论文标题
抛物线方程和系统的通用变分框架
A universal variational framework for parabolic equations and systems
论文作者
论文摘要
我们提出了一种差异方法，以解决抛物线方程和系统的库奇问题，独立于解决方案的规则理论。这产生了一个普遍且概念上简单的基本解决方案操作员（也称为繁殖者）的结构，我们证明了$ {l}^2 $ off-Diagonal估计值，这是我们假设的新事物。在弱解决方案局部界限的系统的特殊情况下，这为相应的基本溶液提供了高斯上限。特别是，我们获得了Aronson对实际方程式估计的新证明。 该方案足够通用，可以允许具有边界条件的Sobolev空间上的全空间或二阶椭圆零件上具有高阶椭圆形零件的系统。 另一个新功能是，对低阶系数的控制是在关键的混合时间空间Lebesgue空间甚至混合Lorentz空间内。
We propose a variational approach to solve Cauchy problems for parabolic equations and systems independently of regularity theory for solutions. This produces a universal and conceptually simple construction of fundamental solution operators (also called propagators) for which we prove ${L}^2$ off-diagonal estimates, which is new under our assumptions. In the special case of systems for which pointwise local bounds hold for weak solutions, this provides Gaussian upper bound for the corresponding fundamental solution. In particular, we obtain a new proof of Aronson's estimates for real equations. The scheme is general enough to allow systems with higher order elliptic parts on full space or second order elliptic parts on Sobolev spaces with boundary conditions. Another new feature is that the control on lower order coefficients is within critical mixed time-space Lebesgue spaces or even mixed Lorentz spaces.
