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arXiv · 1904.04593

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

Boumediene Abdellaoui·Ireneo Peral·Ana Primo·Fernando Soria
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Abstract

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminus\Omega)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \inn\Omega,\\ \end{array}\right. $$ where $\Omega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2} \dfrac{1}{1-s}$. This behavior clearly exhibits a deep difference with the local case.

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Boumediene Abdellaoui, Ireneo Peral, Ana Primo, Fernando Soria. 2019-04-09. On the KPZ equation with fractional diffusion: global regularity and existence results. https://arxiv.org/abs/1904.04593

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