We study a one dimensional dissipative transport equation with nonlocal velocity and critical dissipation. We consider the Cauchy problem for initial values with infinite energy. The control we shall use involves some weighted Lebesgue or Sobolev spaces. More precisely, we consider the familly of weights given by $w_{\beta}(x)=(1+\vert x \vert^{2})^{-\beta/2}$ where $\beta$ is a real parameter in $(0,1)$ and we treat the Cauchy problem for the cases $\theta_{0} \in H^{1/2} (w_{\beta})$ and $\theta_{0} \in H^{1} (w_{\beta})$ for which we prove global existence results (under smallness assumptions on the $L^\infty$ norm of $\theta_0$). The key step in the proof of our theorems is based on the use of two new commutator estimates involving fractional differential operators and the family of Muckenhoupt weights.

中文翻译：

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具有非局部速度的一维输运方程的无限能量解

我们研究具有非局部速度和临界耗散的一维耗散输运方程。我们考虑具有无限能量的初始值的柯西问题。我们将使用的控制涉及一些加权 Lebesgue 或 Sobolev 空间。更准确地说，我们考虑由 $w_{\beta}(x)=(1+\vert x \vert^{2})^{-\beta/2}$ 给出的权重族，其中 $\beta$ 是$(0,1)$ 中的实参数，我们处理 $\theta_{0} \in H^{1/2} (w_{\beta})$ 和 $\theta_{0} \ 情况下的柯西问题在 H^{1} (w_{\beta})$ 中，我们证明了全局存在结果（在 $\theta_0$ 的 $L^\infty$ 范数的小假设下）。证明我们定理的关键步骤是基于使用两个新的交换子估计，涉及分数微分算子和 Muckenhoupt 权重族。