非线性扩散方程finite-difference逼近的一致尾估计和$L^p(\mathbb{R}^N)$ -收敛性
Uniform tail estimates and $L^p(\mathbb{R}^N)$-convergence for finite-difference approximations of nonlinear diffusion equations
摘要：
我们获得了新的等紧性和$C([0,T];L^p(\mathbb{R}^N))$ -收敛结果的形式$$ \partial_tu-\mathfrak{L}[φ(u)]=g\qquad\text{in $\mathbb{R}^N\times(0,T)$}, $$的广义多孔介质方程的有限差分近似的$φ:\mathbb{R}\to\mathbb{R}$是连续和非减的，和$\mathfrak{L}$是一个本地或非本地的扩散算子。我们的结果包括慢扩散，强退化Stefan问题，和快扩散以上的临界指数。这些结果改善了以前的$C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$ -收敛在一系列的论文中得到的主题的作者。为了实现等密性和全局$L^p(\mathbb{R}^N)$收敛，需要对$\mathfrak{L}$和$φ$进行一些额外的限制。最常用的对称算子$\mathfrak{L}$仍然包括：拉普拉斯算子，分数拉普拉斯算子和其他具有分数矩的对称Lévy过程的生成元。我们还讨论了非线性可能强退化对流-扩散方程的推广。
英文摘要：
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.
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