论文标题
一类超临界/临界奇异随机PDE：存在，非唯一性，非高斯，非唯一的真人性
A class of supercritical/critical singular stochastic PDEs: existence, non-uniqueness, non-Gaussianity, non-unique ergodicity
论文作者
论文摘要
我们研究表面Quasi geostrophic方程，并具有不规则的空间扰动$$ \ partial_ {t}θ+ u \ cdot \ cdot \nablaθ= -cnablaθ= -v（-Δ）^{γ/2}θ+ζ+ζ $ [0，\ infty）\ times \ mathbb {t}^{2} $，带有$ν\ geq 0 $，$γ\ in [0,3/2）$和$ζ\ in B^{ - 2+κ} _ {\ infty，\ infty，\ infty，\ infty}（\ some \ mathbbbb c}这涵盖了$ζ=（-Δ）=（-Δ）^{α/2}ξ$，$α<1 $和$ξ$ a $ \ mathbb {t}^{2} $上的空间白噪声。根据$γ$和$α$之间的关系，我们的设置在hairer的规律性结构\ cite {hai14}的语言中是亚临界，关键或超临界的。基于从凸集成中的纯分析工具，并且不需要任何概率论证，包括重态化，我们证明在$ l^{p} _ {\ rm {loc}}}}}（0，\ infty; b _ iffty; b _ {\ infty; b _ { c_ {b}（[[0，\ infty）; b^{ - 1/2-Δ} _ {\ infty，1}）\ cap c^{1} _ {b}（[0，\ infty）; b^{ - 3/2-Δ} $Δ> 0 $。我们能够在有限的时间$ t> 0 $的情况下开出初始条件和终端条件，并构建稳态，即独立时间，解决方案。在所有情况下，解决方案都是非高斯的，但是我们可以在某些时候开处方高斯。此外，相对于扰动和初始条件的降低。最后，我们表明我们的解决方案会产生统计固定的溶液作为千古平均值的限制，并且我们获得了无限的许多非高斯依赖时间依赖性的Ergodic固定溶液的存在。我们还将结果扩展到更一般的单数SPDES。
We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }θ+ u\cdot\nablaθ= -ν(-Δ)^{γ/2}θ+ ζ,\qquad u=\nabla^{\perp}(-Δ)^{-1}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, with $ν\geq 0$, $γ\in [0,3/2)$ and $ζ\in B^{-2+κ}_{\infty,\infty}(\mathbb{T}^{2})$ for some $κ>0$. This covers the case of $ζ= (-Δ)^{α/2}ξ$ for $α<1$ and $ξ$ a spatial white noise on $\mathbb{T}^{2}$. Depending on the relation between $γ$ and $α$, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures \cite{Hai14}. Based on purely analytical tools from convex integration and without the need of any probabilistic arguments including renormalization, we prove existence of infinitely many analytically weak solutions in $L^{p}_{\rm{loc}}(0,\infty;B_{\infty,1}^{-1/2})\cap C_{b}([0,\infty);B^{-1/2-δ}_{\infty,1})\cap C^{1}_{b}([0,\infty);B^{-3/2-δ}_{\infty,1})$ for all $p\in [1,\infty)$ and $δ>0$. We are able to prescribe an initial as well as a terminal condition at a finite time $T>0$, and to construct steady state, i.e. time independent, solutions. In all cases, the solutions are non-Gaussian, but we may as well prescribe Gaussianity at some given times. Moreover, a coming down from infinity with respect to the perturbation and the initial condition holds. Finally, we show that the our solutions generate statistically stationary solutions as limits of ergodic averages, and we obtain existence of infinitely many non-Gaussian time dependent ergodic stationary solutions. We also extend our results to a more general class of singular SPDEs.
