We obtain new equitightness and $C([0,T];L^p(\mathbb{R}^N))$-convergence results for numerical approximations of generalized porous medium equations of the form $$ \partial_tu-\mathfrak{L}[\varphi(u)]=g\qquad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $\varphi:\mathbb{R}\to\mathbb{R}$ 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 $L^p$-convergence, some additional restrictions on $\mathfrak{L}$ and $\varphi$ are needed. Most commonly used symmetric operators $\mathfrak{L}$ are still included: the Laplacian, fractional Laplacians, and other generators of symmetric L\'evy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations.
翻译:我们获得了新的宽度和$C( [0,T];L}p(\mathbb{R}N)) 美元(美元), 美元(美元) 和美元(美元) 。 美元( mathbb{R}N), 美元(美元), 美元(美元) 和美元( mathbb{rb{R}N) ), 美元(美元), 美元(美元) 和美元( mathfrak{L} ) 的通用多孔中方方方方程式的数值近似值。 我们的结果包括缓慢扩散, 严重恶化Stefan问题, 以及快速扩散超过一个关键提示 。 这些结果改进了以前的 $( 0. T) ; L\\ text{ loc} p( mathbb{R\ n) 美元(美元), 作者在一系列关于这个主题的文件中获得的 coverglegence: equitradeal and Game $(美元) mostaphal\ pal\ falalalalal) listrational deal) 也使用了某些限制。