论文标题
上和下$ \ dot {h}^{m} $用于抛物线方程解决方案的估计值
Upper and lower $\dot{H}^{m}$ estimates for solutions to parabolic equations
论文作者
论文摘要
在本文中，我们证明了有关抛物线方程统一的均质索波列夫规范的上和下衰减估计的结果。遵循Kreiss，Hagstrom，Lorenz和Zingano的想法，我们最终使用解决方案的定期性直接与物理空间中的平滑解决方案一起工作，引导程序衰减估计值从$ l^2 $ norm到高阶衍生品。除了通过此方法获得上限和下限外，我们还获得了反向结果：从高阶衍生物衰减估计值中，我们推断出$ l^2 $ norm的界限。我们使用这些一般结果证明了某些方程式的新衰减估计，并恢复了一些众所周知的结果。
In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use eventual regularity of solutions to directly work with smooth solutions in physical space, bootstrapping decay estimates from the $L^2$ norm to higher order derivatives. Besides obtaining upper and lower bounds through this method, we also obtain reverse results: from higher order derivatives decay estimates, we deduce bounds for the $L^2$ norm. We use these general results to prove new decay estimates for some equations and to recover some well known results.
