论文标题
一般无界域上的随机对流Brinkman-Forchheimer方程
Stochastic convective Brinkman-Forchheimer equations on general Unbounded Domains
论文作者
论文摘要
开放连接的$ \ Mathcal {o} \ subseteq \ Mathbb {r}^d $（$ d \ in \ in \ in \ {2,3,4 \} $）或Torus在此工作中考虑了随机对流的对流Brinkman-Forchheimer（SCBF）方程$ \ MATHCAL {O} \ subseteq \ Mathbb {r}^d $（$ d \ in \ in \ in \ in \ {2,3,4 \} $）。我们展示了一个路径独特的强溶液（从概率意义上）的存在，该解决方案满足了能量平等（ITô公式）与乘法高斯噪声扰动的SCBF方程。我们利用了线性和非线性算子的单调性能，以及在证明中的薄荷旋转技术的随机概括。通过使用构成紧凑型运算符特征空间元素的近似函数近似溶液来获得能量平等，以使近似值界定并在Sobolev和Lebesgue的空间中收敛的方式。我们进一步讨论了这种强大解决方案在圆环上的全球规律性结果。在这项工作中，还建立了固定溶液的指数稳定性结果（在均方根和路径方向上），以实现大量有效粘度。此外，通过使用乘法噪声获得了随机对流Brinkman-Forchheimer方程的稳定结果。最后，当$ \ Mathcal {o} $是一个有界的域时，我们使用强解决方案的指数稳定性，建立了具有乘法高斯噪声的SCBF方程的独特不变度度量。
The stochastic convective Brinkman-Forchheimer (SCBF) equations in an open connected set $\mathcal{O}\subseteq\mathbb{R}^d$ ($d\in \{2,3,4\}$) or torus are considered in this work. We show the existence of a pathwise unique strong solution (in the probabilistic sense) satisfying the energy equality (Itô formula) to SCBF equations perturbed by multiplicative Gaussian noise. We exploited a monotonicity property of the linear and nonlinear operators as well as a stochastic generalization of the Minty-Browder technique in the proofs. The energy equality is obtained by approximating the solution using approximate functions constituting the elements of eigenspaces of a compact operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. We further discuss the global in time regularity results of such strong solutions on the torus. The exponential stability results (in mean square and pathwise sense) for the stationary solutions is also established in this work for large effective viscosity. Moreover, a stabilization result of the stochastic convective Brinkman-Forchheimer equations by using a multiplicative noise is obtained. Finally, when $\mathcal{O}$ is a bounded domain, we establish the existence of a unique invariant measure for the SCBF equations with multiplicative Gaussian noise, which is both ergodic and strongly mixing, using the exponential stability of strong solutions.
