论文标题
尖锐的Sobolev估计集合扩散方程的溶液浓度
Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation
论文作者
论文摘要
我们考虑漂移扩散方程$ u_t-εδU + \ nabla \ cdot（u \ nabla k^*u）= 0 $在整个空间中，全球时间解决方案在所有sobolev空间中都有界限；为简单起见，我们将自己限制在模型情况下$ k（x）= - | x | $。我们量化了该方程式的径向对称解的质量浓度现象（一种真正的非线性效应），用于在我们以前的论文[3]中研究的小扩散率$ε$，从而获得了Sobolev规范的最佳尖锐上限和下限。
We consider the drift-diffusion equation $u_t-εΔu + \nabla \cdot(u\nabla K^*u)=0$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case $K(x)=-|x|$. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity $ε$ studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.
