In the first part of this work, we establish the existence and uniqueness of a local mild solution to deterministic convective Brinkman–Forchheimer (CBF) equations defined on the whole space, by using properties of the heat semigroup and fixed point arguments based on an iterative technique. Moreover, we prove that the solution exists globally. The second part is devoted for establishing the existence and uniqueness of a pathwise mild solution upto a random time to the stochastic CBF equations perturbed by Lévy noise by exploiting the contraction mapping principle. Then by using stopping time arguments, we show that the pathwise mild solution exists globally. We also discuss the local and global solvability of the stochastic CBF equations forced by fractional Brownian noise.

在这项工作的第一部分，我们通过使用热半群的性质和基于迭代的不动点参数，建立了在整个空间上定义的确定性对流 Brinkman-Forchheimer (CBF) 方程的局部温和解的存在性和唯一性技术。此外，我们证明了该解决方案全局存在。第二部分致力于通过利用收缩映射原理来建立受 Lévy 噪声扰动的随机 CBF 方程在随机时间内的路径温和解的存在性和唯一性。然后通过使用停止时间参数，我们证明了路径温和解是全局存在的。我们还讨论了由分数布朗噪声强制的随机 CBF 方程的局部和全局可解性。