## Global existence of weak solutions to dissipative transport equations with nonlocal velocity. (arXiv:1609.04357v3 [math.AP] UPDATED)

We consider 1D dissipative transport equations with nonlocal velocity field: $\theta_t+u\theta_x+\delta u_{x} \theta+\Lambda^{\gamma}\theta=0, \quad u=\mathcal{N}(\theta),$ where $\mathcal{N}$ is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators: $\mathcal{N}=\mathcal{H}$, the Hilbert transform, $\mathcal{N}=(1-\partial_{xx} )^{-\alpha}$. In this paper, we show several global existence of weak solutions depending on the range of $\gamma$ and $\delta$. When $0&lt;\gamma&lt;1$, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when $\gamma \in (0,2)$.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We consider 1D dissipative transport equations with nonlocal velocity field: $\theta_t+u\theta_x+\delta u_{x} \theta+\Lambda^{\gamma}\theta=0, \quad u=\mathcal{N}(\theta),$ where $\mathcal{N}$ is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators: $\mathcal{N}=\mathcal{H}$, the Hilbert transform, $\mathcal{N}=(1-\partial_{xx} )^{-\alpha}$. In this paper, we show several global existence of weak solutions depending on the range of $\gamma$ and $\delta$. When $0<\gamma<1$, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when $\gamma \in (0,2)$.