论文标题
一类广泛退化抛物线方程的规律性结果
Regularity results for a class of widely degenerate parabolic equations
论文作者
论文摘要
由于对气体过滤问题的应用，我们研究了强烈退化抛物线PDE的弱解决方案的规律性 美元 其中$ω$是$ \ mathbb {r}^{n} $中的一个有界域，对于$ n \ geq2 $，$ p \ geq2 $，$ν$是一个正常数，$ \ weft（\，\ cdot \ cdot \，\ right）假设基准$ f $属于合适的lebesgue-sobolev抛物线空间，我们建立了弱解决方案的空间梯度的非线性函数的Sobolev空间规律性，这又意味着弱的时间衍生衍生物$ u_ {t} $。这里的主要新颖性是，上述方程的结构函数仅满足标准的增长和椭圆度条件，仅在零件以$ν$为中心的球外部。我们想指出的是，一方面，可以将这里获得的第一个结果视为[5]中建立的椭圆形结果的抛物面对应物，另一方面，作为对某些已知结果的强烈退化上下文的扩展，对于较少退化的抛物线方程。
Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-ν)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in $Ω_{T}=Ω\times(0,T)$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$, $ν$ is a positive constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative $u_{t}$. The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius $ν$ centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [5], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.
