论文标题
具有单数初始条件的分数汉堡方程
Fractional Burgers equation with singular initial condition
论文作者
论文摘要
我们考虑$ {\ Mathbf r}^d $，$ nabla（U | U | u |^{（α-1）/β}）$的分数钻头方程$Δ^{α/2} u + b \ cdot \ nabla（u | u | u | u |^{（α-1）/β}）$初始条件，其中包含不属于任何$ l^p（{\ mathbf r}^d）$的功能，$ 1 \ leq p \ leq \ leq \ infty $。接下来，我们将一般结果应用于初始条件$ u_0（x）= m | x |^{ - β} $，$ 1 <β<d $，并显示出自相似溶液的存在，并得出其特性，例如平滑度，双向估计，渐近估计和梯度估计。
We consider the fractional Burgers equation $ Δ^{α/2} u + b\cdot \nabla (u|u|^{(α-1)/β})$ on ${\mathbf R}^d$, $d\geq2$, with {$α\in (1,2)$ and} $β>1$ and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any $L^p({\mathbf R}^d)$, $1\leq p\leq\infty$. Next, we apply the general results to the initial condition $u_0(x)=M|x|^{-β}$, $1<β<d$, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.
