论文标题
动态毛细管方程，并在流形上有随机强迫：一个奇异的极限问题
A dynamic capillarity equation with stochastic forcing on manifolds: a singular limit problem
论文作者
论文摘要
我们考虑一个动态毛细管方程，并在紧凑的Riemannian歧管$（m，g）$上进行随机强迫。 \ begin {qore*} \ tag {p} d \ left（u _ {\ varepsilon，δ}-Δδ u _ {\ varepsilon，δ} \ right） +\ \ operatorName {div} f _ {\ varepsilon}（x，u _ {\ varepsilon，δ}）\，dt = \varepsilonΔu_{\ varepsilon，δ} \，dt φ（x，u _ {\ varepsilon，δ}）\，dw_t，\ end {equation*}其中$ f _ {\ varepsilon} $是一系列平滑矢量字段的顺序，在$ l^p（m \ bbb {m bbb {r} $ p $ p $ p（$ p> 2 $）中，向量字段$ f \ in l^p（m; c^1（\ bbb {r}））$，$ w_t $是在过滤的概率空间上定义的wiener过程。首先，对于$ \ varepsilon $和$δ$的固定值，我们确定了（p）的Cauchy问题的弱解决方案的存在和独特性。假设$ f $是非分类的，并且$ \ varepsilon $和$δ$倾向于用$Δ/\ varepsilon^2 $有限为零，我们表明，在$ l^1_ {ω，t，x，x，x} $ of martingale of distochoperal of distochoctical of distochoctical of distochoctical of distochoctical of distochoctical of distochoctical prow中存在的解决方案的序列有强烈的收敛性。 +\ operatorName {div} f（x，u）\，dt =φ（u）\，dw_t。 $$的证明利用盖尔金的近似，动力学配方以及$ h $量和新的速度平均随机连续性方程式。该分析依赖于使用A.S.〜在某些特定的准派空间中随机变量的表示。此处开发的收敛框架可以应用于随机保护定律的其他奇异极限问题。
We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold $(M,g)$. \begin{equation*}\tag{P} d \left(u_{\varepsilon,δ}-δΔ u_{\varepsilon,δ}\right) +\operatorname{div} f_{\varepsilon}(x, u_{\varepsilon,δ})\, dt =\varepsilon Δu_{\varepsilon,δ}\, dt Φ(x, u_{\varepsilon,δ})\, dW_t, \end{equation*} where $f_{\varepsilon}$ is a sequence of smooth vector fields converging in $L^p(M\times \Bbb{R})$ ($p>2$) as $\varepsilon\downarrow 0$ towards a vector field $f\in L^p(M;C^1(\Bbb{R}))$, and $W_t$ is a Wiener process defined on a filtered probability space. First, for fixed values of $\varepsilon$ and $δ$, we establish the existence and uniqueness of weak solutions to the Cauchy problem for (P). Assuming that $f$ is non-degenerate and that $\varepsilon$ and $δ$ tend to zero with $δ/\varepsilon^2$ bounded, we show that there exists a subsequence of solutions that strongly converges in $L^1_{ω,t,x}$ to a martingale solution of the following stochastic conservation law with discontinuous flux: $$ d u +\operatorname{div} f(x, u)\,dt=Φ(u)\, dW_t. $$ The proofs make use of Galerkin approximations, kinetic formulations as well as $H$-measures and new velocity averaging results for stochastic continuity equations. The analysis relies in an essential way on the use of a.s.~representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.
