论文标题
非局部非线性扩散方程。平滑效果，绿色功能和功能不平等
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
论文作者
论文摘要
我们建立了$ \ partial_t u+（ - \ mathfrak {l}）形式的通用多孔介质方程的解决方案的界限估计值线性，对称和非负操作员。我们考虑的广泛的运营商包括但不限于Lévy运营商。我们的定量估计以精确的$ l^1 $ - $ l^\ infty $ - 平滑效果和绝对界限的形式，其证明是基于问题的双重配方与$ - \ mathfrak {l} $ {l} $和$ i- \ i- \ mathfrak的绿色功能之间的相互作用。 在线性案例$ m = 1 $中，众所周知，$ l^1 $ - $ l^\ infty $ smoothing效果或超包性等同于NASH的不平等。这也等同于热核估计值，这意味着绿色功能估计代表我们技术中的关键成分。 我们在非线性设置$ m> 1 $中建立了类似的方案。首先，我们可以证明，在非线性案例中，超包率的运算符也提供了$ l^1 $ - $ l^\ infty $ - 平滑效果。相反的含义一般不正确。反例由$ 0 $ -Orderlévy运算符提供，例如$ - \ Mathfrak {l} = i-j \ ast $。当$ m = 1 $时，它们不会正规化，但是由于凸非线性，当$ m> 1 $时，我们表明他们这样做了。这揭示了非线性方程的引人注目的属性：非线性允许几乎独立于线性操作员来更好地正规化属性。 最后，我们表明平滑效应是线性和非线性的，这意味着Gagliardo-Nirenberg-Sobolev类型的不平等的家族，并且我们通过使用Moser迭代探索了线性和非线性设置中的等效性。
We establish boundedness estimates for solutions of generalized porous medium equations of the form $$ \partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise $L^1$--$L^\infty$-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of $-\mathfrak{L}$ and $I-\mathfrak{L}$. In the linear case $m=1$, it is well-known that the $L^1$--$L^\infty$-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting $m>1$. First, we can show that operators for which ultracontractivity holds, also provide $L^1$--$L^\infty$-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by $0$-order Lévy operators like $-\mathfrak{L}=I-J\ast$. They do not regularize when $m=1$, but we show that surprisingly enough they do so when $m>1$, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.
