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

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 $\Lambda =(-{\partial }_{xx}{)}^{\frac{1}{2}}$ 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.

21 pages, submitted

References in corpus (2)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Global well-posedness of the critical Burgers equation in critical Besov spaces

Cited by in corpus (1)

• Global ${\dot{H}}^1\cap {\dot{H}}^{-1}$ solutions to a logarithmically regularized 2D Euler equation