy过程的发生器。我们还讨论非线性的扩展 可能会强烈退化对流扩散方程。
We obtain new equitightness and $C([0,T];L^p(\mathbb{R}^N))$-convergence results for finite-difference approximations of generalized porous medium equations of the form $$ \partial_tu-\mathfrak{L}[φ(u)]=g\qquad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $φ:\mathbb{R}\to\mathbb{R}$ is continuous and nondecreasing, and $\mathfrak{L}$ is a local or nonlocal diffusion operator. Our results include slow diffusions, strongly degenerate Stefan problems, and fast diffusions above a critical exponent. These results improve the previous $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$-convergence obtained in a series of papers on the topic by the authors. To have equitightness and global $L^p(\mathbb{R}^N)$-convergence, some additional restrictions on $\mathfrak{L}$ and $φ$ are needed. Most commonly used symmetric operators $\mathfrak{L}$ are still included: the Laplacian, fractional Laplacians, and other generators of symmetric Lévy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations.
