论文标题
具有分数扩散的非局部通量的全球存在和空间分析性
Global existence and spatial analyticity for a nonlocal flux with fractional diffusion
论文作者
论文摘要
在本文中，我们研究一个具有扩散$-ν（ - \ partial_ {xx}））^{\fracα{2}} $的一维非线性方程式，对于$ 0 \ leqleqα\ leq 2 $和$ n \ $ν> 0 $。我们使用粘性拆分算法在空间中获取全局非负弱解决方案$ l^1（\ Mathbb {r}）\ cap h^{1/2}（\ Mathbb {r}）$当$ 0 \leqα\ leq 2 $时。对于亚临界$ 1 <α\ leq 2 $和关键案例$α= 1 $，我们获得了非负空间分析解决方案的全球存在和唯一性。我们使用分数引导方法来改善贝塞尔电位空间中的轻度解决方案的规律性，用于亚临界情况$ 1 <α\ leq 2 $。然后，我们表明这些解决方案是空间分析性的，可以在全球范围内扩展。对于关键案例$α= 1 $，如果初始数据$ρ_0$满足$-ν<\infρ_0<0 $，我们使用用于复杂汉堡方程的特性方法来在某些界限时间间隔中获得我们目标方程的唯一空间分析解决方案。如果$ρ_0\ geq0 $，则该解决方案存在于全球并收敛到稳态。
In this paper, we study a one dimensional nonlinear equation with diffusion $-ν(-\partial_{xx})^{\fracα{2}}$ for $0\leq α\leq 2$ and $ν>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when $0\leqα\leq 2$. For subcritical $1<α\leq 2$ and critical case $α=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<α\leq 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $α=1$, if the initial data $ρ_0$ satisfies $-ν<\infρ_0<0$, we use the characteristics methods for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If $ρ_0\geq0$, the solution exists globally and converges to steady state.
