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arXiv 1904.04593math.AP
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具有分数阶扩散的KPZ方程：整体正则性和存在性结果

On the KPZ equation with fractional diffusion: global regularity and existence results

Boumediene Abdellaoui, Ireneo Peral, Ana Primo, Fernando Soria

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Corrected version; references and remarks added
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摘要：

在这项工作中，我们分析的分数拟线性问题，$$ (P) \left\{ \begin{array}{rcll} u_t+(-Δ)^s u &=&|\nabla u|^α+ f &\inn Ω_T\equivΩ\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminusΩ)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \innΩ,\\ \end{array}\right. $$，其中$Ω$是$C^{1,1}$有界域$\mathbb{R}^N$，$N> 2s$和$\frac{1}{2}<s<1$的存在性。我们将假设$f$和$u_0$是满足一些附加假设的非负函数，这些假设将在后面详细说明。假设$f$上有一定的正则性，我们将证明问题$(P)$对于值$α<\dfrac{s}{1-s}$的解的存在性，以及当$α>\dfrac{1}{1-s}$时这种解的不存在性。这种行为显然与当地情况有很大的不同。

英文摘要：

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-Δ)^s u &=&|\nabla u|^α+ f &\inn Ω_T\equivΩ\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminusΩ)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \innΩ,\\ \end{array}\right. $$ where $Ω$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2}<s<1$. We will assume that $f$ and $u_0$ are non negative functions satisfying some additional hypotheses that will be specified later on. Assuming certain regularity on $f$, we will prove the existence of a solution to problem $(P)$ for values $α<\dfrac{s}{1-s}$, as well as the non existence of such a solution when $α>\dfrac{1}{1-s}$. This behavior clearly exhibits a deep difference with the local case.

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