论文标题
临界空间II部分中的非线性抛物线随机演变方程。爆破标准和瞬时正则化
Nonlinear parabolic stochastic evolution equations in critical spaces Part II. Blow-up criteria and instantaneous regularization
论文作者
论文摘要
本文是该项目的第一部分的延续，在那里我们为具有高斯噪声的非线性随机PDE开发了一种新的本地适应性理论。在当前第二部分中，我们考虑爆炸标准和正则化现象。与第一部分一样，我们可以允许具有多项式生长的非线性，以及来自临界空间的粗糙初始值。 在第一个主要结果中，我们获得了准和半线性随机演化方程的几个新爆破标准。特别是，对于半线性方程，我们获得了锯齿蛋白型爆破标准，该标准将Prüss-Simonett-Wilke（2018）的最新结果扩展到了随机设置。爆炸标准可用于证明全球SPDES。与第一部分一样，最大的规律性技术和权重及时在证明中起着核心作用。 我们的第二个贡献是在时间和空间上引导Sobolev和Hölder规律性引导的新方法，这不需要初始数据的平滑度。爆破标准是基于这些新方法的。此外，在应用程序中，引导结果可以与我们的爆破标准结合使用，以获得有效的方法来证明全球存在。即使在经典的$ l^2 $ - 设置中，这也给出了新的结果，我们为混凝土SPDE说明了这一点。 在准备未来的工作中，我们应用当前论文的结果来获得全球适应性的结果，并针对几种混凝土SPDE进行规律性。这些包括随机的Navier-Stokes，反应扩散方程和Allen-Cahn方程。我们的设置允许将这些SPDE放入更灵活的框架中，在这种框架中，对非线性的限制更少，并且我们能够从关键空间中处理粗糙的初始值。此外，我们将获得更高的规律性结果。
This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth, and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss-Simonett-Wilke (2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical $L^2$-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results, and regularity for several concrete SPDEs. These include stochastic Navier-Stokes, reaction diffusion equations, and Allen-Cahn equations. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher order regularity results.
