On a one-dimensional \alpha-patch model with nonlocal drift and fractional dissipation

TL;DR

This paper studies a one-dimensional fractional PDE with nonlocal drift and dissipation, proving global well-posedness in certain regimes and finite-time singularity formation in others, using a new nonlocal inequality.

Contribution

It establishes the global existence and finite-time blow-up results for a fractional nonlocal PDE, introducing a novel nonlocal weighted inequality technique.

Findings

Global well-posedness for $1-\alpha \le \beta \le 2$

Finite-time singularity formation for $0 \le \beta < 1-\alpha$

Development of a new nonlocal weighted inequality

Abstract

We consider a one-dimensional nonlocal nonlinear equation of the form: $\partial_t u = (\Lambda^{-\alpha} u)\partial_x u - \nu \Lambda^{\beta}u$ where $\Lambda =(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $\nu\ge 0$ is the viscosity coefficient. We consider primarily the regime $0<\alpha<1$ and $0\le \beta \le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $\alpha$-patch models. In the critical and subcritical range $1-\alpha\le \beta \le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 \le \beta<1-\alpha$, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest.