具有无界噪声扩散的随机Cahn-Hilliard方程的适定性和正则性估计
Wellposedness and regularity estimate for stochastic Cahn--Hilliard equation with unbounded noise diffusion
摘要：
本文研究了由乘性时空白噪声驱动的具有次线性增长扩散系数的随机Cahn-Hilliard方程。通过引入谱Galerkin方法，我们首先得到了有限维近似方程的适定性。然后利用半群理论和因子分解方法，证明了逼近过程具有许多令人满意的性质。利用插值不等式和变分方法证明了该逼近过程在一定的巴拿赫空间中是强收敛的。最后，利用逼近过程的强收敛性证明了唯一解过程的全局存在性和正则性估计。
英文摘要：
In this article, we consider the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. By introducing the spectral Galerkin method, we first obtain the well-posedness of the approximated equation in finite dimension. Then with the help of the semigroup theory and the factorization method, the approximation processes is shown to possess many desirable properties. Further, we show that the approximation process is strongly convergent in certain Banach space via the interpolation inequality and variational approach. Finally, the global existence and regularity estimate of the unique solution process are proven by means of the strong convergence of the approximation process.
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